U& DEPARTMENT OF COMMERCE • £ r iwironmen1!al Science S hitfdMH Mathematical Geodesy Courtesy of Foster Morrison Martin Hotine 1898-1968 Martin Hotine's death on November 12, 1968, ended a brilliant career; his energetic leadership in geodesy will be missed. Although seriously ill, his drive and enthusiasm enabled him to complete the manuscript of this monograph, Mathematical Geodesy— a fitting climax to a lifetime of geodetic research and application. During the 50-year span of Martin Hotine's professional career, he provided numerous valuable contributions of lasting significance to basic re- search and practical applications of geodesy. Among the many achievements from his surveying and mapping career in England, two contributions are most outstanding — the retriangulation of Great Britain under his direction from 1935 until its completion in 1962, and the surveys and mapping of underdeveloped countries, initiated and directed by him in 1946 and still continuing with his high standards of accuracy. Martin Hotine was truly a builder of worldwide geodetic networks. A firm belief in international geodetic coopera- tion was one of Martin Hotine's convictions. This was manifested by his leadership of the Common- wealth Survey Officers Conferences from 1955 to 1963, by his intense participation in the general assemblies of the International Association of Geodesy, and most notably by his collaboration with Professor Antonio Marussi of Italy in the formation of three symposia on three-dimensional and mathematical geodesy. He was heavily involved in the program planning for the fourth symposium to be held in May 1969. Many significant theoretical contributions to the science of geodesy were made by Martin Hotine. He expanded the classical theoretical limitations of the current geodetic horizon by insisting on a unified three-dimensional approach to geodetic measurements and principles, and by applying the most relevant mathematical tools, such as the tensor calculus, to exploit these con- cepts properly. Many of the papers on these sub- jects never appeared in print. However, by being presented at various international meetings, they were well publicized and proved very influential in their impact on other geodesists. It was thus fortunate that while employed at ESSA, Martin Hotine was able to combine and expand these ideas, formulated over the years, into this treatise on mathematical geodesy. In recognition of his service to the United States Government, Martin Hotine was awarded posthu- mously the Gold Medal of the Department of Commerce "for highly distinguished and productive authorship of exceptional quality and extraordinary importance to science: for outstanding leadership in assisting ESSA in formalizing its geodetic research program." Mrs. Hotine accepted the award at the American Embassy in London on January 24, 1969. MONOGRAPH Mathematical Geodesy by Martin Hotine *> — JTZ — 7 U.S. DEPARTMENT OF COMMERCE, Maurice H. Stans, Secretary ENVIRONMENTAL SCIENCE SERVICES ADMINISTRATION, Robert M. White, Administrator Washington, D.C., 1969 ESSA MONOGRAPHS Editors William O. Davis, Miles F. Harris, and Fergus J. Wood Consulting Editor Bernard H. Chovitz Editorial Board S. T. Algermissen, Gerald L. Barger, Bradford R. Bean, Wallace H. Campbell, Bernard H. Chovitz, Douglas D. Crombie, Frank A. Gifford, Steacy D. Hicks, George M. Keller, William H. Klein, David G. Knapp, Max A. Kohler, J. Murray Mitchell, Vincent J. Oliver, Feodor Ostapoff, John S. Rinehart, Frederick G. Shuman, Joseph Smagorinsky, Sidney Teweles, Herbert C. S. Thorn, J. Gordon Vaeth, David Q. Wark, Helmut K. Weickmann, Jay S. Winston, and Bernard D. Zetler Editors of the ESSA Monographs, Environmental Science Services Administra- tion, Rockville, Md. 20852, invite readers to submit to them all errors or omissions noted in the text for correction in future reprints of this book. UDC 528:51(021) 528 Geodesy 51 Mathematical (021) Comprehensive manual Library of Congress Catalog Card Number: 73-602618 For sale by the Superintendent of Documents, U.S. (.overnment Printing Office Washington. D.C. 20402. Price $5.50 Foreword In 1963, Martin Hotine completed a distinguished career of government service — both military and civil — in Great Britain. He attained the military rank of Brigadier, and later as a civil servant he was Director of the Directorate of Overseas Surveys and Advisor on Surveys to the Secretary of the Department of Technical Cooperation. In November 1963. he accepted an invitation of Rear Admiral H. Arnold Karo, Director of the U.S. Coast and Geodetic Survey, to join his scientific staff as a research geodesist. When ESSA was formed in 1965. Martin Hotine became a member of the Earth Sciences Laboratories in Boulder, where he remained until his return to England in August 1968. During these 5 years in the United States, Martin Hotine devoted his attention to new concepts in the geodetic sciences and continued the work that led to his recognition as one of the world's foremost authorities on geodesy. To compile scientific thought within a particular specialty of any discipline is never an easy task. Only an individual who has a proficiency in his field gained through years of practical experience and one who is dedicated to the advancement of science would undertake such a difficult task. Martin Hotine was such an in- dividual, and the result of his efforts provides a foundation in basic theory and current thought in mathematical geodesy and another step from which the science of geodesy can progress. ESSA is highly honored and extremely fortunate to be able to include this volume in its monograph series. The purpose of this series is to add authoritative information to the depository of total scientific knowledge. Mathematical Geodesy is such a treatise. Robert M. White Administrator vn Preface This book is an attempt to free geodesy from its centuries-long bondage in two dimensions. This does not mean that any geodesist, from Eratosthenes to modern contenders for the title, has ever considered the Earth to be flat; the two dimensions, such as latitude and longitude, have always been non-Euclidian and have been taken as coordinates on a curved reference surface. It has been usual, nevertheless, to project points from the topographic surface of the Earth to the reference surface and thereafter to work entirely between points on the reference surface. The third dimension of height above the reference surface is, after all, small compared with the mean radius of the Earth; this fact has made it possible to avoid any precise definition of the actual process of projection or of the exact location and orientation of the reference surface in relation to points on the topographic surface. The main process of projecting the line of observa- tion into curves of normal section on the reference surface (usually a spheroid or ellipsoid of revolution), combining these curves into a spheroidal geodesic, and solving geodesic triangles does give sufficiently accurate results from fairly simple formulas over short lines. Unfortunately, the process involves an element of indiscipline which could bring the subject into disrepute; for example, the author's own interest was aroused some years ago by an argument in print between two leading European geodesists on the correct application of Laplace azimuth adjustment, between points not located on the reference surface, which showed that neither geodesist had clearly defined what he meant by a geodetic azimuth at points in space. The classical process could not, in any case, deal with the longer lines of observation in flare triangulation, stellar triangulation, and now satellite triangulation without excessive complication; it is actually simpler to consider the line of observation as a line in three dimensions and to carry out all computations and network adjustments in three dimensions. It can be said that one form of the classical process was first introduced for the reduction of a survey of Hanover, Germany, by the celebrated Karl Fried- rich Gauss who also introduced the differential geometry of curved surfaces. There is little doubt that Gauss, faced with modern geodetic problems, would have antedated Ricci and others by extending his differential geometry to three or more dimensions. The first geodetic application of these extended methods was made in 1949, far too many years after Gauss, by Marussi of the University of Trieste. (See, Marussi (1949), "Fondements de Geometrie Differentielle Absolue du Champ Potentiel Terrestre," Bulletin Geodesique, new series, no. 14, pp. 411-439.) Cartesian coordinates in three dimensional space are not suitable for all geodetic processes. We are led inevitably to consider more general curvilinear systems, and to publish a book requiring the differential geometry of such systems without using the tensor calculus (including vector calculus in index notation) would indeed be an archaism. Unfortunately, very few geodesists have yet studied x Mathematical Geodesy this important branch of mathematics, and the older generation is now unlikely ever to do so. Geodesists are by no means alone in their conservatism. Most new and ad- vanced texts on mathematical physics are still being published in the old dot-and- cross boldface-type vector notation, which is peculiarly suitable for only a very few applications; this notation is much more restricted than the use of index notation even for vectors considered as first-order tensors. Index notation is a practical necessity for tensors of higher order than the second: for the derivation of results, particularly those involving differentiation, which are true in any coordinate system; for generalized curvilinear coordinate systems and other applications requiring a mixture of both vectors and higher order tensors; and for the notion of curved space required not only in relativity but also in such applica- tions as generalized conformal transformations. Nevertheless, physicists have still to acquire the no less difficult dot-and-cross boldface-type vector and dyadic notation for use in the more elementary applications. For more advanced work, they also need index notation which would serve all purposes. The waste of effort involved in using one notation for first-order tensors and an altogether different notation when tensors of higher order are required should be avoided in geodesy, which already requires the use of higher order tensors in quite elementary applications. It is still possible to obtain a master's degree in mathematics at most uni- versities without any knowledge of the tensor calculus, but we may expect less conservatism in the future now that the subject is being taught to undergraduates in some universities and is being included in a growing number of special courses in applied mathematics. Moreover, many simplified texts have been made avail- able since Eddington in 1923 sought a wide English-speaking audience with his Mathematical Theory of Relativity. Part I of this book attempts to introduce tensor calculus to geodesists and to cover the ground required for present and foreseeable future geodetic applica- tions. It has been written only after searching the readily available literature in the hope of recommending instead a single text containing all the required material and written by someone with teaching experience. It is not surprising that none suitable for the purpose could be found among the many excellent books which are now available. Many of these books are naturally written to cover in outline a wide range of applications, and those that specialize are usually relativity-oriented. Moreover, most books on the subject have been written by mathematicians who are compelled to treat the subject rigorously; whereas the geodesist, who has to keep up to date in many other areas, is prepared to take much on trust, and is able to do so because he deals only with such well-behaved functions as Newtonian potentials in free space or with very regular functions suitable as coordinates. Even so, the treatment in Part I, necessarily compressed in a book which is re- quired to cover even in outline the entire ground of theoretical geodesy, may prove too difficult for the beginner. It is recommended that he read a more ele- mentary account of the broad basis of the subject first; for example, the first 83 pages of Spain's (1953) Tensor Calculus or Chapters 2 and 5 of Lawden's (1968, 2d ed.) An Introduction to Tensor Calculus and Relativity. It is always better to read two books on a subject, one more general than the other, instead of one specialized book twice. Much, but by no means all, of the subject matter of Part I is covered by McConnell's (1931) Applications of the Absolute Differential Calculus (also published in a 1957 Dover edition as Applications of Tensor Analysis). The reader who requires a more elegant and rigorous treatment — and some geodesists demand rigor— might read Guggenheimer's (1963) Differential Geometry. Part I was first drafted as a collection of formulas to save the reader from the annoyance of continual reference to several other books and also to include some Preface xi formulas which are not to be found readily, if at all, in books or papers. By the time the formulas and the notation had been explained, the manuscript had reached perhaps half of its present size; it was then decided to derive, or at least indicate how to derive, the results and to expand the explanation of some points likely to prove difficult. In the writer's experience, for example, most geodesists shy at the notion of covariance and contravariance, which seems to be the counterpart of the Euclidian pons asinorum, perhaps because geodesists usually acquire some knowledge of statistics in which covariance means something quite different. In these days of aids to rapid reading, the expert need lose no time over such elementary exposition, but he is, nevertheless, advised to skim through Part I, if his knowledge is rusty, to get the feel of the notation and the conventions. The temptation to include indefinite metrics, requiring little more explanation and leading straight into relativity, has been resisted. Apart from measurements based on the position of stars, geodetic measurements have not yet been made beyond the Moon and relativistic corrections for high velocities in the solar system can be, if necessary, applied quite simply without much knowledge of relativity theory. (See, for example, Walker, in a letter to Nature, v. 168, December 1, 1951, pp. 961-962.) The methods used in relativity, like the tensor calculus itself, may, nevertheless, become important to the research geodesist who, if he knows or acquires Part I will have no difficulty in extracting keen enjoyment from Synge and Schild's (1949) Tensor Calculus, to prepare himself for Synge's two master- pieces on relativity. Some consideration has also been given to including in Part I a short account of more general deformations of space than the conformal transformations of Chapter 10. This will come, together perhaps with some geodetic excursions into non-Riemannian geometry, but the geodetic application of this subject is still young and publication in book form would probably be premature. Meanwhile, some account has been given in §30-19 of a method of systematically deforming one member of a general family of surfaces into another member of the family for a particular application. Part II deals with coordinate systems of special interest in geodesy. In Chapter 12, the properties of a general class of three-dimensional systems are developed from a single-valued, continuous and differentiable scalar N which serves as one coordinate, while the other two coordinates are defined by the direction of the gradient of N. In Chapters 15 through 18, the scalar N is restricted to provide simpler systems, whose properties can then be derived at once from the general results of Chapter 12. Transformations between members of the general class for different values of /V are treated in Chapter 19. Another advan- tage of treating the subject in this way is that the scalar N can also be given a physical meaning (for example, the gravitational potential in Chapter 20) so that Chapter 12 also provides the geometry of the gravitational field. In case it should be required to transfer the values of point functions from a point in space to a particular /V-surface, which is the rigorous counterpart of sev- eral operations of classical geodesy, methods of transfer along the isozenithals (the /V-coordinate lines) and along the normals to the TV-surfaces are worked out for each coordinate system in Part II, following a general discussion in Chapters 13 and 14. The process is connected intimately with Gaussian spherical representa- tion, which is developed in this context, following a more general discussion in Chapter 11, and is extended to nonspherical representation in Chapter 13. Such methods of projection are seldom any simpler than three-dimensional methods, although they are put to occasional special use in Part III, but it is as well that the process should be more fully understood in the future. Part III deals with the main geodetic applications of the mathematics in Parts I and II. Geometry, which used to mean literally the science of Earth measure- xii Mathematical Geodesy merits, is no longer confined to geodesy, but there is, nevertheless, still a con- siderable overlap, more so perhaps than the overlap with physics, and we cannot expect a rigid division between the two subjects. For example, the differential geometry of Chapter 12 contains all the metrical properties of the gravitational field used in geodesy if we restrict one coordinate in accordance with a physical law. As another example, the transformation between two members of a class of coordinate systems in Chapter 19 includes the process of switching between geodetic and astronomical systems. Part III simply attempts to show how these mathematical concepts can be used today in attacking the main problems in geodesy. The treatment is not complete; for example, nothing is included on the formation and solution of normal equations in least-square adjustments, which are adequately treated in existing literature. Nor does the treatment cover all pos- sible applications; few geodesists have so far worked on these fines, and future developments may be considerable. For example, the reader cannot expect to learn all about so-called physical geodesy (which in fact is again mostly geometry) from Chapters 29 and 30, although it is hoped that he will acquire a clear idea of the basic theory which will enable him to follow the considerable literature of the subject more easily and critically. The same applies to satellite geodesy in earlier chapters. Manipulative skill in any branch of mathematics cannot be obtained by reading alone. In most cases, the work has been shortened by omitting several steps leading to a result, but full references are given to enable the reader to fill in the missing steps, if he so desires. It is hoped that this procedure will serve the purpose of the examples and problems in textbooks which would be quite out of place here. The experts, no doubt, will omit the whole procedure and will take the results on trust. References to other publications are given only as required by the text. They do not provide anything like a complete bibliography or any indication of priority or relative importance. For example, Marussis classical paper noted earlier in this preface is referenced only once in the text, although it can be considered the foundation stone of modern theoretical geodesy. However, the reader who looks at the references, particularly those to books, will soon find that he has access through them to a considerable bibliography. The question of credits and priorities is particularly difficult in this subject. Classical results are given a nametag to help identify them in the literature, but the name is that normally associated with the result in English, without attempt- ing to assess priority between, for example, Gauss, Green, and Ostrogradskii. Some of the named results seem almost trivial when derived by modern methods, but it is hoped this will not dim the luster of great men who unearthed them with less serviceable tools. Credit is also given, when known, for particular recent results, but such credits are few because not many geodesists as yet have worked in this area. To offset what must seem like cavalier treatment, no priority is claimed for any results, although it is believed that some are new, either in con- tent or in presentation. The title of the book requires some explanation. An attempt has been made to cover only the basic mathematical discipline of geodesy, excluding such spe- cialized matters as routine computer programs and including only such references to instrumentation and field (or laboratory) procedures as may be necessary to a full appreciation of the underlying theory. The book accordingly bears much the same relation to the whole of geodesy as numerous books entitled "Mathe- matical Physics" do to the whole of physics. Various alternatives have been con- sidered and rejected; for example, the title would be some variant of "Higher Geodesy" if published on the European continent, but the content of the book is quite different from any other book bearing that title. Preface xiii The terminology and symbolism used in the book cannot be expected to com- mand universal acceptance. For example, there is a growing tendency in geodesy to call an ellipsoid of revolution simply an ellipsoid and to reserve the term sphe- roid for an equipotential surface of the standard gravitational field. This conven- tion can cause confusion whenever reference is made to mathematical literature in English where a spheroid is defined geometrically as an ellipsoid of revolution and an ellipsoid in general means a quartic with three unequal axes; for example, the treatment of spheroidal and ellipsoidal harmonics in Hobson's standard work on the subject is based on this definition, which is clearly stated in the Van Nostrand (1968, 3d ed.) publication, Mathematics Dictionary, edited by James and James. In a mainly mathematical book, it has accordingly been decided to retain the mathematical convention, which incidentally is also used by most English-speaking geodesists. In much the same way, the physical sign convention has been used for a Newtonian potential, although the fact that the potential is invariably negative in terrestrial applications has led most geodesists to change the sign. No good can come through willfully discarding scientific conventions universally accepted in a parent subject which has every right to prescribe the convention. Adoption of the physical convention for potential not only facilitates reference to the literature of physics, but also accords better with the geometrical basis of this book. Most geodesists use the symbol A for longitude. However, in a book using vectors, there is an overriding need for an orthogonal triad A,-, fx, , v r frequently used in mathematical literature. In the geodetic applications, A r is associated with longitude, but it is not the gradient of the longitude as the use of A for the scalar longitude would imply. The symbol a>, usually associated with a rotation, is accordingly used for longitude. Whenever possible, however, the symbolism most generally adopted by the best literature in a particular branch of the subject has been used to facilitate wider reading although this often results in using the same symbol for different purposes in different chapters. The Index of Sym- bols at the end of the book indicates the general use of a symbol, any departure from which is invariably noted in the text. For example, a and f3 are generally used for azimuth and zenith distance, which differ in different coordinate systems, but the context will show which coordinate system is being used. The same applies to latitude and longitude, and this arrangement enables us to dispense with special symbolism for particular coordinate systems, such as spherical (geo- centric) and spheroidal (geodetic) systems. Following standard mathematical conventions in English, right-handed systems are used throughout the book, and sign conventions are adopted to conform. In general, some warning or com- ment is given in the text whenever there is a departure from standard mathe- matical or physical conventions in the geodetic literature; for example, the use of left-handed systems imported from photogrammetry into satellite triangula- tion. To facilitate reference, summaries of main formulas are collected as a Sum- mary of Formulas at the end of the book. In some cases, a particular chapter suggests a particular arrangement; for example, some formulas in the summary for Chapter 17 are obtained by specializing the results of earlier chapters at sight and are not given in the text of Chapter 17, although they do apply to the subject matter of Chapter 17. The best way of using the Summary of Formulas is to look first at the chapter headings or subheadings for the required subject matter. Each equation in the index carries a reference to the text which gives the derivation and sym- bolism. Back references in the text are always to the text, but a reference to the Summary of Formulas may be sufficient and quicker; however, if the back refer- ence is not given in the index, it will be necessary to refer to the text. It is difficult to make adequate acknowledgment covering a lifetime of study, discussion, and collaboration. The author's main source of inspiration in the sub- xiv Mathematical Geodesy ject of this book has been Professor Antonio Marussi of the University of Trieste, not only for the range and originality of his ideas but also for continual advice and encouragement. The book and its writer owe much to the two official reviewers, Mr. Bernard H. Chovitz of the Earth Sciences Laboratories of ESSA and Professor Ivan I. Mueller of the Ohio State University, for careful reading and checking and for many improvements. In addition, specialist reviews and information have been freely provided by Professor Arne Bjerhammar and his associates of the Royal Institute of Technology, Stockholm; Mr. Robert H. Hanson of the Earth Sciences Laboratories of ESSA; Dr. Karl-Rudolf Koch of the Ohio State University; Professor Helmut Moritz of the Technical University of Berlin; Mr. F. Foster Mor- rison of the Earth Sciences Laboratories of ESSA; Mr. Allen J. Pope of the Coast and Geodetic Survey of ESSA; Professor Erik Tengstrom of the University of Uppsala; Dr. Moody C. Thompson of the Institute for Telecommunication Sciences of ESSA; and Mr. John Wright of the Directorate of Overseas Surveys of Great Britain. None of these distinquished men, especially neither of the official re- viewers, is responsible for any remaining errors and omissions. The difficult and unrewarding task of editing such a specialized book has been successfully undertaken throughout by Mr. John R. Bernick. The index has been compiled by Jean S. Campbell. The production coordination of the publication has been accomplished by Mr. Edward W. Koehler and the manuscript has been marked for printing by Miss Lila Paavola and Mrs. Helen Hoener. Last, but far from least, the manuscript has been typed and retyped most expeditiously and efficiently by Mrs. Nancy Durazzo and Mrs. Judy Shore. August 1968 Martin Hotine Contents Page Foreword vii Preface ix PART I Contents 1 Chapter 1. Vectors 3 Chapter 2. Tensors 9 Chapter 3. Covariant Differentiation 17 Chapter 4. Intrinsic Properties of Curves 21 Chapter 5. Intrinsic Curvature of Space 25 Chapter 6. Extrinsic Properties of Surfaces 31 Chapter 7. Extrinsic Properties of Surface Curves 39 Chapter 8. Further Extrinsic Properties of Curves and Surfaces 43 Chapter 9. Areas and Volumes 49 Chapter 10. Conformal Transformation of Space 55 CHAPTER 11. Spherical Representation 63 PART II Contents 67 Chapter 12. The (q),4>,N) Coordinate System 69 Chapter 13. Spherical Representation in (a>, , N) 89 Chapter 14. Isozenithal Differentiation 93 Chapter 15. Normal Coordinate Systems 103 Chapter 16. Triply Orthogonal Systems 113 CHAPTER 17. The (to, 4>, h) Coordinate System 117 Chapter 18. Symmetrical (a>, , h) Systems 125 Chapter 19. Transformations between A/-Systems 131 XV xvi Mathematical Geodesy PART III Contents 139 Chapter 20. The Newtonian Gravitational Field 143 Chapter 21. The Potential in Spherical Harmonics 153 Chapter 22. The Potential in Spheroidal Harmonics 187 Chapter 23. The Standard Gravity Field 199 CHAPTER 24. Atmospheric Refraction 209 Chapter 25. The Line of Observation 227 Chapter 26. Internal Adjustment of Networks 239 Chapter 27. External Adjustment of Networks 261 CHAPTER 28. Dynamic Satellite Geodesy 269 CHAPTER 29. Integration of Gravity Anomalies — The Poisson-Stokes Approach 309 Chapter 30. Integration of Gravity Anomalies — The Green-Molodenskii Approach 327 Index of Symbols 347 Summary of Formulas 353 General Index 399 CONTENTS Part I Page Chapter 1 — Vectors 3 Cartesian Vectors 3 Vectors in Curvilinear Coordinates 4 Transformation of Vectors 7 Chapter 2 - Tensors 9 General Rules 9 Tensor Character 11 The Associated Metric Tensor 12 The Permutation Symbols in Three Dimensions 13 Generalized Kronecker Deltas 13 Vector Products 14 The Permutation Symbols and the Met- ric Tensor in Two Dimensions 15 CHAPTER 3 - Covariant Differentiation 17 The Christoffel Symbols 17 Covariant Derivatives 18 Differential Invariants 19 Rules for Covariant Differentiation 19 Chapter 4 — Intrinsic Properties of Curves... 21 Curves in Three Dimensions 21 Curves in Two Dimensions 22 Chapter 5 — Intrinsic Curvature of Space... 25 The Curvature Tensor 25 Locally Cartesian Systems 26 Special Forms of the Curvature Tensor. . . 26 Curvature in Two Dimensions 27 Riemannian Curvature 28 CHAPTER 6 — Extrinsic Properties of Surfaces. Forms of Surface Equations The Metric Tensors Surface Vectors The Unit Normal Surface Covariant Derivatives The Gauss Equations The Weingarten Equations The Mainardi-Codazzi Equations The Gaussian Curvature CHAPTER 7 — Extrinsic Properties of Surface Curves The Tangent Vectors Curvature Torsion Curvature Invariants Principal Curvatures Chapter 8 — Further Extrinsic Properties of Curves and Surfaces The Contravariant Fundamental Forms . . Covariant Derivatives of the Funda- mental Forms Relation Between Surface and Space Tensors Extension to Curved Space CHAPTER 9 — Areas and Volumes Elements of Area and Volume Surface and Contour Integrals Volume and Surface Integrals Page 31 31 32 32 33 33 35 35 35 36 39 39 39 10 41 41 43 13 U 15 15 49 49 V) 51 306-962 0-69— 2 Mathematical Geodesy CHAPTER 10 — Conformal Transformation of Page Space 55 Metrical Relations 55 The Curvature Tensor in Three Di- mensions 56 Transformation of Tensors 56 Curvature and Torsion of Corresponding Lines 57 Transformation of Surface Normals 58 Transformation of Surfaces 59 Geodesic Curvatures 60 Extrinsic Properties of Corresponding Surfaces in Conformal Space 60 Chapter 10-Continued Page The Gauss-Bonnet Theorem 61 Chapter 11 — Spherical Representation 63 Definitions 63 Fundamental Forms of the Surfaces 63 Corresponding Surface Vectors 64 The Principal Directions 64 Scale Factor and Directions Referred to the Principal Directions 65 Christoffel Symbols 65 Representation of a Family oi Surfaces. . . 66 CHAPTER 1 Vectors CARTESIAN VECTORS 1. Geometrically, a length in a certain direction defines a vector OP. In ordinary three-dimensional space, we can, for instance, take as the origin of a rectangular Cartesian coordinate system and specify the vector completely by the three coordi- nates of P. Or, if we wish to define a number of vectors at different points in the space, we can take a fixed origin and define the vector by the differences in rectangular coordinates over the length OP, that is, by the orthogonal projection of OP on the co- ordinate axes. These three quantities, known as the rectangular Cartesian components of the vector, will depend on the choice of coordinate system; but the sum of their squares will be the square of the length OP, which does not depend on the coordinate system. If the vector is of unit length, or if we divide the components by the length, the components become the direction cosines of the direction OP, and the vector is known as a unit vector. 2. The matter becomes more complicated when we consider inclined coordinate axes. For the present, we shall continue to consider a Cartesian system; that is, a system in which the coordinates are actual lengths along straight coordinate axes. For ease of illustration, we shall consider a vector OP in relation to coordinate axes OX, OY (fig. 1) in two dimensions, but similar conclusions will apply in three or more dimensions. We can still specify the vector by its orthogonal projections OQ. OR on the coordinate axes, in which case the components of a unit vector in the direction OP will still be the direction cosines of OP. We call these covariant components and write l^=OQ = OP cos 0, 1.01 k = OR = OP cos 02, making use of index notation l\, l-> for the com- ponents. 3. Alternatively, we could specify the vector com- pletely by taking the differences in coordinates OS, OT as components, which we shall call the contravariant components. We distinguish them from the covariant components by using super- script indices and write n=OS = OP sin 0,/sin (0, + 2 ) 1.02 [i = OT= OP sin 0,/sin (0, + d>). Figure 1. We can no longer square and add either set of components as a means of obtaining the length or magnitude of the vector, but the above formulas I Mathematical Geodesy lead at once to the result /'/, +PU = Of*. As a form of shorthand whose value will become more apparent later, we can write this as 1.03 l a l a =OP 2 in which we use the summation convention. When- ever a superscript and a subscript index are the same, we assume that this index takes all possible values (in this case a=l, 2), and the results are then summed. 4. Next, suppose we have two vectors OL, OM (fig. 2), and that the angles giving the direction of equally well in three dimensions. A simpler method is to assume that Figure 2. OM are distinguished by overbars. We have Pm ] + I 2 m 2 =l a m a = OL ■ OM (sin 0> cos 0, + sin 0, cos 2 ) sin (0, + 0,) = OL ■ OM cos 0, and we can obtain the same result from l a m a . We call this the scalar product of the two vectors and write 1.04 l a r l a m a = OL -OM cos d. Or, to phrase this in words, the scalar product is the product of the two magnitudes and of the cosine of the angle between the two vectors. The scalar prod- uct of two perpendicular vectors is clearly zero. Also, Equation 1.03 is a special case of Equation 1.04 in which the two vectors coincide. 5. The reader with an inclination for spherical trigonometry can verify that Equation 1.04 holds I' m, (r=l, 2, 3) has the same value in all coordinate systems — or, in other words, is invariant under coordinate transfor- mations—as we found l a m a to be in two dimensions, and to evaluate the expression in a special coordi- nate system. We choose OX to coincide with OL and leave OY, OZ arbitrary. In that case, l x — OL and / 2 , I 3 are both zero because the y- and z-coordi- nates do not change in the direction OL. Conse- quently, we have l'm r = OL ■m 1 = OL-OM cos 0. By choosing a coordinate axis along OM, we find that l,-m r is the same so that we have 1.05 /' l r m r =OL-OM cos 0. 6. Throughout this book, we shall adopt Greek indices for the two-dimensional components of vectors and Roman indices for three dimensions. The index notation for a vector /' need not be con- fused with the rth-power of a quantity /. The context will usually distinguish between the two without explanation, but in cases where confusion could arise, we shall use and shall describe special nota- tion for a power index. In the same way, numerical subscripts will often be used to distinguish certain quantities. Covariant vectors will usually have a literal subscript; but if a numerical subscript has to be used for a particular component, attention will, if necessary, be called to the fact. 7. It will be clear from the definitions of the covariant and contravariant components of a vector that the two sets of components are equal in rectan- gular Cartesian coordinates, but are not equal in inclined Cartesian coordinates. By introducing the two sets of components, however, we have been able to ensure that such results as Equation 1.05 apply in both rectangular and inclined Cartesian coordinates. VECTORS IN CURVILINEAR COORDINATES 8. We have now to generalize the matter still further by considering curvilinear coordinate sys- tems. Through each point in some region of three- dimensional space, there will still be three unique coordinate lines along each of which only one co- ordinate varies, the other two being constant; but the coordinate lines may be curved as well as Vectors inclined, and will not, as a rule, be parallel to the directions of the corresponding coordinate lines at other points. The space itself may be curved, like the surface of a sphere in two dimensions, and in that case, the space can only be described in curvilinear coordinates; we should be unable to find a Cartesian system which would give the posi- tions of points in an extended region of the space. Finally, a curvilinear coordinate will no longer necessarily be an actual length measured along a coordinate line, as in the case of Cartesian coordi- nates, although lengths and coordinates must obviously be related in some way because a dis- placement over a given length in a certain direction must involve a unique change in coordinates. 9. This relation, which may vary from point to point, is expressed by the metric or line element of the space; the square of an element of length (Is in a small neighborhood of a point can be expressed in terms of the changes in coordinates dx r over the element of length by a relation of the form 1.07 1.06 ds 2 = g„ dx'dx* (r, 5 = 1 , 2 , 3). We assume that the summation convention is used in this formula, which accordingly may contain nine coefficients gn in three dimensions to go with all possible combinations of the coordinates. We do not need, however, more than six and can take g,s as symmetrical so that we have gw — gzi, for example. We can then expand Equation 1.06 as ds 2 = gt i(dx x )- + gn(dx 2 ) 2 + #«(<& 3 ) 2 + 2g n dx l dx 2 + 2gndx l dx* + 2g 23 dx 1 dx i . Throughout this book, we shall use only what are known as positive-definite metrics; that is, for any real and nonzero displacement dx r , the value of the quadratic form in Equation 1.06 is positive and not zero. Only in this way can the form represent the square of a real element of length. Relativity metrics in four dimensions, on the other hand, are usually indefinite, in the sense that ds 2 may be zero without all the dx r being zero. 10. The numbers g rs (totaling nine, of which six may have different values) will vary continuously from point to point, but will be defined uniquely at each point for a particular coordinate system; in other words, they will be functions of the coordi- nates x r , or functions of position. This array of numbers is known as the metric tensor, for reasons which will appear later. In rectangular Cartesian coordinates, the metric must reduce to the Pythag- orean form ds- = (dx)- + (dy)- + (dz)- = (dx 1 r' + Uix 2 ) 2 + (dx :i ) in which case we have g„=\ (r=«); (r^s). In inclined Cartesian coordinates, the g rs (r ^ 5) are functions of the angles enclosed by the coordinate axes and are therefore the same at all points, but are not zero. 11. As a simple example of curvilinear coordi- nates, we take spherical polar coordinates (to, c/>, r), defined by x = r cos (/> cos to y— r cos sin to z = r sin )dto' 2 + r 2 c/c/> 2 + dr 2 , and the components of the metric tensor are gu = r 2 cos 2 (f> ; g 22 = r 2 ; g 33 = 1 g rs = (r^s). The co-coordinate lines, along which c/> and r are constant, are circles parallel to the .vy-plane and centered on the z-axis; the (^-coordinate lines are circles centered on the Cartesian origin whose planes contain the z-axis; and the r-coordinate lines are radial lines from the Cartesian origin. Alter- natively, we can say that the co-coordinate surfaces (over any one of which o> is a constant) are planes containing the z-axis, the ^-coordinate surfaces are cones whose common axis is the z-axis. and the r-coordinate surfaces are spheres centering on the Cartesian origin. In a Cartesian system, all the coordinate lines would be straight and all the co- ordinate surfaces would be planes. 12. Over short distances, we can, nevertheless, consider that the coordinate lines are straight in a curvilinear system. By analogy with the Cartesian definition, we still can say that a small change in coordinates dx' (r=l,2,3) represents the three contravariant components of a small vector of length ds, and that in the limit, the ratios dx r ds 1.08 (r=l,2, 3) are the contravariant components of a unit vector Mathematical Geodesy l r . Although we are no longer dealing with finite lengths, it is easy to see that this definition of a contravariant unit vector agrees with the Cartesian definition. We can also define a nonunit vector of magnitude A, in the same direction as the unit vector l r , as 1.09 1/ = \l r without contradicting the Cartesian conception, although we may no longer be able to interpret A as a finite length. 13. The covariant components, however, need lurther consideration because they were defined in Cartesian coordinates as lengths along the axes. By dividing Equation 1.06 by ds 2 and substituting Equation 1.08, we have 1.10 grsl r l S =h To preserve the Cartesian conception of a covariant vector as far as possible, we may use Equation 1.05 and write for a unit vector 1.11 W s =i. However, if both Equations 1.10 and 1.11 are to hold for all directions at a point, that is, for arbi- trary values of the contravariant components I s , we must have 1.12 k = grsl r as the definition of the covariant components of a unit vector. From Equation 1.09, we have also g rs L r L s = X l g rs l r l s =K 2 . However, to preserve the Cartesian conception corresponding to Equation 1.05, this must equal L,L r so that a general covariant vector can be written as 1.13 L, = g rs L s = A/,.. Comparing this with Equation 1.09, we see that multiplication by g rs and use of the summation convention have lowered the indices of the vector Equation 1.09. It is easy to see that the same opera- tion would lower the free (not summed) index in any vector equation. 14. We now consider whether the above definition of a generalized covariant vector agrees completely with the Cartesian conception, in the sense that a Cartesian system provides a special case. For ease of illustration, we shall again consider the case of two dimensions. In figure 3, we take a small dis- placement of length ds, made up of displacements of length va\ X dx x and va^dx 2 along the coordinate axes, obtained, respectively, by making dx 2 = and (/x' = in the metric ds 2 = an(dx> ) 2 + 2a v ,dx l dx 2 + a 2 >(dx 2 ) 2 . From the figure, we have at once ds 2 = au(dx l ) 2 + a 2 >(dx 2 ) 2 + 2Va~7iVa^>dx ] dx 2 cos (0, + 2 ). By comparing these two forms of the metric, we have 1.14 cos (0, + &) = «i2/Va„a 22 . Figure 3. Using the generalized definition of the covariant components l a of OP and evaluating dx l /ds, etc., from triangles in figure 3, we have /. = a,^ = a„ ( dx'/ds )+a v >( dx 2 /ds ) 1.15 Von sin 6-2 sin (0, + 2 ) «n cos + a v . sin 6\ /gZ sin ( 0, + d-2 ) on substitution of an from Equation 1.14 and expan- sion. In the same way, we have 1.16 I, an cos 02- If the coordinate system were Cartesian (fl n = « 22 =l), this would agree exactly with Equations 1.01 for a unit vector. We can obtain the same result in three dimensions. We can accordingly claim to have generalized the conception of contravariant and covariant vectors for a general curvilinear coordi- nate system and to have shown that previous results in Cartesian coordinates are merely special cases. Vectors 15. It will be noted that we use the symbol (tan (a,/3=l,2) for the metric tensor in a two-dimensional space instead of the three-dimensional adopt this convention as standard. We shall 16. We are now able to conclude in much the same way that Equation 1.05 holds equally well in curvilinear coordinates. If L'\ M r are two vectors in the directions of unit vectors /'. m' and of magni- tudes A, fM, we can write L,M' = g rs L r M s = \/xg rs l r m s = kfjc/'m, 1.17 = K/jl. cos 6 where 6 is the angle between the two vectors. This generalizes the scalar product of two vectors, which again is zero for two perpendicular vectors. TRANSFORMATION OF VECTORS 17. We now consider the effect on the components of a vector when the coordinate system is changed. We shall denote the new coordinates x r and the new components by overbars. For a contravariant unit vector, we have at once 1.18 J r _dx^_ (W dx" _ dx? ds dx s ds dx s in which, of course, the summation convention is applied to the index 5, and we have used the chain rule of elementary calculus. The same formula clearly will apply to nonunit vectors L'\ L s . 18. In the case of a covariant vector, we form the scalar product with an arbitrary vector A'. The result is an invariant, which has the same value in either coordinate system, because it depends only on the magnitudes of the two vectors and the angle between them so that we may write 7 r A '' = /,. A s = /,. (dx s /dx r )A r , using Equation 1.18 for the vector A r ._ Since this relation holds for any arbitrary vector A'\ we must have 1.19 8 X s dx r which is the required transformation. The same formula will apply to nonunit vectors L r , L s . 19. We could define a vector as a set of three quantities (in three dimensions) which transform in this way. To illustrate the point, we take a con- tinuous differentiable scalar N; that is, a real num- ber which has a unique value at all points of a region of space and can therefore be considered a function of the coordinates. The scalar /V is also an invariant whose value at a particular point is the same what- ever the coordinate system. Most physical quan- tities, such as potential or gravity, are scalar invariants. We differentiate N with respect to each coordinate x r and write N r = dN/dx r . But because N is an invariant {N = N), we can write - _ SN _ dN _ dN dx s _ fix" A • £* W / V /• _ — ~ — n — z dx r dx r dx" dx r dx r v s so that N r transforms like a covariant vector and can be taken as a covariant vector. It is called the gradient of N. Because TV is differentiable, there will be some directions /'' in which N is constant so that we have N r l r =dN/ds = 0. The gradient of N is accordingly perpendicular to all such directions. If /V is constant over a surface, its gradient is perpendicular to all surface direc- tions at a point and is therefore in the direction of the unit normal v, to the surface. We can then write 1.21 N r = nv r where n is the magnitude of the gradient vector. In this discussion, we have, of course, assumed that at least some of the derivatives of /V exist, even though N itself may be zero; otherwise, the gradient of N and therefore v, would be undefined. The assumption is justified in the case of surfaces dealt with in this book. 20. If, in three dimensions, we know the com- ponents in both coordinate systems of three mutually perpendicular unit vectors A.,, fJ-r. v r . we can derive the set of transformation factors from the formulas dx r ldx s = X r X s + fx'fJLs + xFvs 1 .22 dx r ldx s = \ r K + n r jZ s + v r v s . To verify these formulas, we multiply the first equa- tion by A v , for example, use Equation 1.11 and the fact that the scalar product of two perpendicular vectors is zero, and so recover Equation 1.18. 8 Mathematical Geodesy 21. We can also show that the scalar product of any two vectors is an invariant, 1.23 74 r B< = d ^A^B<=8l4 >i B< = A = Ajf s B'C s = A'^Bi'C", so long as we do not confuse them with the free indices (r in this example). 8. It is evident that any scalar formed by tensor contraction will be an invariant, whose value will be the same in any coordinate system. 9. A tensor is said to be symmetric in two indices, both upper or both lower, if it remains the same on interchanging those two indices. For example, if At = A r then the tensor is symmetric in the second- and third-covariant indices. If its value remains the same but the sign changes, for example, B> xlu = -B< xu „ then it is said to be skew-symmetric or antisymmetric in the two indices. These properties are retained on change of coordinates because, for example, all components of the tensors (A^-A^ t ) or (fl;„, + £;;„) are zero in one coordinate system and must there- fore be zero in any other. 10. Any second-order tensor can be expressed as a sum of a symmetric and a skew-symmetric tensor, as is evident from the identity A ,„ = ±(A „ +A sr ) + \{A ,„ - A sr ), the first tensor within parentheses being symmetric and the second being skew-symmetric. 11. If we contract the product of a symmetric tensor A rs and a skew-symmetric tensor 5 jrs on the symmetric and skew-symmetric indices, the result will be zero because B> rs A r8 = — BJ S 'A , T = — BJ rs A rs on interchanging the dummy indices so that we have B^An-O. 12. The relations, Equations 2.04, 2.05, and 2.06, are examples of tensor equations. If we take any such equation relating the components of tensors in one system of coordinates and multiply across by the transformation factors for the free indices as was done, for example, with Equation 2.04, we see at once that the same equation holds between components in the transformed coordinate system. In other words, if a tensor equation is true in one coordinate system, it is true in any coordinate sys- tem. This fact is of fundamental importance in all applications of the subject, particularly the physical applications, because a physical law must, from its very nature, be independent of a man-chosen coordinate system and so is best expressed in tensor form. We can very often set up a tensor equation in a simple coordinate system, for instance Car- tesian, and immediately can assert that it is true in a complicated system, whereas it would be very difficult to find it or to prove it in the complicated Tensors 11 system alone; we have merely to make quite sure that all the terms in the equation are tensors. TENSOR CHARACTER 13. Tests for tensor character are for this reason most important. Ultimately, these must require the set of quantities in question to obey the transforma- tion law, but we can derive some simple rules to avoid having to resort to the transformation law in each case. If, for example, the given set of quan- tities form an invariant when contracted to a scalar with arbitrary nonzero vectors, then it will be a tensor. In that case, we have, for example, a rs A r B s = a rs A r B s = a jk AW k in which a rs is the set under test and A'\ B s are arbitrary vectors. Transforming the vectors, we have _ dx J dx K '\ «rs-ajk——)A >B » = 0. dx' dxr / In three dimensions, this is an equation with nine arbitrary coefficients A l B 2 , etc., connecting the nine components of the matrix within parentheses. Nine or more of these equations containing different values of the arbitrary coefficients can only be satis- fied if each component of the matrix within paren- theses is zero, that is. a r . *jk Bx> dx k dx r dx s which proves the tensor character of a rs - We could not say this if A' , B* were the same vector because we should have then only six independent coeffi- cients connecting the nine components of the matrix. We could, however, interchange the indices r, 5 and add the result to provide an equation of the form ^ _ dxJ dx k _ dx J dxJ V ' dx r dx" ] dx s dx r A r A s = in which there are now six distinct components of the matrix and six arbitrary coefficients (A 1 ) 2 , A 1 A'-, etc. We now can say that a rs + a sr = a.jk t~z dxJ dx" . _ dxJ dx k (Ojk dx r dx s dx* dx r dxJ dx k dx r dx s ik}) on interchanging the dummy indices j, k in the last term. This shows that {a rs + a S r) is a tensor, and so is a rs if a rs and dji; are symmetric, that is, if a r s = a sr in all coordinate systems. In that case, a rs is a tensor if it forms an invariant with only one arbitrary nonzero vector. 14. It is evident from the working that, instead of two arbitrary vectors A'\ B\ we could equally well have used an arbitrary tensor C rs ; and that this could be an arbitrary symmetric tensor when a rs is sym- metric in all coordinate systems. Moreover, it is not necessary that the operation of contraction should result in an invariant. It is sufficient if contraction with an arbitrary vector results in a tensor, but the proof of this, on much the same lines as above, is left to the reader. 15. We must now prove that the metric tensor is in fact a tensor. From Equation 1.17, we can say that if L r , M s are arbitrary vectors, we have g rs L r M s = g rs L r M s because the magnitudes of the vectors and the angle between them are obviously unaffected by the choice of coordinate system. Therefore, #,„ forms an invariant with any two arbitrary vectors (even though not the same invariant for different vectors), and is accordingly a tensor. Again, the square of the line element ds 2 is clearly independent of the coordinate system so that g ls dx r dx s is an invariant for an arbitrary small vector dx r . Because g rs is symmetric in all coordinate systems, it is therefore a tensor. Yet again, we have from Equation 1.13, in the case of an arbitrary vector L r , g rs L s =L r , and this again shows that g rs is a tensor. 16. The Kronecker delta is a mixed tensor because dx" dx* _ dx" dx r _ dx" __ j, s dx r dx," ~ dx r dx" ~ dx" ~ 9 straight from the transformation law. 17. Now suppose that we have a mutually or- thogonal triad of unit vectors (V, fi r , v r ) and con- sider the tensor X r \ s + fX'/JLs+P'l's. In rectangular Cartesian coordinates whose axes 12 Mathematical Geodesy are in the direction of these vectors, their com- ponents are V or \ s (1,0,0) fx r or fi s (0, 1, 0) v r or v s (0, 0, 1), and we can see in this Cartesian system that 2.07 V'\, + ix'ix* + v r v s = 8J . But this is a tensor equation because we have seen above that the right-hand side is a tensor and the left-hand side is formed by the multiplication of vectors. Consequently, this equation is true in any coordinates for any orthogonal triad of vectors. THE ASSOCIATED METRIC TENSOR 18. If, in the same Cartesian system, we con- sider the tensor k,k s + /U,r/X s + VjVs, we find that it is equivalent to the metric tensor grs, which in this system is unity for r — s and is zero for r # s. Consequently, we can say that the tensor equation 2.08 g rs = k r k s + IXrjXs + V, Vs is true for the metric tensor in any coordinates for any orthogonal triad. If we know the components of such a triad in any coordinate system, we can find the components of the metric tensor in the same coordinates at once. 19. Using the same triad of unit vectors, we now inquire what meaning should be attached to the tensor 2.09 k'k' + fx'ix' + v r v' If we multiply Equations 2.08 and 2.09 and remem- ber that the vectors are unit perpendicular vectors so that k'Ki = 1, A''/u.,=0, etc., we have g rt g rs = A'A.s- + fi'fJLs + v'v x = 8L Next, we multiply this equation by G ks , the cofactor of gk S in the determinant formed by the components of the metric tensor which we shall denote by g. Using the ordinary rules for expanding a deter- minant and applying the summation convention, we then have so that 2.10 g r '8 k r g=8 t sG ks = G k S*' = G*7* which enables us to calculate all the components of this tensor from the components of the metric tensor. We see from Equation 2.09 that g*' is a tensor and is symmetric. It is called the associated or conjugate metric tensor. We can easily show that the determinant of the associated tensor is 1/^. In deriving these results, we have assumed that g is not zero. It can be shown ' that in the case of the positive-definite metrics used throughout this book, g is positive and never zero. 20. We can use the associated tensor to raise the index of a vector and to determine its contravariant from the covariant components in the same way as we use the metric tensor to lower the indices. An arbitrary vector L r , whose Cartesian components relative to the axes (k r , fx, -, v r ) are (a, 6, c), can be written as L, — ak r + bfJL, + cv r or L r =ak r + b/jL r + cv'\ both of which are vector equations true for any coordinates. If, in a general coordinate system, we multiply the first of these equations by Equation 2.09, we have 2.11 g rl L r =ak' + bix' + cp' = L'. which raises the index of the vector. 21. The process is not confined to vectors, and we can raise or lower the indices of tensors in the same way. By the ordinary multiplication rules for tensors, we have, for example, g r *A rt = £?, where B is some tensor of the type and order indi- cated. If we multiply this across by g S k and sum. we have 8kArt = A kt = g sk Bs t =C kt , for instance, in which all components of A and C are equal so that they are the same tensor. The re- sult of raising an index and then of lowering it again is similar to recovering the original tensor; there- fore, we are justified in considering B as simply another form of A, just as the covariant and con- travariant components are considered as describ- ing the same vector. We may accordingly write g rs A rl = A s . t . But. because A n is not, in general, the same as 'Levi-Civita (1926), The Absolute Differential Calculus, 90. Tensors 13 At,-, we must be careful to leave a space or a dot to show from where the raised index came so that it may be returned later to the right place. If there is likely to be any confusion, it is best to write any tensor so that no superscript is vertically above a subscript. THE PERMUTATION SYMBOLS IN THREE DIMENSIONS 22. We now introduce a system e rs t or e'*' in three dimensions, defined as follows: (a) When any two indices are the same, the sys- tem is zero — for example. e m = 0. (b) When the arrangement of indices is 123, or the cyclic order 231 or 312 — that is, an even per- mutation of 123 — it is + 1 . (c) In all other cases, that is, an odd permutation of 123, it is — 1— for example. e~ ,:i = — 1. In short, the systems are skew-symmetric in any two indices. 23. If A\. is a term in any third-order determinant, the superscript being the row and the subscript the column, it is not difficult to verify that the value of the determinant A is given in terms of these e- systems by the formula 2.12 Ae rst = e ijk A\. A{A^, using, as always, the summation convention. If the terms of the determinant are the tensor transfor- mation matrix, this is 2.13- dx 1 ' dx" erst — ?ijk <-")%' ()X J dx dx 1 ' dx s dx 1 i, But the values of the e-systems are the same in all coordinates and, in consequence, the left-hand side cannot, in general, be e rs i- The e-systems are accordingly not absolute tensors, although systems which transform like Equation 2.13 are often called relative tensors. 24. We now take the unbarred coordinates to be rectangular Cartesian. The metric tensor of the transformed space is _ _dx» dx" ox' ox'' By taking the determinant of this and using the ordinary rule for the multiplication of determinants ith = 1, we find that 2 Throughout this book, side-line notation will be used, as here, for determinants. In a few cases, which will be clear from the context, side lining may indicate an absolute value. dx 1 ' dx r = Vg. Consequently, if we write 2.14 e,,, = V^fe,,,, we can make Equation 2.13 into trsl dx' dx j dx 1 ' dx'' dx s dx' so that for this transformation from Cartesian to general coordinates, the covariant e-system be- haves like an absolute tensor. But if e rs / is a tensor in one general coordinate system, it is a tensor in any other. In much the same way, using the fact that the determinant of the associated tensor g rs is l/g, we can show that 2.15 e rs '=e rs e rs >A ir Aj S A klt which shows that if A,,- is a tensor, then A\g is an absolute invariant which has the same value in any coordinate system. 26. We can also write cofactors A"' of the deter- minant in the form 2.17 2lA ir =e''J k e rs 'A JS A kt , which can easily be verified from the ordinary rules. Equation 2.17 shows that if Aj S is a tensor, then A' r lg is an absolute tensor. We have met one ex- ample of this in the metric tensor itself. GENERALIZED KRONECKER DELTAS 27. Next, we introduce a generalized Kronecker delta formed by multiplying e-systems and defined as 2.18 llllll : 583 28. If we contract on. for instance, the indices (/. /). we have yet another form of the Kronecker delta defined as 2.19 8T = d's'ljl" = d'M ' + 8'sW 2 + 8'M :i in which we have, of course, applied the summation convention. By combining Equations 2.18 and 2.19 and using the rules for the e-system, we can verify without difficulty that Equation 2.19 equals 14 (a) + 1 when (m, n) and (5, i) are the same two numbers in the same order (m ¥> n and t¥>s), for example, 811=81! =+ 1: (1)) —1 when (m, n) and (s, t) are the same two numbers in the opposite order (m^n and t^s), for example. 8i? = 8?i = -l: and >1\ "13 (c) otherwise zero. If we contract a tensor with this 8-system, it is not difficult to verify such results as 2.20 §TAmn,>=A st ,-A tsl) 2.21 8% n A s .[ p =A™-A™. 29. We can further contract Equation 2.19 into the ordinary two-index Kronecker delta, but, in this case, to square with the previous definition in Equa- tions 1.24, we shall need a factor of (V2) so that 2.22 8r=j8j|3i"=H8lfl 1 + 8SS* + 8a?), which can easily be verified. VECTOR PRODUCTS 30. We shall often meet a contracted product of the e-systems with two vectors, and shall now con- sider what this means. We revert to the mutually orthogonal triad of vectors (A.,, fx r , v>) discussed above, and again take these temporarily as rectangular Cartesian axes as in § 2-17. Then the tensor equations \ r ' — e™ 'fisVt M = e rs 'v s k< : v r = € rs 'K sf x, 2.23 are evidently true in these coordinates and are accordingly true in any coordinate system, as we have seen in § 2-12. We now take a unit vector /u.. s in the plane of /Lt. s and v s , and making an angle with v s so that JJL K = IJL* sin 6 + v s cos 0, we evaluate e rs 'jj e rs 'JA s vt = K r sin 0. in which (/> is the angle C r makes with the plane of A s , B,. The expression on the left of Equation 2.25 is known as a scalar triple product. For the product to be positive, the three vectors must be right- handed in the order of the e-system indices. If any two of the vectors have the same direction, the scalar triple product will be zero because either or 4> will be zero. This also follows from §2-11. 32. It is evident from Equation 2.25 that, if (A,, jtir, v r ) is any right-handed mutually orthog- onal triad of unit vectors, we have 2.26 e nt \ r mvt=l. The sign of this product will be changed if any two of the vectors are interchanged; the product will be zero if any two of the vectors are the same. We can accordingly express the e-systems as products of the three vectors as follows, e rs > = k'( /a s v> - V s ix l ) + i* r ( v s k' - AV ) 2.27 +v r (k s f jL t -fj. s k t ), with a covariant equation obtained by simply lower ing all the indices. 33. By multiplying two tensors of the form Equa tion 2.27 and contracting with the metric tensor we have g rifi rSt e ijk - ( ^.sy _ ptfj/l ) ( jjjjjl. — yjp a- ) + two similar terms. Multiplying this out and using Equation 2.09, w« have finally 2.28 v^ci'ttpUk = psjptk — gskgjt !,-,€ e :gVg- Tensors 15 with a similar equation obtained by raising or lower- ing each index. THE PERMUTATION SYMBOLS AND THE METRIC TENSOR IN TWO DIMENSIONS 34. One advantage of the tensor calculus is that if we have a tensor equation in three dimensions, for instance, then it is likely that a similar equation exists in two or four or any number of dimensions. In many cases, the equation will be exactly the same with the Greek indices of two dimensions as it is in the Roman indices of three dimensions. The reason for this is that the defining and trans- formation equations of tensors are of the same form in any number of dimensions. Thus corresponding to Equation 2.01, for example, we should have 229 A °>=»M> An (a, /3, y, 5=1, 2). 35. There will, of course, be fewer components in two dimensions because fewer numbers can be ' assigned to the indices, and this may affect the form of the tensor. For instance, we cannot have e aliy (a, /3, 7=1,2) 1 defined in the same way as the e-systems in three dimensions because all its components would be j zero. We can, however, have e u/j , e.,0 (a, j8=l, 2) defined in the same way, that is. e aP: P a/3 2.30 e°" 5 = e Qp /V« : e„(j= V(k>„ p in which a is the determinant of the two-dimensional metric tensor a a n and the e a p- or ^-systems are defined as equal to (a) zero if a = /3, (b) +1 if (a, /3) = (1, 2), and (c) -1 if (a, /3) = (2, 1). 36. By analogy with Equation 2.26, we should ' expect I 2.31 6°*A«tfi/B=l, if (k a , /JLfs) are any two mutually orthogonal vectors I in the order of the coordinates (1, 2), just as the I triad (A,, fx r , v r ) in three dimensions is arranged in order of the coordinates (1, 2, 3) to give a right- i handed system. The rotation of \ Q to jx^ must be in | the same direction as the rotation of the x'-co- I ordinate line to the x 2 -coordinate line. We may also ! expect, using the same arguments as for Equation 2.27, that the following tensor equations should hold. -.«H: AV'-iU,"\0 2.32 37. Both Equations 2.31 and 2.32 can easily be verified, remembering that as tensor equations we have only to verify them in one particular coordi- nate system. If we take A", yfi as unit vectors in the directions of the coordinate lines in the orthogonal metric ds 2 = a n {dx l ) 2 + a 22 {dx 2 ) 2 , their components are k a =(llVa7u0) fX a =(Q,! 2.33 A„=(V^.O) /a„=(0, vfl 2 and Equations 2.31 and 2.32 are verified at once. 38. In this same coordinate system, defining a a P as the cofactor of a a ji in the determinant |a u #|, divided by the value of the determinant a — ana>>, we can at once verify that 2.34 2.35 2.36 a a ji = KAn + ij-cx/j-h ,«£: A"A« + fjL a yfi a at3 auy — k a ky + IA a lXy = 8" correspond to the three-dimensional Equations 2.08 and 2.09. Since these are tensor equations, they are true in any coordinate system and for any pair of orthogonal vectors. It should be noted that we have not appealed to Cartesian coordinates (in the plane) in order to prove them. 39. The equations in Equations 2.32 are of par- ticular importance because, given a surface vector A„, we can define an orthogonal surface vector in terms of it as 2.37 Mjtf = ^nk a , etc. 40. As in three dimensions, we can form general- ized Kronecker deltas from products of the e- systems, that is, 2.38 6^=6°%^, and we can contract this to 2.39 8« a § = d\§ + m = 8$, which defines the ordinary Kronecker delta (Equa- tions 1.24), that is, Sg=l (a = /3) 2.40 8g = (a#/8). 16 Mathematical Geodes 41. Corresponding to Equation 2.20, we have also, A/a is a surface invariant, having the same valu for example, in any coordinate system. The cofactors are given b 2.41 SftAaitor^Aysvr—Asypir 2.43 A e #=ef**eP*Ay», in which A is any tensor. which shows that if ^4ys is a surface tensor, then s is A a ®\a. If Ayh is the metric tensor, then we hav 42. Corresponding to Equation 2.16, we have for 2.44 a a& = e^e^ays. the expansion of a second-order determinant , . , • i n a tensor equation in which we can raise and low< 2.42 2U = e ay e iih A al iAy h ^ indices to obtain also which shows that if A a n is a surface tensor, then 2.45 a u n = e u ye#ga Y8 . CHAPTER 3 Covariant Differentiation THE CHRISTOFFEL SYMBOLS 1. We have considered a tensor as a set of point functions denned at a number of discrete points in space, and we must now consider how its compo- nents vary from point to point — in other words, how to differentiate a tensor with respect to a small dis- placement of the coordinates dx r . The differentials of a general tensor must clearly involve the dif- ferentials of the metric tensor whose components will also vary, in general, from point to point. We shall see that the analysis inevitably leads to the following grouping of differentials of the metric tensor, requiring the special symbols on the left, 3.01 3.02 #, *] _ i ( dgjk dgik dgij 2 \dx' dx> dx k These special symbols are known as the Christojfel symbols of the first and second kinds, respectively. We note that both Christoffel symbols are symmetric in (i,j). In Cartesian coordinates, all components of the ! metric tensor are constants and therefore all the ' Christoffel symbols are zero. 2. In the case of a transformation from Cartesian i coordinates (overbarred), we have ftA' = g» dx'" dx" ' dxJ dx k in which the g m >i are constants; and by direct dif- ferentiation and substitution, we find that 3.03 r .. ,-,_- d-x"' dx" l l J • "J glllll - ;-, : ^ ..• dx'dx J dx'~ 306-963 0-69— 3 Multiplying this across by n -,J X '' dx ' 6 dx"dx' J ' we have after some simplification d-x'" dx 1 3.04 r/f- J dx'dxJ dx' 3. We can now take a field of parallel unit vectors A r whose Cartesian components (still denoted by overbars) are the same at any point in space and are given by A'" = A' — - ox' If we differentiate this equation with respect to each of the coordinates xJ in turn, then no matter what the corresponding change in the Cartesian co- ordinates may be, the differentials of the constant Cartesian components on the left will be zero; we may write the complete set of resulting equations as a A' dx'" . .. B 2 x'" A — r : + A 1 — t-—. = 0. dxJ dx' dx'dx> If we multiply this result across by dx'/dx'" and use Equation 3.04, we have 3.05 — +n.A' = o, dxJ 'J which are the differentia] equations of a set of par- allel vectors A 1 in general coordinates. In much the same way, by differentiating Ai = ——A m , dx' 17 18 Mathematical Geodesy ha\ we nave dA;_ r) 2 x'" -7 ._ d 2 X m dx l dx j dx'dx> dx'dxJ dx m so that 3.06 ^-r/x=o, dxJ which is the eovariant form of Equation 3.05. COVARIANT DERIVATIVES 4. Next, we take a general vector field A' (defined in some way at all points of a region of space), which may vary in both magnitude and direction from point to point. We also define an arbitrary set of unit parallel vectors A; over the same region as, for instance, a field of unit vectors all parallel to a Cartesian axis. We differentiate the scalar product of the two vectors with respect to each coordinate x j and use Equation 3.06 to give a.rJ dxJ J dxJ jA / on changing the dummy indices. But we have al- ready seen in § 1—19 that the differentials of an in- variant form a eovariant vector, so that the left-hand side of this equation is a eovariant vector as is also the right-hand side. Because A\ is arbitrary, this means that the expression within the parentheses is a mixed tensor. This we call the eovariant de- rivative of the contravariant vector A.' with respect to the coordinate xK which we write as 3.07 dxJ Jk In exactly the same way, we can find the eovariant derivative of the eovariant vector A, as 3.08 dxJ n,xz. Covariant derivatives are sometimes distinguished from other tensors by writing a comma or a bar before the index of differentiation, thus A„ hlJ- But we shall not do this where the context clearly indicates that the tensor has been formed by co- variant differentiation, or where the distinction is immaterial. 5. We can similarly derive an expression for the covariant derivative of a tensor of any order or type by reduction to an invariant with a number of arbitrary parallel vector fields. For example, or- dinary differentiation of the invariant k' sl A,B s O will show that the covariant derivative of the tensor is 3.09 r)A'' ul .y ■ The rule is to place each index of the original tensor inside a Christoffel symbol at the same level. The sign of the Christoffel symbol is positive for a trans- ferred contravariant index and is negative for a transferred covariant index. The place of the trans- ferred index is taken by a dummy (summation) index (J) , which must also be inserted at the oppo- site level in the Christoffel symbol. The Christoffel symbol is completed with the derivative index (u in the above example). 6. The covariant derivative of the gradient of a scalar (f) can be written as 3.10 dx' dx s which is evidently symmetric in (r, 5) because the Christoffel symbols are symmetric in these indices and the ordinary derivatives commute. We can accordingly write in this case 3.11 r,s—s,r. 7. Compared with ordinary differentiation of the separate components, the great advantage of co variant differentiation is that it results in a tensor If we differentiate a tensor equation covariantly, we get another tensor equation which remains true any coordinate system and retains all the oth advantages of working in tensors. m er 8. The Christoffel symbols are not tensors, evei though their addition to the ordinary derivatives which are not tensors either, produces a derivec tensor of a higher order. From Equations 3.0] and 3.02, it is clear that the Christoffel symbols are all zero in Cartesian coordinates; if they were components of a tensor, they would have to be zero in all coordinate systems. That this is not so we can observe from Equation 3.04. The fact that all the Christoffel symbols are zen in Cartesian coordinates implies that covarian derivatives become ordinary derivatives in Car Covariant Differentiation 19 tesian coordinates. This fact is apparent at once from Equation 3.09. which are usually known as the curl of the original vector. Accordingly, we take Equation 3.15 as the curl of a vector in ■• ^HiMr V(F,G)=g>-s FrGs 3.14 dF dx *Mm) +(£)(f 11. Again, if F, is a vector and we expand the icontravariant vector 3.15 ■ rs, F ts in rectangular Cartesian coordinates {x ] we have a vector whose components are dF :i dF, dx 2 dx :i dF\ dx s dF-, dx 1 dF, dF, dx* dx 2 RULES FOR COVARIANT DIFFERENTIATION 12. A few rules for covariant differentiation may be noted rapidly. Since all components of the tensor are zero in Cartesian coordinates where the €,-si are constants, they must be zero also in any co- ordinates, which means that the e-systems, covariant and contravariant, behave as constants under covariant differentiation. For the same reason, the metric tensor and its associate and the Kronecker deltas behave as constants. 13. Expansion of these results leads to a number of useful formulas. For example, we have ens, « = = ~^T ~~ r f«e**s - r&eua - r£ u e m - =^f-v^(n„+ri u +r 3 3 u ) so that 3.16 a (In yj) dx" All' which enables us to write the divergence of a vector F i in the form 3.17 —.(V^F'), Vg* or the Laplacian of a scalar F in the form 3.18 af=4=t L (V^tf.,). y/ldx 14. The sum or products of tensors can be dif- ferentiated covariantly by the same rules as those for ordinary differentiation. To establish this fact, we have only to remember that covariant differ- entiation is the same as ordinary differentiation in Cartesian coordinates. Thus, the product A p q B$, differentiated covariantly, is (Ai'Ri)„ = A>> Ri+A>>Ri 20 Mathematical Geodesy This, with ordinary differentials, is clearly the By raising and lowering indices in the first term correct result in Cartesian coordinates and must we find that also be true in any coordinates because it is a tensor equation. 3.19 l,,sl'=0. Similarly, if /', j r are two unit perpendicular vec 15. If /' is any unit vector, we have from Equation tors ' then we have 1.11 /'7, = 1; differentiating this covariantly, we have l'j,=0 and 3.20 lr,sj r = -jr,sl r . These two simple equations will be in constant use CHAPTER 4 Intrinsic Properties of Curves CURVES IN THREE DIMENSIONS 1. We have determined the covariant derivatives with respect to each coordinate, which means that the tensor being differentiated must be defined in space. If the tensor is merely defined along a line, we can use the same formulas (as if we were dealing with a family or congruence of lines in space) and can restrict their application to a particular line by contracting with the unit tangent vector of the line. For example, the differential of a tensor A r st along a curve whose unit tangent is /" is 4.01 A r sltU l" = A> s ,, u (dx"/dl) where dl is the arc element of the curve. This is known as the intrinsic derivative of the tensor, with respect to the arc length of the curve, and is written as -(A'',). In place of the arc element, we could use equally well any parameter (q) defined along the curve because this parameter would be some function of the arc. In that case, 8q (A r s( ) would be the intrinsic derivative with respect to the parameter. 2. The intrinsic derivative of the unit tangent itself is I, ,1 s and is called the vector curvature of the line; it represents the arc rate-of-change in the tangent vector along the line and so is a generalization of the notion of curvature for plane curves. If the vector curvature is zero throughout, the curve is said to be a geodesic of the space — that is, a straight line in flat space — although it would not necessarily be "straight'" in a curved space, such as a curved two-dimensional surface. 3. We can write the vector curvature as 4.02 /,•*/* = xmr in which m, is a unit vector, known as the principal normal to the curve. The magnitude of the vector curvature is the scalar invariant X = /,.s-m'7*. and is known as the first or principal curvature, or simply the curvature of the curve. If we multiply Equation 4.02 by g"l, and use Equation 3.19, we have Xg rt ltm r = l rs l r l s = which shows that unless X = 0< tne principal normal is perpendicular to the unit tangent. If x = 0, tne curve is a geodesic, and its principal normal is indeterminate. 4. In the case of a curve in three dimensions, we can associate another unit vector n, with the curve, such that (/,, m r , n,) form a mutually orthogonal right-handed system. This third vector, known as the binormal, is perpendicular to the osculating plane of the curve defined by /, and m, and will therefore remain parallel to itself along a plane curve. However, if the curve is not a plane but a twisted curve, the binormal will not remain parallel 21 22 Mathematical Geodesy to itself but will have an intrinsic derivative which is a vector and can therefore be written in the form 4.03 n rs l s = — rp r where t is the magnitude of the unit vector p, (the negative sign is simply a convention). If we take (/,-, m r , rii) as temporary coordinate axes, it is clear that p, must be expressible as Pr^Alr + Bnir + Cn, in which A, B, C are the components of p r . Using Equation 3.20, we then have — tA = — Tp,l' = n,J r l s — — l rs n '7 s , which from Equation 4.02 is zero because n' is perpendicular to m r ; also we have — tC '= — rp r n r = n, s n r l s , which is zero from Equation 3.19. Because r.is not, in general, zero, we have and thus A = C = p, •= m, , both being unit vectors. We may accordingly re- write Equation 4.03 as 4.04 n rs l s = — rm r , The magnitude t of this vector is called the second curvature or torsion of the curve. 5. The variation of the principal normal along the curve is settled by the variation of the tangent and binormal because, by definition, the principal normal remains perpendicular to both. We cannot, therefore, obtain an independent expression for the variation of the principal normal, but it is, nevertheless, useful to express the variation in terms of Equations 4.02 and 4.04. Proceeding on the same lines as above, we write m rs l s =Cl r + Dm r +En r in which D is zero from Equation 3.19 and C = m rs l s l r = — Irsm '/"' = — x from Equations 3.20 and 4.02. Also, we have E — m, s n r l s = — n, x m '7 s ' = t, so that we have finally 4.05 m rs l s =— xlr + rn,: The three Equations 4.02, 4.04, and 4.05, two of which are independent, are known as the Frenet equations of the curve and are collected for easier reference as /,,/* = X m < m,-J s = — X^' +Tft, 4.06 n rs l s =—Tm r . CURVES IN TWO DIMENSIONS 6. In the case of a curve contained wholly on a surface, the vector curvature can similarly bt defined as 4.07 la^= /" and use the two-dimensional form of Equation 3.19, we find that the unit surface vector j a is perpendicular to the unit tangent l a and is known as the normal to the curve. The magnitude cr oi the vector curvature is called the geodesic curvature of the curve. If cr is zero, the curve is called £ geodesic of the surface, paralleling the definitior of a three-dimensional geodesic in §4—2. We mus remember, however, that the curve is also a curve in the surrounding space and will have a first principal curvature in three dimensions as wel as geodesic curvature in two dimensions. The curve will not be a geodesic of the surrounding space unless its principal curvature is zero. W< shall see later that the principal curvature an< geodesic curvature are related; but for the present we shall consider only the intrinsic curvatun properties of surface curves and shall defer con sideration of them as curves in the surrounding space. 7. If we confine our attention to the surface alom it is clear that the curve can have no surface b normal because there is no surface directh perpendicular to both l a and j a - The only indepenc ent Frenet equation is accordingly Equation 4.01 We can, however, derive a useful dependent equc tion in much the same way as we derived the secon Frenet equation in 3-space from the other twt We write jafiP^Ala+Bj* and note at once that B = from the two-dimension; Intrinsic Properties of Curves 23 form of Equation 3.19. Further, we use Equation 3.20 and find that from Equation 4.07, so that finally we have 4.08 /<*/* = - u,. Because A/ is an arbitrary vector and the left-hand side of Equation 5.02 is a third-order tensor, then it follows that Equation 5.03 is a fourth-order tensor, known as the Riemann-Christoffel or curvature tensor. If the space is flat, there exists a Cartesian coordinate system in which all the Christoffel sym- bols in Equation 5.03 are zero: therefore, all com- ponents of the Riemann-Christoffel tensor are zero. All components of this tensor are then zero in any coordinate system. The vanishing of the Riemann-Christoffel tensor is accordingly a neces- sary condition for flat space, and it can be shown that the vanishing of the tensor is also a sufficient condition. 4. From Equation 5.03, we can see at once that the tensor is skew-symmetric in (/, k) so that 25 26 Mathematical Geodesy we have 5.04 K?y*=- R>. ikj. Further, by straight substitution, we can also show that 5.05 R l . m +R l . jki +R l . kij in which the three lower indices are given a cyclic permutation. 5. There is also a covariant form of the Riemann- Christoffel tensor, obtained by lowering the super- script and written as 5.06 Rmijk — glm Rl ijk With a little manipulation, the covariant tensor can be written in either of the following forms, R 5.07 or m ijk =7-7 [ik, m] — — [ij, m \ dx J dx K + r\j[mk,l\-r\ k [mj,l] R m Ijk d' 2 gmk . d 2 gij d 2 g, dh 5.08 dx'dxi dx"'c)x k dx i dx k dx'"dxJ g™{[mk,p][ij,q]-[mj,p][ik,q]} in which the Christoffel symbols are given by Equations 3.01 and 3.02. The covariant form has the same properties as the mixed form in Equations 5.04 and 5.05 with the superscript lowered. In addition, the covariant form is skew-symmetric in the first two indices (m, i) and symmetric with respect to the two pairs of indices, that is, 5.09 R 111 ijk— Rjl LOCALLY CARTESIAN SYSTEMS 6. In earlier sections, we have derived a number of results such as Equations 2.23 by assuming a Cartesian coordinate system; the question arises whether these results are true only in flat space. It is apparent from Equation 5.03 that the curvature of the space enters the question only when we dif- ferentiate the Christoffel symbols — that is, when we compare their values at different points in space. There is nothing to stop our choosing a co- ordinate system in which the Christoffel symbols are zero at one particular point; it is only when we insist on these symbols remaining zero at all other points that we require the space to be flat. A co- ordinate system in which all the Christoffel symbols are zero at one point of the space is known as i locally Cartesian system. In such a system, th( curvature tensor would be Ri =JL Y' ——T> •'•>* dxJ ik dx k & but only at the origin, or the point where th( Christoffel symbols are zero. If the space is curved the symbols are not, in general, zero elsewhere and we should use the full formula of Equation 5.0^ for the curvature tensor. 7. Clearly, any result obtained by applying ; Cartesian system to tensor point functions, sue! as those in Equation 2.05, is valid because we couk have obtained the same result by choosing a localb Cartesian system at the point under consideration Any results containing the first covariant derivative; of a tensor (or the second covariant derivatives of i scalar) are valid because they do not contain de rivatives of the Christoffel symbols. In short, al results, given prior to this chapter, are valid it curved space. We cannot, however, verify a tensoi equation containing higher covariant derivatives by an appeal to Cartesian coordinates unless the space is flat. 8. In a locally Cartesian system, the first ordinar derivatives of components of the metric tensor arc zero at the origin of the system because the Christof fel symbols and the covariant derivatives of thi metric tensor are zero. We can accordingly say tha the system is Cartesian to a first order, or in tin immediate neighborhood of the origin where thi Christoffel symbols are zero. 9. Fermi 1 has proved further that a locall Cartesian system need not be confined to the neigh borhood of one point assigned beforehand; it i possible to choose a Cartesian system in curvei space which applies in the immediate neighbor hood of all points of a given line assigned before hand. This extension is sometimes useful. SPECIAL FORMS OF THE CURVATURE TENSOR 10. It can be shown 2 that the number of inde pendent components of the covariant curvatur tensor in a space of N dimensions is 1 Levi-Civita (1926), The Absolute Differential Calculus, 16' 2 See for example. Synge and Scliild (corrected reprint of 1964 Tensor Calculus, original ed. of 1949. 86. Intrinsic Curvature of Space 27 5.10 /V 2 (/V 2 -l)/12. This means that in three dimensions there are only six independent components; thus, all the curvature properties should be expressible in terms of a simpler symmetric second-order tensor formed by contracting the full curvature tensor. 11. One such contraction, known as the Ricci tensor, is formed by contracting the first and last indices, thus we have 5.11 R ij = g"< k R mi j k = R':ij k . Using the symmetrical and skew-symmetrical properties of the curvature tensor, this can also be written as g mk Rimkj =g mk Rhjim = Rji, which shows that the Ricci tensor is symmetric. The tensor has therefore six independent compo- nents and can represent all the curvature proper- ties of 3-space. By direct contraction of Equation 5.03 on the indices (/, A;) and use of Equation 3.16, we can write the Ricci tensor as 5.12 Rt,~£ j\~ r< ij+ njp-u- IW in which the first and last terms are ml d S (iny5)_ r a02v£ = ^ dx'dxJ ,J dx m J dx'dx* In this last expression, g is the determinant of the j metric tensor which is obviously not an invariant, | although in a particular coordinate system, the I determinant will be a function of the coordinates; and we can accordingly take its gradient and second covariant derivative. 12. Another contraction of the curvature tensor in three dimensions is 5.13 S"^ = W"" i e ( 'J k R ll ,ijK: If we multiply this by tprstqtu and use Equation 2.20 and the skew-symmetrical properties of the curvature tensor, we have an alternative form 5.14 epr S e q mi ev k Rj kmi = S<&, showing that the tensor is symmetric with six in- dependent components, which again can represent all the curvature properties of 3-space. We shall call this the Lame tensor because, when all of its components are set equal to zero in flat 3-space, the tensor gives the well-known six Lame equations which must be satisfied by the metric of any co- ordinate system in flat 3-space. 13. We shall finally relate the Ricci and Lame tensors from Equations 5.11 and 5.14 as Rij=g mk e pmV e gjl jSP9 = g" lk e lllip e kqj SPi = {giqgl>j-gijgl>q)S"", using Equation 2.28, so that if S is the contraction gpqSn, 5.15 we nave R'j — Su — Sgij. CURVATURE IN TWO DIMENSIONS 14. The idea of curved space is difficult because we are accustomed to think of "space" as the ordinary Euclidian flat space of three dimensions. We are more familiar with curved spaces of two dimensions, or curved "surfaces," because we can measure the curvature from the outside. As far as the tensor calculus is concerned, there is no essen- tial difference between spaces of two- and three- or ra-dimensions, except in the number of components which tensors can have in such spaces. A curved space of two dimensions has intrinsic curvature properties which do not depend on outside measure- ments. We can define the curvature tensor of two- dimensional curved space, as in Equation 5.03. by simply substituting Greek indices for Roman and restricting them to the numbers (1, 2). However, reference to Equation 5.10 will show that in two dimensions, the curvature tensor has only one inde- pendent component. The intrinsic curvature prop- erties of a two-dimensional surface can accordingly be completely exhibited by an invariant, just as those of a 3-space can be completely specified by the six independent components of a symmetric second-order tensor. We denote this invariant by K and call it the Gaussian or specific curvature of the surface, defined by the following contraction of the curvature tensor, 5.16 K= W^Rapys, corresponding to Equation 5.13. 28 Mathematical Geodes 15. We could, of course, have contracted the curvature tensor in another manner and so have defined K differently; but, if we had used a sym- metric tensor, such as a a ® in the same way as e"' 3 , the result would have been zero because of the skew-symmetry of the curvature tensor. By substi- tuting for the e-systems from Equations 2.30 and using the skew-symmetry of the curvature tensor, we can reduce Equation 5.16 to 5.17 K=Rm 2 la, which, in conjunction with Equation 5.07 or 5.08, enables us to calculate K for any given metric. The sheer labor of substitution is lightened if we choose orthogonal coordinates, so that ai> = and the metric is ds 2 = a xx (dx') 2 + a T1 (dx i ) 2 . In that case, if we form the Christoffel symbols directly from the definitions of Equation 3.01 and substitute in Equation 5.08, we can express the result as K = 5.18 1 2 Va c) { 1 da->., \ _d_ /_!_ dan dx 1 VVa dx 1 I dx 2 \Va dx 2 16. Multiplying Equation 5.16 by e (p e, rT and using Equations 2.40, we have 5.19 R< as an alternative expression which is sometimes useful. If we contract this equation to form the Ricci tensor in two dimensions and use Equations 2.32 and 2.34 for two arbitrary orthogonal vectors k a , /jL a , we have Rpa = Ka tT { k t fJL f> — IXtk,, ) ( ka-fJh — (J^rK) = — K(\ p Kr+ fJiplAo) 5.20 =-Ka, t-ptr- We could accordingly have defined K by contract- ing the Ricci tensor as 5.21 a»»R (lir = -K8l = -2K, which gives us another way of calculating A.' from the two-dimensional equivalent of Equation 5.12. 17. Corresponding to Equation 5.02, we have for an arbitrary vector A a , 5.22 A a , py — A„, y^ = ksR° a ^y= K^RfraPY- If we substitute Equation 5.19 and use Equation 2.36, we have 5.23 A«. py — k a . yp— KfJL a €py in which ix a is the usual vector orthogonal to A Q . we multiply this by e^ and use Equation 2.38, w have after some interchange of dummy indice 5.24 ^ y k a ,py = Ktl a . These equations enable us to interchange indice in the second covariant derivatives of surfac vectors. 18. It should be noted that in this chapter w have derived only properties of a surface whic depend on the metric tensor and its derivative: Such properties are called intrinsic. They usual] have counterparts in the intrinsic properties < spaces of more than two dimensions, which is on of the great advantages of the tensor calculus, surface can also have extrinsic properties, derive from the space in which it is embedded. We sha consider these properties later. RIEMANNIAN CURVATURE 19. We can simplify the notion of curvature of general space by considering the curvature of su faces within it. We take a pair of unit orthogonal vectors K r , /x a point P in the space and let the pair define section of the space, so that any other unit vect in the section is given in terms of a parameter 6 the relation 1 l r =X r sin + i* r cos d. The geodesies of the space in all these direction /', will form a definite surface whose Gaussic curvature is called the Riemannian curvature oft! space for the section defined by A.'', fx 1 . If the spa< is flat, all the geodesies would be straight lines; ar the Gaussian curvature of all the section planes zero, so that the Riemannian curvature for all se tions would be zero. Working from this definition, it can be shown 3 th the Riemannian curvature of the section, define by the unit orthogonal vectors (A', fl r ) , is given 1 5.25 C = R miJk K'"iJL i kJ t jL k . 20. In two dimensions, the only "section" of tl space in this sense is the space itself; and the ge desic surface formed by geodesies of the space that is, by geodesies of the surface — is again tl 3 Levi-Civita, op. cit. supra note 1, 196. Intrinsic Curvature of Space 29 surface itself. Accordingly, we may write the Rie- mannian curvature as Evaluating this invariant for the special coordinate system of Equations 2.33 gives C = K=R m »la t which agrees with Equation 5.17. This result does not, of course, prove the more general formula of Equation 5.25, but does demonstrate the consistency of Equation 5.25. 21. Now suppose that in three dimensions we complete the orthogonal triad with a third vector v r , such that (A.', fi r , v r ) is a right-handed system. If we multiply Equation 5.14 by X r /j, s X t fJL u and use Equations 2.23, we have Swv p v q = RrttuK r H*KlL u so that the Riemannian curvature in three dimen- sions can be written in terms of the Lame tensor as 5.26 C = SPiVpVq = S^i'i'i'". It should be noted that the geodesic surface, whose Gaussian curvature is C, is now formed by all the geodesies perpendicular to the direction v**. It can be shown 4 that, in general, there will be three mutually orthogonal principal directions at a point which give rise in this way to stationary (usually maximum or minimum) values of the Riemannian curvature known as the principal curvatures. The analogy with a curved surface will become clear later. We can also consider the Riemannian curva- ture as analogous to inertia or strain, the only difference being the nature of the tensor S pq . 4 Ibid., 201. kl CHAPTER 6 Extrinsic Properties of Surfaces FORMS OF SURFACE EQUATIONS 1. We have considered the intrinsic properties of surfaces as two-dimensional spaces in their own right. We have now to consider the properties of the same surfaces when embedded in space of three dimensions. 2. The link between the two sets of properties i will be an infinitesimal displacement on the sur- i face, which can be described either as dx r in the ! space coordinates or as dx" in the surface coordi- nates, following the convention introduced in § 1—6 and §2-34. The two are related by the ordinary formula for total differentiation 6.01 dx r dx r = t— dx a dx a ! in which the partial derivatives are considered as \ known from the equations of the surface, so that i each space coordinate is expressed in terms of the ! two surface coordinates — either explicitly or im- plicitly. For example, the equations of a spherical surface of constant radius r are given in terms of latitude ((f)) and longitude (co) as x = r cos (f) cos co y= r cos 4> sin o) z = r sin <£>. In these equations, the x r are (x, y, z) and the x a are (o», (/>). We can obtain the dx r ldx a by direct dif- ferentiation as, for example, a*. d<}>' r sin (/) cos u>. 3. These partial derivatives occur so often that 1 it is usual to give them the special symbol 6.02 dx r Bx a ' Evidently, the set of these quantities will transform like a contravariant space vector for each value of a and like a covariant surface vector for each value of r. This last point can be illustrated by con- sidering each space coordinate as a scalar defined over the surface, in which case the corresponding x' a becomes the surface gradient of the scalar. 4. The equations of a surface in relation to the surrounding space may be given in one of three forms. The first, or Gauss' form, expresses each space coordinate as some function of the two sur- face coordinates (it 1 , u 1 ) . In symbols, this form is usually shown as 6.03 x'=x'{u\u 2 ) (r=l, 2, 3), much as the equations of a sphere are expressed above. 5. The second, or Monge's form, expresses one space coordinate as a function of the other two as, for example, 6.04 x*=f(x\x 2 ), which similarly imposes a restriction on what points of the space can form the surface. We could take (x 1 , x 2 ) as surface coordinates, in which case the form is equivalent to the Gauss form x :! =/(«', u 2 ) X' = w 31 32 Mathematical Geodesy If, for example, the surface is given by z=f(x, y) in rectangular Cartesian coordinates, then y or u 2 is constant over the xz-plane, and the x- or u 1 -surface coordinate lines are accordingly the intersection of the surface with the xz-planes. We then have x 3 1 = dfldu 1 = df/dx and similarly x : l = dfldu 2 = 8fldy, while the other components are given by *£ = $; (r=l,2). It should be noted, however, that this last equation is not a tensor equation, but is merely a relation between some components of the tensor x' a in a particular coordinate system. We cannot manipu- late this last equation as a tensor equation by, for example, taking its covariant derivative. 6. This device of taking two of the space co- ordinates as surface coordinates often leads quickly to simple results, and we shall use this device throughout Part II. We lose no generality by doing so, but we must check the results for tensor charac- ter, as in the case of x' a above, before manipulating the results further as tensors. 7. The third form of surface equation expresses some functional relation between the three space coordinates, which are accordingly restricted in value at points on the surface. In this case, we may write 6.05 f(xKx 2 , x 3 )=N in which TV is a constant over the surface. By as- signing different values to N, we should have differ- ent surfaces which would, nevertheless, have some properties in common, dictated by the form of the function/. This third form, or its equivalent fix 1 , x 2 ,x\N) = 0, is accordingly most useful when we are required to express a family of surfaces. If we are not given the surface coordinates in terms of the space co- ordinates, we could, as in Monge's form, take x 1 , x 2 as surface coordinates. By partial differ- entiation of Equation 6.05 over the surface with x 2 and x 1 , respectively, constant, we then have ^-+^x 3 = dx 1 dx 3 ' ^ + -^ = 0, dx 2 dx 3 2 which give the x 3 ; the other components are givei as before by 8: (r=l,2), 8. Finally, we could take N in Equation 6.05 as one of the space coordinates. The other two space coordinates, which could be adopted as surface coordinates on the family of constant /V-surfaces must then be chosen in such a way that they car vary independently of N and of each other; this implies that the gradient of each coordinate must be perpendicular to the other two coordinate lines This arrangement is adopted for Part II, where i will be explained in greater detail. 9. The functions in the three forms of surface equations and their derivatives must satisfy certair conditions if the functions are to represent a rea nondegenerate surface, and even then there ma} be singular points on the surface. 1 This need not present too much of a problem because the sur faces with which we shall be dealing will eithei satisfy these conditions or will be prescribed as existing surfaces by the physical conditions. THE METRIC TENSORS 10. We can easily relate the space and surfaci metric tensors, g n and a a 0, by considering a smal surface line element ds. Considered as a displace ment in space, this is ds 2 = g rs dx r dx s = grsX^dxfdx in which we have used Equations 6.01 and 6.02 But considered as a displacement on the surface it is ds 2 = a a (idx a dx li , and because the two invariant displacements ar< the same for any arbitrary dx" and the tensor multiplying the dx" are symmetric, we must have as in § 2-13, 6.06 a a p = g rs x^x%. SURFACE VECTORS 11. If we suppose that the changes in coordinate in Equation 6.01 take place over an arc length ds i 1 See, for example, Kreyszig (revised reprint of 1964), Diffe ential Geometry, English ed. of 1959, 1-117. This is a fre translation of "Differentialgeometrie," printed in 1957 in Math matik und ihre Anwendungen in Phvsik und Technik, series I v. 25, 1-143. Extrinsic Properties of Surfaces 33 the direction of a unit surface vector whose space components are /' and whose surface components are l a and then divide Equation 6.01 by ds, we have 6.07 l r = x'J a . which relates the space and surface contravariant components of any unit surface vector. If we mul- tiply Equation 6.07 by gnXfc and use Equation 6.06, we have 6.08 lsxf ) = a a pl a =lf i , which relates the covariant components. 12. We have seen that the x' a are equivalent to the contravariant space components and the co- variant surface components of a surface vector and must therefore be expressible in terms of any two mutually orthogonal surface vectors /' or l a and/ or j a . We can easily verify from Equations 6.07 and 6.08 that this expression is 6.09 x r a = l'i a +j'j a . We note that the two vectors in this tensor equation are quite arbitrary. If we know the space and sur- face components of any two orthogonal unit surface vectors in a particular coordinate system, then we have all the x' a in the same system. THE UNIT NORMAL from a closed surface, that is, away from the region of space enclosed by the surface. We shall consider that the two surface vectors in Equation 6.09 form a right-handed orthogonal triad with v T in the order (/', j'\ v r ) , and the rotation from /' to/ is in the same sense as the rotation of the positive direction of the ^-surface coordinate line toward the x 2 - surface coordinate line. The diagram (fig. 5) illus- trates the situation if the paper represents a tangent plane to the surface and if the unit normal points toward the reader. 14. We are now able to obtain a relation between the contravariant metric tensors corresponding to Equation 6.06. Using Equations 2.09 and 2.35, we have g rs = l r l s +j r j s + v r v s 6.10 =o"%^+vV. 15. Next, we shall express the unit normal in terms of the x' a . Using Equations 2.27 and 2.32. we have V) . e rst = isjt -jslt = ^ ( laf -jap ) 6.11 Multiplying this by e, Kt and using Equations 2.19 and 2.22, we have 13. We shall normally be dealing with closed surfaces, and we denote the unit vector normal to the surface by v r and define its direction as outward (increasing) 6.12 . = i &» 5 G up e JM (X'X&, l'orl 1 (increasing) Figure 5. showing that p,„ besides being a covariant space vector, is a surface invariant: its components do not change if the surface coordinates are transformed independently of the space coordinates. SURFACE COVARIANT DERIVATIVES 16. All the formulas in Chapter 3 on covariant differentiation can be obtained in exactly the same way in two dimensions, in regard to the differentia- tion of tensors which are defined only on the sur- face. We have only to form the surface Christoffel symbols from the metric a ( ,# instead of g n and to restrict the indices (a, (3) to (1, 2). In cases where we used locally Cartesian coordinates x' in the course of a proof, we now use locally Cartesian coordinates x a in two dimensions, when all com- ponents of the metric tensor a ati will be constants at the point considered. The metric tensor a a $ and 306-692 0-69— 4 34 Mathematical Geodesy its associated tensor a a/3 , the e-systems e 01 ® and e a /3, and the Kronecker deltas all behave as constants under covariant differentiation in two dimensions, just as their counterparts did in three dimensions. We can immediately write down, for instance, the counterpart of the Equation 3.16 as d (In Vo) 6.14 6.13 dx a rg*. 17. In the case of tensors defined in space, the procedure is much the same. First, we differentiate the tensors covariantly with the respect to the space metric g, s in order to discover the variation of the tensor for a change in the space coordinates dx r ; and then second, we restrict this change to a dis- placement on the surface, just as we did along a line in §4-1. For example, the change in a space tensor A rs for a change in the space coordinates dx' is A,s,t dx'. But if the change in the space coordinates results from a displacement on the surface corresponding to a change dx a in the surface coordinates, this is ^ rs > t X a O?A . We call the new tensor Ars, I X l a , the surface covariant derivatives of the space tensor A,-s with respect to the surface coordinates x a . 18. It is at once evident that g rs , g rs , e' s ', e r gt and all the Kronecker deltas formed from the three- index e-systems behave as constants under surface covariant differentiation. For example, the surface covariant derivative of g, s is grslX' a = because g, s t is zero. 19. As an example, we take the surface covariant derivative of x' a with respect to .r , which we shall write as x' a p, from the tensor Equation 6.09, xr = l r l a +fj a . We then have x af] = l r S X%l a + l'l«lS + j r s XJij a +fj a f}. By expanding the covariant derivatives and re- arranging terms, this expression becomes £p (l r l a +j%) + r^ t (in a +j%)x^-Typ(l r ly+fJy) so that finally we have X a0 dx a ()X li *t X a X h ' 2/3-*> y in which the space Christoffel symbols formec from g rs are given Roman indices; and the sur face Christoffel symbols formed from a a/ 3 are giver Greek indices. It should be noted that x' ali is sym metric in the Greek indices. 20. The rules for surface covariant differentiation of mixed space and surface tensors are illustrated by Equation 6.14 for the tensor x' a . To obtain the terms containing the space Christoffel symbols, we simply treat the tensor as a space tensor with re- spect to each of its Roman indices and hold the Greek indices fixed. If we are differentiating with respect to the surface coordinate x 7 , we complete the term with x y in which u is a dummy index ap- pearing also in the space Christoffel symbol. The terms containing surface Christoffel symbols are obtained by treating the tensor as a surface tensoi with respect to each of its Greek indices while hold- ing the Roman indices fixed. Thus, the surface covariant derivative of Agfa is l ap. y 4- V r //'« Y »+ Vs Art Y u > l tu /1 a)3 x y ' l tU ^afi x y Toy^s! /3T^ aS' 21. It is of no consequence if a space tensor i< defined over one surface only, such as the surfac< vector whose space components are /'', or the vector v r normal to one given surface. We can a ways suppose that the given surface belongs t some family of surfaces in which case, for example the v r would become a unit vector field, differen tiable in any direction in space. When we multipl by x' a , we restrict the variation to displacements o one particular surface, and we can forget the othe members of the family. We have already used thi device to find the variation of tensors defined alon a line in Chapter 4. 22. We shall often denote surface covariant dii ferentiation,with or without a comma, by simply adding a Greek subscript, particularly when all tb Roman indices are superscripts, for example, V a V a V -S x a v s x a . The commas will be dropped if it is clear from tb context or from the usage that covariant differ entiation is involved. Extrinsic Properties of Surfaces 35 THE GAUSS EQUATIONS 23. If we take the surface covariant derivative of Equation 6.06, we have g rs x r ay x%+ g rs x r a xfa, = 0. By cyclic permutation of the free surface indices (a, /3, y), we have also gr S X r p a Xy+gr S X^Xy a = gr S Xy X s a +g rs XyX s a0 =O. Adding the first two, subtracting the third, and remembering that x„p is symmetric in (a, /3) and thatg,. s is symmetric, we have 6.15 g rs X r a yX%=0. If we consider the space coordinates, it is apparent from Equation 6.09 that xfe is an arbitrary surface vector: therefore. x%y must be a space vector in the direction of the normal. We can then write 6.16 X' n y=b«yv' in which b a y is evidently a symmetric surface tensor ! like x' a y. These equations are usually known as the Gauss equations of the surface (not a very distinc- i tive name in this subject), and the tensor b u y is j known as the second fundamental form of the sur- face. (The metric tensor a a y is sometimes known as the first fundamental form.) We shall see later that the second fundamental form settles the extrinsic curvatures of the surface. THE WEINGARTEN EQUATIONS 24. We take next the surface covariant derivative of Equation 6.10 as = a" [x r a yX% + X^Xfa) + V'yl'X + V'Vy. If we multiply this by v x and use Equation 3.19, we shall have V s Vy = V s V\Xy = and, from Equation 6.09, ^| = 0, |so that finally, using Equation 6.16, we have 6.17 v y =- a a Vbpyx' a . These equations, giving the surface derivatives of the normal, are known as the Weingarten equations )f the surface. They give rise to a third fundamental form of the surface which we define as If we substitute Equation 6.17 and use Equations 6.06 and 2.36, we have also Caii = a yh b„ybn?,. 25. We shall see later that the three fundamental forms a (I| 8, 6 a #, c a p are not independent; correspond- ing components are connected by a linear relation. 26. If we contract the Weingarten Equation 6.17 withg,,,jtg, we have gi*l'yX$, = — a^bjiya^f, = — 8fbny = — byf> as an alternative expression for the second fun- damental form. Comparable expressions for all three fundamental forms are collected here for easy reference as ««£ = grsX r a X% b a fS = — grsX r a V% 6.18 c u/s = g rs v r a v%= a^baybps. THE MAINARDI-CODAZZI EQUATIONS 27. We have so far not considered the sort ot space in which the surface is embedded: it could be either curved or flat. As we have seen, this question involves the second covariant derivatives of a space vector, for which we shall take the unit normal. 28. We start with the relation i>,x' a = 0, obtainable from Equation 6.09. because v v is perpendicular to all surface vectors. Taking the surface covariant derivative and using Equation 6.16, we have 6.19 Vrsx r a x s p = -b a p; differentiating again, we have 6.20 v,stx' a x s (i Xy+ VrsX' a v s bny- >atiy in which we have used Equations 3.19 and 6.16. We shall now interchange (/3, y). In the first term, we can also interchange (5, /) if the space is flat because, as we have seen in Equation 5.01, v n Vris, so that the first term remains unchanged. The second term also remains unchanged because buy is symmetric. We conclude therefore that if the 36 Mathematical Geodes space in the immediate neighborhood of the surface is flat, we have 6.21 bafjy — b a yp- These are known as the Mainardi-Codazzi equa- tions. Owing to the symmetry of the 6 a #, there are only two independent equations, namely, 6112 = 6121 6*u = 62*1. We have shown that the Mainardi-Codazzi equa- tions are necessary conditions for the surface to be embedded in flat space. They can take various forms, which we shall derive later, sometimes by considering the second covariant derivatives of space vectors other than v, ■; but all these forms are equivalent to the simple relation in Equation 6.21 between the surface covariant derivatives of the b a p. 29. It should be noted that while the b a (iy are first covariant derivatives of the surface tensor bafi, it is evident from the Gauss Equation 6.16 or from Equation 6.20 that they are connected with the second derivatives of space vectors and, for this reason, are affected by the curvature of the surrounding space. In flat space, the surface tensor b a py is symmetric in any two indices because of the Codazzi Equation 6.21 and also because b a p is symmetric. 30. However, if the surface is embedded in space whose curvature tensor is Rurst, we can use Equa- tion 5.02 and make the necessary modifications in working from Equation 6.20 to show that the "Mainardi-Codazzi" equations would then take the form 6.22 b a py — b a y$ — R urs tV u X r a X ! j i Xy This equation reduces to Equation 6.21 when the space is flat. If the curvature tensor is specified, as it usually will be by the conditions of the problem, then these equations, although different, are just as restrictive as Equation 6.21. THE GAUSSIAN CURVATURE 31. We shall see later that the b a p determine the curvatures of the surface, so that there must be a relation between the b a p and the intrinsic curvature of the surface considered as a space of two dimen- sions—that is, the Gaussian or specific curvature which we defined in Equation 5.16. We start with Equation 6.08 for an arbitrary uni surface vector / = / x 1 and take its surface covariant derivative 6.23 l a p = l rs x r a x$,+ {l,V)b a0 , the last term being zero because /, is perpendicula to v r . Again, we differentiate and have 6.24 l u py = IrslX'aX^x'y + l rs X r a Xpy + l rs x s pV'b a y. If we interchange (/3, y), the first term on the righ remains the same if the surrounding space is fla because, in that case, we have l r st = lrts from Equa tion 5.01. The second term remains the same any way because x%y is symmetric in (/3, y). We thei have Lpy — hyp = lv S v r x%b a y — IrsVxfybap 6.25 =- p rs x^x%b a yl s + v r!i x' h x y b ali l\ using Equations 3.19 and 6.07. If we now introduc Equations 6.19 and 5.22, we have R?>«iiyl h = {b a ybah — bhybap)l b ■ Because / s is arbitrary, we have also 6.26 KesaCpy = Rbapy — (6„y6#s — b?>yb a p) , the only nonzero form of which, introducing Equc tion 5.17, is 6.27 aK=Rin 2 = 6,1622- (6, 2 ) 2 = b. In Equation 6.27, we write b for the determinar of the b tt 0, while a is as usual the determinant the metric tensor a a p. This remarkable result relate the b a ji to the expression of K or Rim in terms differentials of the a a p (for example, Equation 5.18 This result is again due to Gauss and is in fac equivalent to his "theorema egregium." The fori b = R\2\2, when expanded, is sometimes known as the Gam characteristic equation. 32. It should be noted, however, that this rest is true only if the surface is embedded in flat spac If we make the necessary modifications and fro Equation 5.02 use /,,,-/,,, = /?,„,,/». we find that for a surface embedded in space who curvature tensor is Rurst, the combination of Eqi Extrinsic Properties of Surfaces 37 tion 6.26 with Equation 5.19 would be K.€&a£0Y — /?fia/ay — baybps ■ 6.28 bf,yba/i + R U rstX*X*X l$*-y. The Gaussian curvature K, being intrinsic to the surface, is the same whether the space is curved or flat. We conclude therefore that the b a n must change with the curvature of the space. We shall consider this further in §8-19 through §8-26. 33. According to a theorem of Bonnet, any six quantities a a ^ and &„#, together with their deriva- tives, which satisfy the Gauss characteristic equa- tion and the two Mainardi-Codazzi equations, determine a surface uniquely except for its position and orientation in space. The theorem is usually proved for a surface in flat space, 2 but is obviously true also in curved space, provided the curvature tensor is specified and the full Codazzi and Gauss equations, Equations 6.22 and 6.28, are used. We cannot expect therefore to derive any other in- dependent properties of a surface; indeed, some of the quantities we have already derived, such as the c a 0, cannot be independent. They are, nevertheless, useful tools, so long as we do not expect them to unearth a completely new result which could not be obtained otherwise. 2 See, for instance, Forsyth (reprint of 1920), Lectures on the Differential Geometry of Curves and Surfaces, original ed. of 1912,51. CHAPTER 7 Extrinsic Properties of Surface Curves THE TANGENT VECTORS 1. We shall now investigate the properties of surface curves considered both as curves on the surface and in space. The unit tangent to the curve will be either /' or /", depending on whether we consider the unit tangent to be a space or a surface vector, and the orthogonal surface vector will be j r or j a as in figure 5 (see §6-13). As before, we shall also consider, as we can do without any loss of generality, that the two vectors are the unit tangents, respectively, to a family of surface curves and to their orthogonal trajectories, defined in some way over a finite region of the surface, in which case we can differentiate the vectors with respect to the surface coordinates without confining our attention to one particular curve. CURVATURE 2. As in Equation 6.07, the space and surface components of the unit tangent are connected by l r = x r J a . We differentiate this with respect to the surface coordinate x 13 and use Equation 6.16 to obtain l r s x$ = v r (b a 0l a ) + xrl<$. In the last term, we substitute Equation 6.09 for x' a and introduce the (intrinsic) geodesic curvatures cr, a* of the /„-curves and of their j a -trajectories from Equations 4.11. We then have 7.01 l r s x&=v r (b ali l a )+j r ( = t\a\\\a so that we have 7. 16 b= {kk* - t-)ana-,-> = (kk* - t-)a and finally 7.17 K=kk*-t*. We conclude that the right-hand side of Equation 7.17 must be the same for any pair of orthogonal directions because K is an invariant which has the same value in any coordinate system, not only in the temporary system used above. 11. The same temporary coordinate system ap- plied to Equation 7.14 gives c n = (k 2 + t 2 )a u c v > = 2HtVci7iVa^ C22 = (k* 2 + t 2 )a-, 2 , bii ■ leading to the determinant c = (kk*-t 2 ) 2 a = aK 2 . Because K is a surface invariant, so is c/a which accordingly has the same value in any coordinate system. We can then write 7.18 K=bla = c/b = (kk*-f 2 ) and can assert that these relations are true in any coordinate system and for any pair of orthogonal directions. 12. An alternative formula for the mean curvature can be found at once from Equations 7.03 and 2.09 as 7.19 2H = -v[ r , which is the negative of the divergence of the unit normal. 13. The components of the three fundamental forms are not independent, but are related by means of the curvature invariants. From Equations 7.12, 7.13, and 7.14, we find at once that 7.20 Ka a p — 2Hb a + Cap = 0. PRINCIPAL CURVATURES 14. We consider next the maxima and minima of the normal curvature for different directions around a fixed point. For this purpose, we take a pair of fixed unit orthogonal surface vectors A a , B a at the point. If l a makes an angle a with A a , we can write l a = A a cos a + B a sin a j a = — A a sin a + B a cos a; and if we differentiate the components with respect to a as a parameter, we have dl a —r- — — A a sin a + B a cos a=j a . da Now we differentiate the normal curvature k=b uli M>K keeping b a p constant because we are merely going to alter direction, not position. For stationary values of A', we must have b a 0J a ll 3 +b a f}l a f = O, or, because b a p is symmetric, we have b afi j a l e =t = 0. But if t is zero in the /"-direction, it must also be zero in the /"-direction because we have seen that 42 Mathematical Geodes the sum of the geodesic torsions in the two directions is zero. We conclude, in general, that there are two orthogonal directions in which the geodesic tor- sions are zero, and the normal curvatures are either a maximum or a minimum. 1 Moreover, proceeding on the above lines, or differentiating (2H) which is the same for all pairs of directions, we find that cPk da 1 ' d 2 k* which shows that the normal curvature is a maxi- mum in one direction and a minimum in the other. We call these directions the principal directions u cv , v a , and the corresponding normal curvatures the principal curvatures K\, /o>. 15. Because t = for the principal directions, the curvature invariants can be expressed as 7.21 7.22 2H= K] + K-2 K= K\K-i. 1 For a more rigorous solution, not confined to two dimensions, see Levi-Civita (1926), The Absolute Differential Calculus, 204. The three fundamental forms become 7.23 a c <0=UaU 1 j + v a viJ 7.24 b a 0= KlU a Up~\- K'VaVp 7.25 C a p= K%U a U0+ K%V a V0. From Equation 7.09, we can also write 7.26 V rs U s = — K\U r 7.27 v, s v s = — K 2 v r . 16. The curves which are tangential to the princ pal directions throughout their length are known , lines of curvature. If v r , v r are the unit surfa( normals at two points separated by a short distanc <& along a line of curvature u s , we have to a fir order v r = v r + v,sU*ds . v, ■ — K\U r ds which shows that the three vectors v r , v r , and u r ai coplanar. Consequently, successive surface norma along a line of curvature intersect. In the case of ar other curve, they would generally be skew. CHAPTER 8 Further Extrinsic Properties of Curves and Surfaces THE CONTRAVARIANT 804 FUNDAMENTAL FORMS 1. We now consider a set of quantities b 01 ® de- fined as the cofactors of b a p in the expansion of the determinant \b a &\ divided by the value b of the determinant, in the same way as the associated metric tensor a a $ is related to the determinant \a a $\. The b a(i can also be considered as consti- tuting the inverse of the matrix b a p- We shall show that b ali is a surface tensor, although it is not the tensor formed by raising the indices of b a p, that is, 2. From the definition and Equation 2.43, we have bb a H=e« Y e l3S bys; 8.07 and dividing this by a, we have 8.01 K6^ = e"^ 8 6y S , which shows at once that b a ^ is a surface tensor o.Oo because K is an absolute invariant. We can expand this last equation from Equations 2.32 and 7.12 as 8.02 Kb"i 3 = k*l a li 3 -t(l a f+j a lf i ) + kj a f, reducing, if we take /", j a as the principal directions 8.09 u a , v a , to 8.03 Kb^=-KiU a u^+Kiv a v^. 3. If we define c ali in the same way as the cofactor of c«ts in the expansion of the determinant \c a p\ divided by c, or as the inverse of the matrix c a p, then we have similarly 43 K 2 c afi = e aY e ps cys, which shows that c a/j is a tensor and, using Equation 7.14, expands to K 2 c^=(k* 2 + t 2 )l a ^-2Ht(l a f+j a lP) + (k 2 + t 2 )j a f 8.05 =Kfu Q ^+KJV'?A 4. We have already found an expression in Equa- tion 2.35 for the contravariant first fundamental form (the metric tensor) as 8.06 a"® = l a iv +j a j li = W'u^ + v«v li and can derive the contravariant form of Equation 7.20 by simple substitution as , a/3. 2Hbrt + Kcrt = Q. 5. From the definitions, we have, as for any matrix and its inverse, b a Pb ay =8% and crt Cay = 8^, which enable us to switch between the fundamental forms. For instance, if we contract Equations 6.18 with b^ and rearrange indices, we have b a > 3 ci 3 y = a a > i b ay- Thus, Weingartens formula in Equation 6.17 can be written either as 8.10 b a0 v > = — artx'p or as 8.11 c^^-i^. 44 Mathematical Geodes If we contract this last equation with g rs Xy and use Equations 6.18, we have 8.12 C^bffy=b a ^affy, which is a reciprocal form of Equation 8.09. 6. Use of the above formulas, together with Equations 7.20 and 8.07, gives us without difficulty the following alternative formulas for the curvature invariants, 8. 13 2H = a^bafi = b^Ca,! = KC#bafi = Kb^aan 8.14 a^c ali = K 2 c a f*a ali = (4H 2 -2K) = k't + k\. 7. The main advantage of the formulas in this section is that one or the other of the forms may be simple in a particular coordinate system, or may be constant under some transformation, such as spherical representation, in which case we can often achieve a simple result quickly by switching into the favorable form. COVARIANT DERIVATIVES OF FUNDAMENTAL FORMS THE 8. The covariant derivative of the first funda- mental form (the metric tensor) is zero as we have seen in § 6—16. Consequently, by differentiating Equation 2.34, we have 8.15 ( UaUjj ) y = — ( V a Vn ) y for any pair of perpendicular unit surface vectors, although we shall use this equation only for the principal directions {u a , v a ) to simplify differentia- tion of the other fundamental forms. 9. We now differentiate Equation 7.24, use Equa- tions 8.15 and 4.11, and obtain after some manipu- lation baffy = ( K\ ) yU a U.$ + ( Ki. ) yV a V$ 8.16 + (#C] — K-z) (, o - *) are, respectively, the prin- cipal curvature and geodesic curvature of the lines of curvature u a , (v a )- We may note that the lines of curvature are defined at any point on the surface, other than at singular points (such as an umbilic where the normal curvature is the same in all directions and the principal directions are accord- ingly indeterminate) or on special surfaces (such as the sphere where all points are umbilics). Con- sequently, Ki, K-i are functions whose values are, in general, defined at every point and may accordingly be differentiated with respect to the surface coordi nates: for example, in Equation 8.16, we have (Ki)y = 3ici/a*r. 10. Differentiating Equation 7.25 in the same way we have Caliy = (lC?) yU a Un + («!) yVaVji 8.17 + (Ki — Ki) (-2K)y b a »C a fry=(4H)y c a ^y = 2(ln K)y. 12. We can obtain more complicated expression: in terms of any pair of orthogonal vectors (/ a , j a ] defined in some way over the surface, by differ entiating Equation 7.12, etc.; but we shall find more convenient to obtain particular contractions such as when required. 13. The Codazzi equations for flat space (EqiK tion 6.21) can be rewritten as 8.21 * y b a py = b al jy ( U?V y ~ ^« Y ) = because ba&y is symmetric in (/3, y). If we contrac Equation 8.16 accordingly and separate the resul ing vector equation, we have 8.22 (k\ — K>)a= (ki )yv y (k\ — K>)cr* — (K2)yu y , which are an alternative form of the Codazzi eqm tions. We can obtain another form in terms of an two orthogonal unit vectors /<,, j a by differentiatin Equation 7.12, and shall do so later by a differer method. The result. 8.23 (T(L-k*) = (k)yjy-(t)yl y -2t(T* a* (k-k*) = (k*)y/ y -(t)yjy + 2t(T, Further Extrinsic Properties of Curves and Surfaces 45 is merely stated at this place for the sake of com- pleteness. In these formulas, k, t, ) + F rs v r v s connecting the space and surface divergences of the vector F,. If F is a scalar, F' r and F% are its space and surface Laplacians. We can rewrite the last term as 8.28 F rs v r v s = {Frv^sif-Frivlv') in which VgV 8 is the vector curvature of the normal. The vector curvature must be a surface vector be- cause its normal component v r vlv s is zero from Equation 3.19, so that we may write v'sV s = X w> in which x IS the curvature of the normal and w r is a unit surface vector. If F is a scalar and ds is the arc element along the normal, Equation 8.28 becomes F,.,VV = d-F/ds- - x ( F r W) , so that Equation 8.27 can be written as 8.29 AF = ~KF-2H(dFlds)+d-Flds 2 - x (FrW) where the surface Laplacian (taken with respect to the surface metric) is given an overbar. If F is con- stant over the surface, the last term is zero and the surface Laplacian is also zero. 18. We can connect the space and surface in- variants of the type in Equation 3.14 by using Equation 6.10. We have V (F, G) = g rs F r G s = a^x^F r G s + v'^Ff^ 8.30 = V(F, G) + dF\ (dG yds J \ds assuming in this case that both F and G are scalars. EXTENSION TO CURVED SPACE 19. In Equation 6.28, we derived an equation connecting the intrinsic curvature K of a surface with the baps and the curvature tensor of the sur- rounding space. We deferred further consideration of this equation until we had developed the con- nection between the 6„^s and the extrinsic curva- ture of the surface and of surface curves. 20. We take the usual pair of orthogonal surface vectors l a , j a , together with the normal curvatures A-, k* in those directions and the geodesic torsion t in the direction /", and contract Equation 6.28 with Using Equations 2.31, 6.07, and 5.25, we then have 8.31 K= (kk* - 1 2 ) + C in which C is the Riemannian curvature of the space 46 Mathematical Geodes for the section (l a „ j"). Because K and C are in- variants which are independent of the particular surface directions, so also is {kk* — t 2 ). Indeed, we have already proved in Equations 7.16 and 7.18 that 8.32 (AA*-f 2 ) = 6/«=(c/«)^. The last two members of this equation are independ- ent of direction and from §2-42 are also invariants. We are, however, no longer entitled to equate Equa- tion 8.32 to the intrinsic curvature of the surface if C is not zero. Nevertheless*, it is clear from Equa- tion 8.31 that all the equations containing K, which we have derived in Chapters 7 and 8, are still true provided that we write (K — C) for K. 21. From the definition of Riemannian curvature in §5—19, we know that C is the intrinsic or Gaus- sian curvature of the surface, formed by the space geodesies which are tangent to our surface at the point under consideration. Equation 8.31 then sug- gests that the normal curvatures of the surface are in some way connected with space geodesies tangential to the surface. We will investigate this suggestion. 22. We use a rectangular locally Cartesian co- ordinate system with an origin at the point P under consideration, x :i -axis in the direction of the surface normal v'\ and x'-axis tangential to the surface in the initial direction of a surface curve /'. The unit tangent to a space geodesic, emanating from P initially in the same direction, is g r . The situation is shown in figure 7 which represents the (x 1 -, x :i -) *~x geodesic surface surface Figure 7. coordinate surface. The /'"- and ^'-curves, initially in this surface, will not, however, remain in the coordinate surface. We shall determine now the coordinates of neighboring points to P on these curves. For a small displacement ds along the /' curve, we have the Taylor expansion 8.33 dx r =l r ds + Hl';J s )(ds) 2 in which it is understood that the coefficients an to have their values at P. In the Cartesian system the coefficient of ?(ds) 2 is, of course, d-x r ds 2 ' but it will be simpler to retain the general tenso notation. The change in coordinates along the space geodesi for an equal distance ds is to a second order dx r = g r ds . . . + because g'^g s = 0. Because we have made g r =l at P, the difference in coordinates to a second orde is dx'-dx r = r(/f, /•)(&)*. The difference in x'-coordinates is then Hv,l'; s ! s )(ds) 2 p,j>i°)(dsy 2 =H(dsy 2 where k is the normal curvature of the surface i the direction /' as defined in Equation 7.03 and use throughout this book. If j r is as usual the surface vector perpendicular t /' at P, then the difference in the x 2 -coordinates i I (./,■/:;• I s )(ds) 2 =+M i rs jn s )(ds) 2 =i) of the prin- cipal normal curvatures, without affecting either K or C in that equation. 26. Special forms of the Codazzi equations, which have been obtained from the flat space form in Equation 6.21, would need restatement to include the extra term in the full Equation 6.22. It does not seem, however, that any general conclusions can be drawn from Equation 6.22 without knowl- edge of the curvature tensor in particular cases. We shall provide an illustration in §10-29 of the use of a particular form of the curvature tensor. CHAPTER 9 Areas and Volumes ELEMENTS OF AREA AND VOLUME 1. We shall require an expression for the area of the small near-parallelogram formed by succes- sive coordinate lines on a surface. For short lengths ds\, ds-z along the coordinate lines, the area is (dsi)(ds 2 ) sin 6 where 6 is the angle between the coordinate lines j as shown in figure 8. But, if the unit vectors in the 9.01 dS=Vadx ] dx Figure 8. coordinate directions are Aft,, A$ ; and if fi a is a unit vector perpendicular to K" u , then we have Sin d=/Jiii\f 2) =€ali\(' 1) k^) = V~a(dx'lds l )(dx 1 lds- 1 ), using Equations 2.37 and 2.30, so that finally the element of area is 306-962 0-69— 5 2. For a similar element of volume in three dimen- sions, we have {ds\)( ds-z ) ( dsa ) sin 6 sin (J) where (f) is the angle which the A. ( ' 3) -coordinate line makes with the plane of A ( ' n and A. ( ' 2I . But in this case, it is clear from the expression in Equation 2.25 for a scalar triple product that we have sin sin = €rs are two scalars, we have 9.03 dU 2 dUi dxdy (Uidx+U-zdy) is \ dx dy i jc in which the double integral on the left is taken ' The reader with no previous knowledge of this section should read Springer(1962), Tensor and Vector Analysis, 147-199, where the elementary theory is clearly explained. The treatment in this section generally follows Brand (1947). Vector and Tensor Analysis translated, with variations, into index notation. 49 50 Mathematical Geodes over a closed region of the (x-, y-) plane and the line or contour integral on the right is taken around the closed boundary of the region. If ds is an element of the length of the contour, if dS is an element of area, and if we take Ui, U-i to be the components of a vector F a , then this formula can be written in the tensor form 9.04 [ € a $F li . a dS=[ FJ a ds in which /" is the unit tangent to the contour. It is clear that in this invariant form the equation holds true in any coordinates — not necessarily Cartesian. Moreover, because only the first co- variant derivative of the vector is involved, it is immaterial whether the space is flat or curved, even though we derived the result from a plane formula in Cartesian coordinates. The formula accordingly holds for any curved surface, provided that F a is defined and its covariant derivatives exist over the closed region S. The same conditions relating to connectivity (which are usually satisfied in the geodetic applications) must apply on the curved surface as on the plane. 4. To obtain the correct signs in either formula, we must describe the contour in such a direction that I" generally rotates in the same sense as from the x- to the y-coordinate line (or, in general coor- dinates, from the u l - to the ^'-coordinate line). The sense of description is as shown in figure 9, which Figure 9. also shows our usual convention for the perpendicu- lar vector j". The normal to the surface (v r ) is toward the reader. 5. If we expand Equation 9.04 in general surface coordinates u\ ur, we have, using Equation 9.01 and the symmetry of the Christoffel symbols, fdF t dfV 905 ' "i? dU-2 duhhr [F l du l + F 2 du 2 which is the same form as Equation 9.03. 6. To extend these results to three dimension we consider the following expression and us Equation 6.11. The tensor Ttj k can be of any ord< and type, but we shall assume Cartesian spac coordinates so that covariant and ordinary deriv tives are the same, and we then have e lmn viT ijk , m = e a Px%x%T Uk , m = (llVa)e a ^d(T ijk )ldu a 1 f dx" dTjjk dx" c)T ijk \ \/a \du 2 du 1 du 1 c)u 2 I 1 f d fdx" \ 3/3*" \\ V^W'W Iljk ) au»W y */J Next, we multiply by dS=vadu l du 2 and integra over some region S of the surface bounded by closed contour C. Using Equation 9.05, we ha^ 9.06 = | T Uk l»ds in which /" is the unit tangent vector to the contou The boundary contour can be any closed curve space spanned by any surface, subject to the usu conditions. We have proved this formula f Cartesian space coordinates and for the comp nents of T,j k in Cartesian coordinates, which meai also that the space must be flat. If, however, v reduce the equation to an invariant form contai ing only first covariant derivatives, then both the: limitations will disappear. For example, if Tij k is space vector F n , we then have p,e lm »F„, m dS= F„l"ds 9.07 in which the two integrands are, respectively, tl normal component of the curl (Equation 3.15) the vector and its component along the bounda contour. This is the tensor form of Stokes' theoret true in any coordinate system — in flat or curv< three-dimensional space. Moreover, we can expe a similar formula to hold true in any number dimensions. In four dimensions, we should ne< different forms of the permutation symbols, whi< Areas and Volumes 51 we shall not discuss in this chapter. 2 However, in two dimensions, we should expect e a VF« a dS= FJ a ds in which F a is now a surface vector, and if we use Equations 6.11 and 8.25, we find at once that this is so. We have in fact recovered Equation 9.04. 7. If the surface is closed, it can be considered as divided into two parts by a closed contour. The contour integral will have the same value but opposite signs for the two parts of the surface, so that over the whole closed surface, we have 9.08 vie lmn F n ,mdS = Q. 8. If we are prepared to continue working in Cartesian space coordinates, we can derive a num- ber of other formulas from the basic Equation 9.06. We can. for instance, multiply by e nM which are constants in Cartesian coordinates and can therefore go under the integral signs. The surface integrand is then Ft'"'l>iT-; Ojjnfl-i ijk, in v„T, ii * ijk- , v'dijk, ,, — Ujk, i>. But from Equation 6.10. we have If we substitute this and Equation 7.19, the surface integrand becomes - 2Hv lt U }k ~ a^g^x^Ujk, q . We now multiply this result by g 1 ' 1 and can do the same to the contour integrand because the g 1 ' 1 are constants in Cartesian space coordinates. The final result is 9.11 £ (2Hv'U Jk + a^x^Ujk, Q )dS= - j fU jk ds which, because Ujk is a general tensor not neces- sarily of the second order, is just as general as either Equation 9.06 or 9.09. 11. As an example of the use of this last result, we take U;i,- to be the gradient of a scalar '' ) + a^xfrfa,, „} dS= -j j'(f>,ds. Using Equation 8.25, we can further reduce this to 9.12 A(t>dS = - cpu'ds in which the Laplacian is taken with respect to the surface metric. In the same way as we obtained Equation 9.08, we conclude that over any closed surface 9.13 /. A(MS = 0. VOLUME AND SURFACE INTEGRALS 12. We shall now consider the triple integration of a tensor Tijk, m over a closed region I of 3-space bounded by a closed surface S. Again, we assume Cartesian coordinates in flat space, and we suppose that an arbitrary field of parallel unit vectors A'" is defined over the region in much the same way as one of the Cartesian coordinate vectors would be de- fined. We suppose further that A'" is the axis of an elementary prism of constant cross-sectional area dcr running through the region, and that dl is an element of length in the direction A'". We then 52 Mathematical Geodei have by integrating along the prism, all components of A" 1 being constant, J T ijk , m dV=j T m , T m ,mdV=\ : dT Uk dx> dx m dl dlda 9.14 J [Tijkl'idcr in which the integrand is now the difference in values of Ty* on the boundary surface at the two ends of the prism. But, if an element of area of the boundary surface is dS and if the exterior or out- ward-drawn unit normal to the boundary surface is i>m, we then have d r of the gradient of a scalar $ and if ds is an element of length along the surface normal, then the last equation becomes 9.17 f (A<}>)dV= j (dlds)dS, which seems to have been given originally t Causs. 14. Again, if we make F'" = g'""((}>l}jn) where c/>, i// are any two scalars, then Equation 9.j becomes 9.18 J {V(c/>, ^)+cf)Ai)j}dV= J 4>(dMds)d which is usually attributed to Green. If we inte change $ and t// and subtract, we have 9.19 J rH »- m} *r-lfc*-,*)4 which is a form of Green much used in potenti theory, where one of the scalars is often taken < the reciprocal of the radius vector. 15. The intrinsic invariance of the Gauss equatk (Equation 9.17) suggests that we could also wri in two dimensions Ac/>dS = 4>«v a ds in which v a is now a surface vector, normal ar outward-drawn to the contour, and the Laplach is taken with respect to the surface metric. In fac we have obtained this result as Equation 9.12 which j 1 or j a is the inward-drawn normal to tl contour (fig. 9) and is therefore the same as mini v a . The two-scalar forms of Equations 9.18 ai 9.19 are similarly valid in two dimensions between surface and contour integrals. 16. If we are prepared to continue working Cartesian coordinates, then, as in the case surface and contour integrals, we can obtain mai other formulas by giving the basic tensor Tijk Equation 9.15 special forms. An instructive examp is to give it the form €""' S F.,, ,-, in which case the closed surface integral vanish because of Equation 9.08. The volume integr therefore vanishes over any arbitrary volum which means that its integrand t r s, rm Areas and Volumes 53 must be zero. Although this is an invariant which case the tensor is symmetric in the two covariant allows us to use any coordinate system, we cannot indices. We then have generalize the result to curved space because it mTsF _,„„., F _ c ,. msF -_ cmrs v . . j j • .• TV/ fc r s, mi t rs.nir — c r s, mi — c r s , mi-, contains second covariant derivatives. We must therefore consider that the space is flat, in which so that the volume integrand is zero, as it should be. ha CHAPTER 10 Conformal Transformation of Space METRICAL RELATIONS 1. We now consider the transformation of a space whose metric is ds 2 to another space whose metric is | 10.01 ds 2 = m 2 ds 2 in which m is a scalar function of position — contin- uous, single-valued, and differentiable over some finite region. The function m must also be an in- I variant because ds 2 and ds 2 are invariants in j Riemannian space. We shall call this function the j scale factor because it multiplies infinitesimal lengths in the one space to obtain the corresponding lengths in the other. 2. We shall also assume that there is a one-to-one correspondence of points over some region of the two spaces. This relation means that the coordinates of points in one space are single-valued functions of the coordinates of corresponding points in the other; for instance, we have x= f(x,y,z), which implies further that the x r can be transformed to the x r and are therefore possible coordinates in the overbarred space. We shall take the coor- | dinates to be the same in both spaces. In that case, if Equation 10.01 is to hold true for all corresponding directions around a point, the two metric tensors will be related by 10.02 grs=m-g r We then have the following relations between the determinants of the metric tensors and between the associated tensors, 10.03 \grs\ = m 6 \g rs \ in three dimensions, and 10.04 g rs = m- 2 g rs . 3. We can also relate the Christcffel symbols straight from the definitions, m '' [ij, k] = [ij, A] + gin ( In m ),■ + g Jk (In //; ) , 10.05 -gtjQn m) k f (j = r' u + 8! ( In m )j + 8j ( In m ) , - gi jg"< ( In m ) k 10.06 in which 8', etc., are Kronecker deltas and (In in), is the gradient of the natural logarithm of the scale factor. 4. Finally, we can relate the two curvature tensors straight from the definition in Equations 5.03 and 5.06, and after some manipulation, the result will be m-' z R q rst — Rqrst = mg QS ( 1/m) ,-, — mg ql ( 1/ m ) , .,. — mg rs (\lm) q t+ mgrt (1/m) (/ ., 10.07 +m 2 ( g rs g ql - g rt g qs ) V ( 1 lm ) in which V(l//n) is the differential invariant from Equation 3.13, that is, S7(llm)=g rs (llm),.(llm) s ; and the expressions (l/m) r t, etc.. are second co- variant derivatives of (1/m). The equations in Equation 10.07 are known as the Finzi equations. 1 1 Levi-Civita (1926). The Absolute Differential Calculus. 229-232. 55 56 Mathematical Geodes THE CURVATURE TENSOR IN THREE DIMENSIONS 5. By contracting the curvature tensors in Equa- tion 10.07 with g Q ' or m^g 9 ', we obtain a relation between the Ricci tensors, which, as we have seen in §5-11, are sufficient to describe the curvature of 3-space. The result, introducing the Laplacian Am = g rs m rs and using the identity 10.08 Am = 2m 3 V(l//n)-m 2 A (1/m), is 10.09 R rs -R rs = -m(llm)rs+ (Ilm)(Am)g rs . 6. We can also relate the Lame tensors in three dimensions by means of Equation 5.13. Using the identity 10.10 A(\n m) =- mMl/m) + m-V (1/m) and lowering indices, the result is 10.11 S rs —Srs= — m(llm) rs — ( A In m )g ls . 7. If both spaces are flat, then the left-hand side of the last equation is zero, and the scale factor must satisfy six second-order differential equations. Using rectangular Cartesian coordinates and sub- stituting Equation 10.10, we see that three of these equations are of the form a 2 (1/m) dxdy and three are of the form d 2 (l/m) , BHl/m) Wd(llm] ~~ m c)x- a r 2 + d(l/m)\ 2 + dx / \ By d (1/m ) VI dz It can be shown 2 that the only nontrivial transfor- mations which satisfy all six equations are inversions with respect to a sphere. If the curvatures of both spaces were to be specified without being zero, the scale factor similarly would have to satisfy six equations, and the choice of scale factor would similarly be restricted so that very few transforma- tions would be available. We shall usually be com- pelled by the nature of the problem to make one 2 See, for example, Forsyth (reprint of 1920), Lectures on the Differential Geometry of Curves and Surfaces, original ed. of 1912,428. space flat, but there is no need to make the othe space flat or to specify its curvature. Nor do w have to attribute any physical significance to th other space; we can consider it simply as a mathi matical device. We can then take the scale facte to be any continuous differentiable function an let it settle the curvature of the space in accordanc with Equations 10.09 and 10.11. If, for instance, th unbarred space is flat, then we have S r .s = 0, and th metric tensor and covariant derivatives on the rigl of Equation 10.11 are all taken with respect to th metric of the flat space. For the present, howeve we shall keep the discussion quite general and n< assume that either space is flat. TRANSFORMATION OF TENSORS 8. Unit contravariant vectors in correspondir directions can easily be related because we ai using the same coordinates for both spaces. We hai 10.12 l'-- dx r ds 1 dx '' -Mr m ds and, for the covariant components, 10.13 I r = grJ s = {m 2 g rs )(m-H s ) = ml r . 9. It is evident that the scalar product of ai two unit vectors remains the same on transform tion so that angles between corresponding directioi are preserved. Small corresponding figures will 1 similar, differing only in scale, which, howeve will vary from point to point. The transformatic is called conformal for this reason. 10. In the case of a nonunit vector field, we con say that the magnitude is a function of the ( ordinates and remains the same on transformati so that nonunit vectors would transform in t same way as Equations 10.12 and 10.13. A tense which can be expressed as a sum of products vectors, would also transform in the same w; but the power of m would, in accordance wi Equations 10.12 and 10.13. be the number of ( variant indices less the number of contravaria indices, for instance, A? st =m 2 A? gt . But it must be noted that all this refers only tensor point functions. It does not apply to c variant derivatives which involve a difference the values of a vector or tensor at two points whe m may have different values. Covariant derivativ must accordingly contain derivatives of m and m Conformal Transformation of Space 57 also contain derivatives of the magnitudes of nonunit vectors. 11. We can relate the covariant derivatives of unit vectors by differentiating Equations 10.12 and 10.13 and by using the relation between the Christof- fel symbols in Equation 10.06, the results being 10.14 m- 1 7,., , = /,-, , - (In m) , I, + g rs (In m) ,/< 10.15 /»/;;. = l r tS + d r s ( In m) t l'-g''(\n m) t l s . The second equation can be shown to be equiv- alent to the first by multiplying the equation by (m--g l( ,) = grq- Higher derivatives can be related in the same way as required. 12. If (f) is a scalar defined to have the same value at corresponding points, such as the scale factor or a common coordinate, then the second covariant derivative of the scalar will be 4>r = rs — 4>r ( In m) s — 4>* ( In m ) ,• + g rs *7 ( In m , ) , 10.16 the differential invariant V being as defined in Equation 3.14. We multiply by m' 1 g rs = g rit to obtain the Laplacian invariants 10.17 m-&(p = A(p + (8? - 2) V (In m,0). Note that in two dimensions the last term is zero, whereas in three dimensions, we have 10.18 m-A = A(/) + V(lnm,<£). CURVATURE AND TORSION OF CORRESPONDING LINES 13. We shall now consider a curve whose unit tangent, normal, and binormal are /,, p r , q r . In the transformed space, the unit tangent, normal, and binormal are l r , n r , b r . The two tangents /, , l r are corresponding directions because the two curves correspond. However, we cannot say that n r , b r , corresponding to n r , b r , will be the same as p r , q, because we have no reason to suppose that the normals and binormals are corresponding direc- tions. From the conformal or angle-true properties of the transformation, we can, however, say that n r , b, will be perpendicular to each other and to /, . The uncertainty in the correspondence is thereby reduced to one angle between n, and p,, which we shall have to determine. The situation is as shown in figure 10, which represents a plane (or transformed normal binormal t ransformed binormal r 'r Figure 10. "section" 3 in curved space) perpendicular to /, . 14. The vector curvatures of the corresponding lines l r , A are related by Equations 10.14 and 10.12 as 10.19 l rs l s = l rs l s - (In m), + {(ln m)t /'}/,, We can also write (In m),= {(ln m),l'} /,.+ { (In m),p<}p, +' {(In m) t q t }q r , and if x- X are tne IWO principal curvatures, we then have Xn,=mxn, ={x~ (hi m),p'} p r — {(\n m),q'}q r . 10.20 But from figure 10, we have n,— (cos 6)p, + (sin 6)q, and so mx cos 9 = x~ (In rn) t p' 10.21 mx sin 0=- (In m),q', which determine both x and 0. 15. If the transformed curve is a geodesic, we then have x~0- The first equation of Equations 10.21 then shows that the principal curvature x 3 A "section" in curved space is defined by a pair of vectors. p r , q, for instance, and is such that any other vector in the sec- tion can be expressed linearly in terms of p, . q, ■ If /, is per- pendicular to p r , q r , then all vectors in the section are perpen- dicular to l r . See also §5-19. IB 58 Mathematical Geodei of the curve in the unbarred space is the arc rate of change of (In m) in the direction of the principal normal to the curve. The second equation of Equa- tions 10.21 shows that the scale factor remains constant for a small displacement in the direction of the binomial to the curve. 16. In regard to the torsion of a general curve, we differentiate the equation n r p r — cos 6 along the line and have, if ds is the arc element, n rs p r l s = — n r pr s l s — sin 9(d6/ds) = -?!,(- xl'+rq 1 ) - sin d(dd/ds) = - t sin 0- sin d(ddlds), using the Frenet equations in Equations 4.06 and introducing the torsion r. Next, we transform the expression on the left to the barred space by means of Equations 10.14 and 10.12 to have {m _1 rc /s -f- (In m) ,n s — g rs {\n m)tn t }p r l s = mh rs p r l s = m(-yjr + rb r )p r — — mf sin #, again using the second Frenet Equations 4.06 and introducing the torsion f of the transformed curve. We have finally 10.22 mf = r+ (dd/ds). By differentiating Equations 10.21, we have after some manipulation mxidd/ds) = — sin d(dxlds) +T(\nm),n' 10.23 +m(llm) rs b r l s ; and by eliminating (dd/ds) with Equation 10.22, we have m-xf = XTCOS 6 — sin d(dxlds) + m ( l/m),- x 6 r / s . 10.24 We have also the following equation connecting the arc rate of change of the two curvatures along the line, m 2 (dx/ds) = cos 9{c>xlds) — r(ln m),b' 10.25 + m(l/m),,n'K 17. The curvature of space enters the equations for the torsion (but not for the curvature) because (dx/ds) involves second covariant derivatives. Tl second covariant derivative of the scale factor ah involves the curvature of the two spaces. We ha\ in fact from Equations 10.09 and 10.11 - m{\ jm ) ls b>/« = (R rs ~ R rs ) b'l s = (§„-s n )bn: 18. A useful way of checking results in a co respondence between two spaces is to interchan^ the spaces. We can transfer the overbars in e equation, such as Equation 10.14, provided v write (1/m) for m; and we then have /»/,,,= /,.,+ (In m)rT, — gn(ln m),l', which is easily shown to be equivalent to tr original equation. In the case of Equation 10.2 we shall also have to change the sign of 6 becaus the rotation from the normal to the transforme normal has the opposite sense in the barred spac Moreover, instead of //, we must write the norm in the barred space, that is. h': and instead of <; we must write the binomial in the barred spac that is, b'. We then have (l/m)X cos Q~X~^~ ( m m )th' (l/m)x sin 0=— (In m) t b* or m X = X cos $ — ( m '")'"' 10.26 x sin 0=— (In m)tb', which are equivalent to the original equation A check on Equations 10.23 and 10.24 is mo difficult, but can be applied using only results whic have already been given — such as Equation 10.1 TRANSFORMATION OF SURFACE NORMALS 19. A continuous differentiable scalar TV in space of three dimensions will define a family surfaces, over each of which N is constant. F example, N=f(x,y,z) defines a surface containing all the points f which /V has a particular constant value: differe members of the family will be obtained by assignii different constant values to N. But /V is consta in directions perpendicular to its gradient, so th the gradient of /V must lie in the direction of tl normal to that surface of the family which pass through the point under consideration. Excludii Conformal Transformation of Space 59 singular points where the vector, we can write ijradient of /V is a mil 10.27 N r nv r in which v r is the unit surface normal and n is the magnitude of the gradient vector. We take the covariant derivative of Equation 10.27 and divide by n to give (lln)N rs = v rs + v r (lnn) s . Because N is a scalar, its second covariant deriva- tive is a symmetric tensor; thus, we can interchange (r, 5) and subtract to find that 10.28 V n v sr -\- v s (\n n In n If we contract this equation with v s and use Equation 3.19, which makes v S rV* zero, we have i 10.29 v rs v s = ( In n ) r — v r ( In n) s v s I Finally, we compare this result with Equation 10.19 I and conclude that a scale factor of n will transform J the space conformally to another space in which I the surface normals become a family of geodesies. J Because of the conformal property of the trans- formation, these geodesies will be normal to the transformed /V-surfaces. Moreover, an element of length along a transformed geodesic will be ds = nds = {N r v')ds = N r dx' dN. The length of a geodesic intercepted between two /V-surfaces Ni and N-> will accordingly be (N-> — N\) and will be the same for all geodesies between these two surfaces. For this reason, the transformed /V-surfaces are known as geodesic parallels. 20. Conversely, if there exists a family of geo- desies and geodesic parallels, expressible by a scalar /V, in a conformal transformation with scale factor n, then we can say that the relation N r = nv r jmust hold true between the corresponding lines land surfaces in the untransformed space. 21. If we write n = m.f(N) in which f(N) is an arbitrary, continuous, dif- ferentiate, and nonvanishing function of N and if we substitute in Equation 10.29, we have VrsV s — (In m) r — tv(ln m) s v s AN) Vrf{N) AN) (N,v). The last two terms cancel by virtue of Equation 10.27. Comparing this result with Equation 10.29, we conclude that the scale factor can well be n multiplied by an arbitrary function of /V. TRANSFORMATION OF SURFACES 22. It should be noted that we have nowhere assumed that either space is flat; the curvature of the space does not arise until we assign particular values to the curvature tensor or until we introduce the second covariant derivatives of vectors. More- over, all the above properties are intrinsic to the space, being based solely on the metric tensor and its derivatives. We can accordingly use all the above tensor formulas with Greek indices for transformations between curved surfaces con- sidered as two-dimensional spaces, provided we do not use results, such as Equation 10.09, which are valid only in three dimensions. We have, for instance, a a i3=m-a LXli \a„p\ = m 4 \a a (j\ 10.30 ««* = m -a a/3 and all of Equations 10.05, 10.06, 10.07, 10.12, 10.13, 10.14, 10.15, 10.16, and 10.17 are valid. 23. Because there is only one component of the curvature tensor in two dimensions, we can simplify the Finzi equations of Equation 10.07 which are. in two dimensions, m--R u py?>— Rafjy?,= ma a y(llm)p8— ma a s{llm)py — mafiy{l/m)as + maps{ilm) a y + m - ( a,fjya a 8 — a^a a y ) V ( 1/m) where the invariant V is now taken with respect to the surface metric a a p. We contract with a ay =m-a ay and use Equations 5.19 and 2.45 to have Kaps — Kaps = mamMl/rn) — m-a^V( 1/m I. If we substitute a/38 = m-aps and use the identity A(ln m)=-mA(l/m)+m 2 V(l/m). we have finally as the sole curvature equation 10.31 A(ln m) = K-m l K in which, of course, the Laplacian is taken with respect to the unbarred surface metric. This is a 60 Mathematical Geodesy well-known formula in the theory of conformal map projections, attributed by Marussi 4 to Souslow (1898). If the Gaussian curvatures of the two surfaces are given, then this formula is a differential equation which the scale factor must satisfy. Alternatively, we could choose the scale factor and one surface, in which case the Gaussian curvature of the other surface is settled by Equation 10.31. If one surface is a plane, then we have 10.32 M\nm) = K in which the Laplacian and, of course, the Gaussian curvature refer to the curved surface. GEODESIC CURVATURES 24. In two dimensions, Equation 10.19 becomes a-j a = orja — (In m) a + { (In m)pl®}l a in which cr, & are, respectively, the geodesic curva- tures of the curve L and of its transform; andj a ,j a are perpendicular to l a and to its transform in the usual sense of figure 5 in § 6-13. But in this case, ja, y u , which are both surface vectors, must corre- spond because of the conformal property of the transformation. (Note that in three dimensions, we could not say that the two principal normals cor- respond.) So, from the two dimensional form of Equation 10.13, we have ja= mj a , and the above equation reduces to 10.33 a—ma=Qsim)cj a . If =-Ka a iil a P=-K, which is the same for all directions at a point, so that the spherical representation would be con- formal. But in this case, we are restricted to a special class of surfaces whose mean curvature H is zero. Such surfaces are known as minimal surfaces. They are of considerable importance in the physics of soap bubbles and in the minima of double in- tegrals, but do not appear to have any present application in geodesy. 10. If the principal directions of the given surface are u a , v& (the principal curvatures are K\, Kj. re- spectively), then t = for these directions, and the scale factor for the //"-direction reduces to 11.17 ds/ds = V KJ. We shall consider that corresponding elements of length are in the same sense so that the scale factor is essentially positive. Throughout this book, we shall be dealing with convex surfaces whose radii of normal curvature run inward in the opposite sense to the outward-drawn normal and will there- fore be numerically negative when com [tilted in accordance with the usual sign conventions from formulas given in this book. Consequently, we must take the negative square root in Equation 11.17 and write 11.18 lis I (Is = — K\. The scale factor in the /'"-direction will similarly be — k>. In this case, Equations 11.12 and 11.13 reduce to 11.19 11.20 Kiir — Ki V ua- vv- UplKi V/jIk-2 SCALE FACTOR AND DIRECTIONS REFERRED TO THE PRINCIPAL DIRECTIONS 11. If the unit surface vector /" makes an angle t// with the principal direction u", then an alternative expression for the square of the scale factor ( in ) in the direction l a is from Equation 7.25 m' 2 = (dslds)- — c a fjl a l li = K\ cos- t/> + kt, sin 1 ' \\i. 11.21 From this equation and from Equations 11.12 and 11.19, we can obtain expressions for i//. the angle between the spherical representations of /". //". as follows. cos \\i = /"/"/„ = — k i l"u a ( ds/ds ) _ — K\ COS l// 11.22 \k\ cos- \\t-\-K7, sin- t//)'- and similarly, we have — ki sin (// 11.23 sin(//=/"fv k] cos- i/> + K.7 sin 2 (//)'-' from which 11.24 tan t//= {k>Ik\ ) tan i//. The sense of ' and some rearrangement of indices give finally 11.30 T^-T^ = -b^b m . The Christoffel symbols in the spherical representa- tion are usually very easy to evaluate in a given coordinate system, so that we have now a compact formula for the Christoffel symbols of any given surface, which we shall have frequent occasion to use. We note from Equation 11.04 that because b a p = — a a t), we must have 6 a ^g = 0, so that Equation 11.30 reduces further to the statement that the quantities H.31 ryp+ b^bapa are unaltered on spherical representation. REPRESENTATION OF A FAMILY OF SURFACES 13. If we have a family of surfaces defined over a certain region of space, for example, by assigning different values to a scalar N which is constant over each surface as discussed in §10-19, then the surface normals will also be defined over the region. In general, as we shall see in the next chapter, there will be a family of lines — to be known as the isozenithals — along any one of which the surface normals are parallel. The spherical representation of an isozenithal is accordingly a point. We can, moreover, draw a figure on any one of the /V-surfaces and project it down the iso- zenithals onto the other surfaces of the family. The original figure and its isozenithal projections will all have the same spherical representation. Moreover, any set of quantities, such as those in Equation 11.31 which have the same values at corresponding points in the spherical representa- tion, will also have the same value at isozenithally projected points. Their differentials along the iso- zenithals will be zero. We shall carry the question of spherical representa- tion further in the next two chapters by using a special coordinate system, which, nevertheless, produces quite general results. CONTENTS Part II Chapter 12 -The (w, <£, TV) Coordinate Pa e e System 69 Definitions 69 Sign Conventions 69 The Base Vectors 70 Relations Between Base Vectors 71 Derivatives of the Base Vectors 72 Contravariant Components of the Base Vectors 73 Covariant Components of the Base Vectors 74 Curvatures of the TV-Surfaces 75 Geodesic Curvatures 76 The Metric Tensor 77 Second Fundamental Form of the TV- Surfaces 78 Third Fundamental Form of the TV- Surfaces 78 The Coordinate Directions 79 Laplacians of the Coordinates 80 The Christoffel Symbols 81 The Mainardi-Codazzi Equations 82 Alternative Derivation of the Mainardi- Codazzi Equations 84 Higher Derivatives of the Base Vectors. . . 85 The Marussi Tensor 86 The Position Vector 86 CHAPTER 13 — Spherical Representation in (a>,4>,N) 89 General 89 Curvatures and Azimuths 89 Geodesic Curvatures 90 Chapter 13 — Continued Page Covariant Derivatives 90 Expansion in Spherical Harmonics 91 Double Spherical Representation 91 CHAPTER 14 — Isozenithal Differentiation 93 Definition 93 Differentiation of the Fundamental Forms 93 Differentiation of Surface Christoffel Symbols 94 Differentiation of b a py 95 Differentiation of Vectors Defined in Space 95 Isozenithal and Normal Differentiation... 96 Differentiation of the Curvature Param- eters 97 Differentiation of the Principal Curva- tures 97 Projection of Surface Vectors 98 CHAPTER 15 — Normal Coordinate Systems... 103 General 103 The Metric Tensor 103 Components of the Normal and of Sur- face Vectors 104 The Christoffel Symbols 105 Variation of the Metric Tensor Along the Normal 105 Space Derivatives of the Normal 106 The Mainardi-Codazzi Equations 106 Normal Differentiation 107 Normal Projection of Surface Vectors... 110 67 68 Mathematical Geodesy Page Chapter 16 — Triply Orthogonal Systems... 113 General 113 The Darboux Equation 113 Solutions of the Darboux Equation 114 Chapter 17 — The (co, , h) Coordinate Sys- tem 117 General Description of the System 117 The Fundamental Forms 118 The Base Vectors 119 The Principal Directions and Curva- tures 120 The Christoffel Symbols 120 Laplacians of the Coordinates 120 Change of Scale and Azimuth in Normal Projection 121 The ^-Differentiation 121 Examples of ^-Differentiation 122 The Position Vector 124 CHAPTER 18 — Symmetrical (to, , h) Sys- Page terns 12J Definition 12! Principal Radii of Curvature 12! Collected Formulas 12! Surface Geodesies 12i The Spheroidal Base 12! Chapter 19 — Transformations Between N- Systems 13 General Remarks 13 Directions 13 Base Vectors 13 Azimuths and Zenith Distances 13 Orientation Conditions 13 The R and S Matrices 13 Tensor Transformation Matrices 13. Parallel Transport of Vectors 13> The Deflection Vector 13i Change in Coordinates 13' CHAPTER 12 The (co, <£>, TV) Coordinate System DEFINITIONS 1. We shall now consider a special, but quite general, coordinate system, generated by a con- tinuous differentiable scalar function of position TV in three-dimensional space,. Points having a particular value of N, for instance C, will lie on a surface N=C; for different values of C, we shall have a family of surfaces. We take A as one co- ordinate of the system. But, if N is specified through- out some region of space, then so is the magnitude {n) and direction (v r ) of its gradient (AV) because, by definition, we have 12.001 N r = nv r The direction of v r in relation to three fixed Car- tesian axes in flat space will define two independent scalars, which can take the form of longitude (co) and latitude {(f)). We shall take these as the other two coordinates. Each of these scalars generates a family of surfaces distinct from the /V-surfaces and from one another. The position of a point in space can accordingly be defined as the intersection of three surfaces, one from each of the to, (/>, and A families over which each of the three coordinates has an assigned value, in much the same way as the position of a point in Cartesian coordinates (a, 6, c) can be defined as the intersection of three planes x — a, y—b,z — c. In the more general case, the coordinate surfaces are curved; each coordinate^ line — that is, the line of intersection of two coordi- nate surfaces along which only the third coordinate varies — will also be curved. The three coordinate lines passing through a point will not, as a rule, be orthogonal, nor will they be parallel to the coordinate lines at any other point. It will be assumed throughout this chapter that (&>, c/>) are the A-surfaee coordinates as well as two of the space coordinates so that, with the notation of Equation 6.02, we have x2 = 0: x li = 8 li In some cases, this leads to results which are clearly only true in this coordinate system because they relate only some of the components of tensors. In other cases, we shall derive relations connecting all the components of tensors. These will accord- ingly be tensor equations, true in any coordinates, which can be differentiated covariantly and manipu- lated generally as tensors even though they were derived in a special coordinate system. SIGN CONVENTIONS 2. There is some advantage in making the (co, (f), N) system right handed in the sense that (x, y. z) is conventionally right handed, as discussed in §1-22. If we look along the positive direction of an TV-coordinate line, then the positive direction of the c£-line is to the right of the positive direction of the cu-line; a similar rule applies to the cyclic permutations (c/>Aw), {Noi(j>). A positive rotation about the A-coordinate line will be clockwise when we look outward along the positive direction of the A-coordinate line. We could say therefore that the co-line can be rotated positively about the A-line toward the (/>-line. 3. We shall later identify A with the gravitational potential, or geopotential, or some standard poten- tial. The A-surfaces will be the level or equipotential surfaces: the n will be the gravitional force "g." M 70 Mathematical Geodesy The positive direction of A, following the ordinary physical convention, will be toward the zenith even though this will make N negative in the geodetic applications. For Equation 12.001 to hold true — and in such applications as conformal transformation it is desirable that the equation should hold true in this positive form — we must also draw the positive direction of the normal to the A-surfaces toward the zenith; this accords with the usual mathematical convention of an outward-drawn normal to a closed surface. We have finally to make the (oj, $, A) system right handed in that order; to do this, we must make longitude positive toward the east if we are to adopt the almost universal convention of making latitude positive toward the north. This accords with the European convention and with astronomical conventions for right ascension and local time (but not hour angle, which is reckoned positive toward the west). It also makes longitude a positive rotation in the mathematical sense about the northward axis of rotation of the Earth. It does not accord with geodetic practice in the United States where it is customary to make west longitudes positive, no doubt for reasons of historical develop- ment, although some Agencies in the United States adopt the more usual eastward convention. On the whole, the balance of advantage seems to lie with positive longitudes east. Any country using the opposite convention has merely to change the sign of longitude, or difference in longitude, wherever it occurs in any formula in this book; the same applies to south latitudes. 4. In the proposed convention, longitude will be the first coordinate, whereas the almost universal convention is to list latitude first. However, this should cause no confusion. We consider longitude to be the first coordinate in a right-handed system (at, 0, A = 1,2,3) in the derivation of mathematical formulas, but the results can, of course, be listed in any convenient order. 5. A positive rotation about the zenithal direction iv r ) in the mathematical sense will be from north to west, whereas the almost universal geodetic convention for azimuth (a) is from north to east. The only way of reconciling the two would be to adopt an inward-drawn normal to the TV-surfaces; this could lead to serious confusion in cases where formulas are taken straight from standard mathe- matical works. However, we can avoid confusion by giving azimuth its own convention and by re- membering that azimuth is a negative mathematical rotation in cases where it is derived that way. 6. In another geodetic application, we shall neec to identify A with "height." By universal conven tion, this is positive in the zenith direction; this then agrees with the proposed convention for A, 7. The conventions which will be adopted through- out this book are illustrated in the diagram (fig. 12); north positive (CO, N constant) east co positive (, N constant) Figure 12. the zenith direction (v r ) or the gradient of A is toward the reader. In an unsymmetrical field, the u>- and -coordinate lines will not run exactly to the east and north, but they will, nevertheless, run in those general directions. In the same way, the A-coordinate line — that is, the direction in which a>, 4> are constant — will not coincide, in general, with v r - The other two vectors A.,, /a, on the diagram lie in the plane of the paper and will now be defined. THE BASE VECTORS 8. Next, we set up three mutually orthogonal unit parallel vector fields A'\ B'\ C r to serve as the axes of a right-handed Cartesian coordinate system (x, y, z). This assumes that we are working in flat three-dimensional space because such a coordinate system would not otherwise be possible. In Carte- sian coordinates, the components of these vectors would be A r = (1. 0. 0) 12.002 B r = (0, 1, 0) C' = (0. 0. 1). but we shall often require their components in other coordinate systems. The vectors are constant The (o>, 4>, N) Coordinate System 71 in the sense that their Cartesian components are the same throughout the region of space considered. Their components will not be the same at all points in other coordinate systems, but because they are parallel at all points, their covariant derivatives, from Equations 3.05 and 3.06, will be zero in all systems. We shall later identify these vectors physically — for example, in some applications C r will be parallel to the axis of rotation of the Earth with its positive direction toward the north — but for the present, the vectors simply provide a fixed Cartesian reference system. 9. We also introduce a local system of mutually orthogonal unit vectors k'\ fx r , v'\ right handed in that order. As before, v r is the zenith direction or the outward-drawn unit normal to the A-surface passing through the point under consideration. We define /jl 1 ' as coplanar with v T and a parallel to C r , and call it the direction of the meridian; the positive direction of /x' will be roughly in the direction of the (^-coordinate line, and because /x'' is perpendicu- lar to v'\ it will be tangential to the /V-surface. The vector X r simply completes the orthogonal triad. It will also be an TV-surface vector, roughly in the direc- tion of the aj-coordinate line, and will be called the parallel direction to accord as nearly as possible with ordinary geographical terms. It is easy to see that A.' will be parallel to a plane containing A r and B' because it is perpendicular to the plane of fx r and v r , and is therefore perpendicular to C r . 10. Next, we define longitude (co) and latitude (c/>) in terms of the direction cosines of the unit normal v r by means of the following scalar products, 12.003 12.004 12.005 COS = v r C'\ The arrangement is illustrated by figure 13 in which the meridian plane is the plane of the paper, except for the vectors A'\ B' . 11. We define azimuth (a) as a rotation about v T from r x r toward A.', as shown in figure 12. A unit A-surfaee vector /'' in azimuth a is accordingly given by 12.006 /' = V sin a+/x' cos a. The use of the term azimuth suggests that the A-surfaces are level in the geodetic sense; in the main geodetic applications, this will be so. We do not yet, however, identify the A-surfaces with level Figure 13. or equipotential surfaces: and in this chapter, the term azimuth is to be understood in a wider sense. With the same object of avoiding multiplication of terms and on the same analogy, we shall sometimes refer to the direction of the normal v r as the zenith and to an angle measured from the zenith as a zenith distance. A unit vector in azimuth a and zenith distance B will be given by the vector equation /'=A/ sin a sin /3 + yu.' cos a sin B+v r cos B, 12.007 which can easily be verified from the direction cosines of /'' relative to the (A/, jjl'\ v r ) axes. RELATIONS BETWEEN BASE VECTORS 12. We can now express one set of vectors in terms of the others, through their direction cosines, as follows. \ r — ~A r sin oo + B, cos to r i r = — A r sin 4> cos co — B r sin c/> sin oo + C r cos cf) v r = A r cos (/) cos oo + B, cos c/> sin oo + C r sin cos to + v r cos cf> cos to y r = B r —k r cos w — /JL, sin sin to + v r cos cfi sin a> 12.009 Z r =C r — fir COS ), is the position vector of the (\ r , fi'\ v') origin. Thus, Equations 12.008 give x = — (x — xo) sin to+ (y — yn) cos to y = — (x — xo) sin cf> cos to — (y— Vo) sin sin to + (z — za) cos (/) i= {x — xo) cos cos to 12.010 + (y — Jo) cos 4> sin w+ (z — Zo) sin , and Equations 12.009 give (x — xo) ~ — x sin to — y sin cos to +z cos cos to (y— yo) — x cos a» — y sin sin o» +z cos $ sin to (z — Zo) —y cos + z sin . 12.011 15. We may also note that the (A.', fx'\ v r ) system can be obtained from the {A'\ B'\ C r ) system by the following rotations: First, (i7r + to) about the 3-axis C' which brings A r into parallelism with A.'; and, Second, (i^r — (f>) about the new 1-axis A.'' which brings C' into parallelism with v r . Accordingly, we may substitute the following matrix equation for Equations 12.008. /K\ l\ W- sin to cos to { \l A \ \ /JL, \= sin fx r )■ \CrJ \ 1/ \0 COS 4> Sill cf) J \vr) 12.013 A very convenient, special notation for rotation matrices will often be found in the literature. A positive rotation of d about each coordinate axis — positive in the usual mathematical sense illustrated in § 12-2 and § 12-5 -is denoted by /I OX Ri(0)= cos 6 sin \ -sing cos e) I cos d -sin 6\ R 2 (6»)= 1 \ sin 6 cos d J I cos 9 sin 6 \ R3(0)=(-sin0 cos 6 ] \ 1 / Using braces notation {A r , B r , C r } for column matrices, Equation 12.012 would then be written {A,, fir, v r } = Ri(br-){Ar, B r , C r }. 12.012A In these formulas, the axes are rotated and points in the space are held fixed; if the axes were fixed, the rotations would have opposite signs. To avoid any possible confusion, the few rotation matrices required in this book will be written in full. DERIVATIVES OF THE BASE VECTORS 16. If we take the covariant derivative of the first equation of Equations 12.008 and remember that A r , B r , C r are constant under covariant differ- entiation, we have krs~ (~A r COS CO — B r sin to)to s 12.014 = (fl r Sin 4> — V r COS (/))to. s The (o», (/>, N) Coordinate System 7:>, by substituting the other equations of Equations 12.008; in the same way, we have 12.015 firs — — sin x 12.016 Vrs = cos \ r 0)s + fi r (f> s - In these expressions, w. s , s are the gradients of the coordinates considered as scalars. They are not necessarily surface vectors. But if we take (w, ) to be the /V-surface coordinates as well as two of the space coordinates, then the (1, 2) components Wo, 4> a will be the surface gradients. In the (w, (/>, N) system, we have ,■=82 0s ~°1, but if we do not make this substitution, then the above tensor equations are true in any coordinate system. 17. By covariant differentiation of the basic gradient equation we have 12.017 N r — nv r Nrs~ n s v r -\- nv sr . But because N is a scalar, N r s is a symmetrical tensor by Equation 3.11. Interchanging r, 5 and subtracting, we have 12.018 n s v r + n V,-s — Jtr Vs + 'I v sr . Multiplying by v s and noting that v sr v s = because v s is a unit vector (Equation 3.19), we have 12.019 n VrsV* —n r —{ n s v s ) v r . But the vector n r is expressible in terms of three orthogonal vectors as n r — (n s \ s )X, + (n s fi s )(ir+ (ris^Vr so that Equation 12.019 reduces, after division by n, to 12.020 v rs v s = {(In ra) s \ s }Ar+{(ln n).,^ 8 }^, showing that the principal normals to the v r are N- surface vectors. We shall write the arc rate of change of (In n) in the parallel and meridian direc- tion as yi, y 2 , respectively, so that this last equation can be written as 12.021 VrsV s — 7l A/- + 72 Air, showing that the curvature of the normal is (7i + 7f) 1/2 The principal normal to the curve is in azimuth arctan (71/72); the binomial, along which n is constant, is a surface vector in azimuth arctan (-72/71)- 18. It should be noted that k rs is a space tensor taken in relation to the space metric. It will, never- theless, have (1, 2) components which can be written as A a /3= (fJ-a Sin )(Dti. Again, if (a>, ) are surface coordinates, we know from Equation 8.25 that k a (i is also the correspond- ing surface tensor. We shall see later that v a — in (a», (/), TV) coordinates so that we have 12.022 k a fi— /JLaOifi Sin (f), whether it is considered to be a surface tensor or the (1, 2) components of a space tensor. In the same way, 12.023 rla/3 — — k a (j)fj Sin (f> is either a surface tensor or the (1,2) components of a space tensor in (co, <£, TV) coordinates, provided (oj, (f>) are taken as surface coordinates. 19. We can see from Equation 6.19 that the (1, 2) components of the space tensor v r s, again in the (to, (/>, N) system with (oj, 4>) as surface coordinates, are given by 12.024 v a p = — b af j where 6 Q /3 — the second fundamental form of the surface — is a surface tensor. Here again, we could say that v a $ is a surface tensor because b a p is a surface tensor, and the v a $ are also the (1, 2) com- ponents of a space tensor. Equations 12.022 and 12.023 are, however, surface tensor equations, but Equation 12.024 is merely a relation expressing some components of the space tensor v rs . If we want to manipulate Equation 12.024 further, we should have to generalize it first as VrsX'aXg — ~ b a 0. CONTRAVARIANT COMPONENTS OF THE BASE VECTORS 20. If we differentiate the defining Equation 12.005 covariantly and remember that C r is a con- stant vector, we have (COS (f))(f)s— v rs C r = v rs (/JL r cos (f) + v r sin 4> ) = W r .,/X r COS (/>. in the derivation of which we have used Equations 74 Mathematical Geodes- 12.009 and 3.19, with v r a unit vector, so that finally we have 12.025 4>s = VrsH r = - VrsV r - 21. In the same way, by covariant differentiation of Equations 12.003 or 12.004, we have 12.026 (cos )aj s =v r sk r = -k rs v r ; by repeating Equation 12.001 to complete the series, we have also 12.027 N s = nv s . 22. In addition, we have already found in Equa- tion 12.021 a formula for the vector curvature of the normal to any family of /V-surfaces, p rs i' s = {(In n) s k s }k r + {(In vI^/jl,- 12.028 =y l K r + y-,fjL r . 23. Now, if {dk) is an element of length in the A. r -direction, then the contravariant components of A r in the (oo, , N) system are, by definition and by using Equations 12.026, 12.025, and 12.027, A r = (aw/ax, d/dk, dN/dk) = (w s \ s , s k s , N s k s ) = (seC V rs k r k s , Vrsk s /JL r , 0) 12.029 =(-ki sec , -t u 0) where k\ is, from Equation 7.03, the normal curva- ture of the iV-surface in the direction of the parallel, and where t\ is, from Equation 7.08, the geodesic torsion of the /V-surface in the same direction. (The geodesic torsion of the A^-surface in the direc- tion of the meridian is, of course, minus t\.) 24. In the same way, we have fi r = (dcoldn, d4>ld/x, dN/dfi) = (sec (f) Vrsk r fX, s , V r slJi r fl s , 0) 12.030 =(-h sec, -fe,0) where k% is the normal curvature of the TV-surface in the direction of the meridian. 25. To complete the triad, we need similarly to evaluate the components of v r . Writing (ds) for an element of length in the direction of the normal, we have from Equations 12.026, 12.025, 12.027, and 12.021 12.031 d doj/ds = VrskV = (In n) s k s = yi 12.033 dNlds = N s v s = n so that we have finally 12.034 v r —(j\ sec (/>, y 2 , n). 26. Without any loss of generality, we can tak< (oj, 4>) as coordinates in the TV-surfaces as well ai two of the space coordinates. We shall as usual us< Greek indices restricted to the values (1, 2) fo surface vectors and tensors; it is then evident fron the definition that the components of A", /jl", con sidered as surface vectors, are 12.035 k a = (-ki sec , -h) 12.036 fi a = (-?, sec (/>, -A,). 27. Collecting results for easier reference, we havi k' = (-Ai sec (j>, —tu 0) /jl' = (—ti sec . -k>, 0) 12.037 v r = (yi sec {^), y%, n) , with the same (1, 2) components for the surfao vectors A", /x" in (w, $) coordinates. 28. All contravariant and, as we shall see, al covariant components of the base vectors cai accordingly be written in terms of the five second order quantities k\, &2, h, yj , y L >. which we shall cal the curvature parameters of the space or of the field We have seen in Equation 12.021 that yi, y 2 defini the curvature of the normals: we shall see in th< section commencing with § 12-36 that ki, A 2 , t completely define the curvature properties of tin A^-surfaces. COVARIANT COMPONENTS OF THE BASE VECTORS 29. Next, we find the covariant components fron Equation 2.07 k'k s + IJL r IJis+V r Vs = 8's in which Si" is the Kronecker delta. For r=3. we have at once nv s — Sis, which gives the components of the normal as 12.038 v s =(0, 0, 1/n). 30. For r= 1, 2 ; s — 1, we have the two equation — (Ai sec )ki — (t x sec 4>)jjl x = 1 — t\k\ — A"2ju.i =0: and writing K for (AiA 2 — t'i), which we have seen Equation 7.17 is the Gaussian or specific curvatur of the A-surface in flat space, we can solve these las equations to provide the 1-components as follows ki = — ki cos 4>/K /JLi = + t\ COS (f>/K. The (co, c/>, N) Coordinate System 75 31. In the same way for r=l, 2 ; 5 = 2, we have the equations — (/,i sec <£) k>— (t\ see )^t-j = — / 1 \j - A2JU2 = 1 - which can be solved for the 2-components k-, = + tJK lx> = — kilK. 32. For r— 1. 2 ; s = 3, we have — (/i sec (/))A;s — (/i sec 0)/u,n + yi sec $/h = 0, — *,\ 3 - ^3 + 72/n = 0, from which we have A.A :! = (tt 2 y\ —tiy 2 )ln 12.039 K t i 3 =(kiy-2-t 1 y i )ln; or, substituting for yi, y. from Equation 12.021. we have 12.040 K\3 = - (lln) s {k 2 k s -t lf x s ) Kfi s = - (lln) A- tik s + ki/J, s ) Again, substituting the above values for Ai, /u-i, etc., we have A.( cos = (l/n) s (kik s + fAi fJL* + ^1 ^ s ) = (l/n).8f = d(lln)lda> and jU-3 — (lln)s(k-2 k s + fX-2 fJL S + V-i V s ) = (l/n),8S = d(l/n)/d(f>. 33. Collecting results, we have 12.041 Kk r =(—k 2 cos . + ti, /v sec <£ d(l/n)/dw) 12.042 K/jL, = (+t x cos (/>. -/,,. Kd(lln)ld), with the alternative expressions in Equations 12.039 and 12.040 for the 3-components, and 12.043 v r = (0, 0, 1/n). 34. We can similarly find the covariant compo- nents \ a , jtta, considered as surface vectors from the two-dimensional formula and have finally 12.044 Kk a =(-ki cos , +/,) 12.045 Kju, Q = ( + / 1 cos 0, - A , ) . which are the same as the (1, 2) components of the space vectors. 35. The gradients of the coordinates can now be expressed in terms of the base vectors A., , /x r , v, by means of the following formulas, 12.046 (cos 4>)o), ■ — — kjk, — ti/jL, --\-y\v r 12.047 <*>,- = - t x k r - kii* r + yn'r. and we have also 12.048 (In n) l =yik l +yiHr+{(\n n) s v s }v r ; because these gradients are vector equations, not merely relations between some components of vectors in a special coordinate system, they are true in any coordinates — provided w, <£> are con- sidered to be scalars. CURVATURES OF THE iV-SURFACES 36. The three quantities A,, k-i, and t\ — respec- tively, the normal curvatures of an A-surface in the direction of the parallel and the meridian, and the geodesic torsion in the direction of the parallel — enable us to determine the normal curvature and geodesic torsion in any azimuth (a). A unit surface vector in this azimuth will be Ifi—ku sin a + (jlu cos a: a unit vector in the perpendicular direction, ob- tained by a positive right-handed rotation of /# about the normal, will be jii — —kij cos a + fxp sin a. The normal curvature in the direction /#, using space coordinates, will be k = -Vrsl'l s = — v rs k r k s sin 2 a —2p,^'f^" si' 1 a cos a — V r sfL r fL s cos 2 a = k\ sin 2 a + 'lt\ sin a cos a+ k-> cos 2 a: 12.049 the geodesic torsion in the direction /# will be t = -v rs l r j* = Visk r k s sin a cos a — v r sk r fi s (sin 2 a — cos 2 a) { k-i — A 1 ) sin a cos a — t\ ( cos 2 a — sin 2 a) 12.050 76 Mathematical Geodesy 37. The geodesic torsion in a principal direction is zero so that the azimuth {A) of the principal directions is given by 12.051 tan 2A=+2t 1 /(k 2 -k 1 ). 38. The principal curvatures (ki in azimuth A and K-i in azimuth (A — \tt)) are then given by Equation 12.049 as K\ = A i sin- A + 2t\ sin A cos A + A 2 cos- /i K2 = k\ cos- /4 — 2?i sin A cos /4 + A 2 sin 2 ,4 12.052 so that the mean curvature is 12.053 ff=i(Ki + K4)=i(ifc, + fe) f as we should expect, because it is the same for any two perpendicular directions. 39. We have also ( K, — k_> ) = ( hi — k\ ) cos 1A + 2ti sin 2A = (/.-,-/„) sec 2A 12.054 = 2;,cosec2//, using Equation 12.051. 40. The Gauss or specific curvature of the surface is then K=KiK 2 = Uki + fa) 2 -Uh-k 2 ) 2 sec 2 2A 12.055 =hk 2 -tl as we should expect from Equation 7.17. 41. We can also recast Equation 12.049 to give the normal curvature in any azimuth (a) as 12.056 k = Ki cos 2 {A-a) + K-i sin 2 (A -a) and the geodesic torsion in azimuth (a) as 12.057 t = H Kl -K 2 ) sin2(A-a). By putting a = ^TT, or zero, in these last two equa- tions, we have also A i = K\ sin 2 A + k-i cos 2 A ki — K\ cos 2 A + k> sin 2 A 12.058 u= (ki — k 2 ) sin A cos A, showing that, instead of the three curvature param- eters k\, k 2 , h, we could equally well use ki, k>. A. 42. If k, t, a are the normal curvature, geodesic torsion, and azimuth in the direction of a general unit surface vector /' and if A*. — t, {a— %rr) refer to a perpendicular unit surface vector/, then from Equations 12.049 and 12.050, we have k = k\ sin 2 a + 2t\ sin a cos a + A'2 cos 2 a t= (ko — ki) sin a cos a — Mcos 2 a — sin 2 a) k* — k\ cos 2 a — 2ti sin a cos a + k-y sin 2 a. 12.059 From these equations and Equations 12.046 an 12.047, if dL dj are elements of length in the tw directions, we easily derive (cos )d(o/dl=— ky sin a— t x cos a = — A' sin a + t cos a d(f)/dl — — t\ sin a — k> cos a = — A cos a~t sin a (cos cf>)do)/dj= A', cos a — fi sin a = A cos a— £ sin a d4>/dj—t\ cos a — A"2 sin a 12.060 = — A* sin a — f cos a. which enable us to rewrite Equations 12.046 an 12.047 as (cos 4>)cOi = (— k sin a + t cos a)/,- 12.061 + (A* cos a — t sin a)j r + y\Vr , N) Coordinate System 77 to the first equation of Equations 4.11, we find that 12.064 crly + (j*jy— toy sin 4> — ay: if the arc element in the direction l y is dl. this reduces to 12.065 (d so that the geodesic curvature of the parallel trace is given by 12.066 . 45. Similarly for the meridian trace, we have a = and da/d/ = 0: while from Equation 12.030, we have d(o/dl — — ti sec (j) so that the geodesic curvature of the meridian trace is given by 12.067 o-> = -ti tan . 46. We can now express the geodesic curvature of any surface curve l y in azimuth a in terms of o~i , cr-> as cr= (sin (f>)coy(k y sin a + /x y cos a) — (da/dl) = CTi sin a + cr> cos a — [dajdl) 12.068 in which dl is the arc element in the direction l y . 47. By equating Equation 12.065 or 12.068 to zero, we have the differential equation of the surface geodesies. It has usually been assumed in classical geodesy that the form of the equation derived from Equation 12.065, that is, da— (sin ) (A r cos a> + B, sin ai) , its principal normal is accordingly — [A r cos (o + B r sin to) , and its binormal is C, which is a constant vector. The angle 6 between the principal normal and the surface normal is given by Equations 12.009 as cos 6 = — cos (f) so that we have 0=7T- 0. Finally. Equation 7.08 gives the space torsion as h - ( dd/dk) =h+( d) as both surface and space coordinates and substituting the vector components from Equations 12.041, etc., we have #n = «ii = (kl + t'i ) cos- /K- ftvi = (i\i = — 2Hti cos 4>/ls.- gl2 = 0,22= (kj+fi)/K 2 #13 = h 3(l/n) ti cos d{\/n) K do; K 34) = -[yi{k'i-\-fj)-2Hy 2 ti]l{nK 2 sec ) _ £i sec d(lln] t d(Hn dto / V dIK 2 . 52. Components of the contravariant space metric tensor are g u = {k 2 + t 2 + y 2 ) sec- a vl = 2Ht\ sec (f> 12.072 a 22 =(k 2 . + t 2 ). 54. The determinants of the associated tensors are 12.073 ri 2 K 2 sec- ; \a a ^\ —K 2 sec' 2 $, which are, as they should be, the reciprocals of the covariant determinants. SECOND FUNDAMENTAL FORM OF THE iV-SURFACES 55. By contracting Equation 12.016 with x r a x\ and using Equation 6.19, we have 12.074 — b a [3 = (COS (\))ka(tili-\- IXafylS from which, assuming as usual that o», (b are also surface coordinates, we have b a ji = — (COS ku £M, M'2> 12.075 = (fa cos 2 IK,-ti cos /K, fajK). 56. The determinant of the form by direct cal- culation is from which the contravariant form (from §8-1) is 12.077 b afi = (/u sec 2 0, h sec . fa). 57. We have already seen in Equation 12.024 that 12.078 b a li — —Vali in which it is understood that v a /i are components of the space tensor v rs taken in relation to the space metric. 58. By combining Equation 12.075 with Equations 12.044 and 12.045, we can write 12.079 b\a — ~ (cos 4>)k a \ b-ia — — Mq, which are frequently useful relations; also, we have 12.080 b u > = -(sec(f>)\ a ; b' la =-fi a . 59. Yet, another useful formula can be obtained by noting from Equation 12.016 that in these co- ordinates we have v,:i = 0. We then have from Equations 12.020 and 12.024 12.081 12.082 (In n) a =Vr S v s x r a = — bajjV 13 v a = — b ali {\n n)^. 12.076 b = cos 2 (j>IK THIRD FUNDAMENTAL FORM OF THE IV-SURFACES 60. There are several ways of computing the third form c a p; perhaps the simplest being from the formula in Equation 7.20 Can = 2Hb a ii — Ka a u so that we have cu= {h + fa)fa cos 2 4>IK- (k'i + t 2 ) cos 2 <})IK = cos 2 (b cn = —{ki + fa)ti cos IK + 2Ht 1 cos 4>IK = c 22 =(A, + A2)A I //v-(A- 2 + ? 2 )/A' = 1, and collecting results we have 12.083 c a /3=(cos 2 tf>, 0, 1). The determinant is 12.084 c=cos 2 (/>. The (to, c/>, N) Coordinate System 7<> agreeing with Equations 7.18 and 12.076; the con- travariant form is accordingly 12.085 c«0=(sec 2 0, 0, 1). It may be noted that if we take the determinant of the defining Equations 6.18, by the ordinary rule for multiplying determinants, we have b 2 = ac, a relation which is accordingly true in any coordi- nate system, as we may see also from Equation 7.18. It can easily be verified from the (w, (j>, A) values in Equations 12.070, 12.076, and 12.084. THE COORDINATE DIRECTIONS Longitude 61. From the metric, an element of length in the to-coordinate direction (dfj>, dN zero) is vgudco. The contravariant components of a unit vector in this direction are accordingly 12.086 i r =(l/Vfti, 0, 0), and its azimuth a\ will be given by cos ai = i r fAr = hi (k'i+ t\y t2 12.087 sin ai = i r k r = -hl(kl + tl) 1 l 2 . 62. Using Equations 12.058 in which A is the azimuth of the Ki-principal direction, we find without difficulty that (Af+*?) 1/2 = (k! sin 2 A + k'{ cos- A) 1 '- — m-2, for instance, I so that we have cos a.\ = ( K\ — Ko ) sin A cos A/nio 1 12.088 sin ai = — (k 2 sin 2 A + ki cos- A) /mo sin (A — oc\ ) = Ki cos A\m% 12.089 cos (A- at) =- Ki sin A/m. Latitude 63. In the same way, the contravariant com- ponents of a unit vector in the (^-coordinate direc- tion are 12.090 /=(o, i/V^, 0), and its azimuth a 2 will be given by cos a 2 =j r (jL, - = — /.,/(/. i + /i ) ,;2 12.091 sin a 2 =j r \ T =W(A? + *i) 1/2 - Again, using Equations 12.058, we have ( k'i + t\ ) "- = ( k\ sin- A + k\ cos 2 A ) ll - = mi, for instance, cos a-i — — (k\ sin 2 A + k-i cos 2 A) /mi 12.092 sin a>= (ki — k>) sin A cos A/nii sin (A — a>) = — ki sin A/nii 12.093 cos (A — a>) = — k 2 cos A\m x . 64. Now consider the spherical representation of the A-surface in which the surface coordinates (to, (/>) will be the same because the normals at corresponding points are parallel. It is evident that the ^-coordinate line is represented by the spherical meridian, which is parallel in space to the meridian direction fx' on the surface. We have also seen in Chapter 11 that a principal direction and its spherical representation are parallel in space. Consequently, the angle (A — a->) on the surface corresponds to A on the sphere; from Equation 11.24, we have tan A = (k 2 /ki) tan (A — a 2 ) , which verifies Equations 12.093. 65. In the same way, the angle (A — at) on the surface corresponds to (A— iir) on the sphere so that we have — cot A= (k-j/kti) tan (A — ai), which verifies Equations 12.089. 66. The fact that the (w, 4>) coordinate lines are represented by the spherical meridians and parallels again shows that the representation is not as a rule conformal because the coordinate lines are not, in general, orthogonal. It is clear from Equations 12.087 and 12.091, or from a 12 in Equations 12.069, that the coordinate lines will be orthogonal if, and only if, £i = 0, corresponding to the axially sym- metrical case. 67. The metric of the spherical representation in these coordinates will be 12.094 ds 1 = cos 2 4> dto- + d'\ so that the scale factor (dsjds) in the direction of 80 Mathematical Geodes the to-coordinate line will be V(cos- la u ) = Kl ( kl + t\ ) 1/2 = Kim, and in the direction of the (^-coordinate line will be V(l/« 22 )= Kj{k\ + t\yi* = Kim, in which m\, m> have been defined in connection with Equations 12.092 and 12.088. The Isozenithal 68. A unit vector in the /V-coordinate direction (a). (j> constant) is similarly given by 12.095 ft r =(0, 0, 1/Vgfc), and its azimuth (a) and zenith distance (j8) will be given by sin a sin fi= k r k,= {sec d ( lln)ldio}lvga3 cos a sin /3 = k r fJL r = {d(l/n)/d}/Vg33 cos /3= k r v r = {Iln)/Vgs3 12.096 or sin a tan (3 — — (see are constant along this line, it is evident from Equations 12.008 that the V at all points along the line are parallel; and so are the (A 1 and v r . The whole triad of vectors can be trans- ported parallel to itself along the line, which we shall call the isozenithal because the zenith direc- tion v r is the same at all points along any one such line. Another way of expressing the parallel trans- port of these vectors is to state that there is no intrinsic change in their components along the line, or in tensor notation 12.098 \rgk s = fl r gk* = v rs k s — 0. These tensor equations are, of course, true in any coordinates. and differentiate it covariantly as 12.099 N rs =n s vr + nv rs . The Laplacian of N in space is then A/V = g rs N rs = n*v s + ng rs v rs and the last term from Equation 7.19 is equal {—2Hn), so that we have finally 12.100 &N=dnlds — 2Hn in which ds is an element of length along the norms This last equation will be recognized as an exa> form of a formula usually attributed to Bruns i applications where /V is the geopotential and n gravity, but we see that it is simply a geometric property of any family of surfaces. 71. From Equations 12.025 and 12.026, togethc with Equation 12.099, we have — without difficulty - the following generally useful relations, 12.101 ( cos <\>) o) s = ( 1/re) N rs \ r 12.102 4),=(l//2)/V,, M '- 12.103 n s = N rs v r \ differentiating the first covariantly. we have wit some substitution (cos (f))o) s t= (sin ) u),s(t>t — - (In n),(cos )co s + (Hn)N rst \ r + (sin )<]> s cot — (In n) K ( cos 4>)a)t. We note that because /V is a scalar in flat space, i third covariant derivative is symmetrical in any tw indices so that we have ^■'/V,,,,= (^W,,),= (A/V),. We also introduce the symbol V for a differenti invariant from Equation 3.14, such that we ha^ V(a>, ) =g' s (i),(f)s : V(oj) =g rs a),(Os. etc. and finally obtain (cos c/>)Aw = 2 sin (f> V(oj, 4>) —2 cos (f> V (w. In / 12.104 -f- (!/«)( A/V ) r K r . LAPLACIANS OF THE COORDINATES 70. For some applications, we need formulas for the Laplacian of each coordinate, particularly that of N and its derivatives. We start with the gradient equation N r =nv r 72. In the same way, we have A<£ = — 2V(. In n) —sin cos (f)V(a)) 12.105 +(1//0(A/V), / Lt''. These last two equations are of particular val in this form in applications where A/V is a constc because the last terms are then zero. The (o>, (/>, N) Coordinate System 81 73. From Equations 12.103, 12.099, and 12.016, we have also An = re{cos 2 (f> V(o>) + V(0)}+ ( A/V),V. 12.106 74. We can easily find the V invariants from Equations 12.046, 12.047, and 12.048 in terms of the five parameters of the space, but first we need to find an alternative expression for the third component of (In n) r , taking account of Equation 12.100. We have 12.107 (In n) r v r = (l/n)dnlds = 2H+ (AN)/n so that Equation 12.048 becomes {lnn)r=yikr+y2fU + {2H+(AN)ln}v r . 12.108 We then have from Equations 12.046, etc., 12.109 cos-' 4> V( V((o, (f>)=2Ht l + y 1 y 2 cos V(o>, In n)=-k 1 y l —tiy* + 2Hyi+ (yiAN)ln 12.112 =A'2y,-/,y,+ (yiAN)ln V(<£, In n)=-f,y,-Ayy 2 + 2//y2 + (y 2 AN)/n 12.113 =A' 1 y 2 -/ 1 y 1 +(y 2 A/V)//j. We have also cos- 4> V(w)+V({/>) = (A 1 + A 2 ) L '-2(A 1 A-2-^) + y i + yi = 4//--2A'+ (y?+yl) 12.114 =Ki + Kl+(yf + yl), which is the sum of the squares of the principal curvatures of the /V-surface plus the square of the principal curvature of the normal, all at the point under consideration. An alternative expression for the Laplacian of n is accordingly (Hn)An = W 1 -2K+ (yf + yf) 12.115 + (l/n) (AN) r v r . 75. All the previously mentioned formulas in this section refer to the space invariants. We can easily find the surface invariants of w and (b (de- noted by overbars) from the (1, 2) components of 306-962 0-69— 7 Equations 12.026 and 12.025 12.116 (COS )o> u y — b a py\& — b a u(x^a)y sin c/> so that we have (cos 0)A&)= (sin <£)V((u, #) -a^ba/fyk + (sin (b)V(a), , cb)-(2H) a K a with 12.119 V(o),; similarly, we have 12.120 A^ = - (sin cos 0)V(a) - {2H) u ^ a with 12.121 V(ft)) = (A? + tf) sec 2 <£. 76. We cannot differentiate Equation 12.081, (In n) a = — b a iiv®, in the same way because this equation is simply a relation involving selected components of the space vector v r in a special coordinate system; it is not a surface tensor equation because v 13 is not a surface vector. We shall, however, find in Equation 14.28 an ex- pression for the surface Laplacian of n, which can easily be put into the following form, comparable with Equation 12.115, as (Hn)An~ = (4H 2 -2K) + 2(yi + yj)-d{2H)lds 12.122 in which 5 is again the arc length of the normal. It should be noted that this, unlike the space invariant, does not depend on AN. 77. The surface Laplacian of N is, of course, zero because ^V is constant over the surface. THE CHRISTOFFEL SYMBOLS 78. We can compute the Christoffel symbols straight from the definitions and the components of the metric tensor or from transformation formulas; but, because we know the components of the base 82 Mathematical Geodesy vectors and have formulas for the covariant deriva- tives of the coordinates, it is possible to take various shortcuts which are more instructive. 79. For example, we can express the tensor N rs as /v,,=-ryv,=-n s ; by covariant differentiation of the gradient equation N r = n v r , we have 12.123 N n = n gVr +nv n . We have also, from Equation 12.016, i/,3 = 0; from Equation 12.024, we have v a p = — b a is- By simple substitution, we can then obtain all the distinct Christoffel symbols with superscript 3 as follows, Tin = nbap ; 12.124 TL = n 3 d(ln n)ldN. 80. To evaluate the symbols which have a sub- script 3 but no superscript 3, we shall make use of a device which is frequently useful in other directions. We can express a Christoffel symbol in terms of the components of any three mutually orthogonal vectors by means of the following formula, which can easily be verified by multiplying kj, fXj, Vj in turn. 12.125 kl = 77 ^ J + 777 f* + 77 v J dx l dx' ^ dx' — ( kklk> + /Ji k IIJL J + V k lV j ) . If (X r , fjb r , Vr) have their usual significance in this chapter and 1 = 3, then the whole term within parentheses vanishes because of Equations 12.014, 12.015, and 12.016; thus we have yet 1 fc3 " dN dN * dN 12.126 dk a d/JL a dv a ' dN Vk, the last line being obtained by differentiating the identity k k k a + t X k /JL a +P l< V a =8 k \ For A = j8(^3) we have from Equations 12.079 and 12.080 1 /33 ^7-b la + —Fr b- a dN dN dN 12.127 : — buy db ay dN' We shall show in Equation 12.144 how this symbol can be expressed in terms of n and iV-surface tensors. 81. For k—S, using Equations 12.041, etc., we have r„ dHl/n) t1 „ dHl/n) ,., d(lln) , ,. x r -=-^v b]a ~^dW b M ^ (ln n) >> which simplifies without difficulty to d 2 (ln n) 12.128 r? 3 = (l/n)6^ BjfidN 82. The remaining symbols are all of the form _ dkq ~ dx® 12.129 dxP ix y — a)p sin (f)(/Jiak y — k a fx y )+ ba^v" on substituting Equations 12.022, 12.023, and 12.024. In evaluating this expression, we can make use ol the symmetry of the Christoffel symbol in the sub- scripts. For example, if either a or /8 = 2, then we can eliminate the whole of the third term by taking )6 = 2. The expressions on the right, obtained b> interchanging a and /3, can be made identical by using the Mainardi-Codazzi equations of the /V surfaces, which we shall consider in the nexl section. 83. We can apply the general formula of Equatior 12.125 in two dimensions and write 12.130 lfl * dx* dx* ^+-^' k a pk y — fJLaplJ? in which the Christoffel symbol must now be take: in relation to the surface metric; k a p, fx a p are sui face tensors. By subtraction from Equation 12.12? we have Tl (space) 12.131 Tin (surface) =— v a $v = b a &v y because, as we have seen in Equations 12.022 am 12.023, the tensors A a/3 , ^t a/3 can be considered eithe as surface tensors or as components of space ten sors in these coordinates. This last result is o frequent use. THE MAINARDI-CODAZZI EQUATION* 84. The two Mainardi-Codazzi equations of a The (co, 4>, N) Coordinate System 83 surface may be considered as conditions of integra- bility, or from § 6-27 as conditions for the surface to be embedded in flat space. In either case, use must be made of the fact that the Christoffel symbols are symmetrical in the two subscripts because this is a distinguishing mark of Riemannian geometry, arising from the nature of the metric tensor. 85. If we are given a set of functions a a p, b a p-, does a surface exist for which these functions are the appropriate fundamental forms? To prove that a surface does exist in the neighborhood of a point where the a a and b a $ are given, we must be able to integrate the Weingarten and Gauss equations (Equations 6.17 and 6.16) v r a = — a^b ay x r p x^=b u0 v r : it can be shown that the necessary conditions for this are the Mainardi-Codazzi equations. For our purposes, we shall always start with a family of surfaces — definable in nature over finite regions by other means — so that these equations may be considered as properties of the geometry or of the space. 86. If we take the surface covariant derivative of a surface vector X a , we have then a necessary condition for the Christoffel symbol to be symmetrical in a, /3 is 12.132 \afi ~ kite = dKJdxfi ~ dXpldx?. For a given superscript, there is only one Christoffel symbol in two dimensions with dissimilar subscripts and therefore only two such symbols in all. It will accordingly be sufficient to satisfy Equation 12.132 for one other independent vector fjL a so that we have 12.133 llafi — flpa = dfJLjdx 13 — dflpldx a . Both equations are satisfied identically unless a and /3 are different; so it will be sufficient to make B'=l, )3 = 2, and to substitute Equations 12.022 and 12.023 to obtain — /jl-2 sin <£ = d\i/d<£ — dAo/dto X.2 sin — d/x 2 /do>, which reduce on substitution of Equations 12.079 to db\\ld — dbvilda) + bw tan + 6 2 2 sin (b cos = 12.134 It should be noted that in deriving these formulas, we have used Equations 12.022 and 12.023, which were themselves derived on the assumption that the space is flat through use of the Cartesian vectors A r , B r , C r . Equations 12.134 and 12.135 are the Mainardi- Codazzi equations of the /V-surfaces in (w, cb, N) coordinates. They can be expressed in several other equivalent forms, but for the present, we shall be content with them as they stand. 87. If, instead of the surface vectors A.,, fx, , we take the space Cartesian vectors A,, B r . C r whose covariant derivatives are zero, then, so far as the TV-surfaces are concerned, we have to satisfy the following equations to ensure that the appropriate Christoffel symbols are symmetrical. dAJd^dAilda) : dBJd^dBzlda) : dd/d^dCz/do) in which A\, etc., are components in (oj, cb, N). If we obtain A\, etc., from Equations 12.009 by substi- tuting the (o>, (/>, N) components of \ r , etc., from Equation 12.041, then these conditions give exactly the same results as Equations 12.134 and 12.135 — no more and no less. Moreover, it is evident from Equa- tions 12.009, etc.. that the above conditions are equivalent to d 2 x d 2 x 'cry by d z z o-z 12.135 dbvzld — dbz-ildu) — b V i tan (f) — 0. debdo) dtod4> " d4>d(t) d(od(f> ' defcdeo Swdcf)' which are well-known integrability conditions for the existence of the Cartesian coordinates (x, y, z). This demonstration goes part of the way toward justifying the statement made in § 12-84 and § 6-27 that the Mainardi-Codazzi equations are conditions for a given surface to be embedded in flat space. If the surface is embedded in curved space, the Mainardi-Codazzi equations or integrability condi- tions take the different form of Equation 6.22. 88. We have so far considered only the /V-surfaces, but there must similarly be two equations for each of the other coordinate surfaces. We need not, how- ever, consider these surfaces specifically. We shall derive the same answer more easily if we form equa- tions similar to Equation 12.132 for three independ- ent space vectors and if we substitute such relations as Equations 12.014, 12.015, and 12.016 which apply only in flat space. 89. First, we consider the equation 12.136 Vrs~ V S r = d V r /dx s — d V s jdx r , and then substitute Equation 12.016 and the (w, (f>. 84 Mathematical Geodesy TV) components of v r - For r— 1, 5 = 2, the equation is satisfied identically. For r= 1,5 = 3 and r=2, 5 = 3, we have A 3 cos 4> = d(l/n)lda) ; /M3 = d ( l/n)/d<£, obtained before in Equations 12.041 and 12.042. 90. The equation 12.137 k rs - ksr = d k r ldx s - d k g /dx r for r=l, 5 = 2 gives Equation 12.134, and for r— 1. 5 = 3 gives — /A 3 sin + 1^3 cos = dkildN — dk:ild(D, which, on substitution of Equations 12.079, 12.042, and 12.043, reduces to db u d 2 (l/n) , , , d(l/n ) cos 2 "Trr = ; — ~, r" Sin rf) COS or 12.139 a6 12 a 2 (i/«; — tan (f) d(lln) dco dN d(od(f) both of which are new. 91. The equation P-rs — fJLsr = d/Xrldx s ~ d/X s ldx r for r=l, 5 = 2 gives Equation 12.135, and for r—1, 5 = 3 gives Equation 12.139. For r=2, 5 = 3, we have V;i = dfJi-z/dN — dfJLsldcf) or 12.140 db-22 d 2 (l/n) 1 d 2 n 92. There are accordingly only three independent Mainardi-Codazzi equations for the space in addi- tion to the two for the /V-surfaces — a total of five out of a maximum of six. The coordinate system is, nevertheless, perfectly general, except that the /V-surfaces are generated by a scalar, which means that the equation N rs — n s V r + TlVrs ~ H-rVs + n,V sr must apply because /YVs is symmetrical in r and s. This symmetrical relation serves to satisfy Equation 12.136. We are not therefore missing one of the six equations; we have already included it. 93. Next, we shall put the Equations 12.138, 12.139, and 12.140 in tensor form. From Equation 11.03, the metric of the spherical representation of an /V-surfaee in (oj, ) coordinates is 12.141 ds 2 = c a fidx a dxV = cos 2 dco 2 + d<$> 2 , using Equation 12.083. It is easy to show by direct calculation from the definitions that the only non- zero Christoffel symbols in this metric are 12.142 T\ = sin cos ; r} 2 = -tan. By inspection, we can now write the Equations 12.138, 12.139, and 12.140 in the form io uo db ati ^ d 2 (l/n) — d(l/ra) c a/3 lJ " 1 * 6 dN~ dx?dxP + L <# d x y ~T ; substituting Equation 11.30, we have dbali 12.144 dN L+^-a-T in which the second covariant derivative of (1/ra) is taken with respect to the metric of the /V-surface. Each term of the right-hand side of this equation is a surface tensor; therefore, the left-hand side must be a surface tensor. 94. The foregoing analysis has been given in some detail because it is important to ensure that we have not overlooked any essential relation in the differ- ential geometry of the space, such as an omitted Mainardi-Codazzi equation. Moreover, we require Equation 12.144 to show how the Christoffel sym- bols of Equation 12.127, dbpy r% 3 = b°y dN can be expressed in terms of n and surface tensors — as in the case of all other Christoffel symbols with a fixed 3-index. ALTERNATIVE DERIVATION OF THE MAINARDI-CODAZZI EQUATIONS 95. In view of the fundamental importance of the three additional space equations in Equation 12.144 we shall now approach them from a different direc tion and, at the same time, shall derive some gen erally useful formulas. We take one particular /V-surface and draw the tangent plane at a point P (fig. 14). We drop a perpendicular OQ on the tangent plane from the Cartesian origin 0, and denote the length of this perpendicular by p. The vector OQ is according!) of magnitude p and of direction v r , while the vectoi OP is the position vector p r . The coordinates (&>, , (/>, N) Coordinate System 85 isozenithal Figure 14. We have at once 12.145 p = p r v r ; taking the surface covariant derivative of this, we have also 12.146 Pa = grsX a V r + p r V r a = PrV a , the remaining term being zero because of the orthogonality of x s a and v T as space vectors. We have also used the fact that the tensor equation prs = grs is true in Cartesian, and therefore in any coordinates. Again, taking the surface tensor derivative of Equation 12.146, we have PaH = grsX s V r a + p r V r aji = — b a (j + p r {b yb bal3bV^ — C a $V r ) in which we have used Equations 11.26 and 8.10. With some slight rearrangement and use of Equa- tions 12.146 and 12.145, we have baii = — Pali + b y8 b a o py — pC a fi 12.147 dx a dx^ + rZi}Py-pca0 where we have used Equation 11.30; the (overbarred) Christoffel symbols of the spherical representation have values from Equations 12.142 in (o», <£) coordinates. 96. Next, we differentiate this last expression along the isozenithal at P. The tangent plane moves parallel to itself because the direction of the normal is unaltered; for the same reason, the spherical representation remains unaltered. Consequently, the Christoffel symbols in Equation 12.147 remain constant, as is otherwise obvious from Equations 12.142, because they are functions of cf> only. Again, the c a ji from Equation 12.083 are constant during the differentiation. If the unit isozenithal vector is k r and if the displacement along the isozenithal is ds, then we have dN = N r k r ds — nv r k r ds = n cos (3 ds = ndp so that we may write 12.148 dp = l dN n This result also could have been obtained from the third component of the space covariant derivative of Equation 12.145 in (gj, , N). Ordinary partial differentiation of Equation 12.147 accordingly gives us dba0 dN ' 12.149 3S(1/n) + TzJ±) -* d^dxP a/3 n/y which is precisely the same as Equation 12.143 or 12.144. In deriving this equation, we have made use of the properties of the Cartesian position vector and of the constant components of Cartesian vectors during spherical representation. In other words, we have assumed that the space is flat, but have as- sumed nothing else; this again illustrates the two ways of considering the Mainardi-Codazzi equations. 97. We can also show by ordinary partial differ- entiation of Equation 12.147, with respect to surface coordinates, that we have d bqjj d b u y dx y dxP ' 12.150 r^bys+rtybps. which on expansion is easily shown to be equivalent to the Codazzi equations of the A^-surface in Equa- tions 12.134 and 12.135. Accordingly, we can say that Equation 12.147 is an integral of all five Codazzi equations, which are automatically satisfied every time we use Equation 12.147. A more compact form of Equation 12.147 is 12.151 —b a = Pat} + pC a p in which the overbar indicates that the second covariant derivative of p is taken with respect to the metric of the spherical representation of the A-surface. HIGHER DERIVATIVES OF THE BASE VECTORS 98. Now that we have formulas for the Christoffel symbols and for the Mainardi-Codazzi equations, we can without difficulty find expressions for the higher derivatives of the base vectors in these co- ordinates. First, however, we shall collect some formulas for the first derivatives. 86 From Equations 12.014, etc., we have at once 12.152 \ r 3 — ,Atr3 = VrZ = 0. The only nonzero components containing a 3-index are accordingly k 3a , etc.; by substitution in Equa- tions 12.014, 12.041, etc., we have at once 12.153 A: la — sin djlln) cos \ s] 3(/> n 12.154 ^-tan^ftjjj doj 12.155 i*«=(l/ii) a . 99. The only other nonzero components have been obtained before in Equations 12.022, 12.023, and 12.024, but are collected for easy reference as follows, 12.156 kafi— (J-aOJp sin 12.158 Va0 — — b a f). 100. Components of the second covariant deriva- tives may now be obtained straight from the defi- nition. For example, we have Kp3 = dkap/dN— r^Kfj — r^ 3 \ ar = (up sin 4>) (dfia/dN) — r% 3 /jLy(O0 sin (/> — r^ 3 /ji a o)y sin in which the first two terms cancel because /x a3 = 0. In the same way, using the fact that the second and third indices are interchangeable in flat space, we have K#a = A. a3 /3 = -(sin 4))iLab ly (dbpyldN) &a&i = l*aW = (s,m 4>)k a b ly (dbffyldN) 12.159 Poj93 = V*30 = d bap/dN in which we can substitute Equation 12.143 or 12.144 for dbpy/dN. In much the same way, we find Aa.i.3 = - n 3 k a y =fjL a (l/n) tan A s {a 2 (ln n)/dx s dN} fia33 = — Tl 3 fJi a y=— k a (l/n) tan A 6 {r*-(ln n)/dx s dN} i'a-s-s =-n 3 v a y=a/n){dHln n)/dx a dN}. 12.160 We can also find by direct covariant differentiation and by use of Equation 12.131 that 12.161 v a $y = — b a 0y — 6/jy(ln n) a ; other components can be found similarly when required. Mathematical Geodesy THE MARUSSI TENSOR 101. It is now clear that the second and higher order metrical properties of the system can be written in terms of the five curvature parameters (Ai. ki, t\, yi, jz) and their derivatives. But the entire system has been generated from a single scalar A whose covariant derivatives must be related to the curvature parameters. To show this, we have onl) to contract the tensor Equation 12.017, Nrs=n s V r + nVrs, with the base vectors to obtain N rs k r k s = -nk! N rsjJ-'' jJL S = ~ Tlk% N n \ r fi , = -nti N r sk r v s =nyi NrsfSv* =rey 2 12.162 N rs v'v s = n(ln n ) s v s in which we have used only definitions and Equa tion 12.028. Apart from the factor n, all the param eters on the right are accordingly the components of the symmetric tensor N rs . This fact was firsl noticed by Marussi ' in the case where TV is a gravi tational potential as well as a generalized coordinate 102. As we shall see later, the case of a Newtonian gravitational field simply involves assigning a par- ticular value to the Laplacian of A\ AN = N rs (k r k s + IU, r /JL s + V r P s ) 12.163 =-n(k 1 + h->) + n(\n n) s v s , so that the law of gravity eliminates one of the components of N rs , leaving us with the other five. In a local Cartesian system (x, y, z) with axes (k'\ jx'\ v r ), we have N rs k r k s = d-N/dx 2 12.164 N rs k r ix s = d 2 Nldxdy. etc. The parameters are usually given in this par ticular form in the literature, except that the x-axis is sometimes /x' . THE POSITION VECTOR 103. We have seen in § 12-95 that the perpen dicular p from the Cartesian origin to the tangen plane of an /V-surface is of special significance h 1 Marussi (1949), "Fondements de Geometrie Differentieli Absolue du Champ Potentiel Terrestre," Bulletin Geodesique new series, no. 14, 411-439. The (oj, , N) Coordinate System 87 this coordinate system. Because p is the scalar product of the position vector p, and the unit nor- mal v'\ the question naturally arises whether we can express the other components of the position vector in terms of p. We can express any vector in terms of the orthogonal triad k, ., p, r , v r , and so can write 12.165 p,-= qk,+ rfjL,- + pv r in which the scalars q, r have to be determined. 104. In rectangular Cartesian coordinates, the components of p r are (x, y, z): it is easy to verify from Equation 1.07 that in these coordinates 12.166 & which is a tensor equation true in any coordinates. If we take the covariant derivative of Equation 12.165 and substitute Equations 12.014, 12.015, and 12.016, we have grs= {qs~{r sin (/j)o). s + (p cos (fj)o>. s }A.,- + {r s + (q sin 0)w. s + p)(o s -~r(f) s }v r ; contracting this in turn with A' , pf, v'\ we have the equivalent three vector equations k s =q s — (r sin (j)))) o) s — r„. 105. Evaluation of the third of these equations in (oj, (/>, TV) coordinates gives at once dpldw = q cos ; dpld=r ; dp/dN=l/n ; 12.168 substitution of these values in the first two equations of Equations 12.167, together with the (co, )(dpld(0)k r + (dpld(l>)p, r +pv r . 106. The same result could have been obtained from Equations 12.145 and 12.016, but it is of some interest to obtain the result by this alternative route, and at the same time to verify Equation 12.147. 107. If the equations of one of the /V-surfaces are given in the Gaussian form of Equation 6.03 as x r =x r ((o, (f>) where the x r are Cartesian space coordinates (x, y, z), then we can easily find p and its derivatives from the formulas p — p,-v ' = x cos 4> cos co + y cos 4> s ' n &> + z sin cos o»— y sin (f) sin co+z cos (/> 12.171 (sec )dp/da> = pA '= — x sin ut + y cos o> 12.172 in which we have used Equations 12.008. 108. Otherwise, if a surface is given in the form 12.173 N=f(x, y, z) = constant, then by evaluating the gradient Equation 12.001 in Cartesian coordinates, we have n cos 4> cos (x> — df/dx n cos (/> sin aj = dfldy n sin (f> = dfjdz 12.174 with n 2 = (df/dx) 2 + (df/dy) 2 + (dflBz) 2 . These equations are sufficient to express p and its derivatives in terms of (x, y, z); together with Equation 12.173, these equations may serve to ex- press (x, y, z) in terms (a>, = p> + sl>+h 2 (l r s l s ) + }sHW s )J'+. ■ ■ 12.176 is true in Cartesian coordinates where it reduces to a Taylor expansion for each Cartesian coordinate. If p r is interpreted as drawn through the barred point parallel to its current direction and length so that its Cartesian components remain the same during the parallel transport, then Equation 12.176 can be considered as an equation between vectors all at the barred point. It is accordingly true between such parallel vectors in any coordinates, provided, of course, that the Taylor expansion is valid. We may also lower the r-index by contracting with grk, in which case pk become the covariant com- ponents of the parallel vector. 110. If the line is straight, then Equation 12.176 reduces to 12.177 p r = p'+sl>\ which is an elementary vector equation either be- tween the Cartesian components or between the components in any coordinates of vectors drawn equal and parallel to p r , p r , 7 r through any point in space. 111. The expression in Equation 12.169 of the position vector in terms of the base vectors is im- portant because we are usually concerned with the terminal azimuths a and zenith distances /3 of the line. For example, if we contract Equation 12.177 with /,■ and note that l r = l r for a straight line in Car- tesian coordinates and in the invariant scalar products, we find that the length s is equal to the difference in the values of (sec(f))(dpldco)sma sin/3+(dp/d$)cos a sin/3+p cos/3 at the two ends. This depends on knowing the value of p and its derivatives for the two TV-surfaces. The problem then arises how to transfer such functions from one /V-surface to another, usually along the isozenithals. We shall see how to do this in later chapters, both in a general (a>, (f), N) system and in simpler coordinate systems which can be used to linearize the problem. CHAPTER 13 Spherical Representation in (, N) GENERAL 1. Some properties of the spherical representa- tion of surfaces, over and above those derived in Chapter 11, can be obtained most simply in the special coordinate system of the last chapter; we are now able to do this. CURVATURES AND AZIMUTHS 2. We have seen in § 11-8 that a principal direc- tion of the surface is parallel in space to its spherical image. The meridian planes at corresponding points are parallel because they contain the parallel nor- mals and parallels to the common C'-axis. Accord- ingly, the meridian directions at corresponding points are parallel, and therefore the azimuth A of a principal direction is unaltered in the spherical representation. 3. If a, a are, respectively, the azimuth of a line on the surface and the azimuth of the corresponding line on the sphere, and if i//, ifj are the angles (in the sense of fig. 11, Chapter 11) between these corre- sponding directions and the principal direction whose azimuth is A, then we have 13.01 (A — «) = «// ; [A — a) = «/>. 4. The normal curvature A in azimuth a is then from Equation 12.056 k = K\ cos ijj sin (A — a) + k- 2 sin i)j sin (A — a) — — m cos i/J cos (A — a) — m sin ip sin (A — a) =—m cos (A — a) cos {A— a) — m sin {A — a) sin {A — a) = — m cos (a — a) 13.02 in which m is the scale factor for the direction a, that is, {k 2 + t 2 ) 112 from Equation 11.21, and we have used Equations 11.22 and 11.23. 5. Similarly, from Equation 12.057, the geodesic torsion in azimuth a is 13.03 m sin [a — a) which shows that the two azimuths are the same only if the direction considered is a principal direction. 6. Direct expressions for the azimuths are easily obtained from the last two equations as m cos a = — k cos a — t sin a = — ko cos a — 1\ sin a 13.04 =d/ds, in the second line of which we have used Equations 12.060 while ds is the arc element in azimuth a; similarly, we have m sin a — — k sin a + t cos a 13.05 = — ki sin a — t\ cos a = (cos 4>)dwlds. 89 90 Mathematical Geodesy These equations give us the spherical azimuth and scale factor in terms of the three curvature param- eters k u k>, tt of the surface. 7. For the (^-coordinate direction, we have a = a 2 and a = so that Equation 13.05 gives 13.06 ta.na 2 =—tilki — tlk, agreeing with Equations 12.091, which, substi- tuted in the Equations 13.02 and 13.03, give k=Kk 1 /(k 2 l + t 2 1 ) t = -Khl(k\ + t\) 13.07 m = KI(kHt 2 ] V 12 . 8. For the co-coordinate direction, we have a = d\ and a = ^7r so that Equation 13.04 gives 13.08 tan ai = -k 2 lt\= — k/t, agreeing with Equations 12.087, which, substituted in the Equations 13.02 and 13.03, give k=Kk 2 l{k\ + t\) t = KtJ(k 2 + t 2 ) 13.09 m = KI(k 2 + t 2 yi°-. GEODESIC CURVATURES 9. The geodesic curvature of a surface curve whose unit tangent is /" is derived from Equation 12.065 as — an)^: for the corresponding curve in the spherical repre- sentation, (to, c/>) remaining the same, we have — au)l e . But if m is the scale factor in the direction l a , that is, {k 2 + t 2 ) 11 ' 2 , we have from Equation 11.12 and finally ma — a — (a-a)tsft 'P\ d(t/k 13.10 dl in which dl is an element of length in the direction /". The last line in this equation is obtained by differ- entiation of tan (a — a) — tjk from Equations 13.02 and 13.03. 10. We see at once that a geodesic of the surface (cr = 0) cannot correspond to a great circle (cr = 0) unless (t/k) is constant along the curve. This would usually imply that f = so that the curve would also have to be a line of curvature. Even in the symmetrical case when the meridian geodesies are lines of curvature, they would, in general, be the only geodesies to correspond with great circles. 11. If we multiply Equation 13.10 by the element of length ds=(l/m)ds of a closed continuous contour and then integrate Equation 13.10 around corre- sponding contours, we have I ads— I ads — because the total change in azimuth around each contour is 2 77-. We conclude from Equation 10.47 that KdS is the same over corresponding areas; we shall see in Equation 13.14 that this is true. 12. Equation 13.10 reduces in the case of the lines of curvature (t = 0) to a — ma. If as usual the lines of curvature are U a , O-', K U A v a , a", K2, (A —jtt) then we have 13.11 a =Kicr ; Kia COVARIANT DERIVATIVES 13. We suppose as usual that l a are the unit tangents to a family of surface curves and thatja are tangential to their orthogonal trajectories. This involves no loss of generality in dealing with one particular curve because any given curve can be considered a member of some family. For example, it is well known that any surface curve can generate a family of geodesic parallels, in which case the j a would be tangential to a family of geodesies. From Equation 12.063. we have laii=j\x(o)n sin — au) and the corresponding equation Iat3=fa{a>i3 sin — dtp) in which j a is perpendicular to l a on the sphere, but Spherical Representation in (to, $, N) 91 does not necessarily correspond toj a - Nevertheless, from Equation 11.15, we have j a =(m/K)j a ; the required relation follows at once as (mlK)l a p = l a f) +ja(ct — a)p 'k-\ d(tlk) 13.12 — latf+ja dx* 14. If F is a scalar defined over a region of the surface, it must be some function of (to, (j>) and can be regarded as having the same value at the corre- sponding point on the sphere. For its second co- variant derivative, we have F aP = ;>-F -IV: and Using Equation 11.30, we then have 13.13 Fap-Fa^b^bafisFy. EXPANSION IN SPHERICAL HARMONICS 15. If K is the Gaussian curvature of an /V-surface and if dS is an element of area of the surface, then in (a;, (}>) coordinates, we have 13.14 KdS = KVa do)d = (cos (j>)da>d), it can be considered as having the same value at corresponding points of the spherical representation where (to, (f>) are the same. It can accordingly be expanded in spherical har- monics u,i of (to, (/>) as 13.15 F= V a„u,i in which the coefficients a„ are constant over the sphere or over the /V-surfaee. Moreover, since all points on the same isozenithal will have the same spherical representation and the same (to, (/>), F can be a scalar defined over some region of space, in which case the a„ will be functions of N at most, always assuming that the resulting series is convergent. 17. We can also write 13.16 FIK=^b, l u„ in which case the coefficients b„ for a particular /V-surface can be obtained in the usual way by integrating over that surface and by using Equation 13.14. All the operations of spherical harmonic analysis, usually carried out in spherical polar co- ordinates over a sphere, can be generalized in this way for a family of /V-surfaces. Ordinary spherical harmonic analysis is, in fact, a particular case (K= 1) of this generalization. DOUBLE SPHERICAL REPRESENTATION 18. We shall now consider the case of two sur- faces having a common spherical representation, which implies that the surface normals are parallel at corresponding points on the two surfaces. This definition would enable us to represent one surface directly on the other without a spherical inter- mediary; but if we retain the conception of a common spherical representation, we shall be able to use all the spherical results without having to rederive the geometry again. As in ordinary spherical represen- tation, we use the same surface coordinates and the same Cartesian space system. 19. As an illustration, suppose we draw a figure on one of the /V-surfaces of a (to, 4>, N) system and then project it down the isozenithals to another N- surface of the same family. The two figures will clearly have a common spherical representation, and are accordingly in this form of correspondence. We shall call this process isozenithal projection. 20. In the more general case, not restricted to two surfaces of the same family, we denote quan- tities related to the second surface with a star, and can then write equations corresponding to Equations 13.04 and 13.05 as 13.17 m* cos a = — A? cos a* — t* sin a* 13.18 m* sin a = — k* sin a* — t* cos a* in which the same spherical azimuth a is retained 92 Mathematical Geodesy for the common corresponding direction on the sphere. Division of these equations into Equations 13.04 and 13.05 then gives 13.19 13.23 m k 2 cos a + 1 1 sin a m* k* cos a* + t* sin a* Ai sin a + ti cos a k* sin a* + t* cos a* in which m/m* is the scale factor multiplying an element of length on the unstarred surface to obtain the corresponding length on the starred surface. Solution of these equations gives us 13.20 13.21 where tan a = - tan cr a + b tana* c + G?tan a* a + c tan a b + d tan a a=(k 2 tt-t l k$) ; b=(k 2 kf-titt) c=(htf-kiki) ; d^ihki-kttt) 13.22 ad-bc = KK*. It is easy to verify from Equations 13.06 and 13.08 that the coordinate directions satisfy these formulas, and so are corresponding directions. 21. The unstarred surface will often be a refer- ence surface which can be taken as symmetrical about the Cartesian z-axis, in which case fi = and the remaining curvature parameters become the principal curvatures #ci, k-i. In that case, we have Ki _ t* + k* tan a* ki k* + t* tan a* which is a simple generalization of the formula for the spherical azimuth a, obtainable directly from Equations 13.04 and 13.05 as t* + k* tan a* 13.24 tan a = k* + tf tan a* 22. It should be noted that the functions a, b, c, d are the same for all directions at a point, but vary from point to point. Without a knowledge of the curvature parameters, either by calculation on a given surface or by measurement, the transforma- tion cannot be effected. Once we have calculated the corresponding azimuth a*, the scale factor follows from Equation 13.19, with the following alternative formulas connecting the scale factor and corresponding azimuths, (m/m*)K* sin a* = — (a cos a + c sin a) (m/m*)K* cos a* — (b cos a + d sin a) (m*lm)K sin a = {a cos a* + b sin a*) 13.25 (m* I m)K cos a = — (ccos a* + of sin a*). 23. All the spherical formulas in Chapter 11, which depend on the scale factor or on direction, can now easily be modified for the more general case. Tensor point functions, such as Equations 11.08 and 11.31 which are unaltered on spherical representation, will also have the same value on a more general surface, provided, of course, that the metric of that surface is used in (a>, (/>) coordinates. CHAPTER 14 Isozenithal Differentiation DEFINITION 1. Chapters 11 and 13 dealt only with integral relationships between two surfaces having a com- mon spherical representation. In the case of iso- zenithal projection of TV-surfaces, this meant that the two /V-surfaces could be separated by any finite distance measured along the isozenithals. For example, the scale factor multiplying an element of length on the unstarred surface would be k 2 + f 1 k** + t* whatever the separation of the two surfaces. How- ever, such formulas are not often of much practical use because we do not know the curvature param- eters of both surfaces. We may know the curvature parameters on one surface and may have to derive them on another by means of a Taylor series; the same applies to any other metrical quantities de- fined or measured on one of the surfaces. For this purpose, we need to know the derivatives of these quantities along the isozenithals — or what amounts to the same thing, their ordinary partial derivatives with respect to /V— because the other two coordi- nates (c«j, 4>) will be constant during the change. 2. In this chapter, we shall obtain such deriva- tives for most of the metrical quantities of the surfaces. The geodetic applications, such as projec- tion from points on the topographic surface to the geoid, are not likely to be carried over considerable distances along the isozenithals; for this reason, we shall find first derivatives only. Higher deriva- tives could be obtained in much the same way, but would naturally be far more complicated. 3. Any quantities in the common spherical representation of the /V-surfaces would, of course, be unchanged during the process, and their iso- zenithal derivatives are accordingly zero. For example, we have at once, from the definition of the representation or from Equation 11.01, provided the space coordinates are Cartesian. DIFFERENTIATION OF THE FUNDAMENTAL FORMS 4. We have already seen in Equations 12.143 and 12.144 that three of the five Mainardi-Codazzi equations of a system of A-surfaces can be written in the form of isozenithal derivatives of the second fundamental form dban ^ d 2 (l/ft) — djl/n) c ali dx a dx (i Iff dx y n dN 14.01 r b^ba, us V tl/y n in which the overbarred Christoffel symbols are taken in the metric of the spherical representation; the only nonzero values from Equations 12.142 are 14.02 r^= sin (/> cos ; rj 2 = — tan $. We shall find that the isozenithal derivatives of most other metric quantities can be expressed in terms of dbapldN, and thus stem from the Codazzi equations. 5. We begin with the metric tensor of an A-surface whose components are seen from Equations 12.069 to be the same as the (1,2) components of the space 93 94 Mathematical Geodesy metric tensor in (a>, <£, N) coordinates where the surface coordinates are (&>, (f>). We then have da afi ldN=dg a ijldN = g^s + r^ 3 g r0 + r^ 3 g- Qr . Because all components of the covariant derivative of the metric tensor are zero and because H^ is zero in these coordinates, this reduces to da a0 /dN= rv 3 af3y+rv 3 a ay 14.03 = by s afsy(db aS ldN) + bv*a ay (db^ldN) , using Equation 12.127. 6. From the ordinary expression for the derivative of a determinant, we have also 5 (In a)ldN=a a ^(da ali ldN) = 2b^(dbafildN) = 26 (In b)/dN. But the specific curvature of the A^-surface from Equation 7.18 is K = b/a- substituting the logarithmic differential of this, we have finally 14.04 d(ln a)/dN=2d(ln b)/dN=-2d(ln K)/dN. We can verify this result by noting from Equation 12.070 that Kb — K 2 a — cos 2 , which is constant along an isozenithal. As in Equation 9.01, an element of surface area is dS = Va dojd(f) ; because the coordinates w, are constant along the isozenithals, we have 14.05 d(dS)_ d(\nK) dN dN dS, using Equation 14.04. This shows that KdS is constant under isozenithal differentiation, as we should expect from § 13-15. 7. By differentiating the identity we find without difficulty that da a ' i ldN = - a^a^days/dN 14.06 =- {a a yb^ + a^b aS )(dby S /dN); and, to complete the picture in regard to the second fundamental form, we have 14.07 db^ldN=-b a yb^{dby S ldN). 8. The third fundamental form is easy because all its components are, at most, functions of latitude only, and are constant along the isozenithals so that we have 14.08 dN = dc ali ' dN' 9. We shall also require derivatives of the surface permutation symbols e„/j = Vae«)5; € ati =e a0 /Va. Using Equation 14.04, we have at once 14.09 de„itldN =-e a fid (In K)/dN 14.10 Be a ' 3 ldN = + e ali d(\n K)ldN. 10. Note that Ke a & and e a/3 /A. behave as constants under isozenithal differentiation: because the specific curvature of a sphere is unity, the follow- ing relations hold true in spherical representation, 14.11 e a n = Ke aa ; i^ = e^/K where the overbars refer to the metric of the sphere. DIFFERENTIATION OF SURFACE CHRISTOFFEL SYMBOUS 11. If we are working in flat space, the most direct way of obtaining derivatives of the Christoffel symbols is to equate to zero certain components of the Riemann-Christoffel space tensor. We have, for instance, 14.12 _d_ ~ yy * fiY — a ^ y 1 jiZ l py l m3~ l 03 1 ay where we have dropped from the summation those symbols which are zero in (co, 4>, N) coordinates. All the symbols in this expression are space sym- bols; we need to replace those containing only Greek indices by surface symbols, denoted by an overbar, from the relation in Equation 12.131 so that we have 14.13 ^h = ^h + hyv a - To differentiate this expression, we use d{v a )ldN=v%-Yhv r because v§ = 0. Using Equations 12.124 and 12.127, we have 3( y ) --r^rg 3 -r^(r^-r^)+rg 3 (rg y -rg y ). From this equation and Equations 14.13 and 14.12, Isozenithal Differentiation 95 we then have -i n i_fo = fa —f^r r« _J_ r« per But we have already seen from Equations 12.127 and 12.144 that T^. A is a surface tensor, and the right-hand side of this last equation is its surface covariant derivative with respect to x y . We may accordingly write the last equation as dNjy symmetrical in any two indices; we have finally, with some rearrangement of indices, 14.14 ' I 6/38 dN 12. This remarkable result shows that although the surface Christoffel symbols are not themselves surface tensors, their isozenithal derivatives are surface tensors. In the final result, we have dropped the overbar because there is no longer any confusion with the corresponding space symbols, but we must remember that we are differentiating the surface symbol in Equation 14.14. It is evident that Equation 14.14 is symmetrical in (/3, y) so that we have 14.15 (rg 3 K=(r? 3 )0. DIFFERENTIATION OF b a p y 13. We interpolate now, because it follows directly from the last section, a result which will be required later. We have seen in Equation 11.31 that the quantities r% y + b a6 b f m have the same values at corresponding points on a surface and in its spherical representation, which imply that this expression is constant under iso- zenithal differentiation. Using Equations 14.14 and 14.07, we have , (db af >\ , /db aS \ — O/tifiy -~TT — Oflfi — — \dN The third term is •b m b°w° d ^+ baS d j^=o. db*' 7 db a8 + b m b°>>b pa rjfi-=b (i y S -^-, which cancels with the first term because bum is 14.16 db a yy , , /db>'° dN bapbfja dN in which the final index denotes surface covariant differentiation. DIFFERENTIATION OF VECTORS DEFINED IN SPACE 14. We take a unit surface vector l r in azimuth a which is defined in space, such as the meridian direction or a principal direction of the TV-surface through the point under consideration. The usual perpendicular surface vector j r in azimuth {a — \tt) is defined as perpendicular to /', and must therefore remain perpendicular to /'" after differentiation. Because the space vector equations /,■ = X r sin a + /jl, cos a j r = — k r cos a + jXr sin a are to remain true after the process, we may differ- entiate them covariantly along the isozenithal, that is, with respect to N. Remembering from Equation 12.098 that X. r3 = fir S = 0, we then have lr3 = -jr{daldN) 14.17 jr, = l r (daldN) with similar contravariant equations. The change in azimuth in these equations refers to changes in the vectors as defined in space; it does not refer to the change of direction which would be obtained by projecting the two ends of the vector down the iso- zenithals. We shall consider this case in § 14-25. 15. We could expand Equations 14.17 with r — /3 and substitute Equations 12.124, 12.127. and 12.128 for the Christoffel symbols, thus deriving expressions for the differentials of the components. However, in this case, we are able to use covariant differentiation; we shall find it simpler to do so as a means of obtaining changes in the normal curvatures A' (of l r ) and k* (of j r ), together with the change in the geodesic torsion t (of /, •). We have, for example. vrsl r j s = -t; differentiating this covariantly along the isozenithal gives, with Equations 14.17, Vrs3l r j s -v T sj r j s (daldN) + pJ r i s (da/dN) = -Bt/dN. In (oj, (/>, N) coordinates and using Equations 12.159, the first term is (Bb a nldN)l a f so that we have 14.18 (db a ^/dN)I a p =(k- k* ) (da/dN) -dt/dN: 96 Mathematical Geodesy in much the same way, we have 14.19 (db a vldN)I a lt i = -2t(dal8N)-dkldN 14.20 (db a i3ldN)j a f = + 2t(daldN)-dk*ldN. 16. Before substituting for db a pldN from the Mainardi-Codazzi Equation 14.01, we need to work on the middle term of the latter, that is, on bv s b a ps(lln}y = (lln)b a j}8V s , using Equation 12.082. We have WV" = (baiil a j' 3 )s ~ Ms/ ~ b ali l a j§ = dtldx s + (k-k*)((TU + a*js) 14.21 =dt/dx 8 +(k- k*)( m sin (b- an) in which cr, a* are the geodesic curvatures of / a ,y'^, respectively; in deriving the equation, we have used Equations 4.11, 7.08, and 12.064. Similarly, we have 14.22 bonsM* = dkldx 5 - 2t(al s + a*js) 14.23 b aPS j a P= dk*ldx 8 + 2t(a-ls+a-*js). 17. Now, if F is any scalar or component of a tensor, we have (dF/dx?)v s = (dF/dx r )v r - (dFldN)v* 14.24 ={dF/ds)-n(dFldN) in which ds is the arc element in the direction of the normal; so then we have from Equation 14.21, using Equation 12.032, by s b a0S l a p ( 1 In ) y = ( 1/n ) ba f ul a j f 'v* = (Hn)(dtlds-ndt/dN) + (1/n) (*-**) X (y, tan (b-da/ds + nda/dN). Substituting in Equations 14.01 and 14.18 and using Equation 7.14, we have finally dtlds = n{\ln) ali l a f + 2Ht 14.25 — (A- — A-*)(y, tan -da/ds); similarly from Equations 14.22 and 14.23, we have d k/ds = n(lln) aflW + ( k 2 + t? ) 14.26 +2t(yi tan Ht), the second by (k* 2 + t 2 ), and the third by (k 2 + t 2 ), using Equation 8.04 and adding, we have 14.30 d(2HIK)lds = -nc a min) ati -2. ISOZENITHAL AND NORMAL DIFFERENTIATION 18. The last six equations, giving variations along the normals, are somewhat simpler than the corre- sponding variations along the isozenithals. We can, however, relate normal and isozenithal differentiation by Equation 14.24 or by means of the following formula. If F is any scalar or particular component of a tensor, defined in space and there- fore also on the A^-surfaces. we have dF/ds = F,v r = F 3 v :i +F a v a = n(dFidN)+ yi sec ). 14.31 Or, if we use Equation 12.082, we have 14.32 dFlds = n(dF/dN) -W a (ln n)p. In applying these formulas, it is important to realize that the /V-surfaces must be the same for both oper- ations. If we use Equation 14.31, then the (co, (j>) coordinates must also be the same; we are compar- ing the variation in F along two different fines (the isozenithal and the normal) in the same (oj, (/>, N) system. If we use the second Equation 14.32, the /V-surfaces must still be the same; but the surface coordinates need not be the same because the last term is a surface invariant, unless F is a com- ponent of a surface tensor, in which case we must use the same surface coordinates. On this basis, for example, we have from Equation 14.28 14.28 d(2H)/ds = nM\ln) + (4H 2 -2K) d{2H)ldN = A(lln) + (\ln)()u)h sin cb b«(j^ a ki 3 = dk i ldx d — 2tim sin 14.35 b a ps(M c '^ = =dkildx f ' + 2t l (x}f ) sin 0, giving the variations of the curvature parameters along the isozenithal and reducing, as in the last j section, to the following variations along the normals, BtilBs = n{lln) a pk a fjfi + 2Ht i — y.fAi — A: 2 ) tan Bkjds = n(l/n) a0 k a k» + (kj + t\) + 2y, t, tan dk-ilds=n(\ln) aJ nx a iJLi i + (kl + t\) — 2y l t\ tan (/>. 14.36 The equations for the invariants 2//, K, 2H/K are, of course, the same as Equations 14.28 through 14.30. 20. We have seen that db a p/dN is a surface tensor so that we can express it as a sum of products of surface vectors. From Equations 14.34, we have at once -db a ii/dN= (dkll8N)k a k^+ (BtJSN) (K t fl0 + (Lakp) 14.37 +(dkt/dN)iLafip. Accordingly, for example, we have d(2H)/dN = - a<*(dbapldN) 14.38 = Mlln)-b^(2H) a (lln)p + UM(4# 2 -2/0, using Equation 14.01 which requires the Laplacian to be taken in the surface metric. This agrees with Equation 14.33. 21. The remaining two parameters y%, y-i are best differentiated from y, = (ln n),k r 72 = (In n) r (JL r , 306-962 0-69— 8 from which we have at once, because \£ — M'3=0, r')y,/d/V=(ln ri)cak a 14.39 dy 2 ldN=(lh n) aS fi a . Alternatively, we may take the covariant derivative of Equation 12.021, t , r.s^ S= 7lA,+y-/Lt,-. to derive dyijdN= v rs zk r v s = v r zsk r v s ; using the fact that iv:s = 0, this formula reduces with the help of Equation 12.016 to 14.40 9y | I8N = - T% s V s Vrak lr = - P, s ^' COS (j) and similarly to 14.41 ay,/d/V = -fV s . We could have obtained the same results by ordinary differentiation of the (1, 2) components of v r in Equation 12.034. DIFFERENTIATION OF THE PRINCIPAL CURVATURES 22. If I", j& in Equations 14.18, etc., are tangent to the lines of curvature u c \ v&, then we have t=0 throughout; Equations 14.18 through 14.23 become (k, - k>) (dA/dN)= {db a0 /dN)u a v^ dK 1 /dN=- {db a0 ldN)u a u^ 14.42 dK->ldN = - (dbapldWiPvP b a 08U a V P = {k, — K 2 )(a>s sin —As) balihU a lft= c)K\ld\ h 14.43 b a p8V a v l3 =dK-zldx s , leading to the following variations along the normals 14.44 (k, - k 2 ) (y, tan lds= n(lln) a fjV a v li + K$ in which the covariant derivatives are taken in the surface metric. Equations 14.43 could have been obtained by contracting Equation 8.16 and by using Equation 12.064. 23. The surface tensor db a pldN can be expressed as db ali /dN = - (dKildN)uaUp + (Ki — K t )(dAldN)( u a vn + v a U0 ) 14.47 - (dK 2 ldN)v a vv. 98 Mathematical Geodesy the equations for the invariants can be obtained directly from this as d(2H)ldN = -a a Hdb a$ ldN) 3 (In K)ldN = -b a Hdb al ildN) 14.48 d(2HIK)ldN=c a Hdb a vldN). PROJECTION OF SURFACE VECTORS 24. In the last few sections, we have considered vectors and the parameters associated with them, which can be considered as point functions in that the vectors are uniquely defined at all points of all the TV-surfaces within a region of space; we have arranged for the vectors to retain their definition after isozenithal differentiation. For example, the principal directions are, in general, uniquely defined at points in space by the form of the TV-surfaces, that is, by the form of the scalar point-function TV and its derivatives. We have found expressions for the change in azimuth and curvature in the principal directions on the assumption that they remain principal directions during the change. 25. We now consider surface vectors which are not so defined; we shall obtain expressions for the changes associated with these vectors as they are projected an infinitesimal distance down the iso- zenithals. The two ends of a vector subjected to this process each move down isozenithals to a neighbor- ing TV-surface, the surface coordinates (ai, (/>) of both ends remaining unchanged. The length of the vector will change as a rule, but we shall find it convenient to correct for this and to find expressions connecting unit vectors in the projected directions. 26. We must be careful not to differentiate such expressions as containing two related vectors, because this would tend to hold the relation during the change with the result that neither would, in general, be projected in the sense we are considering. We should differ- entiate expressions containing only the one vector which we wish to project, together with point func- tions. For example, we could recast the preceding formula with the help of Equations 2.32 as 14.49 t^-baftPeMy before differentiating. We can, of course, restore the related (perpendicular) vector j® after differentiation. 27. The process will involve ordinary partial differentiation with respect to TV of various compo- nents of tensor functions. We could use covariant differentiation only if it were possible to write the formula in terms of space components. For ex- ample, covariant differentiation of a a a with respect to TV is meaningless, but we can replace a a n in these coordinates by g a n and can take the covariant derivative g a 03, as indeed we did in deriving Equa- tion 14.03. We must, of course, use one sort of differentiation throughout an operation. 28. Because the spherical representation is unchanged by this form of projection, we can take any formula connecting elements on an TV-surface and on the unit sphere, and then can differentiate with the spherical elements fixed. We have had an example of this in § 14-3. As another example, we found in Equation 11.08 that if the space co- ordinates are Cartesian and the surface coordinates are the same for both surfaces, then b al3 x' a is unaltered on spherical representation. Differen- tiating with respect to TV, we have (db a>3 ldN)x r a + b af3 (dx r a ldN) = from which we have dx' a ldN = -b a ^{db^ldN)x , y = b B y(db ae ldN) Xy 14.50 =1^7, using Equation 12.127. Again, if the space coordinates are Cartesian, we know from Equation 11.02 that v' a is unaltered on spherical representation, and therefore we have 14.51 Bv r aldN=0. We can also write rv$ V'afJ = dv'aldX 13 - in space Cartesian coordinates because the space Christoffel symbols are then zero. Differentiating this and using Equation 14.14, we have 14.52 dv r a pldN=-(rHa)i3Vy. Length 29. If 8s is the length of a small TV-surface vector, the corresponding length in the spherical represen- tation (scale factor m) is from Equation 11.11 8s = (k?+t 2 yi 2 8s = m8s in which k , t are the normal curvature and geodesic torsion in the direction of the vector. Differentiating this isozenithally, Ss remaining fixed, we have Isozenithal Differentiation 8 (In 8s) _ 3 (In to) _ dN dN 8{\n (k 2 + t 2 )^} dN 14.53 30. An alternative expression may be obtained as follows. If the element 8s corresponds to a differ- ence of surface coordinates 8x a in the direction of the unit vector /'", we have 8s' 1 = a a fj8x a 8xP; differentiating this, we have d (In 8s) _ , ddafidx 01 dx 13 dN - BN ds ds i da«£ , a m 1 dN 14.54 = l a b^yly(db af ildN) using Equations 14.03 and 12.127. 31. We have seen in Equations 14.48 that -d(laK) IdN = b^ ( dbap/dN) = Tf,., , using the contraction of Equation 12.127. We may write further - f)(ln K)/dN=Tfa8<* = rZ S {l a ly+j a jy), if j" is the usual surface vector perpendicular to /" and if we remember that r& 5 is a surface tensor. If we add this last result to 14.55 3(liim)/3#=-r&(Z a Zy), obtainable from Equations 14.53 and 14.54, we have 14.56 d{ln(mlK)}ldN=Ty a3 j a jy. But it m* is the scale factor in the ^"-direction, we have also 14.57 and so 14.58 3(ln m*)ldN=-ry 3 j a j y d{lnmm*IK)ldN=0. But this last relation is true only to a first order. We cannot differentiate it again or assert that (mm*/K) is constant over a finite length of the isozenithal because the two directions do not, in general, re- main perpendicular when projected down the isozenithal. Contravariant Components 32. If we use the same surface coordinates, for instance (w, ), for the /V-surfaces and the sphere, we have found in Equation 11.12. relating the con- 99 travariant components of a unit surface vector and the corresponding unit vector in the spherical rep- resentation, that (1/to)/<*= 7". Differentiating this, we have at once dl a /dN=l a {d In m/dN) = /"{, 4>) only, and are therefore constant under isozenithal dif- ferentiation so that we have dl r ldN={\ r cos a — fi' sin a){c)ajdN) 14.60 =-j r (da/dN). 34. We now differentiate the equation l' = l a xL using Equations 14.59 and 14.50. and find that -j r (da/dN)=l r (d In m/dN)+r&il a Xy. Multiplication by /, gives Equation 14.55 again, and multiplication by j r gives 14.61 (daldN)=-T&l a jv. Note that this is not the same as Equations 14.17. The change in azimuth in the perpendicular direc- tion /'' is similarly 14.62 (da*ldN)=ry 3 j-I y and d(a + a*)ldN= r^(ey fi /«/ fi + € yfi ./"/ 8 ) 14.63 = rz 3 ey S a aS . Note that this is a point function which is the same for all directions at a point. Consequently, the change in mean azimuth of a pair of perpendicular directions is the same for all pairs. 100 Mathematical Geodesy Covariant Components 35. If we differentiate the equation in which the space coordinates are Cartesian, in the same way we have dlaldN = -jr(daldN)x r a + Ty 3 x r y l r 14.64 =-j a (daldN)+n 3 ly in which we can substitute Equations 14.61 and 12.127 and ultimately Equation 14.01 to show the result in terms of (1/rc) and surface tensors. How- ever, we shall usually be content to leave the results in terms of the Christoffel symbols rg 3 or in a form which can readily be translated into these symbols. Curvatures 36. The simplest way of differentiating the curva- tures in the direction l a is to differentiate Equa- tions 13.02 and 13.03 in which a, the azimuth of the spherical representation of I", is held fixed. We have at once 14.65 d'kldN=k{d(\n m)ldN}-t(daldN) 14.66 dtldN=t{30n m)/dN} + k(daldN) in which we can as usual substitute Equations 14.55 and 14.61. We have also an alternative expression for the variation in azimuth from these equations, 14.67 da _ A 2 d(tlk) dN~ m- dN 37. It may be emphasized again that the ex- pressions give the changes in k, f, etc., between a direction on an /V-surface and the projected direction on the next surface. Suppose, for ex- ample, that we start with a principal direction (t = 0). Equations 14.65 and 14.66 then give the change in normal curvature and geodesic torsion resulting from projection of the principal direc- tion down the isozenithals onto the neighboring surface. If the projected direction is to remain a principal direction, then we must also have df/d/V = 0, in which case we have da/dN=0 from Equation 14.67. From Equation 14.61, we then have r%3U a vy = in which u", v a are as usual the principal direc- tions. With the help of Equations 12.127 and 8.03, we can show that this is equivalent to 14.68 (dbanldN)u a v^ = as the condition for the principal directions tc project as principal directions. It is clear from Equation 14.01 that this condition implies a spe cial relationship between n and surface tensors which would restrict the form of N. We must conclude therefore that principal directions, ir general, do not project isozenithally as principa directions. 38. To obtain the variation in geodesic curva ture of isozenithally projected curves, we car differentiate Equation 13.10 in the form &- (l/m)cr= (1/m) (a-'a)^, holding the spherical elements cr, a constant The result, after some simplification, is Bct _ d (In m) dN~^ 14.69 d 2 a a dN dx 8N P in which we should substitute Equation 14.55 anc the differential of Equation 14.61 or 14.67. Tc verify this, the reader is invited to obtain the same result without spherical representation but with greater labor, by differentiating (T = -l ali e a n y P- Covariant Derivatives 39. To find the variation in the surface covarian derivatives of isozenithally projected curves, w» can similarly differentiate Equation 13.12. Alter natively, we can differentiate Equation 12.063 ii the form lap — — € a yl y ((i)p sin « Fy. 41. If F is a scalar, defined in space and there- The result in either case is fore over any /V-surface, we can find the isozenithal variation of its second surface covariant deriva- i/i 71 dr „/j _ [dF\ 1471 "TaT - \Jat ) ~i r L)0Fy- tive Fay by differentiating either Equation 13.13 or dN \dN/ a p CHAPTER 15 Normal Coordinate Systems GENERAL 1. In the (to, , N) system, the (at, (j>) surface coordinates are not, in general, constant along the normals to the A-surfaces so that the normals are not the /V-coordinate lines. The gradients of (co, (/)), considered as space vectors, are not contained within the TV-surfaces as is evident from Equations 12.046 and 12.047. We shall now consider how to overcome these complications by adopting surface coordinates, which are defined to be constant along a normal, so that the normals are the /V-coordinate lines. The geometry of the system will be simpler and can often be used to derive quickly results in the form of invariants or tensor equations which are true in any coordi- nates. However, the surface coordinates are not directly measurable throughout a region of space; the system is accordingly of direct practical use only when we are interested in the immediate neighborhood of one particular surface. 2. As usual, we start with a scalar A', single- valued, continuous, and differentiable through- out some region of space, and make it one coordi- nate of the system. The family of A-surfaces, over each of which /V is a constant, is accord- ingly one family of coordinate surfaces. The gradient of A is 15.01 N r - nv, in which v r are unit tangents to the normals or the orthogonal trajectories of the /V-surfaces. We have seen in § 10-19 that a scale factor of n will transform the space conformally to a curved space in which the normals become a family of geodesies and the A-surfaces become a family of geodesic parallels. An element of length along the geodesic normals will be dN: it is well known ' that the metric of the curved space can then be expressed in the quasi-Pythagorean form ds 1 = d at idx a dx iiJ rdN- (a, 0=1, 2) in which the dx a are the other two coordinates. We transform this expression back to the original space with scale factor {1/n) and have as the metric of the original space 15.02 ds- = a a udx a dx* + ( UnYdN 1 in which we have retained the same coordinates x a , whatever they may be. This merely demonstrates the possibility of a metric in this form; we have now to examine it and to find all we can about the x a . THE METRIC TENSOR 3. We can write the metric tensor in the abbre- viated form 15.03 g rs = (a a p, l/« 2 ), the determinant of which is 15.04 g=(lln°-)a. See. for instance. Eisenhart (1926). Riemannian Geometry. 57. 103 104 Mathematical Geodesy leading to the associated tensor 15.05 g n = (a l #, n 2 ), as we may easily verify from the definition in § 2-19. 4. The cosine of the angle between the gradients of x 1 and /V is proportional to gr'xlNs = g 13 = 0. Consequently, the gradient of x 1 is perpendicular to the gradient of N, that is, toy r , and must be there- fore a surface vector. Similarly, the gradient of x 2 is a surface vector. Both ^"-coordinates are thus constant along the normals, which must therefore be the /V-coordinate lines. 5. In much the same way, we can prove from the absence of ^-components in the metric tensor that the % "-coordinate lines are perpendicular to the /V-coordinate fine, and must therefore lie in the /V-surfaces. 6. As usual, we take the x a as surface coordinates and as two of the space coordinates so that 15.06 8 r 7. One possible way, and indeed so far as we know the only way, of defining the £ "-coordinates further is as follows. Through a point P in space, we draw a line or trajectory which is normal to all the TV-surfaces. The intersection of this line with a particular /V-surface, which we shall call the base surface, is Q. The coordinates of Q on the base surface are taken as the x "-coordinates of P. Evi- dently, all points on the same normal have the same x a , and the x a can be used as surface coordinates on the other /V-surfaces. In this way, we can meet all the preceding requirements of a metric in the form of Equation 15.02. We shall assume that co- ordinates have been chosen in this manner, but we shall leave open for the present the particular choice of coordinates on the base surface. 8. It should be noted that we have defined the ^"-coordinates as functions of position only on the base surface. We can transform them in two di- mensions on the base surface, in which case their values will be settled at any point in space. We can also transform the ^"-coordinates at any point in space by taking the /V-surface through the point as base surface. We cannot, in general, choose latitude and longitude, defined in §12-10, as co- ordinates in this system because they are not constant along the normals, unless the normals are straight. This would be a special case with which we shall deal in Chapter 17. We could, however choose latitude and longitude on one particulai base surface, even if the normals are not straight. 9. Because we have not so far specified the actua surface coordinates, even on the base surface, we cannot specify the surface metric tensor a a n. Noi can we determine the second and third fundamental forms b a p, c a fi of the /V-surfaces as we did in the (a>, (j), N) system where all three coordinates were completely specified in space. These three forms must vary between surfaces and must therefore be dependent on N. We shall derive expressions later for this which will enable us to calculate the a a p, b a p, and c a /j on any surface from the cor responding components on the base surface. COMPONENTS OF THE NORMAL AND OF SURFACE VECTORS 10. From the basic gradient equation for N that is, N r = nv r , we can find the covariant components of the unit normal at once because, whatever the (1, 2) coordi nates, TV does not change when differentiated witl respect to them. We then write 15.07 IV = (0,0, 1/ra). The (1, 2) contravariant components must also be zero because the (1, 2) coordinates do not chang( along the normal. Also, we have v r v r = 1 because v r is a unit vector so that we can write 15.08 W = (0,0, n). 11. We shall also require expressions for the surface tensor derivative of the unit normal. Wein garten's formula in Equation 6.17 becomes 15.09 v' a = ~ ai i ^baiix r y =- a*vb a pty so that we have 15.10 v l = 0- v% = - a<*v b fjy = - b a y c vy , using Equation 8.09. 12. Because any surface vector l r is perpendiculai to v r , we may write t r v r = and l r v r = 0; expanding these formulas from Equations 15.0' and 15.08, we see at once that both the covarian and contravariant 3-components of any surface vector are zero in this system. Normal Coordinate Systems 105 THE CHRISTOFFEL SYMBOLS 13. Because g a p = (ia0 and the x "-coordinates are the same on the surface and in space, it is evident that the Christoffel symbols of the first kind, [«0,y], are the same whether they are computed from the space metric or the surface metric. In regard to the Christoffel symbols of the second kind con- taining no 3-index, we consider the space symbols r& =g ry [<*P, r] = ^ 8 [«/3, 8] = a > 8 [«/3, 8] - T& taken in relation to the surface metric. In deriving this result, we have used g 3y = and g yS = a yS . Consequently, all Christoffel symbols containing no 3-index are the same whether they refer to the space or to the surface metric, and we have no need to differentiate between the two. Their actual values, as in the case of the metric tensor from which they are derived, will depend on what co- ordinates are adopted for the base surface. Once the Christoffel symbols are settled for the base surface, we should be able to find them at other points in space; we shall later derive formulas for this. 14. Symbols containing a 3-index can be obtained by expanding Equations 15.10. We have and = dv 3 /dx a + Y? a v r = dnldx" + nY§ a a ay bpy = dv a /dx^ + r# v r = nTfe . Because the covariant derivatives of all com- ponents of the metric tensor are zero, we have ga3P = = dgcjdx 13 - Y Sugar ~ Y^g rZ = -Y y li a a y-Y^g zz = (l/n)ay%sa a y-T^(Hn 2 ), using one of the preceding results and gas — 0, so that we have r«0 = nbaji. In the same way from g 333 , we obtain r 3 3 3 ; from £a33, we obtain Y§ 3 . Collecting results, we can list all the symbols containing one or more 3-indices as r 3 <3 f (lln 2 )a ^(ln n ) p ■ Yfe = -( 1/n )a**bfry T 3 = nb afi ; Yl = - (In n ) a ; r 3 3 3 = - d (In n )/dN . 15.11 15. In the last symbol, if 3 a aY = da a pldN+(lln )a> 8 6 a8 a^+ (1/n )ay% s a a y = da a( ildN+2(lln)baf} so that 15.13 da u plds = — 2b a f). Given the second fundamental form of the base surface, we can accordingly extend all components of the surface metric tensor along a normal by a Taylor expansion — at any rate to a first order. We shall show later how to obtain the higher derivatives. 17. By expanding g°g 3 similarly or by differentiat- ing a^a a i3 = 8y along the normal, we find that da a(i 15.14 ds la^a^b ys- Using Equations 8.07 and 8.09, the last equation can be written in the alternative form 15.15 da af: ds = ma^-2Kb^. II 106 Mathematical Geodesy 18. By the ordinary rules for differentiating a determinant, we have also 15.16 ^™) = a <*^ = -4//. ds 3s 19. We can now differentiate the permutation symbols e«/3 = e a p V a ; :«fl = e afl /Va to have 15.17 de a p ds ds — — 2He a ji = + 2He a P, which should be compared with the results in Equations 14.09 and 14.10 for differentiation along the isozenithals. SPACE DERIVATIVES OF THE NORMAL 20. In the (a>, , N) system, we found that the covariant derivatives of the meridian, parallel, and normal vectors were most useful. In this system, we have not yet defined any surface vectors in relation to the surrounding space, but we can now find the covariant derivatives of the normal by sub- stitution of the preceding results in the defining formula v rs = dv r ldx s -ain)r? s . We find that 15.18 Vad — ~ b a /3 , as it should, because this result in Equation 6.19 depended only on making the surface coordinates x a two of the space coordinates. Also, we have Vaz =— (l/«)a 15.19 ^33 = 0. By substitution in Equation 11.26, we have in these coordinates 15.20 15.21 08. 21. In the (w, c6, N) system, we obtained all the Mainardi-Codazzi equations by considering the first covariant derivatives of three vectors in Hat space. In this system, we have defined only one vector field, the normal; its first derivatives given in the last section do not include any condition that the space must be flat. We can, however, apply the alternative condition that the second covariant derivatives of an arbitrary vector must be symmetri- cal in the last two indices in flat space, as in Equa- tion 5.01. In this case, we have 15.22 Vrst = V rts . At this stage, we use this formula as a necessary condition without asserting that it is sufficient to use only one vector field. 22. We consider first the components containing no 3-index, V a py='dV a ^'dX y — r&yVrP — T r pyV ar =-db a0 idxy+Ti y b sp +r^b aS +r^(i/n) a = — b a 0y — b t r/(\n n) a . 15.23 We interchange (J3, y) and subtract in order to satisfy Equation 15.22 so that 15.24 b a py=b a yi3, which are the most general forms of the Codazzi equations for the /V-surfaces in Equation 6.21. 23. JNext, we evaluate l'cxji:i = dVapldN—Y^VrH — FfeVar =-Bb a0 idN+ri.,by li +ry 3 b a y+ri l3 (iin) a = -BbafildN- (l/«)a y8 6as^- (Vn)a?*bi}8b a y — (In n) (lln) a = — dba0ldN — 2capln — (In n)p(lln) a and also Vasfi = — d 2 {l[n) ldx a dxP — T r a/3 v r3 — T'^Va,- = -d z ain)d^dx^ + ry p (lln)y + ry(lln) a + Tfobay = — {1/ n) a/3 — c an/ n — (In n)p(l/n) a in which the overbar implies that the covarian! derivative has been taken with respect to the sur face metric. Applying Equation 15.22, we have finally THE MAINARDI-CODAZZI EQUATIONS 15.25 ds n/aii ■C a 0, which should be compared with the corresponding Codazzi equations for the (o>, , N) system ir Normal Coordinate Systems 107 Equation 12.144. This gives us, in general, three independent equations, making a total of five with the Codazzi equations for the /V-surfaces in Equation 15.24. As shown in the (w, <£, N) system, five is all we can expect in the case of a family of coordinate surfaces derived from a given scalar N. Indeed, we can derive no fresh relations from Equation 15.22 for any other values of (r, 5, t). We conclude there- fore that the relation in Equation 15.22 is sufficient to ensure that the space is flat. We could directly ensure that the space is flat by equating to zero all the Riemann-Christoffel symbols of Equation 5.03 for the metric ds 2 = a a0 dx a dx' i + ( l/n 2 )dN 2 ; or, what amounts to the same thing in three dimen- sions, we could equate to zero the six independent components of the contracted Ricci tensor of Equation 5.12. This gives us six equations, but we find that only five are independent in this metric and that they are the same as the five equations obtained far more simply above. NORMAL DIFFERENTIATION The Fundamental Forms 24. We have already shown how to differentiate a a n and b a n along the normals and incidentally how to obtain the second derivative of a a #. To carry the expansion further, we need to find an expression for dCaplds in terms of the fundamental forms of the base surface and of n. If we differentiate the ordi- nary formula Can = a^baybps and use Equations 15.14 and 15.25, we have Scalds = 2a^a b ' T b p i _ ds - b" y b^ dbyh ds 15.27 — -nb ay b^(- + ( ,ali. \n/ ys similarly, we have 15.28 — = -nb ay c^(-) -nb^c^l- ds \n/vft \n/ys which completes the differentiation of all the fun- damental forms. 26. The differentials of the surface invariants are now easily found. Using Equation 6.27, we have d(\nK) _ d(lnb) d(\na) ds ds ds ds = nb a f>(-) +2H. \'l/al3 15.29 Also, we have d(2H) d(a a »b aii ) ds ds da"* db a t = ba0— — +a ali — — ds ds 2a a < i c a 8+nA(lln)-a a l 3 c a is 15.30 = nA(TM+(4tf 2 -2K) in which the overbar implies that the Laplacian of (1//;) is taken with respect to the surface metric. In addition, we have d(2H/K) dib^aap) ds 15.31 ds ,a/3 a/3 27. These last three invariant equations are, of course, true in any surface coordinates, and can be evaluated, if we wish, by substituting (a>, ) values of «"", b a/i , c ttfi , and (l/n) a0 . In fact, we have already obtained them from (o», , N) coordinates as Equations 14.28, 14.29, and 14.30. The Christoflfel Symbols 28. Using Equation 15.13 for the normal deriva- tive of the surface metric tensor, we can differ- entiate the equation which defines the surface 108 Mathematical Geodesy Christoffel symbols, that is, a a8 I>=[j3y, 8] _ j_ ( daps days _ dapA _ _2 \^ 7 ^ a* 8 /' Alternatively, as we did in the (co, (/>, N) system, we can equate the n iuy components of the Riemann-Christoffel tensor of the flat space to zero and can use expressions ap- propriate to this (normal) system of coordinates for the Christoffel symbols. In either case, we ob- tain after some manipulation dry d S 15.32 -a a6 bpy 6 + a aB b 0b {\n n)y + a a8 fey 8 (hi n)ts — a a& bpy{\n n)i The more compact expression obtained in Equa- tion 14.14 for (oj, (f>, N) coordinates does not, how- ever, hold in normal coordinates. The bafiy 29. Nor can we obtain d(bauy)lds in these coordinates by the shortcut used in (a», (/>, N) because points along the normals, as distinct from the isozenithals, do not have the same- spher- ical representation. We can, however, differentiate the defining equation , _ d(baii) g OapV — 7 1 ayOS/3 _ 1 pyOSa, after some manipulation and use of the identity c a py=a Se b a sybp ( + a bl b a bbp t y, obtained by covariant differentiation of Equations 6.18, we find that d(b a py)/ds = n(i/n)ai3Y — c a p(ln n) y — Cpy{\n n) a — cy a (ln n)p 15.33 +a 8€ (ln n) e (b a yb s p + bpybsa) in which the overbar denotes covariant differentia- tion with respect to the surface metric. 30. If we interchange /3 and y and subtract the result from Equation 15.33, the left-hand side be- comes zero by virtue of the Codazzi equations, and we have At first sight, this shows a relation between n and surface tensors which is required in order to ensure that the Codazzi equations are satisfied on all the /V-surfaces. However, we can see from Equations 5.22 and 6.26 that it is an identity true not only for n, but also for any scalar. 31. From Equation 15.33, we can derive a«y*Z=n a{ y iln)} -(*H*-2K){\nn)y ds ox y b<*^=nb<*W\ -2H(\nn,y ds \nlapy lf e*b£L =vr ..,:[ • i _ 2 (lnnK ds n/<*$y 15.34 n(l/ n) a py — n (I In) a y p = a 8 * (In n) e (bafibys — b a ybp?,). which can be verified by differentiating Equations 8.20 and by using formulas already given in this chapter. Other Point Functions 32. There are no equations for the differentials of the point functions X'a ; V a ; V' a p in these coordinates corresponding to Equations 11.08, 14.50, 14.51, and 14.52 because the v r are not constant along the normals, even in Cartesian space coordinates. If required, differentials of these quantities should be obtained in normal space coordinates by differentiating Equations 15.06, 15.10, 15.20, and 15.21, using formulas which have now been given. The differentials of any other functions which are defined in space, such as can be obtained by covariant differentiation along the normal and by evaluation either in these coordi- nates or in (o>, 0, N) because the result will be an invariant true in any coordinates. Differentiation of Vectors Defined in Space 33. We take as usual a pair of unit orthogonal surface vectors l'\ j r (l'\j'\ ^'right-handed) which are defined in space as vector functions of position. They could, for example, be the meridian and paral- lel directions defined in the usual way from the normal to the /V-surface and from the Cartesian vectors; but in this case, the latitude and longitude Normal Coordinate Systems 109 would be functions of position and not coordinates, except possibly on the base surface. The two vec- tors could also be the unit tangents to the lines of curvature of the /V-surfaces, which similarly are uniquely defined at a point and, in consequence, constitute a vector field in space. 34. If a, k, t are the azimuth, normal curvature, and geodesic torsion in the direction l r and if k* is the normal curvature in the direction f, then all these quantities are point functions; Equations 14.25 through 14.27 hold equally well for changes in these functions along the normals, even though they were obtained in (w, \ T — y-iV r , obtainable from Equations 12.015 and 12.034 in {(o, , /V) coordinates, is true in any coordinates. If we expand it in normal coordinates, we find that the equation for r=3 is an identity; we are left then with ds yi tan (/> A a + nF^/x^ = — y\ tan (j> Kx — bapfJL 13 = — (y, tan + ti)\ a — k 2 fi, a , which gives us the ordinary differential of the meridian vector along the normal. The meridian vector remains the meridian vector and is not projected down the normals to the next /V-surface. In the same way and collecting the last equation, we have d^I'ds = - ( y i tan -/,) A" + k 2 fi a dfJLalds = — (yi tan (f> + /, ) A„ — k>n. lx Bk a /3s={y l tan + /, )/x" + A,A" 15.35 dA a /ds=(y, tan -/, )//,„- A, A„. We can now differentiate the equation A, = b aP k a \^, using Equation 15.25 and the preceding formulas, and can obtain the second equation of Equations 14.36. The other two equations of Equations 14.36 follow similarly. 37. In general, if /, (in azimuth a) and y, (in azi- muth a — jtt) are any pair of orthogonal unit sur- face vectors, we have lr—k, sin a + /ji,- cos a j r — — k, cos a + fx, sin a, which are defined in space and have to preserve their identity under normal differentiation; or, if /,■ is such a vector and \i j, is defined perpendicular to it, then we may write IrsV* — A rg v s sin a + fJL r sV s cos a—j r (da/ds) —j r (y\ tan 4> — da/ds) — v, \y x sina + y-j cos a), using Equations 12.014 and 12.015. Proceeding as for Equations 15.35, we find that dl a lds = — kl a +j a (yi tan — t — da/ds) 15.36 dl a lds^kl a +j a (y ] tan (f> + t — da/ds) in which k, t are the normal curvature and geodesic torsion in the /"-direction. These two equations cover all four equations of Equations 15.35 as the special case da/ds = 0. 38. For the principal directions (f = 0), u a (in azimuth A. principal curvature Kj), and v" (in azimuth A— In. principal curvature k->). we have at once dujds — — K\U a + i' ( ,(yi tan (fc — BA/ds) du a /ds = +K 1 u a + i> a (yi tan — dA /ds) dv a /ds = -K-,v a -u a (y ] tan — dA/ds) 15.37 dv a /ds = + K-,r a -u a (yi tan -dA/ds) • From these equations, we can derive Equations 14.44 through 14.46 by normal differentiation of K\ = bapu ^ k> = baptPvf* = b a pu a vf t - 110 Mathematical Geodesy NORMAL PROJECTION OF SURFACE VECTORS 39. We have now to consider, just as we did in isozenithal projection, the effect of projecting a surface vector from one /V-surface to another down the normals. In this case, we shall not be able to derive any assistance from the spherical represen- tation, unless the normals happen to be straight, because projected figures will not have the same spherical representation. Some of the formulas are, nevertheless, simpler. We shall not be able to obtain closed integral formulas any more than we could in isozenithal projection, and we shall again derive the first-order changes only. Changes of a higher order can be obtained when required by successive differentiation and substitution in a Taylor series. Length 40. A first-order element of length 81 on an TV- surface is given by (5/) 2 = fl a/ j8x a 8.^ in which 8x a are changes in the surface coordinates over the element. Because the surface coordinates remain constant along the normals in this system, we may differentiate and use Equation 15.13 to obtain 8(ln8/) ds bat 8x a \ /8*0 81 \8I 15.38 =-k in the limit when 8/ becomes infinitesimal. Of course, A is the normal curvature of the /V-surface in the direction of the length element. 41. An element of area on an /V-surface is given by 8S=Vci:8x a 8x< 3 , which can be differentiated with the help of Equa- tion 15.16 to give 15.39 a(ln 8S) ds 2H. Components 42. The change in surface coordinates over a small length 81 of a unit surface vector /" is given by 8x a = l a 8l; differentiating this with the aid of Equation 15.38. we have 0=8l(dl a /ds-kl a ). Because 81 is not zero, we have dl a 15.40 ds kl a - 43. In regard to the covariant components, the simplest course in these coordinates is to differ- entiate and to obtain dla, ds' L = a a pl ■2baf,P + ka a eP = -2(kla+tja)+kl a 15.41 =-kla~2tja in which t is the geodesic torsion of the surface in the direction /„, andy a is as usual a unit surface vec- tor perpendicular to l a - It should be noted that j a will not, in general, project as a perpendicular vector. We cannot therefore differentiate Equation 15.41 again and use a formula corresponding to Equation 15.41 for the differential of j a because to do so would require j a to be defined as perpen- dicular to /„, not only on one surface, but on pro- jection to the next surface. As in the case of iso- zenithal projection, we need to recast Equation 15.41 in the form 15.42 ^ = -kla + 2teafiP ds before differentiating again. In this way, we re- tain j a as an auxiliary perpendicular vector, but do not project it. Azimuth 44. To obtain the change in azimuth a of the vector /" on projection, we can differentiate cos a = l^fJL/j in which /xp is the meridian direction. We must not, however, project the meridian direction by using Equation 15.41, but must ensure that it remains the meridian direction by using Equations 15.35. With that proviso, we have — sin a(da/ds) = k cos a — (yi tan + ti) sin a — k-z cos a, which, with the help of the second equation of Equa- tions 12.060, reduces to 15.43 dac/ds = t + y l tan Normal Coordinate Systems 111 where as usual t is the geodesic torsion in the direction /„. This result may be obtained in a variety of other ways, for example, by differentiating sin a = afiyl^k 7 . 45. If we project a principal direction (t — in Equation 15.43), the projected direction is not necessarily a principal direction of the new A surface. If it is, then we may write dA/ds = y\ tan — ap). Although this equation was obtained in (co. $, A) coordinates, it is, nevertheless, a surface tensor equation true in any coordinates, provided we treat a), (/) as scalar functions of position and not as coor- dinates. Because we are now dealing with the same A-surfaces and with the same Cartesian axes, it is evident that a>, (/>, as originally defined in the (a>, 0, A) system, will have the same values; we can use Equations 12.032 and 12.031 and write dtr)/ds = yi sec 4> '■ dc/>/ds = -y 2 - 49. We first differentiate the term within paren- theses in Equation 15.47 and note that second derivatives, with respect to x 13 and A, commute. Also in these coordinates, we have ds~"dN so that we have, for example, d ( 1 do)\ d (j\ sec (/> ds ' _daA " dx^\n ds dxP \ Using Equation 15.43, we then have d(cou sin 4> — ap) . d !y\ sec 4> = n sin 7— § its d (t-\-j\ tan (b 15.48 + y-> cos <{) (Da + t(\n n)n — tp on expansion. Using Equations 12.061. 12.062. and 112 Mathematical Geodes" 8.02, we can rewrite the first two terms as — yi l l3 } = kb y Hln n)y€ S [} 15.49 =Qn, for instance, which shows that these two terms are a point function and that the only way a particular direction enters Equation 15.48 is through the geo- desic torsion. Using Equation 8.01. we can also express the last relation as 15.50 Qu = € y »e b ' T b f>,(£,/V) system. As we shall see, the latter system has- the advantage that all three co- ordinates are directly measurable quantities in many geodetic applications, whereas two of them in the normal system must be inferred from their values on a particular A-surface. Nevertheless, the normal system is of considerable theoretical value because it enables us to derive certain general results more simply than in other systems. We shall now inquire whether it is possible to achieve still greater simplification by adopting orthogonal sur- face coordinates in a normal system, in which case all three coordinate lines would be mutually per- pendicular and, in addition, would have the same direction as the gradients of the scalar coordinates. THE DARBOUX EQUATION 2. A small displacement on a surface can be written as 8x a where x a are the surface coordinates because this displacement amounts to a small change in the coordinates x a over the line con- sidered. If it is a displacement along a coordinate line, then a is either 1 or 2, but we prefer to keep the notation general and still write it as 8x a . The displacement 8x a can be considered a small contra- variant surface vector. Now consider two small displacements 8x a , Sx® along the surface coordinate lines and choose orthogonal surface coordinates on the base surface so that, on the base surface, we have 16.01 a a p8x a 8xfi = Q. If the coordinate lines are to remain orthogonal on the next /V-surface, infinitesimally close to the base surface, then there must be no change in this re- lation as we proceed from one surface to the other. In other words, the differential of Equation 16.01 along the normal arc element (is must be zero. During this change, the 8x a , S.f^ remain the same because, by definition, the surface coordinates are constant along the normals in a normal coordi- nate system. Accordingly, we may write {da a nlds)8x a 8x' 3 = 0; from Equation 15.13, this is equivalent to 16.02 b al i8x a 8x0 = O. In this equation, we can replace 8x a , 8x# by the unit vectors u a , v® in the coordinate directions simply by dividing by the lengths or magnitudes of the two displacements so that baliU a V li = 0, which shows that the geodesic torsion in the co- ordinate directions must be zero and therefore the coordinate lines on the base surface must be the lines of curvature. This is a well known, necessary condition, originally due to Dupin, for triply orthog- onal systems. It is not, however, sufficient to ensure orthogonality throughout a finite region of space. For this to be possible, the coordinate lines must re- main fines of curvature on the surface next to the base surface. The situation on this next surface will then be the same as on the base surface; we can repeat the process to build up the entire field. In other words, the differential of Equation 16.02 along 113 306-962 0-69— 9 114 Mathematical Geodesy the normals must be zero; using Equation 15.25, we must have n(\ln Udx^x^ - Ca0 8x a dx & = 0. But because the displacements 8x a , dx® are in the principal directions, the second term is zero by Equation 7.25 so that we have finally 16.03 (l/n) a 0uV = O. We can choose any one of the /V-surfaces as the base surface so that this condition must apply to all of them. Moreover, it is evident from Equation 8.26 that the condition is equivalent to 16.04 [lln) rs u r v s = in space, simply because (1/n) is a scalar. The form of the scalar /V settles not only n at any point in space, but also the principal directions of the N- surfaces. Accordingly, the condition of Equation 16.04 restricts the form of N, which must arise from a solution of Equation 16.04 in order to be one co- ordinate of a triply orthogonal system. 3. The Equation 16.03 or 16.04 is equivalent to what is usually known as the Darboux equation in classical differential geometry; it is given by such writers as Bianchi and Forsyth in several more complicated forms. Forsyth l gives a Cartesian version of the Darboux equation, which is equivalent to the invariant form of Equation 16.04, although he does not derive it in the same way. 4. Referring to Equation 14.44, we find that Equation 16.03 is equivalent to (ki — K 2 )(yi tan(/>-d,4/ds) = 0. Unless Ki = K2,'in which case the TV-surfaces are spheres whose lines of curvature are indefinite, the Darboux equation can accordingly be expressed as 16.05 d.4/ds = yi tan $. 5. In the case of a field symmetrical about the Cartesian C r -axis, we have seen in §12-48 that the meridians are principal directions {A — every- where) and j\ = so that each side of this last equation is zero. Accordingly, the /V-surfaces in such a symmetrical case are certainly possible triple orthogonals, but the Equation 16.05 then is oversatisfied; we may conclude that some non- symmetrical surfaces are also possible triple orthogonals. One such case is a family of confocal triaxial ellipsoids as is well known. SOLUTIONS OF THE DARBOUX EQUATION 6. Because u a , v& must be the coordinate direc tions in a triply orthogonal coordinate system Equation 16.03 can be written as (l/n)i2 = in the surface metric; this can be expanded as Uft , d 2 (l/n) d(lln) , ^ 9 d(lln) dx Y dx- dx 1 dx l This is a second-order linear partial differentia equation in the two independent variables x 1 anc x 2 , known usually in the theory of differentia equations as Laplace's equation (not to be cpnfusec with the Laplace equation used in potential theory) 7. Equation 16.06 is also known as Laplace's equation in classical differential geometry 2 where it arises because three particular solutions of the equation are the Cartesian space coordinates (x, y, 2). We can very easily show, for example that x satisfies Equation 16.03 by using Equatior 6.16 when we have (x) afiU a V P = V X b a ^U a V & = in which v x is the x-Cartesian component of the unit normal to the surface. In the same way, the equation is satisfied by y and z. 8. We can also show that Equation 16.06 is satisfied by r 2 = g rs p r p s in which r is the radius vector and p r is the position vector of a curren point on the surface. By surface covariant dif ferentiation, we have (r 2 ) a =2g rs xZp s ( r 2 ) a u = 2g rs Xlpp s + 'IgrsX^ = 2 ( g rs v r p s ) ban + 2a a p ; and so we have (r 2 ) a/3 uV=0, provided only that u a , v® are the principal directions 9. Moreover, because any function of A is constan under surface covariant differentiation, we can sa; at once that Equation 16.03 or 16.04 is satisfied b; 16.07 a + bx + cy + dz + er 2 in which a, b, c, d, e are arbitrary functions of N This is a very general solution of the equation, bu 1 Forsyth (reprint of 1920), Lectures on the Differential Geometry of Curves and Surfaces, original ed. of 1912, 437. ■Ibid., 69. Triply Orthogonal Systems 115 still not the most general solution which would 10. It may help the reader to find his way through require two arbitrary functions, not of N only but the considerable classical literature on the subject of all three coordinates. This result, first obtained if he realizes that the surface tensor equation by Darboux, is also derived by Forsyth. 3 Anyone r, a li _n who doubts the value of the tensor calculus in such problems should compare the classical derivations in which F is a scalar and the two vectors are the with the derivation given here. principal directions, is called Laplace's equation when F is x, y, or z and is called the Darboux 3 Ibid., 447. equation when F is (l/n). CHAPTER 17 The ( co, <$>, h) Coordinate System GENERAL DESCRIPTION OF THE SYSTEM 1. We have so far considered special coordinate systems in which a given family of /V-surfaces are coordinate surfaces without applying any restriction on the form of the scalar /V, other than continuity and differentiability. We now choose N to make the function n— the magnitude of the gradient of /V — a constant which, without any real loss of generality, we can make unity. The form of the /V-surfaces is then no longer as one chooses, but, as we shall see, one of the surfaces can still be chosen arbi- trarily. This coordinate system is accordingly of value when we are concerned with space in the immediate vicinity of a given surface, which can be chosen to provide a close approximation to actual physical conditions. The geometry of the system is much simpler than that of the more general sys- tems so far considered, and by suitable choice of surface can be made even more simple. We can also use the system to derive easily some valuable properties of surfaces in general. 2. In most current geodetic applications of this system, one of the surfaces is chosen to be a sphe- roid whose minor axis lies in the Cartesian (^-di- rection and whose dimensions are chosen to make it a good approximation to an equipotential surface of the gravitational field. In such a system, it is possible to calculate finite distances and direc- tions by means of closed formulas and so to linear- ize the observation equations for measures which are necessarily made in the less regular gravitational field. The problem usually involves a transformation of one /V-system into another; a spheroidal (o>, $, h) system (known as the geodetic system) into a (cu, (/>, N) system (known as the astronomical sys- tem) in which the /V-surfaces are gravitational equipotential or level surfaces modified by the rotation of the Earth. 3. In the present chapter, however, we shall derive general formulas which do not involve the choice of a spheroid as a special case. The results can then be used for other applications, such as the choice of the geoid as a surface in this coordinate system. Modification of these more general results to the special choice of a spheroid as base surface is a very simple matter which will be treated in the next chapter. 4. The basic gradient equation for the coordinate N with n= 1 becomes 17.01 N T =V r , which can be differentiated covariantly as 17.02 N rs =v rs , showing that the tensor v rs , like N r s, must be symmetrical. The vector curvature of the normals is then 17.03 v rs v s = v sr v s = because v s is a unit vector. See Equation 3.19. Consequently, we find from §4-2 that the normals are space geodesies, that is. straight lines in flat space. If h is a length measured along the normal, we have from Equation 17.01 dN/dh = N r v r =l so that we can take N as /( — measured from one particular /V-surface which we shall call the base surface. It is evident that equal lengths of the 117 118 Mathematical Geodesy straight normals are intercepted by two particular /V-surfaces which are, for this reason, known as parallel surfaces. 5. Because n — 1 , it follows at once from Equations 12.097 that the isozenithals are the same as the normals. The (o», , N) system, will be constant along the isozenithal-normals, we can use them as coordinates — not only on the base surface, but also in space. 6. It is evident that the Darboux equation of Equation 16.04 is satisfied if n is a constant so that we could choose triply orthogonal coordinates. In general, however, we could not use in that case latitude and longitude. We would have to refer the base surface to its lines of curvature and to define the resulting coordinates as constant along the normals, just as we did in the normal system. We have already seen from a V z in Equations 12.069 that the oj- and ^-coordinate lines are not orthogonal unless ri = 0, corresponding to the axially symmetrical system discussed in §12-48. In that case, but not otherwise, the a»- and cMines also would be the fines of curvature, and we could take latitude and longitude, together with h, as triply orthogonal coordinates. 7. For the present, however, we shall retain a general nonsymmetrical form for the base surface; unless otherwise stated, we shall assume that the surface coordinates are latitude and longitude. We can then use all the surface formulas in the (o», (b, N) system, that is, any formula not containing TV or n. In fact, the whole system becomes a special case of the (co, (/), N) system or the normal system with n = l, N=h, and ds = dh. However, we shall find that some useful integral relationships can also be obtained in this special system which are not available in the general (o», c/>, N) system. THE FUNDAMENTAL FORMS 8. The space metric in these coordinates, ob- tained by making n=l in the normal system of Equations 15.03 and 15.05, is 17.04 ds 2 = a afi dx a dx» + dh 2 , with the associated tensor 17.05 g rs =(a a ^l). It is evident that a a/3 and a a n are also components oi the h -surface metric, if as usual we make the surface coordinates two of the space coordinates This will be done throughout this chapter in whicl also, as stated previously, the surface coordinates will be latitude and longitude. 9. The components of the metric tensor wil depend on h. As in the normal system Equatior 15.13, we have 17.06 da a0 ldh=-2b a p; from either Equation 15.25 or 14.01, we have 17.07 dbanldh=—Cap. The third fundamental form, as in Equation 14.08 is constant along the isozenithal-normal so that we Jiave 17.08 dc a pldh=0. 10. If overbars refer to the base surface where h = 0, we have accordingly the following integral relations, 17.09 a a f3 = a a i3 — 2hbai3 + h 2 Ca/3 17.10 bat3=b~af3 — hCal3 17.11 C a 0=C a p, enabling us to find all three fundamental forms al any point in space from values at the foot of the normal on the base surface — that is, from values at points on the base surface having the same latitude and longitude as the point in space. 11. The components of the three surface forms in terms of the three curvature parameters (ki, k>, ti) of the h-surfaces are as given in the (co, $, N) system, namely, an = {k\ + ti) cos- cb/K 2 b n = h cos 2 4>l K a V i = — 2Ht\ cos (b/K- bn = — h cos /K a-i2 = (ki + t\)IK? b 22 = k x IK 17.12 c a/3 = (cos 2 , h) Coordinate System 119 equations and subtracting the square of the second, we have 17.14 l/K=HK-(2HIK)h + h 2 or 17.15 K/K = (1-/;ki)(1-/(k 2 ), using the principal curvatures K\, k-i. Also by adding the first and third equations of Equations 17.13, we have 17.16 2HIK=2H/K-2h, which relate the curvature invariants of the A -surfaces. 13. We can also find without difficulty that 17.17 a^lK 2 = Ti^jK 2 ~2h Pi* IK + h 2 c^ 17.18 b^lK = b^lK - hc^ 17.19 c *v = c«0 from the contravariant components o»= {k\ + t\) sec 2 (f) b n = ki sec 2 a vl — 2Ht\ sec (b b V2 = t x sec , , N) formulas, such as in Equations 12.037, 17.23 k r =(-ki sec (b, -t u Q) 17.24 fx, r =(-t 1 sec p sin )k a (x>n + IA a 4>H- whether the components of ka$. iA a & are taken in the space metric or in the surface metric. The only nonzero components containing a 3-index are from Equations 12.014, etc., 17.32 A.3J = — cos , c/>, N) formulas, we find that the only nonzero symbols containing a 3-index are 1 a/3 bad 17.36 FSB = - b^Cffy = ~ a^bey = V% LAPLACIANS OF THE COORDINATES 25. Because N=h and ra=l, we have at once from Equation 12.100 17.37 Mi 2H- from Equations 12.104 and 12.105, we have 17.38 (cos c/>)Aw = 2(sin 0)V(a>, ) — (2H) a k a 17.39 A = -(sin c/> cos ()))V(o))-(2H) a iJL a with 1 7.40 V(o>, ) = a vl = 2Ht x sec 17.41 V(w) = a n = {k'i + t'i) sec 2 , using Equations 17.20. It should be noted that the space Laplacians in Equations 17.38 and 17.39 are the same as the surface Laplacians, obtained in (a>, c/>, N) coordinates as Equations 12.118 and 12.120. Although we have defined the coordinates {co, c/>) in the same way for both systems, they do not have the same values at any point in space because the two normals do not have the same direction. Consequently, the space Laplacians are different in the two systems. We can, however, choose any one of the TV-surfaces as base surface in the (a>, c/>, h) system; and on that surface, (oj, c/>) will be the same. Consequently, the surface Laplacians will be the same in the two systems. 26. The general Equation 8.29, for converting the surface Laplacian of a scalar to the space Laplacian becomes in these coordinates 17.42 dF d-F dn da- rn which the surface Laplacian is denoted by an overbar. This equation shows again that the space and surface Laplacians of the coordinates (w, c/> are the same in this system. Using Equation 17.22, we may express Equation 17.42 in the alternative form 17.43 &F = AF + K^{- , dF K dh The (&>, (/>, h) Coordinate System 121 CHANGE OF SCALE AND AZIMUTH IN NORMAL PROJECTION 27. Projection from one A-surface to the base surface down the isozenithal-normals requires the surface coordinates (&>, ) to be the same for a point and its projection. We consider a displace- ment on the current surface given by a change dx a in surface coordinates over an element of length ds in the direction of a unit vector / a : the corre- sponding quantities, projected on the base surface, are denoted by overbars. Accordingly, we have l a ds = dx a =dx a =l a ds. If now we multiply the three Equations 17.09, 17.10, and 17.11 by {dx a dx ti ), we have at once (dslds) 2 =l-2hk + h 2 (k 2 + P) (dslds) 2 k=k-h(k 1 + t-) 17.44 (dslds) 2 (k- + f-) = k- + t- in which k, t are the normal curvature and geodesic torsion of the base surface in the projected direc- tion. The last of these formulas applies to isozenithal projection in the general coordinates (w, (/>, N) and could be obtained from the formula for spherical representation, but the other two formulas are peculiar to the (o», $, h ) system. 28. Following Equations 13.04 and 13.05, we can j also relate azimuths on the two surfaces as follows, (dslds)(k cos a + t sin a) = — c)(/>/r')s = (k cos a 4- t sin a) {ds/ds) (k sin a — t cos a) =— cos (f) dco/ds 17.45 = (k sin a — t cos a); if the change in azimuth on projection is Aa = (a — a), we have the equivalent equations (ds/ds) k — (k cos Aa + t sin Aa) 17.46 (ds/ds)t = (—k sin Aa + t cos Aa). The only solution of the first of these equations which will also satisfy the first two equations of Equations 17.44 is 17.47 (ds/ds) sin Aa = — ht (ds/ds) cos Aa = ( 1 — hk in which case the second equation of Equations 17.46 reduces to Combined with the third equation of Equations 17.44, this shows that 17.49 t/(k 2 + t 2 ) is unaltered on projection; we can verify this fact by differentiation, using formulas given in the Summary of Formulas. 29. In all the preceding equations, we can inter- change the overbars; the interchange is equivalent to taking the unbarred surface as the base surface, provided we also change the sign of h and Aa. For example, from Equations 17.47, we have 17.50 tan Aa hi -ht (l-hk) {1 + hky from Equations 17.44, we have 17.51 (ds/ds) 2 =l + 2hk + h 2 (k 2 + t 2 ). This device, applied to the first equation of Equa- tions 17.47, enables us to verify Equation 17.48. THE ^-DIFFERENTIATION 30. Some formulas for differentiating the compo- nents of surface tensors, etc., along the straight normals have been given in previous sections ol this chapter. Many more can be obtained from the collected formulas in the Summary of Formulas for Chapters 14 and 15, whichever is easier, simply by substituting n=l, N = h, ds = dh. This fact results in drastic simplification. For example, from Equations 14.14, 17.07, and 8.09, we have dfr lftY ' •I 17.52 from Equations 14.16 and 14.07, we have dbaffy ■pf>y- 17.53 dh b ap bn,AaP' T )y=0, showing that bapy has the same components at all points along a straight normal. 31. Again, from Equation 14.71 or 15.53, we have for the surface covariant derivatives of a scalar F 17.54 dh -r) +a^F,b a ^ y . fin j a/i 17.48 (dslds) 2 t = t. This last equation contracts to aF «0_T(d? tih \ dh ) 17.55 .<* a; r l+V(2#, F) 122 Mathematical Geodesy in which the overbars indicate surface invariants. Using Equation 15.15, we then have after a little manipulation ^fp= A (jp) + V(2#, F) + 4//AF - 2Kb^F a0 . 17.56 32. In some cases, we could obtain such results quickly and directly in (oi, , , h) system, provided that adjacent normals do not intersect within the region of space considered; if they do, the (a», , h) Coordinate System ] 23 are accordingly equivalent to the Codazzi equations of a surface. In deriving Equation 17.59 from Equa- tion 17.58 for the y a -curves, for which the geodesic curvature and torsion are cr* and minus t and the normal curvature is k* , we must remember that the new y' a -direction is minus l". In this direction of the /"-curves, the geodesic curvature is minus cr, but the normal curvature is + A\ 37. For the lines of curvature {u a , if), t — and Equations 17.58 and 17.59 reduce to 17.60 {K, —K>)(T=(Ki)aV a (Ki-Kl)(T* = (K2)aU a which have been already given in Equations 8.22 as special forms of the Codazzi equations. 38. As another example, consider the two- dimensional divergence theorem in the form of Equation 9.12, 17.61 j ( F a j a ds = j ( F a e a %ds = j F a e a %ds= \ AFdS, in which F is an arbitrary scalar defined on and in the immediate neighborhood of the surface. As before, to ensure that the unit surface vector nor- mal to the contour stays that way after differentia- tion instead of becoming the projected direction, we have written it in the form (e^lp) where /# is the unit vector tangent to the contour. After normal differentiation, the contour integrand becomes (dFldh) a e a f 3 h + 2HF a e a Hi 3 - Fae^ ( kk + 2t//» ) - F a e a %k and the area integrand becomes &{dFldh)+V(2H, F)+2HAF-2Kb<^F aP . The first contour integral cancels the first area integral by the divergence theorem for the scalar (dF/dh). The second contour integral, transformed to an area integral, becomes 1 aF*(2HF*)pdS, the integrand of which is V(2H,F)+2HAF. This last formula cancels the second and third terms of the main area integrand: we are left with Kb^FafidS = [ F a e^ ( kk + tin ) ds = Fat^bfiylvds 17.62 [ = J (-kF a j a +tFJ a )ds =- { KbtFajpds, using Equations 7.12 and 8.02. To verify this, we could differentiate again, using the second form of the contour integral, which does not contain jp explicitly, and remembering that (KdS) is constant under normal differentiation. The result is an identity. We cannot take the third form of the contour integral and transform this by the divergence and Stokes' theorems because k, t refer to the boundary curve only and are not defined over the area. We could ; however, transform the second contour integral by Stokes' theorem. Or, we can transform the last form of the contour integral by the divergence theorem to an area integral whose integrand is a y^Kb^F a ap S )y. Because aps is constant under covariant differentia- tion, this last expression is (K&#Fah=Kbr*Fcfi+ (Kbr*)^ Combining this last equation with the original Equation 17.62 for an arbitrary area, we must have (X6«*)/,F„ = 0; and because F is arbitrary, this means that 17.63 (Kb*)p = Q. Or, using Equation 8.01, we have ( e«^ 8 6y S ) p = = eove^bysp , which is so because 6-ys/s is symmetrical in (6. /3) by virtue of the Codazzi equations of the surface. Again, we have verified the process and have checked a number of other results on the way. The form of the Codazzi equations in Equation 17.63 is sometimes useful and, although easy to verify, might otherwise have escaped notice. If we differentiate Equation 8.02 covariantly, use Equation 4.11, and substitute in Equation 17.63, we obtain the Codazzi equations in the form of Equa- tions 17.58 and 17.59. 124 Mathematical Geodesy THE POSITION VECTOR 39. If p' is the position vector at a point in space and p r is the position vector of the projected point on the base surface, then 17.64 p r = p r +hv r is evidently true in Cartesian coordinates or be- tween parallel vectors at a single point in space. Equation 12.169, written for the projected point on the base surface, is 17.65 p r = (sec dplda>)\' + (dp/d)p r + pv r in which p is the perpendicular from the origin onto the tangent plane to the base surface at the projected point. The base vectors k r , £L r , v r are parallel to X r , p'\ v r (see § 17—19) so that we may drop the overbars on these vectors in Cartesian coordinates. Substituting in Equation 17.64, we have p' = (sec 4> dplda))k r + (dpld(f>)p r + (p + h)v r , 17.66 an equation between vectors at the same point ir space which, although derived in Cartesian coordi nates, is true in any coordinates. Contracting Equation 17.64 with v r =v r , we have also 17.67 p = p + h. CHAPTER 18 Symmetrical (co, , h) system by making the parameter t\ zero at all points of the base sur- face, in which case it is clear from Equations 17.13 that t\ will be zero at all points in the region of space covered by the coordinate system. As we have seen in § 12-48, the meridian and parallel traces on any h -surface are then the latitude- and longitude- coordinate lines which are accordingly orthogonal; it is clear from § 17-6 that the (a>, , h) system is triply orthogonal. The w, (f> coordinate lines are lines of curvature, and the parameters Ai, A 2 are the principal curvatures Ki, k 2 of the h -surfaces. In addition to being the , 0, (R 2 + h)*} 125 126 Mathematical Geodes 18.05 18.06 18.07 18.08 18.09 bap={-(Ri + h) COS 2 , 0, -(R 2 + h)} 17.12 c a p= {cos- <£,G, 1} a^ = {sec 2 , 0, 0} /x,= »,. = {0, (Ri + h), 0} 18.11 Pr = {0,0,1}. The surface components of A', ^x' , A,-, /jl, are the same as the first two space components. 7. For the gradients of the coordinates, we have 18.12 (cos (b)(o r =krKRi + h) 17.29 18.13 (j> r =fJLrl(R2 + h). 17.30 8. A unit vector /'" in azimuth a and zenith dis- tance /3 is /' = A.'' sin a sin (3 + /jl' cos a sin (3 + v' cos /3; its components are , _ (sec (/) sin a sin j3 cos a sin /3 1 /,- = { (/?i + h)cos 4> sin a sin /3. 18.14 (& + A) cos asin/3, cos /3}. Derivatives of Base Vectors 9. The only nonzero components are 18.15 \>i=(R> + h) sin 4> : A :! , 18.16 /u-ii— — (Ri + h) sin $ cos : v 22 = R 2 + h. The components A 2 i. A<-ii have the same values i: the surface and space metrics. Surface Curvatures 10. Normal curvature and geodesic torsion i: azimuth a are 18.18 18.19 -k=l/R = sin 2 al(Ri + h)+ cos- a/(R-, + h) 12.04 t= (&-/?,) sin a cos a (R x + h)(R-, + h) The meridians /u/' or 1/ are geodesies. 12.06 Geodesic curvature of the parallels A'" or u r is 18.20 dR 2 da) 0, 12.13 12.13 because h cancels on substitution in Equation 12.134 and 12.135, and we can accordingly drop th overbars. The remaining Codazzi equations of th space are 18.24 ah ■ = -C a f3, 12.14 which are satisfied by expressing the fundament; forms as in Equations 18.05 and 18.06. 12. Over a particular surface, Equation 18.2 shows that R> is a function of latitude only. Thi implies that all the meridians of the surface, whic we have seen are plane curves, must be supe imposable in much the same way as two circle of the same radius can be superimposed. Thi condition is met if we take the h -surfaces as surface of revolution about the C'-axis passing through th Cartesian origin; it will be sufficient for our purpose to do so, although this restriction is not require by the differential relations. If the /(-surface Symmetrical (w, (/>, h) Systems 127 are surfaces of revolution, then the parallels are circles of radius Ri cos $ (§12-49); Ri over a particular surface is also a function of latitude only. Again, this condition is not necessary to satisfy the differential relations. The other Codazzi Equa- tion 18.22 would be satisfied if #i = g((/>) + sec(f) j{(o) where g{ = — \ R> sin d<}>. The Position Vector 13. In Equation 12.169, we found an equation for the position vector in terms of the perpendicular p from the Cartesian origin to the tangent plane of an /V-surface. If the /V-surfaces are of revolution about the C r -axis through the Cartesian origin, then from considerations of symmetry, we have dpi day = 0; thus we obtain from Equation 12.172 18.26 x sin w = y cos co- in this case, we find from § 12-49 that the radius of the parallel is 18.27 U- + y- ) 1/2 = - 1/ (A-i sec 0)=/?i cos so that from Equations 18.26 and 18.27 we can write •% = /?! COS(f) cos (O 18.28 y=R\ cos (j) sin oj. 14. As we proceed northward along a meridian over the closed surface, we have 18.29 dz = - (cot )d(Ri cos ). Using Equation 18.25 or integrating by parts, we can then express the z-coordinate with a suitable choice of limits in any of the three following forms, z= I Ry cos 4> d$ = — R\ cos 4> cot ~ R i cos 4> cosec 2 dcf> = /?isin0— I (Ri — R>) sec 4> d<$>- Substituting these forms and Equations 18.28 in 12.170 and 12.171, we have p = R\ cos 2 4> + sin 4> I R; cos (j> d J R\ cos 4> cosec 2 (/> dcf) 18.31 =Ri -sin \ (/?,-/?■-) sec d cos + cos 4> I R> cos 4> d(j> — — Ri cot (j) — cos (f) \ R\ cos (f) cosec 2 $ d(f> = — cos 4> I (Ri —R2) sec 4> dcj). 18.32 15. The last four sets of equations apply to any surface of revolution and therefore to the h -sur- faces of any axially symmetrical system. If we distinguish these equations by overbars on /? u R>, and p, they apply to the base surface in a sym- metrical (w, 0, h) system. The position vector of any point in space in this system is then given by Equation 17.66 as 18.33 p r =(dpld(f>)ix>-(p + h)i> r . Christoflfel Symbols 16. The only nonzero symbols are r 2 , = (« 1 + / J ) sin cos l (R 2 + h) ri2=-(R-> + h) tan <£/(£, + A) 18.34 r ., _ aln (R., + h ) 12.129 18.30 all in the surface or space metric, and n\ =-(/?,+/*) cos 2 (/> n,=-(R- 2 + h) ri 3 =n(R t +h) 18.35 rf, = !/(/?, + *). 17.36 Higher Derivatives of Base Vectors 17. Second derivatives of the base vectors can be found direct from Equations 18.15, 18.16, 18.17, together with the Christoffel symbols in Equations 18.34 and 18.35, or by substitution in previous 128 Mathematical Geodes* formulas. For example, we have 18.36 k 2 i 3 =-(R 2 + h) sinl(Ri + h), 12.159 with all other k a p3 zero: we have 18.37 jLt n3 = sin0cos , 12.159 with all other /i, a ^ 3 zero; and we have 18.38 V a f33 = -Cal3 12.159 18.39 Xa33 = /U«33= ^«33 = 12.160 18.40 v a py=-b a py 12.161 Vapy=8l8l(R 1 )y cos 2 <)) + Sld%(R 2 )y 18.41 + (Ri -R>) sin cf) cos 0(8^+6^)8^. 8.16 Laplacians of the Coordinates 18. As in Equation 17.37, we have 18.42 Mi=-2H; also in the symmetrical (a», (f>, h) system, we have 18.43 V(co, 1 dR 2 any surface is obtained at once from Equations 13.0^ and 13.05 as 18.46 (Ri + h)(R 2 + h) (R 2 + h) 3 dt\>' 17.39 The second (alternative) expression may be ob- tained either by manipulating the first, or direct from the defining equation using components of the metric tensor and the Christoffel symbols already given. The surface and space Laplacians of oj and (f) are the same. SURFACE GEODESICS 19. If a is the azimuth of its spherical repre- sentation, the differential equation of any curve on cos (f) dco d(}> tan a — 18.47 &i sin a + t\ cos a k-z cos a + t\ sin a k sin a — t cos a k cos a + t sin a If the curve is a geodesic of the surface, we hav< also from § 12-47 18.48 da = sin dw along the curve so that da k\ sin a + ti cos a cot (}> 18.49 dc/> k-z cos a + ti sin a _k sin a — t cos a k cos a-\- 1 sin a If hi, k-z, t\ are specified as functions of a>, <£ anc if we then assume that the curve belongs to somi family of geodesies defined over some region o the surface, we can integrate this equatioi numerically. 20. In the case we are considering Ui = ki, ki functions of 4> only), the equation become: the ordinary differential equation cot a da= {Ro/Ri) tan d)IR i } tan d$+ tan d<\> = - (1/ 'Ri)XdRil d)d(f) + tan <£ d, using the Codazzi Equation 18.22. This integrates to 18.50 R\ cos sin a = constant or 18.51 (^i + h) cos (/> sin a = constant as the general equation of geodesies on an axiall symmetrical /?-surface. It is a generalization c the result usually known as Clairaut's theorem i classical geodesy. 21. The normal projection of a geodesic, even i an axially symmetrical system, is not, in genera a geodesic on any other h-suriace. For this to b true, it is clear from Equation 15.54 that the gee desic torsion would have to be constant along th curve. 22. The general equation of a surface geodesi whose unit tangent is l a is, from Equation 4.0' 18.52 /„^ = 0. If we evaluate this equation in the symmetries Symmetrical (w, (b, h) Systems 129 (&>, , h) system is a spheroid so that the meridian section is an ellipse of semimajor axis a and eccentricity e. The semi- minor axis b is the Cartesian z-axis; we define the complementary eccentricity e as 18.53 = b/a = +(l-e 2 ) ] 24. The principal radii of curvature in and per- pendicular to the meridian (variously known in the literature as M, N or p, v, respectively) are well known as R->(=M = p) = ae' 1 (l-e 2 sin 1 as Equation 18.54, or any other required function of cb, and have determined Rt from Equation 18.25. 25. From any of the Equations 18.30, 18.31, and 18.32, we now find 18.56 z = ~e-R\ sin ). 306-692 0-69— 10 CHAPTER 19 Transformations Between TV-Systems GENERAL REMARKS 1. Transformations between coordinate sys- tems arise in geodesy mainly from the practical necessity to linearize computations. The general (to, <£, N) system, in which N is interpreted as the gravitational potential and the effect of the Earth's rotation is included, is most useful for theo- retical investigations and is closely related to most systems of measurement; for example, the v r in this system are then the directions of the astro- nomical zenith or plumbline and so enter directly into astronomical observations and into the measure- ment of horizontal and vertical angles. Neverthe- less, we have little numerical knowledge as yet of the curvature parameters in this system; ulti- mately, if they become known in sufficient detail, the curvature parameters will probably be too irregular to provide a practical basis for calcula- tion over finite distances. It is usual therefore to work in the simpler (to, ^>, h) system (N=h) with a regular base surface and to transform the observations accordingly. Moreover, we usually make the base surface a spheroid, which is a close approximation to an actual equipotential surface, so as to ensure that first-order transformation — leading to linear observational equations — shall be sufficient. 2. An alternative would be to use a regular (to, (/>, N) system, representing a standard gravi- tational field in which one of the equipotential TV-surfaces is a spheroid — approximating closely to a selected equipotential surface in nature. Calculations over finite distances in such a sys- tem, although possible, are not as simple as in the (to, , h) system. For certain purposes, we need a standard or model gravitational field, but there seems to be no advantage in making all of the field's equipotentials the coordinate surfaces of the geometric system. Instead, we can calculate the standard gravitational elements at positions given in (to, (/>, h) coordinates, an operation which again amounts to coordinate transformation. 3. We shall continue to assume, as we have done throughout this book, that the TV-systems share a common Cartesian system whose C'- axis is parallel to the physical axis of rotation of the Earth at a particular epoch; we shall derive conditions which ensure this arrangement. It may be thought that an unnecessary and an arbi- trary restriction thereby is introduced, but this is not so. We cannot transform from one system to another without completely relating the two in some way; the adoption of a third system, common to the two, introduces no more conditions than are necessary and sufficient for this purpose. The adoption of a common Cartesian system can also be used to ensure that the space remains flat during the transformation. 4. In addition to specifying the n's (the magni- tudes of the scalar gradients N r ), we shall relate the base vectors in the two systems. We do this by means of vector equations, true in either coordinate system. The same vector equations will then hold between parallel vectors at other points in space because, in that case, the equations will be true in Cartesian coordinates at the new point — and thus in any coordinates at the new point. The same equa- tions will accordingly serve to relate the base vectors and other vectors associated with them, either 131 132 Mathematical Geodesy (a) at the- same point in space after a coordinate transformation, or, (b) at a different point in space in the same coordinate system, in which case the relation will hold true between parallel vectors. The changes in coordinates in case (a) will usually be small in practice; whereas, in case (b), they may be large. To take advantage of both possibilities, we shall accordingly derive quite general trans- formation formulas not confined to first-order changes. Quantities in the second /V-system (or at the second point in space in the same /V-system) will be denoted by overbars. DIRECTIONS 5. We shall first deal with transformation of directions; for this purpose, it will be convenient to define a few auxiliary angles on the spherical diagram in figure 15. A radius vector of the sphere Figure 15. is drawn parallel to a direction in space (such as one of the normals v r ); the point P where the radius cuts the sphere can accordingly be said to represent the direction (v r ). The normal at the other point in space or in the other coordinate system is repre- sented by P. The Q represents a fixed direction l r during transformation or a parallel direction at the overbarred point. The C represents the common Cartesian axis so that the latitudes c6, c/> and the difference in longitude 8a>=(a> — to) are as shown in the diagram. The great circle PP or arc-length a— arc cos {v r v r ) is simply an auxiliary, and so are the angles a*, a*. The azimuths (a, a) and zenith distances (/3, /3) of the fixed direction Q (or of the parallel directions Q) are as shown. 6. The PP' and PP' are_ quadrants so that the vectors represented by P' , P' are ^-surface vectors in the plane containing the normal and C'; these vectors are accordingly by definition fi r , yT. The P'P' defines another auxiliary angle t. The remain ing base vectors A.'", V (not shown in the diagram are, respectively, the poles (to the right in the diagram) of great circles PCP' and PCP'. 7. The following formulas, collected for eas} reference, are obtainable from scalar and vectoi products or by ordinary spherical trigonometry from the triangles CPP, CP'P' , cos cr = sin c6 sin c/> + cos c6 cos c6 cos 8a; 19.01 cos t — cos (/> cos (f> + sin c6 sin c/> cos Soj 19.02 = sin a* sin a* + cos a* cos a* cos cr cos 8co = cos a* cos a* 19.03 +sin a* sin d* cos cr sin (f> sin 8(o = — sin a* cos a* 19.04 +cos a* sin a* cos cr sin <£ sin 8w = cos a* sin a* 19.05 sin a cos a ' cos cr sin cr cos a* = cos cos <£ cos 8a> sin cr cos a* = — sin (/> cos 19.07 + cos (f) sin (f) cos 8co 19.08 sin cr sin a* = cos sin 8io 19.09 sin cr sin a* = cos cf) sin 8co cos (f) cos a* = — sin (/> sin cr 19.10 +cos (/> cos cr cos a* cos cos a*= sin <£ sin cr 19.11 + cos cos cr cos a* cos <£ cos 8 cos cr 19.12 — sin c6 sin cr cos a* Transformations Between N-Systems 133 19.13 19.14 19.15 cos (f> cos 8a> = cos <£ cos cr + sin = cos $ tan <£ — sin cos 8co cot a* sin 8to = — cos tan c6 + sin ^ cos Soo, together with the following differential relations, sin cr da* = sin a* cos cr d4> + cos $ cos a* c/(8oo) 19.16 —cos (/> sec <£ sin a* d<$> sin cr da* = sin a* sec cos c/c6 19.17 + cos (f) cos a*c/(8co) — sin a* cos cr c/<£ dcr = — cos a* c/<£ 19.18 +cos (/> sin a* c/(8co) + cos a* d§. Several of the preceding formulas may be obtained or verified by changing the symbolism between the two ends of the line or between the two co- ordinate systems, that is, by interchanging the overbars and by changing the signs of Sco and cr. BASE VECTORS 8. We can obtain scalar products of the base vectors from the spherical diagram in figure 15. For example, k r k r is the cosine of the angle between the great circles PCP' and PCP' , that is, cos 8co_. Again, v, k r is the sine of the perpendicular from P to the greaj^ circle PCP', that is, sin a* sin cr or sin Sco cos c6. Proceeding in this way, we obtain one set of base vectors in terms of the other set as k r cos 8(o sin $ sin Sco [— sin (/> sin 8co cos t ^cos sin 8(D sin cr cos a* 19.19 — cos 4> sin 8o) — sin cr cos a* cos cr 9. Next, we shall derive this same result in terms of the rotation matrices of § 12-15. Writing 19.20 — cos (f) sin c6 / — sin (i) cos 0) 19.21 Q= -cosfi) -sinco V 1 19.22 Q = sin 8(o —cos (f> sin 8w N QQ 7 =|— sin (j) sin 8(d cos t — sin cr cos a" \cos 4> sin 8a) sin cr cos a* cos cr 19.25 we can easily verify this equation by multiplying out the Q-matrices. For easy reference, we add the full expression for Q, / — sin co cos co \ Q = I — sin cos co — sin sin co cos c6 I. \cos (j) cos a) cos sin (o sin c6/ 19.26 AZIMUTHS AND ZENITH DISTANCES 10. The arbitrary unit vector l r in figure 15 can be expressed in the following alternative forms l r = k r sin a sin/3+ ix r cos a sin /3+ v r cos /3 = k r sin a sin ^3+ /u/cos a sin/3 + v r cos /3: using these forms to contract the vector Equation 19.24, we have {sin a sin /3, cos a sin /3, cos /3} 19.27 =QQ 7 '{sin a sin /3, cos a sin /3. cos /3}. Only two of these three equations for cv. /3 are independent because each term is equivalent to the component of a unit vector so that an identity would result from squaring and adding the equa- tions. 134 Mathematical Geodesy 11. Equation 19.27 gives the azimuth and zenith distance of a vector in the transformed (barred) coordinates. The equations also relate the azimuths and zenith distances of two parallel vectors in the same coordinate system at two different points in space so that it is immaterial whether the changes result from coordinate transformation or from parallel transport — or both. 12. In particular, Equation 19.27 can refer to the straight line joining two points in space in any (oj, (/), A) coordinate system. The equation would enable us to determine any two of the seven quan- tities a, (3, a, /S, , <£, 8a> from the other five. For example, if a, /3, a, f3, (f> are measured or given, we can determine (f) and 8w and so can build up a latitude and azimuth traverse without measuring any more latitudes, although error would be likely to accumulate through the effect of residual (un- corrected) atmospheric refraction on (3, f3. At the other extreme, if a, a, 0, $, 5w are measured, then we could determine /3, (3 and so could evaluate the refraction. Whatever we do, we must take account ol the fact that these seven quantities are related. ORIENTATION CONDITIONS 13. If we transform from one /V-system to an- other, the seven quantities in the last section cannot be independently chosen, but they must satisfy two conditions — equivalent to the two independent equations in Equation 19.27 — to ensure parallelism of the Cartesian axes. The most common case in practice arises when the changes in {(o, = cj) + 8(f> as we have already written to = w + Saj, then it is easy to show to a first order that Equation 19.25 or 19.26 gives 14. If f3 is nearly 90° so that the line is almost horizontal, the first equation reduces to 19.30 8a = sin (f> Sw, sm -sin 8w cos 4> 8cd 8(f) where I as usual is the unit matrix, and Equation 19.27 then reduces to the following two independent equations, connecting the first-order changes in latitude, longitude, azimuth, and zenith distance, 8a — sin(/> 8oj+ cot (3 (sin a 8 — cos a cos (f> 8co) 8(3 —— cos sin a 8w — cos a 8(f). 19.29 which is independent of the direction chosen as a fixed line in the two coordinate systems and is simply a difference in the azimuths of all nearly horizontal lines emanating from the same point. This is the so-called Laplace equation of classical geodesy, which alone is used in the hope of preserv ing orientation of the Cartesian axes. But even il (3 — 90° is on all observed lines emanating from a point, this fact does nothing to satisfy the second condition of Equations 19.29, which does not depend on (3 at all and cannot therefore be satisfied by choosing favorable values of /3. Satisfaction ol the equations of Equations 19.29 for a particulai (a, /3) ensures that the Cartesian components ol the corresponding direction are the same in both coordinate systems; but this fact is not sufficient to ensure parallelism of the Cartesian axes because it would still be possible to rotate either system about the (a, (3) direction without any effect on its Cartesian components. To ensure parallelism of the Cartesian axes, we need to satisfy both Equations 19.29 for two different directions. 15. It is clear therefore that the simple Laplace azimuth Equation 19.30 does not preserve orienta tion at a single point during a change of A-coordi nate systems, such as the change from an astro nomical (o>, , N) components of the base vectors from §12-27 and §12-33, A 1 A 2 A 3 \ y—faseccfr —tj Of; | /x 1 ix 2 /a 3 = I — f, sec - k 2 v 1 v 2 v 3 / \ yi sec (f) y 2 n /Ai A2 ^3N S = l Hi jx 2 fx.i Vi v-i v s/ I— k 2 cos (j>IK tijK sec 3 ( I/72 ) /dco\ = 1 hcosQlK -ki/K d(l/n)/d(() V (1/n) 19.32 Because the base vectors are unit orthogonal vectors so that, for example, A'A r =l, k r (JL r = 0, etc., we have also 19.33 RS T =SR r =I in which I as usual is the unit matrix; thus, we have R -!= S r 19.34 S - x = Rr 17. We also define R= (A 1 , . . .) = (— Ai sec, . . .) and S similarly as the corresponding matrices in the (to, 0, N) system, that is, the matrices of the (to, 4>, N) components of the base vectors of the barred system. It should be noted from Equation 19.24 that OQ r R does not give R; it gives the (to, c/>, N) components of the base vectors of the barred system. To trans- form these components to (to, (/>, TV) components, we use the vector transformation formula of Equation 1.18, equivalent to postmultiplying by the transpose of the transformation matrix of Equa- tion 19.37 in §19-21. To verify this, we have R = OQR(R OQS)' = OQ'RS QO R = R, using Equation 19.33 and the fact that the Q's are orthogonal matrices. 18. If one of the systems is a symmetric (o>, c/>, h) system, the corresponding R and S matrices become diagonal; this introduces a considerable simplifica- tion into all matrix equations containing R and S. The necessary modifications can be made at sight, using the results of Chapter 18. TENSOR TRANSFORMATION MATRICES 19. To transform vectors and tensors between Cartesian and (o», c/>, N) coordinate systems, we need the partial differentials dx/d(o, BN/By, etc. These are all components of the Cartesian vectors A r , etc., in the (to, 4>, N) system. For example, we have from Equations 12.009 A r = (Bx/Bo), Bx/Btf), Bx/BN), while the contravariant components give A r = (Bo/Bx, 3 By/BN ) »Az/dto Bz/B/3x BN/Bx\ /A 1 A 2 A 3 \ B(o/By B(f>/3y BN/By = I B 1 B 2 B 3 = Q/R >Bco/Bz B4>lBz BNlBzl \C l C 2 C*l 19.36 with the Jacobian (nK sec (f>). 21. To transform between (to, , N) systems, we have, for example, dto_dto Bx_ dto dy , dto Bz_ dto Bx dto By dto Bz B(o 136 so that (dcoldu) da)/d dwldN\ dj>lda> d4>ld<)> dfrdN )=(Q r R) 7 lQ 7 S dN/dai dN/d dN/dN/ 19.37 = RTQQT S PARALLEL TRANSPORT OF VECTORS 22. To obtain the (w, (f>, N) components of parallel vectors (/ r , l r ) at two different points in space (one point overbarred), we can use the vector equation at the barred point l r = A r (A s l°) +B r (B s l°) +C r (C s l°), which expresses the equality of Cartesian compo- nents at the two points. We have 7»\ lA^mC^jAtAzAAll 1 ] i*\=Ia*b*c*\IbiB % bAu % l7 3 / \a 3 b 3 c 3 I \ciC 2 c 3 / \i*l = (Q T R) 5r Q I S{/ 1 ,/ 2 ,Z 8 } = R r QO T S{/ 1 ,/ 2 ,/ 3 } 19.38 = R T QQ r {sina sin/3, cos a sin/3, cos/3} if a, /3 are the azimuth and zenith distance of l r . This equation is easily verified from Equation 19.27. The covariant components are similarly given by {l l J 2 J 3 } = s T QQ T R{iui2,k} 19.39 = S r QQ/{sin a sin /3, cos a sin j8, cos /8}. Mathematical Geodesy formation, such as the direction between two ground stations, then the component of deflection in that direction is 19.41 A r / r = cos /3 — cos /3 where /3, (5 are the astronomical and geodetic zenith distances, respectively. This relation is rigorously true even for large deflections. 25. The definition does, however, agree with the usual first-order conventions in classical geodesy. For small changes in coordinates, we have at once from Equations 19.24 and 19.28 19.42 k r =v r -v r =(cos())8a))k r + (i8(t>)fji r in which b<\> = <\> — <\> is the astronomical minus the geodetic latitude; similarly, 6w = o> — a> is the astronomical minus the geodetic longitude of the point under consideration. To a first order, the meridian and parallel components of the deflection vector are accordingly 8<£ and cos 4> da> as in the classical conception. 26. We can express the deflection vector rigor- ously from Equations 19.24 and 19.25 as A r = (cos$ sin 8to)\ r 19.43 + (sino- cosa*)fx r -2 sin 2 (o-/2)i^ r , which holds true also for the change in the v 1 between two widely separated points, if we ust Cartesian coordinates or if we interpret v r in th( usual way as a parallel vector at the unbarred point THE DEFLECTION VECTOR 23. We define the deflection vector A r as the change in the v r on transformation between ./V- systems so that we have 19.40 ^=v r —v J In the usual geodetic application, the overbarred vector will refer to the astronomical (w, , N) system with N interpreted as the geopotential, that is, the gravitational potential with allowance for the Earth's rotation. The unbarred vector will refer to the geodetic system, usually an (to, <£, h) system with a spheroidal base. 24. The definition does not require the change of coordinates to be small. For example, if l r is a unit vector which remains fixed during the trans- CHANGE IN COORDINATES 27. Another way of viewing this question is tc consider the differences in the coordinates them selves, Sto = w — (o 8(f> = 4> — 4> 8N=N-N, as a measure of "deflection," with an appropriat< choice of unit for the /V's. This method is some times useful in considering changes in the "deflec tions" between two points in the field; and for thi purpose, we require their gradients. Transformations Between N-Systems 137 28. Using Equation 19.31, Equations 12.046, 12.047, and 12.001 can be put in the matrix form {(O r , 4> r , Nr} = R r {A r , fl r , V r } = R T QQ T {\ r ,(jL r ,p r } = R T QQ T S{a> r , r ,N r } 19.44 =S- 1 QQ 7 S{o> r ,0 r ,^r} in which we have used Equations 19.24 and 19.34 so that {(8o») r , (8) r , (8N)r}=(S- 1 QQ T S-l){0> r ,r,Nr} 19.45 =(R T QQ T -R T ){K, f jL r , v r }. 29. In evaluating Equation 19.45 for small 8cu, 8(j), we can use Equation 19.28; but there is no guarantee that the changes in the curvatures in the R or S matrices will also be small. 30. If we complete the three vector equations in Equation 19.45 and contract in turn with k r , ix r , v r , then, if elements of length in the direction of the base vectors are dk, d/x, dv, we have fd(8) d(8a>Y M = 19.46 dk dfi dv d(8) dk d(8<{>) dfJb d(8(f>) dv d(8N) d(8N) d(8N) R T QQ T -R T , dk dfi dv giving components of the "deflections" in the directions of the base vectors. 31. By transposing the equation M = R T QQ T -R T , we have 19.47 R + M r = QQ r R=(QQ r ) 7 R, which gives us a relation between the R's; from this, we have a relation between the parameters Ai, kz, ti, ji, 72 and the components of the "deflections." 32. We may also require the components of the "deflections" in the direction of a unit vector l r =k r sin a sin/3 + /it r cos a sin/3 + v r cos ^3 in azimuth a, zenith distance /3, and arc element dl. Contracting Equation 19.45 with / r , we have at once d(8(o) d (80) di8NY] dl ' dl ' dl J = (R r 0Q T— R r ){sina sin/3, cosa sin /3, cos /3}- 19.48 33. It is clear from Equation 19.44 that Q T S{w r , r , N r } =Q r S{(L r , 4> r , N r } is an invariant which has the same value in any (oj, (/>, iV) system; it is useful to inquire what this invariant may be. Using Equation 19.35, we have /AiAzAsUaA Q T S{(O r ,(f>r,Nr} = Ui B 2 B 3 \Ur \C l C 2 cJ\Nrj /o) r (dx/do)) + (f, r (dxld(f))+ N r (dx/dN\ 19.49 = {x r , y r , Z r } = {A r ,B r ,Cr}, using Equations 12.009. The invariant is accord- ingly the common Cartesian system. CONTENTS Part III Page CHAPTER 20 -The Newtonian Gravitational Field 143 Summary of Mechanical Principles 143 The Poisson Equation 146 Geometry of the Field 148 Flux of Gravitational Force 149 Measurement of the Parameters 150 Chapter 21 -The Potential in Spherical Harmonics 153 Generalized Harmonic Functions 153 The Newtonian Potential at Distant Points 155 Rotation of the Earth 168 The Newtonian Potential at Near Points. 169 Analytic Continuation 172 The Potential at Internal Points 173 Alternative Expression of the External Potential 1 74 Maxwell's Theory of Poles 176 Representation of Gravity 179 Curvatures of the Field 180 Determination of the Potential in Spheri- cal Harmonics 184 Magnetic Analogy 184 CHAPTER 22 -The Potential in Spheroidal Harmonics 187 The Coordinate System 187 The Meridian Ellipse 187 Spheroidal Coordinates 189 The Potential in Spheroidal Coordinates. 191 Chapter 22 -Continued The Mass Distribution 193 Convergence 194 Relations Between Spherical and Sphe- roidal Coefficients 194 The Potential at Near and Internal Points 196 Differential Form of the Potential 196 Chapter 23 -The Standard Gravity Field... 199 Field Models 199 Symmetrical Models 199 The Spheroidal Model 200 The Standard Potential in Spheroidal Harmonics 200 The Standard Potential in Spherical Harmonics 201 Standard Gravity on the Equipotential Spheroid 202 Standard Gravity in Space 204 Standard Gravity in Spherical Har- monics 204 Curvatures of the Field 205 The Gravity Field in Geodetic Coor- dinates 207 CHAPTER 24 — Atmospheric Refraction 209 General Remarks 209 The Laws of Refraction 209 Differential Equation of the Refracted Ray 210 The Spherically Symmetrical Medium... 211 139 140 Mathematical Geodesy Page Chapter 24 -Continued Geometry of Flat Curves 211 Arc-to-Chord Corrections 213 The Geodetic Model Atmosphere 214 Arc-to-Chord Corrections — Geodetic Model 214 Velocity Correction 215 The Equation of State 216 Index of Refraction — Optical Wave- lengths 218 Index of Refraction — Microwaves 219 Measurement of Refractive Index 220 Curvature 220 Lapse Rates 221 Astronomical Refraction 223 Measurement of Refraction 225 CHAPTER 25 - The Line of Observation 227 General Remarks 227 General Equations of the Line 227 The Line in Geodetic Coordinates 229 Taylor Expansion Along the Line 231 Expansion of the Gravitational Potential. 231 Expansion of Geodetic Heights 233 Expansion of Latitude and Longitude 233 Astro-Geodetic Leveling 233 Deflections by Torsion Balance Meas- urements 234 Chapter 26 — Internal Adjustment of Net- works 239 General Remarks 239 The Triangle in Space 239 Variation of Position 240 Variation of Position in Geodetic Coor- dinates 241 Observation Equations in Geodetic Coordinates 242 Observation Equations in Cartesian Coordinates 246 Flare Triangulation 246 Stellar Triangulation 247 Satellite Triangulation — Directions 250 Satellite Triangulation — Distances 256 Lunar Observations 257 Line-Crossing Techniques 258 Chapter 27 — External Adjustment of Net- works 261 Change of Spheroid 261 Change of Origin 261 Changes of Cartesian Axes 262 Change of Scale and Orientation 264 Page Chapter 27 — Continued Extension to Astronomical Coordinates.. 265 Adjustment Procedure 265 Figure of the Earth 266 Chapter 28 — Dynamic Satellite Geodesy 269 General Remarks 269 Equations of Motion — Inertial Axes 269 Equations of Motion — Moving Axes 271 Inertial Axes — First Integrals 272 Moving Axes — First Integrals 274 The Lagrangian 275 The Canonical Equations 275 The Kepler Ellipse 276 Perturbed Orbits 281 Variation of the Elements 282 The Gauss Equations 285 Derivatives With Respect to the Ele- ments 285 The Lagrange Planetary Equations 290 Curvature and Torsion of the Orbit 290 The Delaunay Variables 291 First Integrals of the Equations of Motion — Further General Considera- tions 293 Integration of the Gauss Equations 294 Integration of the Lagrange Equations... 298 Integration of the Canonical Equations... 299 Differential Observation Equations — Direction and Range 302 Differential Observation Equations — Range Rate 305 The Variational Method 306 CHAPTER 29 — Integration of Gravity Anom- alies — The Poisson-Stokes Approach 309 General Remarks 309 Surface Integrals of Spherical Har- monics 309 Series Expansions 310 Introduction of the Standard Field 311 The Spherical Standard Field 314 Poisson's Integral 315 Stokes' Integral 317 Deflection of the Vertical 318 Gravity and Deflection From Poisson's Integral 319 Extension to a Spheroidal Base Surface.. 320 Bjerhammar's Method 323 The Equivalent Spherical Layer 324 Chapter 30 — Integration of Gravity Anom- alies—The Green-Molodenskii Approach... 327 General Remarks 327 Part 111 Contents 141 Page Page Chapter 30- Continued Chapter 30- Continued The S-Surface in (oj. . h) Coordinates... 329 The Equivalent Surface Layers 340 Application of Creen's Theorem 333 The Basic Integral Equations in Ceo- Potential and Attraction of a Single detic Coordinates 341 Layer 337 The Equivalent Single Layer 344 Potential and Attraction of a Double Layer 338 CHAPTER 20 The Newtonian Gravitational Field SUMMARY OF MECHANICAL PRINCIPLES 1. In this chapter, we shall show that the geometry of the Newtonian gravitational field can be treated as a special case of a (o», (/>, N) coordinate system in which N is the potential, the TV-surfaces are equipotentials, and the form of N is restricted by the Newtonian law of gravitation. The Central Field 2. In a gravitational field set up by a single particle of mass m, the force of attraction on another particle of unit mass at a distance r from the first particle is, by Newton's law, Gm/r 2 in which G is the gravitational constant. The direc- tion of the force is toward the massive particle along the line joining the two particles. The particle of unit mass is usually known as a test particle because the notion of such a particle serves to materialize the gravitational force and so helps us to explore the field; there must be at least two particles in the field for Newtonian gravitation to have any meaning. 3. The potential is usually defined physically as the negative of the work done by the force of attrac- tion on a test particle of unit mass in moving the test particle from an infinite distance to the distance r from the massive particle or the positive work which must somehow be done against the force of attraction to remove the test particle to an infinite distance. The potential in a field set up by a single particle of mass m is accordingly 20.01 /: Gm , , , Gm — X (-dr)= , which is opposite in sign to the usual geodetic convention. We shall, however, use the physical convention, which accords better with mathematical conventions. The work done by the force of attrac- tion in moving the test particle from infinity is considered to be stored as available energy, known as potential energy, which is accordingly the nega- tive of the potential. 4. The equipotential surfaces in a central field set up by a single massive particle are evidently spheres centered on the attracting particle; the outward-drawn unit normal to the equipotential surfaces is the gradient of r, that is, r s . If we take N as the potential, then by covariant differentiation of Equation 20.01, we have 20.02 N s — nv s = (Gm/r 2 )r s in which /?, the "distance function" of the family of /V-surfaces obtained from Equation 12.001, is seen to be the magnitude of the attracting force whose direction is — v s . Differentiating Equation 20.02 again, we have N st - 2Gm . Gm — r- r s r, + -r- r st 2Gm V s Vf Gm v$t- If we contract this equation with the metric tensor g st and use Equation 7.19, together with the fact 143 144 Mathematical Geodesy that the mean curvature H of the spherical TV- surfaces is (— 1/r), we find that the potential satisfies the Laplace equation 20.03 AN = g rs N rs = which, expanded in Cartesian coordinates, is 20.04 d*N d 2 N d 2 N =(] 8x 2 dy 2 dz 2 It is an essential part of the Newtonian system that the space should be flat and unbounded be- cause the expressions of force and potential require r to be a finite radial distance measured in a straight line; the field must extend to infinity to satisfy Equation 20.01. Accordingly, we can use simple Euclidean geometry and can choose Car- tesian coordinates. 5. If F s is the force vector of magnitude Gm/r 2 and direction — r s or — v s , then Equation 20.02 is equivalent to 20.05 F s =-N s =-nv s , which means that the force vector is the negative gradient of the potential. Accordingly, the field is completely specified if we know the scalar potential N at each point of the field; we shall find that this statement applies also to more complicated fields. Superposition of Fields 6. In dealing with the geometrical properties of more complicated fields, we shall continue to use the symbols /V and n, respectively, for the potential and the magnitude of the gravitational force — in place of the more usual symbols V (or W) and g — because this will enable us to use all of the more general formulas of Part II as they stand. However, we shall also use V (or W) and g in ex- pressing final results or when the physical properties of the field predominate. 7. We can generalize the simple central field set up by a single massive particle to the more compli- cated field set up by any number of massive particles in an attracting body of finite dimensions by invoking the principle of superposition, which simply states that the total effect on the test particle will be the sum of the effects arising from each individual massive particle. The total potential will be the sum of the individual potentials ^ — Gm/r; the total potential will satisfy the Laplace equation because each term in the summation satisfies the invariant form of the Laplace Equation 20.03, regardless of the coordinate system. We do not, for example, require the origin of Cartesian coor- dinates to be at an attracting particle as in § 20-4; the origin could be at the test particle or anywhere else, and the Laplacian property would still hold true. 8. The forces of attraction, unlike the scalar potentials, have direction as well as magnitude and would have to be added vectorially. But it is evident that the vector Equation 20.05 still holds true (and holds in any coordinates) between the gradient of the total potential and the vector sum — or result- ant—of the individual force vectors, even though the potential no longer has the simple form —Gm/r and the magnitude n of the resultant force is no longer Gm/r 2 . The direction v r of the gradient of the potential N is no longer the radial direction from a Cartesian origin, but is the unit normal to the equipotential surface or /V-surface passing through the test particle, as in Equation 1.21. If the attracting body is the Earth, v r is the direction of the zenith at the test particle or at the point under consideration, and — v s is the direction of the plumbline or the direction of the resultant force. The Effeet of Rotation 9. All the previously mentioned conclusions apply to the attraction of a static body. The Earth, how- ever, rotates, which means that particles attached to it, or resting on it, are subject to centrifugal force acting generally against the gravitational attraction. Because the effects of the two forces are, for the most part, indistinguishable, it is usual to combine them into a single force called "gravity.' The scalar whose gradient is equivalent to the re- sultant force of gravity, including the centrifugal force, is known as the geopotential. 10. We shall consider the rotation of the Earth in more detail in § 21-55 through § 21-59; for the present, it will be sufficient to assume rotation with uniform angular velocity o> about a physical axis which we shall suppose is fixed in the Earth. The direction of this physical axis is not fixed in relation to the stars, but that does not at present concern us. It will be shown in § 21-56 that the center of mass, which we shall choose as Cartesian origin, must lie on the physical axis of rotation, which we shall choose as z-axis of coordinates, unit vector C r - The other Cartesian vectors A r , B, (fig. 16) are The Newtonian Gravitational Field 145 *~B r Figure 16. fixed in space, not in the Earth, in the sense that they do not rotate with the Earth. The linear velocity vector of a point P at a distance d from the z-axis is then 20.06 cod(B, cos ayt — A, sin cot) in which t is the time which has elapsed since the point P crossed thexz-plane. The acceleration vector is the intrinsic time derivative of the velocity vector, that is, — co 2 d(B r sin cot + Ar cos cot) = — co 2 (xA r + yB r ) = — co 2 (xx r + yy r ) 20.07 = — 2 0J 2 {X 2 + y 2 )r We can consider this expression to be the force, acting on a test particle of unit mass required to maintain the particle on the Earth's surface. The force is directed inward and must come from the force of attraction whose inward sense is — TtVr=—N r . If we assume that the residual force is derived from a geopotential M with the same conventions as for N, then we must have -M r =-N r +$d> 2 (x 2 + y 2 )r. Integrating this equation, subject to the condition that there is no rotational effect on the axis (x = y = 0), we have M = N-hco 2 (x 2 + y 2 ) 20.08 = N-hco 2 d 2 . The geopotential at P of a particle of mass m at the origin is, for example, — Gm/r— 2(o 2 d 2 . The Laplacian of Equation 20.08 is because A/V = so that the geopotential is not a harmonic function whose Laplacian would be zero. We may note, however, that the Laplacian of the geopotential, in addition to being independent of the coordinate system, is also independent of the location of the rotation axis. 11. Reverting to the original notation, we can say that the basic gradient equation N r = nv r represents the Newtonian gravitational field of the rotating Earth if N is the geopotential, as defined in §20—9, if n is "gravity," and if v r is the outward- drawn normal to the TV-surfaces, that is, the level surfaces of the combined attraction and rotation. The unit normal v r is accordingly the direction of the astronomical zenith as revealed by instrumental spirit levels. The remaining coordinates (co, (f>) of a (co, cf>, N) system are the astronomical longitude and latitude in relation to the physical axis of rotation, which we have assumed is fixed in the Earth, and in relation to an initial meridian plane defined by the physical axis and by the zenith at some fixed point on the Earth's surface. The New- tonian law of gravity is expressed by the condition A/V = -2oo 2 , and this alone distinguishes the system from any other (to, cf>, N) system. Subject to this condition, the general geometry of a {co, cf>, N) system, as developed in Part II, applies in its entirety. 12. In the basic geometry of the (co, cf>, N) system, we consider the Cartesian axes A r , B r to be fixed in relation to all points belonging to the system, that is, fixed in the Earth and rotating with the Earth. We can derive from figure 16 the following relations between the A,,B r axes, revolving like the point P, and the inertial A r , B, axes, which are fixed in space, 20.10 A,—A r cos cot + B, sin wt B,= — A, sin cot + B, cos cot. In these equations, t is the time which has elapsed since the two sets of axes coincided. So far as the condition AN-- 2co' 2 20.09 AM = is concerned, it does not matter whether we consider the point P as moving in relation to the fixed axes A r , B_, or the axes A r , B, as moving in relation to A r , B r because the condition is invariant and is therefore unaffected by the choice of coordinate system. We could say that substitution of the geo- 306-962 0-69— 11 146 Mathematical Geodesy potential for the static attraction has had the effect of reducing the whole system to rest. 13. If we are dealing with an object such as an artificial satellite, which is not attached to the rotating Earth, then, in the absence of any other impressed force, the only force acting on the satellite would be the gradient of the attraction potential — V r , and the Newtonian condition would be AV=0. In accordance with Newton's second law, the equations of motion of a satellite of unit mass relative to fixed axes A r , B r , C r would be 20.11 5 2 p r _ 8v r _ y 8t 2 8t the left-hand member of which is the acceleration vector — that is, the second intrinsic time derivative of the position vector p,. The second member of Equation 20.11 is the intrinsic time derivative of the velocity vector of the satellite relative to the fixed axes A r , B r , C r - The attraction potential at a fixed point in space would not be constant, but would generally vary with time as the unsymmetrical field rotates with the Earth. The coordinates of terrestrial observation stations or tracking stations would also change with time. 14. If we refer the motion of the satellite to rotating axes, A r , B,, C r fixed in the Earth, the equations of motion will include three forces: The force of attraction — V r ; the centrifugal or centripetal force in Equation 20.07 which, being the gradient of a scalar, can be combined with — V r as the force of "gravity" arising from the geopotential; and the Coriolis force which is twice the vector product of the angular rotation vector (a/ = a)C s ) and the apparent velocity vector v r of the satellite rela- tive to the moving axes A r , B r , C, • We have 20.12 8v r 8t Wr-2e rst (cbC s )v' in which W is the geopotential. To offset the extra complication in the equations of motion, the geo- potential would be a function of coordinates in the (A r , B r , C r ) system only and would not vary with time. The coordinates of tracking stations in the same system would also be independent of time. The equations of satellite motion referred to rotating axes are considered more fully in Chapter 28. THE POISSON EQUATION 15. If the test particle or the point P at which the potential is required were to coincide with a massive particle, then the potential arising from the massive particle would be infinite and could no longer be added to the potential set up by other particles. Therefore, all the preceding argument would break down. In particular, we could not say that the La- placian of the total potential at points inside or on matter is zero. The difficulty can be overcome by some limiting process involving the temporary removal of matter to form a cavity whose dimensions are finally reduced indefinitely. However, we shall approach the problem by a different route more in line with modern geodetic applications. This route indicates more clearly what assumptions are being made. 16. We consider first the field set up by a single particle of mass m at a point O (fig. 17) and suppose Figure 17. that is surrounded by an arbitrary closed surface S. At a point on the surface whose position vectoi from O is p s , the force vector is F s -(-r;„ i /r 2 )(p s /r), directed toward O. We shall apply the divergence theorem (§ 9—13) to this vector. If v s is the outward drawn unit normal to the closed surface, then the surface integral in the divergence theorem will be J FrV'dS = - j (Gm/r 2 ) cos y dS = — I GmdCl = — 4tirGm where dfl is the element of solid angle subtendec at O by the element of surface area dS. If the ele mentary cone (dfl) is extended and cuts the surface again, it would have to do so twice more as we car The Newtonian Gravitational Field 147 see from figure 17; the corresponding extra con- tributions to the surface integral would cancel so that the form of the surface is immaterial as long as it is closed. If the mass m is outside the surface, then the elementary cone would cut the surface twice (or an even number of times); again, the con- tributions to the integral would cancel, although the force F r on the surface arising from this external mass would not be zero. Applying the principle of superposition to all the masses inside and out- side the surface, we can accordingly say that 20.13 / F r v r dS = -4mGM. In this result, due to Gauss, F r is the vector sum of all forces at a point on the surface arising from all masses inside and outside the surface S; M is the sum of all masses inside the surface. If, instead of a number of discrete masses, we have a continuous distribution of matter, we can write M I pdv where p is the density of a volume element dv and the integral is taken over the whole volume enclosed by the surface S. Transforming the first member of Equation 20.13 by the divergence theorem and using Equation 20.05, we have g rs F rs dv = -\ ANdv = -47rG pdv or I [bN-4rrGp)dv = 0. \ But the initial closed surface S (and therefore its i enclosed volume) is quite arbitrary, so the inte- ; grand of this last integral must be zero at all points of the volume; we then have i 20.14 A/V = 47rGp in which p is the density at the point where the La- placian of the potential is taken. In deriving this result, which is known as Poisson's equation, we use only the inverse square law of force and the principle of superposition and make no other assumptions at all. 17. Also, we verify that at any point in empty space (p = 0), we have A/V=0. The potential at points attached to a body rotating with constant angular velocity w about the z-axis was modified in Equation 20.08 by subtracting ior^ + y 2 ) from the static potential; we must do the same for points within the rotating body. The full Poisson equation modified for rotation is accordingly 20.15 AM = 4.jtGp-26j- in which M represents the geopotential. 18. If we cross from a region of empty space into a region occupied by matter, the potential must satisfy Laplace's equation on one side of the sur- face—separating the two regions — and Poisson's equation on the other side. We conclude that some of the second derivatives of the potential at least are discontinuous across such a surface. A similar conclusion applies to a surface within the attracting body, if the density is discontinuous across the surface. In that case, we can form Equation 20.15 for two points close to and on opposite sides of the surface and subtract: the discontinuities in the second derivatives are then equal to 4-TrGp where p is the difference in density across the surface. The Newtonian System — General Remarks 19. The Newtonian system has received massive support from observations on the outer planets in the solar system, which indicates that the inverse square law at least is true to within the precision of modern observations. The system does not ac- count for the observed advance in the perihelion of Mercury, the nearest planet to the Sun. How- ever, this discrepancy has been accounted for by high-velocity relativistic effects, which are not at present (1968) measurable and are unlikely ever to be significant in the case of near-Earth satellites. In short, the system has been amply verified in the case of a few near-spherical attracting bodies whose dimensions are small compared with their distances apart, in which case the principle of superposition is involved to a limited extent only. But it has never been demonstrated to the degree of accuracy now attainable that this principle of superposition applies close to, or actually on, a large unsym- metrical mass such as the Earth. An opportunity to do so may arise in reconciling results from satellites with those from ground observations. There are already indications, through the consistency ob- tained in results from satellites at different heights, that the effect of any departure from the principle becomes inappreciable at satellite distances from the Earth. 148 Mathematical Geodesy GEOMETRY OF THE FIELD 20. It is clear from Equations 20.05 and 12.001 that the field can be represented by the coordinate system of Chapter 12. In this coordinate system, N is the potential, the /V-surfaces are equipotentials, n is the magnitude of the resultant force, and v r is the unit normal to the /V-surfaces — the negative direction of the resultant force — and the unit tangent to the lines of force. The unit normal v T is also the apparent vertical and defines the (w, (f>) coordinates, that is, the longitude and latitude of the apparent vertical in relation to Cartesian axes fixed in the Earth in accordance with Equations 12.003, 12.004, and 12.005. The Cartesian z-axis coincides with, or is at least parallel to, the axis of rotation. 21. The Newtonian law of gravity is necessarily and sufficiently expressed by making A/V (a) zero in a static field at points not occupied by matter; or (b) — 2d>- in a field rotating with constant angular velocity di; or (c) ^irGp at points in a static field occupied by matter of density p, G being the gravitational con- stant; or (d) {4f7rGp — 2d)'-) in the rotating field at points occupied by matter. Subject to whichever of these restrictions is ap- propriate in a particular region of space and with the connotation of symbols given in the preceding section, all the geometrical relations in Chapters 12 and 13 apply to the gravitational field; these relations give us at once, for example, the curvature proper- ties of the equipotential surfaces and of the lines of force, together with the properties of lines traced on the equipotential surfaces and in space. There is no need to repeat all the formulas of Chapters 12 and 13; indeed, it will be found that most of the formulas do not contain A/V, and so do not need any modification at all. 22. We shall be concerned mostly with a rotating field in regions of space not occupied by matter, in which case the formula 20.19 Acf)=— sin (f> cos (f> V(w) — 2V((/>, In n) 20.16 A/V = -2d> 2 will apply. The only formulas in Chapters 12 and 13 which contain A/V are Equations 12.100, 12.104, 12.105, 12.106, 12.112, 12.113, and 12.115. Assum- ing that d) and therefore A/V are constant, these equations reduce to 20.17 -2u 2 = (dnlds)-2Hn 20.18 Aw = 2 tan V(o>, 0)-2V(co, In n) {lln)An = cos 2 4> V(w)+V(0) 20.20 =K? + Ki + (y? + yi) 20.21 cos $V(o>, In n)= k 2 ji — t x y 2 — 2wry x \n 20.22 V(0, In n)=k,y 2 -t x y x -2(b 2 y 2 ln. However, the last five of these equations, although useful, are not independent, but can all be derived from Equation 20.17 with the help of other relations given in Chapter 12. For example, by differentiating the logarithmic form of Equation 20.17, that is, 20.23 d(ln n)jds = Q.n n) r v r = 2H-2(b 2 ln in the parallel direction X s , we obtain after some manipulation dyjds =(2H)ak a + 4dry,/n 20.24 +yiy2 tan (/) + y 2 fi — yife, which can be shown to be equivalent to Equation 20.18, using nothing but relations given in Chapter 12. In the same way, by differentiating Equation 20.23 in the meridian direction (jl s , we find that dy 2 lds = (2H) at JL a + 4d> 2 y 2 /n 20.25 — y 2 tan (/> + y^i — y 2 A;i , which can be proved equivalent to Equation 20.19. Differentiation in the normal direction v r leads tc Equation 20.20, although we do not obtain any rela tion in this case which has not already been given in Chapter 12. The remaining Equations 20.21 and 20.22 follow directly from Equation 20.17 withoul differentiation. Equation 20.17 or 20.23 is usually known as Brans' equation if n is interpreted as gravity at a point in a rotating field and if H is the mean curvature of the equipotential surface passing through the same point. 23. We conclude that no independent geometrical relations other than Equation 20.17 have been introduced by applying the law of gravity, and that Equations 20.16 and 20.17 must therefore be equiva- lent. Indeed, we can write -2d>- = AN = {nv r ) r = n r v r +nv r r = (dnlds)—2Hn 20.26 in which we have used Equation 7.19. We can say that either Equation 20.16 or 20.17 is a sufficient expression of the law of gravity. We may note that Equation 20.17 gives us an expression for the variation of gravity along the lines of force: the law of gravity tells us nothing in general about the variation of gravity over an equipotential surface, although we may be able to deduce this geometri- cally in special cases. The Newtonian Gravitational Field 149 24. For example, if the equipotential surfaces are all concentric spheres of radius r(H=—l/r), the gravitational equation becomes 2n r ()n dr ■2dr or Bjnr 2 ) Br = -2wV, which can be integrated to nr 2 = — iforr ! +/(co, ) or dN Br = — #co-r + /(w, 0) and can be integrated again to N = -Wr 2 -f(o). (f>)lr+g(o),(f)). But N is constant over the spheres and must there- fore be a function of r only so that the arbitrary functions /(w, ), g((o. 4>) are at most constants. Gravity (n) is accordingly constant over an equi- potential surface, as it would be in the case of a nonrotating field with spherical equipotentials, although the magnitude of gravity is different in the rotating field. 25. If we know the form of one equipotential surface and the variation of gravity over that surface, we can build the whole field along either the nor- mals or the isozenithals. If we work along the iso- zenithals, we shall need to recast the gravitational equation, with the help of Equation 14.32, into the form d(Un) 1,1 $ 1\3 \n n/p dN 20.27 Next, we differentiate the Codazzi equations in the form of Equation 12.143 with respect to N, using the fact that all the coefficients in Equation 12.143 are constant during the differentiation. Substitution of the gravitational equation in the result of this last operation gives us the second isozenithal derivative of ban in terms solely of surface functions or surface derivatives of (1/n), which are presumed known or calculable on the starting surface. Repetition of the process gives us higher isozenithal derivatives of 6 a /3 in the neighborhood of the starting surface and leads to a Taylor expansion for b a a along the isozenithals. The other fundamental forms and metrical proper- ties of the equipotential surfaces can be expanded similarly from formulas for isozenithal differentia- tion given in Chapter 14. For example, we could obtain successive differentials oib a & from Equation 14.07 and then of a a p from Equation 14.03. The formulas soon become very complicated in the case of a general starting surface, but in practice, it would not be necessary to carry the process very far. The first differentials can be obtained from the Codazzi Equation 12.143 simply by knowing the variation of gravity over the starting surface; the law of gravity enters only in the second and higher differentials. 26. We could similarly expand the elements of the starting surface along the normals instead of along the isozenithals. In that case, we should work in the normal coordinates of Chapter 15. The Codazzi Equations 15.25 now contain covariant derivatives which would have to be differentiated by Equation 15.53; the c a n are no longer constants, but would have to be differentiated by Equation 15.26. Otherwise, the procedure is much the same as expansion along the isozenithals, remembering that in these coordinates _B_ BN~ i a_ t n ds' differentials with respect to N, not with respect to s, commute with differentials with respect to the surface coordinates. FLUX OF GRAVITATIONAL FORCE 27. The common normals, or orthogonal trajec- tories, of the equipotential surfaces are also known as lines of force because the tangent to such a line indicates the direction of the resultant force in accordance with the generalized form of Equation 20.05. A volume of small cross-sectional area 8S, bounded by lines of force, is called a tube of force. The cross-sectional areas of a tube of force, where the tube crosses different equipotential surfaces, are evidently related by the normal projection sys- tem of § 15-39; and from Equation 15.39, we have 20.28 r'Kln 8S) Bs = -2H in which as usual ds is an element of length along the normal and H is the mean curvature of the equi- potential surface. Substitution of this last equation in the gravitational Equation 20.26 gives B{\n(n8S)} = kN ds n ft! 150 Mathematical Geodes or 20.29 d(n8S) ds = (AN)8S. We now introduce a quantity known as the flux of force across the area 8S, and define this quantity as F r v r dS in which F, is the force vector and v r is the unit normal to the element of area 8S. In the case we are now considering, the generalized form of Equation 20.05 shows that the flux is f=-n'8S; we can rewrite Equation 20.29 as 20.30 df/ds = -(MV)8S. In this equation, the positive direction of ds is that of v s in Equation 20.05, that is, against the direction of the force. Consequently, the rate of change of flux along a tube of force in the direction of the force is + (A/V) 8S in which A/V takes one of the Newtonian values given in §20-21, depending on whether the field is rotating and on whether the tube contains matter in the small area under consideration. All of the preceding is simply an alternative statement of the Newtonian law of gravity. In particular, we may note that the flux is constant along a tube not containing matter in a static field; because this is true for any number of adjacent tubes, there is no need for the tube to be of small cross-sectional area. Again, if the tube does not contain matter in a static field, the cross-sectional area of the tube is inversely proportional to the magnitude or intensity n of the force. 28. Another and more usual way of considering the flux is to apply the divergence theorem to a finite length of a tube of force between equipotential surfaces. From Equation 9.16, we have 20.31 F r v r 8S = - ANdV Jv in which v r is now the unit normal to the surface of the tube. The contributions of the sides of the tube to the surface integral are zero because v r at points on the sides is perpendicular to the force vector; we are left with the contributions of the ends. Now suppose that one end of the tube is held fixed and that the other end is extended a short length ds, ending on another equipotential surface. The resulting increase in the area integral is evidently d(F,i> r 8S) ds ds d(n8S) ds ds ds ds, and the increase in the volume integral is -(AN)8Sds. Because the divergence theorem still holds tru for the extended tube, we may equate these tw increases to have (df/ds)ds = -(AN)8Sds; and because ds, although small, is arbitrary, we hav df/ds=-(AN)8S, which is the same as Equation 20.30, obtained sole] by differential methods. MEASUREMENT OF THE PARAMETERS 29. We have seen in Chapter 12 how the geometr of the field depends on the curvature parameter (ki, k-i, t\, yi, y 2 ) and on (In n) r v r , which are direct! related to the six components of the symmetri Marussi tensor N rs by Equations 12.162. The law c gravity expressed by Equation 20.23, which ca be written as 20.32 (In n ) r i/ r = (A'! + A- 2 )- 2w 2 /n, provides one relation between the six parameters the question naturally arises whether we can obtai other relations by direct measurement. One poss bility is the Eiitvos torsion balance, which consist essentially of two masses A, B (fig. 18) suspende o Al m i" r D B Figure 18. at different levels from a horizontal bar. The ba itself is suspended by a wire whose torsion, arisin from the unequal effects of gravity on the tw masses, can be accurately measured. The Newtonian Gravitational Field 151 30. We suppose that the line AB is of length 2/ in azimuth a and zenith distance B and that the two masses (m) are equal. The unit vector in the direc- tion AB is from Equation 12.007 /' = A' sin a sin B + /j.' cos a sin B + v r cos B. 20.33 A unit equipotential surface vector perpendicular to the plane of /'' and v r is, with the usual right- handed convention, 20.34 y = — A.'' cos a + /u,'' sin a. | If TV is the geopotential at B, the force on B is — wN r : the turning moment of this force about CD is - mNrf XDB=-(ml sin B)N r j r . Similarly, the turning moment about CD in the same sense arising from the force on A is + (ml sin B)N r j r in which A^ is the geopotential at A, so that the resultant torque is (ml sin p)(Nr-N r )j r =- (2ml- sin B)N rs j r l* 20.35 because (N r — N r ) can be considered the intrinsic change in N r in the direction (— l r ) over a distance (21). Expanding Equation 20.35 with Equations 20.33 and 20.34 and using Equations 12.162, we have finally the resultant torque as — (2ml 2 sin 2 /3)rc{(fa — fa) sin a cos a + ti(cos 2 a — sin'- a) 20.36 —J] cos a cot B + y 2 sin a cot B}. 31. Measurement in several azimuths will ! accordingly determine (fa — fa), t\, j\, y> and some \ instrumental constants, but will not separate fa j and A L >. To do this, we need an additional form of | measurement. As one possibility, Marussi in 1947 i suggested measurement of the torsion about the | horizontal axis j'\ but no instrument has yet (1968) ! been constructed on these lines. In principle, j Marussi's suggestion is equivalent to the classical method of an inclined balance. The Haalck hori- zontal pendulum is still another possibility which | has not yet materialized as a field instrument. The 1 only practicable method at present seems to be a ] direct measure of the vertical gradient of gravity with a gravimeter, leading to evaluation of (fa + fa) from Equation 20.32, but this has not so far produced results comparable in accuracy with the torsion balance. No doubt, the problem will not remain j unsolved much longer. 32. An alternative expression for the torque can be obtained from Equations 20.35 and 12.017 as - (2m/ 2 sin 0) (n s v r + ni> rs )j r l s 20.37 =- (2mnl 2 sin B) (v rg j r l») because v r andy r are perpendicular. By using Equa- tions 7.08, 10.29, and 20.34, we can express the torque as 20.38 = (2mnl 2 sin B){t sin B-(\n n) r j r cos B\ 20.39 (2mnl 2 sin B){t sin B + y t cos a cos B — y-i sin a cos B} in which t is the geodesic torsion of the equipotential surface in the azimuth of the line joining the masses. We can eliminate the term containing t, leaving only the horizontal gradients of gravity y x , y-> to be determined, by adding an observation in azimuth (1 77+ a). 33. Eotvos himself introduced a double torsion balance with parallel beams and hanging weights at opposite ends, while modern instruments have incorporated photographic recording and automatic azimuth-change. However, the principles remain the same. 34. The torsion balance has been used extensively in geophysical prospecting to determine differences in gravity from measured (yi. y%) and standard values of the vertical gradient, but the instrument has been superseded for this purpose by sensitive gravimeters which are easier to use. Geodesists, other than Eotvos himself who experimented on the Hungarian plains, have never used the torsion balance extensively because the instrument is extremely sensitive to the attraction of masses in the immediate neighborhood, and is accordingly not considered to give sufficiently representative values for the locality. Recent work ' 2 on the interpolation of deflections of the vertical with the torsion bal- ance has, for example, involved all-round leveling of the sites within 100 meters of the instrument. In addition, due precautions have to be taken to exclude the effect of such temporary masses as wandering cattle: the effect of the observer's mass is usually eliminated by photographic recording and automatic operation. 1 Mueller (1963), "Geodesy and the Torsion Balance," Pro- ceedings of the American Society of Civil Engineers, Journal of the Surveying and Mapping Division, v. 89, no. SU3, 123-155. 2 Mueller (1966), "Interpolation of Deflections of the Vertical by Means of a Torsion Balance," Bulletin Geodesique, new series, no. 80, 171-174. CHAPTER 21 The Potential in Spherical Harmonics GENERALIZED HARMONIC FUNCTIONS 1. Suppose that H is any continuous, differ- entiable scalar function of position and that the nth-order tensor 21.001 H rst ...(n) is formed by n successive covariant differentiations of//; the notation indicating that there are rc-indices r, s, t . . .. The tensor equation nils . . . (h) = first in). in which any two indices have been interchanged, is clearly true in Cartesian coordinates when the covariant derivatives become ordinary commutable derivatives, and is therefore true in any coordinate system in flat space. The nth-order tensor Equation 21.001 is accordingly symmetrical in any two indices and has therefore i(*+l)(n + 2) distinct components at most. 2. Next, suppose that H is a harmonic function. The Laplacian of the tensor Equation 21.001 is then gJ k Hrst . . . (»)jA- = (g jk H jk )rst . . . (n) = so that all components of the tensor Equation 21.001 are harmonic functions. 3. We may similarly write 21.002 grsHr St . . . („)= (g rS tf rs )< . . . („> = 0. The contracted tensor in this equation is of order (/; — 2) and has at most H(n-2) + l}{(n-2)+2}=in(n-l) distinct components. When H is harmonic, there are accordingly hi(n-l) relations, such as Equation 21.002, between the components of the original tensor in Equation 21.001, which can therefore have iO+l)(n + 2)-i/i(/i-l) = (2n+l) independent components at most. 4. We now form an invariant 21.003 »//,,, in which the contracting tensor is constant under covariant differentiation; that is, all components of the contracting tensor are absolute constants in Cartesian coordinates, and are the transforms of Cartesian constants in other coordinate systems. The resulting summation will contain at most (2n+l) independent harmonic functions, so that the contracting tensor should be chosen to introduce no more than (2n+ 1) Cartesian constants, and may therefore be chosen in the form 21.004 CL r M*N' . Q M H rst in which C is an arbitrary constant and the L' are n arbitrary fixed unit vectors, each contributing two independent constant Cartesian components. This last result, as an invariant, is true in any coordinate 153 154 Mathematical Geodesy system provided the vectors L' are fixed — that is, provided their covariant derivatives are zero. 5. If the elements of length in the direction of the fixed vectors L r , M s are dl, dm, etc., we can rewrite Equation 21.004 as CM°N< . . . QM(HrL% . . . in) 21.005 = CM'N' . d £><"> dl L \n-\) c dq d d d „ dn dm dl which shows that the same result would be obtained by successive differentiation of H along each of the arbitrary fixed vectors in turn. 6. We have now succeeded in generating (2n + 1) independent harmonic functions from a single initial function H , all of the same order n. The result of adding these functions with (2n + 1) arbitrary constants is to provide a more generalized harmonic function; we can obviously express a still more general harmonic function K by adding similar groups of higher and lower order as 21.006 2 A ' ™Hrst (n), with corresponding expressions for the alternative forms. Equations 21.004 and 21.005, of the constants. We can extend this result into an infinite series, provided H and the components of the contracting tensors are chosen to make the resulting series convergent. The question then arises whether any harmonic function K can be expressed in terms of another harmonic function H. This is true in the special case where H=\jr and where the coordi- nate system is Cartesian, in which case we shall see that the derivatives of H are solid spherical har- monics; we could reasonably suppose, without formal proof, that it would be true in the more general case when fewer restrictions are applied. 7. If K in Equation 21.006 is to be a Newtonian potential, then we can reasonably expect that the leading, or absolute, undifferentiated term in Equa- tion 21.006 would be of the form (1/r) because this is the simplest form of Newtonian potential. In that case, H would be (1/r). This fact was first noticed by James Clerk Maxwell, 1 who showed that n -differentiations of 1/r, as in Equation 21.005, generated all the nth-degree spherical harmonics. We shall derive this result more simply for a New- 1 Maxwell (1881), A Treatise on Electricity and Magnetism, 2d ed., v. I, 179-214. tonian gravitational potential, and at the same time shall provide a physical interpretation of the con trading tensors in Equation 21.006. 8. We shall find that a convergent series for th< Newtonian potential in the form of Equatioi 21.006 with H= 1/r may not always be possible; w< are led to consider an alternative expansion h homogeneous polynomials in the tensor form 21.007 7 = ££,,, (h)P''PV •An) in which p' is the position vector, whose Cartesiai components are (x,y,z), and B rs t ... is a contracting tensor symmetrical in any two indices and witl constant Cartesian components. We notice that th< covariant derivative of the position vector is givei by the Kronecker delta (§ 1-21), that is. P r . k = % in Cartesian coordinates: and because this resul is a tensor equation, it is true in any coordinates Covariant differentiation of Equation 21.007 thei gives 7a = 2 nBrst . . . (»)8/> s 'p' ■ ■ • p"" = 2 nBkst ■ ■ ■ wP s P l ■ ■ ■ P { "~ u '- and the second derivative is Ju = ^n(n-l)Bu, . . . r; and, in that case, we may add to it term-by-term the similar series repre- senting the contribution to the potential at P which arises from the other particles of the attract- ing body. If r > r for all particles of the body, it is evident that P must lie outside a sphere centered on the origin which just contains all the particles: and we shall accordingly call this the sphere of conver- gence for this case. Otherwise, some particles may set up divergent series which cannot be added term-by-term to the other series. In special cases, the final series might be convergent inside the sphere, which just encloses all the matter, because the elementary divergent series cancels in the sum or is otherwise insignificant, but this would have to be proved by considering the convergence of the final series. In any case, we have not said that the final series is necessarily divergent on or inside the sphere of convergence, but only that it is cer- tainly convergent outside this sphere. Because r is the same for all particles, the total potential V at P, after replacing the gravitational constant G, would then be given by V _M 2,mr cos y G~~r~ r 2 ~S.mr"P„(cos y) 4 . 21.011 in which M is the total mass of the attracting body and the summations are carried out over all particles. The Potential in Maxwell's Form 12. We shall first recast Equation 21.011 in the tensor form involving successive differentials of (1/r), and shall relate the coefficients in this expan- sion to the mass distribution. Later, we shall obtain the more usual expansion in spherical harmonics related to a fixed Cartesian coordinate system. For some purposes, one form is more convenient than the other, and we need both. 13. If we take OQ (fig. 19) as a temporary axis of z and use the well-known formula - d" clz" -)"n\ P„ (cos y), we find from Equation 21.011 that the nth-degree term in the potential arising from a single particle rh - Hobson (1931). The Theory of Spherical and Ellipsoidal Harmonics, 15-16. 156 Mathematical Geodesy {-) n rhf n d" (\ is re! dz" \r Because the unit vector v s toward the particle and in the direction of the temporary z-axis is constant during differentiation at P of (1/r), we can rewrite this last expression in the tensor form (-)"mr» /r re! \rjst; v s v'v" -,(«) (HI which is no longer dependent on a particular co- ordinate system involving a single particle, so that we can sum this expression over all particles. We have also V s = x s /f, if x s are rectangular Cartesian coordinates of the particle m in any fixed system with origin 0, so that the reth-degree term in the total potential at P is finally (-)" Jstu n\ \r/stu . . . (>n where 21.012 I stu ■ ■ ■ <" ) = ^mx s x t x u . . . x in> . This last expression is evidently a tensor because it is formed by the multiplication and addition of of position vectors X s . We shall call this tensor the nth-order inertia tensor because its value depends solely on the mass distribution within the attracting body and on the position within the body of the point O chosen as origin. We can similarly define inertia tensors of the first, second, third, etc., orders as 21.013 21.014 21.015 P = ^mx s /.s' = V fhx s x' Jstu — V mx s x'x l in which, of course, the summation is carried out over all particles. The inertia tensor of zero order is simply the total mass (M) of the body 21.016 ^m = M. With these conventions, we can finally rewrite Equation 21.011 as .K = ft± pt u... M (l (»)• 21.017 It is understood that when re = 0, the inertia tensor is M and (1/r) is undifferentiated; in addition, as usual re! is interpreted as unity when re = 0. The expression ... (H) signifies the nth-covariant derivative of (1/r) suc- cessively with respect to the coordinates x s , x' '. x" . . .. In Cartesian coordinates, the covarian derivatives become, of course, ordinary derivatives and are then combined with the Cartesian form (Equation 21.012) of the inertia tensors. 14. So far, the inertia tensors have been definec only at, and in relation to, a particular origin although we shall later derive expressions for theii components at a different origin where (1/r) and its derivatives would also be different. When the inerth tensors are contracted with other tensors, as ir Equation 21.017, we should use values of the con tracting tensors at the origin, or else express the contracting tensors as sums and products of vectors and use parallel vectors through the origin. We should also use values of the metric tensor at the origin in conjunction with the inertia tensors. Nc difficulty arises if we use Cartesian coordinates because the components of parallel vectors and oi the metric tensor are then the same at all points in space. Continuous Distribution of Matter 15. We can consider that the attracting bodj consists of a continuous distribution of matter o density p per unit volume instead of a system o discrete particles. In that case, we have only t( write 21.018 m = pdv for the mass contained in an element of volume di and replace the summation sign by a volume o: triple integral taken over the whole body, so that for example, we have 21 019 Jrst ...<„)= pp r p s p< . . . p 0l) dv in which p' is the position vector of the element o volume — that is, in Cartesian coordinates p r = x r . The density p can, of course, vary from point t< point, but because it is supposed to have a definite value at a point, the density can be considered ; function of position, that is, of (x, y, z). The densit; need not. however, be a continuous function in the The Potential in Spherical Harmonics 157 mathematical sense because we could integrate over subvolumes bounded by discontinuities and add the results: this result will be clear if we consider the original distribution of an aggregate of particles, which need not have been all of the same mass and indeed could have been separated by empty space. In some cases, we shall find it more convenient to deal with a system of particles, and in other cases with a continuous distribution. There is, however, no essential difference between the two cases, which are quite simply related by use of Equation 21.018 and by integration instead of summation, whether we are dealing with inertia tensors or with any other formulas in this chapter that relate to the mass distribution. Successive Derivatives of (1/r) 16. In order to use the basic Equation 21.017 for the potential, we shall require formulas for the successive derivatives of (1/r) which are intimately connected with the unit position vector v s of the point P where the formula gives the potential (fig. 19). The vector v s is in fact the gradient of the radius vector r, so that we have 21.020 v, = r„. and 21.021 @L~@* giving the first d erivatives of (1/r). 17. Throughout this chapter, (oj, (/>) will be the longitude and latitude of the radius vector OP, that is, the geocentric longitude and latitude, unless otherwise stated. We can accordingly consider v s to be the unit normal to the r-surfaces (spheres) in a symmetrical (a>, <£, r) coordinate system. From Equations 18.12 and 18.13, we then have (cos (f))(o s = k s lr (f>s = t^s/r; and from Equations 12.016 and 2.08, we have v st = (\ s \r + /x s /if)/r 21.022 =(gst-v s vt)lr, so that the second covariant derivative of (1/r) is from Equation 21.021 (l/r) s , = (2/r ! )iw-(l/r> s , 21.023 =(Sv s v t -gst)lT*. The third derivative, using the last two equations, is (1/r ).,,„= (-3lr 4 ){3PsP,-gst)i', l + (1/r 4 ) {ZgsuV, + d>g tu v s — bv s v t v u ) 2 1 .024 = (3/H ) (g s tv u + gtuVs + gusvt ~ 5v s v t v u ) ; proceeding in this manner, we find without diffi- culty that the nth-derivative is given by (-)"(l/r) w ,, sr .,, ( „,r" +1 1 -3-5 . . . (2n-l) VpVqVrVgVt V(n) {gpqVrVsVt ■ ■ ■ V(n)} (2/i-l) , {gliqgrsVt ■ ■ ■ V(n)} (2n-l)(2n-3) 21.025 ... in which the symbol {g pq V r V s V, . . . !>(„>} implies that the indices are permuted cyclically in all different ways, allowing for the symmetry of the metric tensor g, iq , and the results are summed as in Equation 21.024. At each successive term in the expansion, we drop two 'Vs" which we replace by one "g" The final term contains one ' V if n is odd, but otherwise all "g's." 18. Equation 21.025 can also be obtained by suc- cessive covariant differentiation of the identity x 2 + f + z 2 = r\ remembering that all components of the tensors X/>q . . ., Yi>q . . ., Zpq . . . are zero in any coordinate sys- tem because all the components are zero in Car- tesian coordinates. For example, we have xx p + yy„ + zz p = p„ = rv p = - r :! ( 1 jr ) ,, , which is equivalent to Equation 21.021. Also, we have XqXp + jqj,, + ZqZp = g P q = 3r 4 ( l/r) /; ( 1/r),, - r :! ( l/r) M , which is equivalent to Equation 21.023, and = -12rMl/r) /) (l/r),(l/r)., + 3rM(l/r) / „(l/r)., + (l/r)„ s .(l/r) / ,+ (l/r) sp (l/r) g }-r 3 (l/r) w , which is equivalent to Equation 21.024. 19. The number of terms in the symbol {gpqV r V s Vt . . . V( n )} is n(n — l)/2, obtained by taking two indices at a time from «, regardless of order. The same opera- 158 Mathematical Geodes tion, applied to the remaining (n — 2) indices in each term, gives {(/i-2)(n-3)/2} Xn(n-l)/2 as the number of terms in the second symbol {gpqgrsVt ■ ■ ■ V( n )}. But half of these terms are the same as the other half, for example, gpqgrsVt ■ ■ . V(n) = grsgpqVt ■ ■ ■ V(n), so that finally the number of dissimilar terms in the second symbol is n{n-l)(n-2)(n-3)l(2 • 4), and in the third symbol is n("-U(rc-2)(n-3)U-4)U-5)/(2-4-6), and so on. 20. We shall usually contract Equation 21.025 with an nth-order contravariant tensor which is symmetric in any two indices. Each term in a par- ticular braces symbol will make the same contri- bution to the resulting invariant, so that in such cases we can rewrite Equation 21.025 as (-)"(l/rW.,, ( /nr"^ 1 -3-5 . . . (2n-l) = VpVqVrVgVt . . . V( n ) n ( n — 1 ) ■ gpqV r V s Vt 2(2n-l) n(n-l)(n-2)(n-3) 2-4(2/i-l)(2n-3) V(n) ■ gpqgrsVt V(n) 21.026 21. As some verification of this last formula, we take the latitude and longitude of the direction OP as ((f), cj) with respect to rectangular Cartesian axes and take O as origin. Then the Cartesian compo- nents of the unit vector v v are as usual (cos (p cos oj, cos 4> sin oj, sin $), and we have also g pq =l(j> = q) ; g l)Q = 0(p^q). Substitution in Equation 21.026 and use of the usual expansion for P„(x) in powers of x then give us at once vr/333 . . . do dz n \rj r" +1 21.027 This equation recovers our starting point in §21-13 In the same way, we have P„(cos (f> COS (X)) © (') _(-)"«! r «+i 21.028 (7) .(b) dy" !') (-)*n ! 21.029 /'/((cos sin co). Corresponding formulas for mixed derivatives are however, less simple as we shall see. 22. Equation 21.026 is a purely geometrical rela tion. If we multiply by r" and note that rv p = p p , the position vector of P whose distance from the origii is r in the direction v,,, and if we also contract wit! an arbitrary constant tensor A pqr ■ ■ ■ (n) symmetru in any two indices, then the first term on the righ becomes a homogeneous polynomial of the nth degree 21.030 /„(*, y, z)=APi r (n) Pl'PqPr POO The Laplacian of this last equation, as we have already seen in Equation 21.008, is kf^nin-^APi' ai) g IIQ pr again taking the Laplacian, we have k% = n(n-l)(n-2)(n-3) x A pqrst ■ ■ ■ < " ) gpqgrsPt P(H-2i; . P(n-A) so that the right-hand side of Equation 21.026 cai be written as 1 r 2 A 2(2/i-l) r 4 A 2 + 2-4(2n-l)(2n-3) while the left-hand side is f n (x, y, 2) (»)/•' 1-3 . We can also write Apqrst . (n) dlr] pqrst (2n-l) ( " ) /n, a*'ay'aJ(r in which /„ is the same function of the operator d/dx, etc., sls f„(x, y, z) is of the Cartesian coord nates of P in Equation 21.030. The final result is classical theorem of very general application du to Hobson, 3 Hobson, op. cit. supra note 2, 127-129. The Potential in Spherical Harmonics 159 RV 1 -3 . . . {2n-\) J "\dx dy dzj \r h d d d\/\ ■+■ r*A 2 21.031 2(2« — 1) 2-4(2n-l).(2n-3) , . . /»(*, y, z). 23. Hobson's formula is frequently useful as a means of expressing the successive derivatives of (1/r) in spherical harmonics. Suppose, for example, we want to express (l/r)n 2 , then the corresponding polynomial is f 3 (x, y, z) = x 2 y; and we have -r 7 (I 1-3-5 — (r 3 /5) (5 cos 3 sin (o — cos sin o») — (r^/S^cos (/> sin w(5 cos 2 (f> — 1) — 5 cos 3 sin 3 w} = (r 3 /5){i cos (f) sin a>(l — 5 sin 2 (f>) + f cos 3 sin 3ct)} = (r75)M/^(sin4>) sin w + ^/^(sin ) sin 3w} so that finally ^(l/r)n2- 2 P\( sin $) sin ) sin 3w. 21.032 We can be certain that the result must be in terms of harmonic functions. The process is assisted if we first convert powers of the sine or cosine of the longitude into multiple angles. The Potential in Spherical Harmonics 24. If we return to the basic Equation 21.011 and figure 19, we see that the coefficient of (l/r" +1 ), that is, V mf n P n { cos y), is a function of the position of the mass point Q in relation to a temporarily fixed direction OP and of the distribution of mass, and so must be expressible in terms of the inertia tensors for a particular origin. We now seek to express this function alternatively jin terms of spherical harmonics. 25. If the latitude and longitude of the mass point are overbarred and of the point P are unbarred, we have cos y = sin c/> sin + cos cf) cos cos ( w — o> ) 21.033 and by the ordinary addition formula P,,(cos y) =P„(sin <£)P„(sin <£) + 2^ in 7 /*!?'( sin ) P"' ( sin ) 21.034 (n+m)\ X {cos mw cos mw+ sin raw sin moi) We can accordingly rewrite Equation 21.011 to give the potential in the form V Y= 2 2 ^'<"( sm ){C»»> cos mw n = o m=0 21.035 +S„sinmaj}/r« +l in which the term independent of longitude is 21.036 C„oPS(sin0) = C„ o P„(sin<£), provided that C,,u = 1 £mf"P n {sm 4>) (n vi \n — ill i : „ 7 . Cnm = 22smr"-— — -7/T(sin ) sin raw in which m can be any integer between unity and n inclusive, and {n — m)\ is interpreted as unity if m — n. The summation in these expressions is not, however, taken over these values of m as in Equation 21.034, but is taken over all mass points in the attracting body. Accordingly, the C's and S's are constants for a particular body and depend only on the mass distribution. Like the inertia tensors, to which we shall relate them later, the Cs and S's can be calculated if we know or postulate the mass distribution. Conversely, a knowledge of the C's and S's or of the components of the inertia tensors, obtained by observation or measurement, will provide information about the mass distribu- tion, although the C's and S's and the inertia tensors do not determine the mass distribution uniquely. 26. If a is a constant, such as the radius of a sphere centered on the origin and enclosing all the matter, we can multiply Equation 21.035 by a" without affecting the convergence of the series, provided we also divide the C's and S's by a". This 160 Mathematical Geodes device will also ensure the convergence of the series in Equations 21.037, and of the correspond- ing integrals in the case of continuous distributions. The size of the C's and S's will depend largely on m, and to render them more readily comparable, it is usual to adopt normalized functions instead; one such scheme, due to Kaula, 4 is to use the following overbarred coefficients C„o = C„ /(2n+l) 1 / 2 t> n m \ (n + m)l 1/2 (C H Jinn 1 _2(2n + l)(n- /rc)!_ \S n 21.038 m^O, which in effect reduce the coefficients to about the same comparable size as their root mean square values over a sphere. The dimensions of the constants need some consideration. The dimensions of the potential, defined as work done on a particle of unit mass, are L-T-. From the formula Gm/r for the potential, the dimensions of the gravitational constant G are L A M T - and the dimensions of V/G are L~ [ M. Consequently, the dimensions of the C„,„, Sum in Equation 21.035 must be L"M, which is verified by Equations 21.037. If, however, we multiply Equation 21.035 by G and alter the constants accordingly, the dimensions of the constants would be L" + 1 {L 2 T~ 2 ). Relations Between the Constants 27. It will be clear from Equation 21.026 that the nth-degree term in the inertial form of the potential. Equation 21.017, consists of l/r" +1 multiplied by quantities which are independent of r; similarly, so does the nth-degree term of the spherical har- monic form of the potential, Equation 21.035. Both forms of the potential must hold for all values of r outside the sphere of convergence; we may accord- ingly equate the nth-degree terms in the two Equa- tions 21.017 and 21.035. That is not to say, however, that individual terms within the nth-degree are equal; only the sums of these individual terms, com- prising the whole of the nth-degree terms, are the same. We conclude that the C's and S's in Equa- tions 21.037 are expressible in terms of inertia tensors of the same nth-degree. 28. The simplest way of expressing this result is to expand the C's and S's as Cartesian polynomials 4 Kaula (1959), "Statistical and Harmonic Analysis of Gravity,' Journal of Geophysical Research, v. 64, 2410. from the usual formulas 1-3-5 . . . (2n-l) P„(sin ) = . 7 n(n — 1) . , „7 8111 ^~2(2^T) Sin ' -'* n(n-l)(n-2)(n-3) . + 2-4(2n-l)(2n-3) sin ( "- 4) (/> 21.039 P'^sin (j>) (2n)! cos'" 2"n\{n-m)\ sin'"-'"^ (n — m ) (n — m — 1 ) ■Jn-m-Z) •4> 2(2n-l) n — m) (n . — m — 1) (n — m — 2) (n — m —3 2-4(2n-l)(2n-3) X sin' Ui-m-4)J, — 21.040 ... _ m(m — \) ,,„_.,,- . 2 _ , cos ma)= cos'"o> — } cos' -'&> sin- w + 21.041 sin ma> = m cos ( '" _1, d; sin d> m(m-\)(m-2) 1! cos (m-3) o) sin 3 a> + 21.042 and to substitute the relations x = r cos (6 cos oi y= f cos sin ti> z = f sin X- -)-y2 + 2 L> =f 2 . For example, we have C; 13 = 2^mr 3 X (1/6!) '6\ cos 3 4>\ -1 [1] [cos 3 to — 3 cos a) sin 2 a>] 2 3 3! =l 27 V m{x 3 — 3xy 2 ) = 2T(/>"-3/ 122 ). 29. In this way, we find that the complete set of relations for the second and third orders ai c 20 =/ 33 - w '+/--) C„=/' 3 S,,=/ 23 C 22 = i(/"-/ 22 ) 21.043 The Potential in Spherical Harmonics and C 3 o=i 333 -!(/ U3 +/ 223 ) C 31 = / 133 -i(/ m + / 122 ) s,,=/- :i:i -i(/" 2 +/- 22 ) C :! , = {(/" :i -/ 22:! ) 21.044 s,,=^(M U2 -r 122 ). 30. To find more general relations, we rewrite the last two equations of Equations 21.037 in the complex form t> a m i lo a in Z (n + m)l (2n)! 2mf<"- m ) P™(sin 0) (re io >)" 2,rh{x + if)" =<«-»i-2) 2 ( "- 1) n!(n + m)! (n — m) (n — m — 1 ) _., 2(2/i - 1 ) r " Z (n — m)(n — m — l)(n — m — 2)(n — m — 3) 2 -4(2n-l)(2n-3) + 21.045 x f4£(n-m-4). on substituting Equation 21.040. The result is a combination of components of the nth-order inertia tensor which can be written down at once after expanding (x + iy) m . Terms containing r 2 will appear as a Jrspqk . . ■ {n) = J\ \}> is restricted by the Laplace equation to (2n + l); the number of components of the inertia tensor is not so restricted. For example, the Laplace equation in Cartesian coordinates applied to terms of the second degree is (l/r)n + (l/r)te + (l/r)» = 0, and we can write the sum of the corresponding terms in the potential as I n air)u + P 2 air} 22 + P 3 airh3 = (/H-/33) (1/r)]l + (/ 22_/33) (1/r ) 22i Consequently, we shall obtain the same result for the sum of the second-degree terms in the potential if we take as coefficients {I n ) = I n -P 3 (/ 22 ) = /---/ :i:i (/.«) = (/'-) = /'- (/23 ) = / 23 (/13) = J13. We do not reduce the number of components of the inertia tensor by this device, but we do reduce the number of separate terms to the requisite five. The same derivation applies to the nth-order terms where the hi(n — 1) relations (l/r)iu«... ( «)+(l/rWu (llr), oo = are introduced by differentiating the Laplace equation (n — 2) times. As we shall see later, the number of components of the inertia tensor can be reduced by a suitable choice of coordinate axes. J06-962 0-69— 12 162 Mathematical Geodesy but in that case, the number of C's and S's is also reduced. 32. For these reasons, it is not possible to express each component of the inertia tensor explicitly in terms of the C's and S's. We can see from Equa- tions 21.037 that the C's and S's are linear combina- tions of harmonic functions; therefore, any linear combination of these terms must also be a har- monic function. Each component of the inertia tensor is, however, a homogeneous polynomial by definition in Equation 21.012, and not all poly- nomials are harmonic functions. The most we can do is to express certain combinations of the com- ponents of an nth-order inertia tensor, which happen to be harmonic functions, in terms of C n o, C„ m , and S„ m . This procedure will reduce the number of independent relations to (2n + l). 33. For example, the third-degree polynomial (y 3 — 3yz 2 ) is harmonic and can therefore be ex- pressed in terms of solid spherical harmonics as (y 3 — 3yz 2 ) = r 3 (cos 3 4> sin 3 u> — 3 cos sin 2 sin ou) = — r3 {fO^ > 3 ( sm 0) sm 3g> + 2P3 (sin (f>) sin a>}; multiplying this equation by the mass of the particle at (x, y, z) and summing with the aid of Equations 21.037," we have / 222 -3/ 233 = -6S 3 3-3S 31 . Proceeding in this way for the other six basic third- degree harmonic polynomials (xy 2 — xz 2 ) , (yz 2 — yx 2 ), {zx 2 — zy 2 ) , (x 3 — Sxy 2 ) , (z 3 — 3zx 2 ) , xyz, we have for the complete third-order set / 122 -/ 133 = -6C 33 -C 31 / 233 -/ 1,2 = -6S 33 + S 31 / 113 -/ 223 = 4C 23 /in- 3/122 = 24C 33 / 222 -3/ 233 = -6S 33 -3S 31 /333_ 3/U3 = C 3 0-6C 3 2 21.048 / 123 = 2S 32 , agreeing with the reverse set in Equations 21.044. The harmonic polynomials are suggested by Equa- tions 21.044. For example, C 32 = i(/» 3 -/ 223 ) shows that (x 2 z — y 2 z) is harmonic, and we obtain two others by permuting x, y, 2. In the same way, C 33 =^(/ m -3/ 122 ) shows that {x 3 — Zxy 2 ) is harmonic with two others by permutation, and finally S 32 = i/ 123 gives the remaining basic harmonic as xyz. The remaining harmonics in Equations 21.044 are linear combinations of the basic set of seven. For example, C 3 „ = / 333 -f(/ 113 + / 223 ) shows that J-Kxh + fz) is harmonic, but this can be expressed as (z 3 -3zx 2 )+i(zx 2 -zy 2 ). 34 For the sake of completeness, we give the reverse second-order set as 7 11 -/ 22 = 4C 22 /22_/33 = _ C20 _2C 22 / 12 = 2S 22 / 13 = c 21 21.049 / 23 = S 21 . Again, the basic harmonics are suggested by Equa- tions 21.043 as (x 2 — y 2 ) and xy with permutations. Invariance 35. If we define the Newtonian potential at a point as the negative of the work done by the force of attraction in moving a particle from an infinite distance to the point P, then it is clear that the potential depends only on the position off in rela- tion to the attracting body, not on the choice of a particular coordinate system. In other words, the potential must be a scalar invariant. We arrive at the same conclusion if we define the potential as a scalar whose gradient is the resultant force of attraction; the attraction vector at P must also be independent of the coordinate system, although its components will, of course, depend on the co- ordinate system. Again, we can define the potential as a scalar whose Laplacian is zero outside matter and which behaves like (1/r) at great distances from the attracting body; we have seen that the Laplacian is invariant, and if it is required to have a defined (zero) value — independent of the coordi- nate system — at all points in free space, then the original scalar potential must also be an invariant. The Potential in Spherical Harmonics 163 That is not to say, however, that every mathematical expression for the potential is necessarily invariant; we should test the expression for invariance and so ensure that it is a valid representation of the physical definitions. For example, we have added the principle of superposition to the physical definition of the potential which arises from a particle in deriving Equation 21.011, and if this were to result in noninvariance, then the principle could not possibly be true. Some of the mathe- matical processes could also introduce nonin- variance, especially when we work in a particular coordinate system. Accordingly, we shall now test the basic Equation 21.017 for invariance. 36. From the tensor form of the potential in Equation 21.017, we can see at once that each group of terms of the same order is invariant under co- ordinate transformations which do not change the Cartesian origin because, in that case, (1/r) does not change and the inertia tensors remain the same even though their components change. Accordingly, Equation 21.017 is invariant for rotations of the coordinate axes. 37. Next, we consider the effect on the potential at P (fig. 20) of shifting the origin from to On. The Figure 20. position vector of 0o in the old system is p ( 'J in a direction making an angle y with OP, and the mag- nitude 00o of the change is m. Quantities related to the new system are denoted by overbars, for example, 0«/ > = r; an overbarred inertia tensor signifies that its values are to be taken at the point () . We must show that we have 3C (_ \ II I 1 £U_ 7 -...(„,' r /Sill ...(«) 21.050 ~ n\ \r Liu ...(»» for arbitrary values of the vector p/,. 38. As in Equation 21.010, we have F-r{l-2(r () /r)cosy, ) + (r„//) 2 } 1 / 2 4 = -{l+ — P(cosy ( >) r r [ r 21.051 r "I ; provided that r < r. We have also to ensure that P lies outside the new sphere of convergence (fig. 19), centered on O , so that both series in Equation 21.050 may be convergent. Because terms of the same order in either series are invariant for rotations of the coordinate axes, we can take 00o as the old z-axis without any loss of generality. We then follow § 21-13 and rewrite Equations 21.051 as 1 = 1 r r 21.052 d (I dz\r + . m\ dz' in which the derivatives refer to virtual displace- ments of the point P. This expression is similar, apart from signs, to a Taylor expansion for (1/r) along 00o. However, it should not be confused with a Taylor expansion, which would require (1/r) to be defined along 00o and would require values of the derivatives at 0. If the unit vector along the z-axis, 00o, is a p , then we have *£©""" pX and because p^ and cr p are fixed during displace- ments of P, we may similarly write dz 1 ■pM- Fo (m) so that Equation 21.052 may be rewritten in tensor form as 1 = 1 r r 21.053 + m< PM- Pi) (in) in which the mth-order term has m vectors, p#, and m successive covariant derivatives of (1/r). These covariant derivatives still refer to displace- ments of P with O, O fixed. We can accordingly differentiate Equation 21.053 further for displace- 164 Mathematical Geodesy ments of P, with p# a fixed vector, and obtain 1\ (V\ which becomes on exchanging dummy indices ',/"/ slu ...(h) \ r /stu . . .in) m=i m: \r/ ,„,.. . {m)stu . . . («)• 21.054 39. We have now to evaluate the inertia tensors at ( ). If p' , p' are the position vectors from O and On, respectively, to a particle of mass m, then we have p s = p*-pg in which pfi is, as before, the vector OUu. The second- order inertia tensor at 0» is then I st =Yjnp s p< =2/" ( p s ~ p» ) ( p' ~ pi> ) 21.055 =/ s '-pg/'-p^+Mpgp|i in which M is the total mass. If later we contract with a covariant symmetric tensor such as (l/f) s t, then this last expression may be written as 2 1 .056 I st = I st - 2pg/' + Mpiipf,. In the same way, the nth-order inertia tensor is Jstu ...00 = ^ m ( p s _ p * ) ( p f _ p Q ( p u _ p „ ) 21.057 . . . (p^-pk"), and if this last equation is to be contracted with a covariant tensor symmetrical in any two indices, it can be written as JStU . . . ill) = JStU . . . ill) — Jlp*J l " ■ ■ ■ ("- 1 > +i/i(/i-l)p8p£/«- -<"- 2) 21.058 . . . (-)"Mpf,p,',p!i . . . pi," 1 . 40. Multiplication and contraction of the two Equations 21.054 and 21.058 now show that we have /•lu... 00 (T) = js„i...i,i) L\ + _ _ \r/stu . . . in) \r/stu . . . in) 21.059 the remaining terms on the right all contain the arbitrary vector p#. The term containing one vec- tor pg is v / pstu - nl tu • • • ( («) \rjstu dx" Jstu . . . ill) r/stu . . . in) + n jtu... i,,-n (± \T)tu . . . in). 21.060 Pi dx" ■)" where — (V/G)(„) signifies the /?th-term in the ex- pansion of Equation 21.017. If the term in Equation 21.060 is to be invariant for arbitrary p{,\ then the term within brackets in the last expression must be constant or zero, which, in general, is not the case. If. however, we multiply Equation 21.059 by (— )"/n ! and sum from 77 = to /z = °o, then the term containing p^ becomes zero, provided that —{VjG) x is zero as it must be because the series for — {VjG) is convergent. The potential given by Equation 21.017 is then invariant, at least to a first order, although each term or group of terms is not invariant. In the same way, the term in Equation 21.059 contain- ing two vectors is d 2 PoPo dx»dx" 21.061 (-)" V) ( V \ I* (V\ 11 G/in) \Ghn- \G/(„_2)J. . ill) which again becomes zero if, and only if, we mul tiply by ( — )"/«! and sum. The same result is ob tained for terms of higher degree, and we conclud< therefore that the potential as given by Equatioi 21.017 is invariant although each sum of terms o the same order is not invariant under change o origin as is the case for rotations. We may note alsi that the proof depends on summing the complet series; if we omit any numerically significant term the truncated series would not necessarily be in variant. We conclude also that the expression the potential in spherical harmonics is invarian to the same extent because groups of terms of th same degree in (1/r) are equivalent in the tw expressions. The First-Order Inertia Tensor 41. We have seen in Equation 21.016 that tb inertia tensor of zero order is the total (scalar mass (M) of the attracting body, and we shall noA investigate some properties of the higher orde inertia tensors. 42. If jcg are the Cartesian coordinates of th center of mass of the body, then by definition of th The Potential in Spherical Harmonics 165 center of mass, we have from Equation 21.013 2 1 .062 A 7 s = 2 mx s = Mo$, or, in terms of position vectors, 2 1 . 062 B I s = X mp s = Mpl If the origin O of Cartesian coordinates is at the center of mass, then p; s j is a null vector and all components of the first-order inertia tensor are zero. In that case, the first-order terms (n = l), -/ s d/r) s , in the Equation 21.017 for the potential are all zero. Conversely, if all these three terms are absent in the expression for the potential, then all components of p% must be zero because the derivatives of (1/r) are not, in general, zero. In that case, the origin of the coordinate system is at the center of mass. I. 5 If the latitude and longitude of p r , with the direc- tion OP, are (, oj), we can at once expand Equa- tion 21.065 as Ioi' = I — P 1 cos 2 (j) cos 2 (x> — P 2 cos 2 4> sin 2 o> — 7 33 sin 2 (/> — 2/ 12 cos 2 (p sin o» cos cos a> — 27 2:i sin sin o». 21.066 45. The off-diagonal components of the inertia tensor 7 12 = Zjihxy / 13 = 2^mxz 21.067 /23=£myz" are usually known as products of inertia. 8 The Second-Order Inertia Tensor 43. We shall next consider some properties ol the second-order inertia tensor 21.063 P s = ^mx r x s = ^mp r p s , and of the corresponding terms in the potential. We shall also relate this tensor to the moments and products of inertia as usually defined. 44. Returning to figure 19, we note first that the invariant g rs I rs , evaluated in Cartesian coordinates (grs is the same at the origin as at all other points), is 2 1 .064 / = g r jrs =2"'£V.sp r p s = >] fhr- and is also P s v r v s =2/7fp r p s tviA f =^m~r 2 cos 2 y. Therefore, the moment of inertia about the axis OP (unit vector v r ) as usually defined is lor =%jh{QRf=^h\r* sin 2 y = P s (g,- S - v r v s ) 21.065 =I rs (krk s +(Jbrfls) where A.,, p., are any orthogonal unit vectors per- pendicular to v r , and we have used Equation 2.08. It follows that the sum of the moments of inertia about any three mutually orthogonal axes through the origin is I n (k r k x + p,p s ) + / rs (p r p., + V r Vs) + P s (VrVs + Kk s ) = 2 gr J r ° = 27, which is another way of considering the invariant 46. The moment of inertia about an axis depends on the position and direction of the axis. In deriving Equations 21.065 and 21.066, we have in fact assumed that the axis passes through the origin because we have used values of the inertia tensor appropriate to the origin. If we transfer the origin to the center of mass, whose position vector is p£, and use Equation 21.055 with I' = Mp' , we have for the moment of inertia about a parallel axis through 5 There is some confusion in the literature as to the definition of the "inertia tensor." Our second-order inertia tensor is the same as McConnell's inertia tensor (see McConnell (Blackie ed. of 1931, corrected 1936), Applications of the Absolute Differential Calculus, or (Dover ed. of 1957). Applications of Tensor Analysis, 233). On the other hand, what Goldstein calls the inertia tensor (see Goldstein (1950), Classical Mechanics. 149) is equivalent in our notation to (Ig rs -I rs ). which, as we can see from Equation 21.065, gives the moment of inertia about an axis whose unit vector is v r by direct contraction with t'rfs- If 5>r is the angular velocity vector, then the angular momentum vector in our notation is (Ig rs ~I rs )lOr. relative to an origin at the center of mass; the kinetic energy of rotation is i(Ig ,s -I rs )(I) r (I)s. Accordingly, the Goldstein convention suits these dynamical operations slightly better, but the McConnell convention is almost mandatory for our present purposes, particularly in con- nection with the higher order tensors. 6 Goldstein's definition of products of inertia (Goldstein. op. cit. supra note 5, 145) is the negative of ours because of the difference in definition of the inertia tensor. The Goldstein convention is. however, unusual. * 166 Mathematical Geodesy the center of mass I rs {grs—V r V s ) =I rs (g rs —V r V s ) —Mp r ^pl{gr S — V r V s ) 21.068 =I 0P -Md 2 where d is the perpendicular distance of the center of mass from the original axis OP, a result that is well known. Because Md 2 is positive, it follows that the moment of inertia about an axis through the center of mass is less than the moment of inertia about any parallel axis. 47. In much the same way as we investigated the maximum and minimum curvatures of a surface in § 7-14, we now consider the directions of axes about which the moments of inertia are a maximum or a minimum, or at least have stationary values. To obtain these directions, we differentiate Equation 21.065 for a change in the unit vector v r , keeping the origin and therefore / and/ rs fixed. The condition for lop to have a stationary value about the axis v r then is l rs A r Vs = in which A r is a unit vector perpendicular to v s . But if the moment of inertia is to be stationary about the axis v s , regardless of the direction in which we shift v s , then A r must be an arbitrary unit vector perpendicular to v s . We may express A r by means of a single parameter 6 in relation to two fixed vectors A.,-, fAr, both perpendicular to v s , as A r = k r cos 9 + p r sin 9 so that the stationary condition becomes I rs k r Vs cos 9 + I rs p r v s sin 9 = for all values of 9; this condition requires both I rs k, .v s = 21.069 I rs firVs = 0. 48. If the moment of inertia about A, is also to be stationary, we must have also I rs V r ks=0 21.070 /'->As = 0, the first of which is automatically satisfied by the previous condition l rs k r v s = because / rs is sym- metrical. From Equations 21.069 and 21.070, we then have I rs k,p s = I rS V,p s = 0, which show that the moment of inertia about the third axis /x s is also stationary. The three perpen- dicular axes about which the moments of inertia are stationary are known as principal axes of inertia, and the corresponding moments are principal moments of inertia. If the principal axes are taken as rectangular Cartesian coordinate axes, then the condition Equations 21.069 and 21.070 are equiva- lent to stating that the products of inertia are zero, that is, I 12 = 2jmxy= 7 23 =Smyi = 21.071 7 13 = Em.ri = 0. In other words, the matrix I rs has been diagonalized by taking the principal axes as coordinate lines. We know that a symmetric tensor I rs in three dimen- sions can always be diagonalized, and we may there- fore reasonably infer the general existence of principal axes of inertia. There are in fact three principal axes passing through any point. If the z-axis is a principal axis, but the other two coordi- nate axes are not, then we still have from Equations 21.069 21.072 /13 = 7 23 = 0; conversely, if these two equations are satisfied, then the 2-axis is a principal axis of inertia. 49. We have seen in §21-45 that the moment of inertia about an axis through the center of mass is less than about any parallel axis; we shall now consider this question further. We can see from Equation 21.055 that the change in the inertia tensor for a small displacement dr in the direction of a unit vector k s is ^ = -k*I'-\'I\ dr which is zero at the center of mass because all components of/' are zero at that point; therefore, all components of the second-order inertia tensor are stationary at the center of mass. At the center of mass, we have also from Equation 21.055 d 2 I"' which is essentially positive when s = t, making these components a minimum, but which can be negative in certain directions for the nondiagonal components 5 ¥=■ t. Next, we take the principal axes of inertia at the center of mass as coordinate axes A r , B,, C r so that we have I st A s B t = I st A s C, = I s %Ct = at the center of mass. These relations must also The Potential in Spherical Harmonics 167 hold at points near the center of mass because the inertia tensor is stationary and A r , B,, C, are con- stant vectors. We conclude that for small displace- ments from the center of mass, the principal axes of inertia remain parallel to their directions at the center of mass. Moreover, there are now no non- diagonal components at or near the center of mass; we conclude that the remaining three components of the inertia tensor are all a minimum at the center of mass, compared with their values at neighboring points. 50. We shall now express the second-order term in the potential Equation 21.017 in terms of moments of inertia. Using Equation 21.023, we have iI st (llr) st =P t (3vsVt-gst)K2r s ) = -I st (3g st -3v s v t -2g st )l(2r>) 21.073 =(2/-3/„/.)/(2r s ) from Equation 21.065. The OP is the radius vector from the origin to the point P at which the potential is taken, and Iqp is the moment of inertia of the at- tracting body about OP as axis. The same result can be obtained less simply by using the second- degree term in spherical harmonics in Equation 21.035 and by substituting Equations 21.043 and 21.066. 51. Equation 21.073 is a generalization of a formula due to MacCullagh. The equation is usually ob- tained in the special case when the origin is at the center of mass; but it will be clear from our method of derivation, which does not introduce the center of mass, that the same result is true for any origin, provided the moments of inertia are taken with respect to axes passing through that origin. 52. Next, we shall suppose that the z-axis is a principal axis of inertia without requiring the other coordinate axes also to be principal axes, and we shall consider what difference this makes to the second-order term in the potential. We have at once from Equations 21.072 and 21.043 21.074 /•- :! =s,,=o. Expressed in spherical harmonics from Equation 21.035, for example, the second-degree term multi- plied by r' is reduced to C 20 P 2 (sin c/>) + CW^(sin 0) cos 2w 21.075 + S-.P 2 (sin ) sin 2w: we can readily verify this result from Equations 21.066 and 21.043. The Czr and S 2] -terms are simply missing, and this is true for any origin. 53. If all three coordinate axes are principal axes of inertia, then, in addition, we have from Equa- tions 21.071 and 21.043 /'- = 2S-w = 0; the second-degree term in the potential (multiplied by r :i ) further reduces to 21.076 C 20 P 2 (sin (/>) +CwP2 (sin 0) cos 2w. In this case, if A, B, C are the three principal mo- ments of inertia, we have from Equation 21.065 A=I x =I-I n =P 2 +I 33 B = I !I = I-I 22 = P" + I" C = h =7-733 = 711 + /22 21.077 I=i(A + B + C); therefore, from Equations 21.043 we have C 20 = (I-C)-iC =h(A + B)-C 21.078 C 22 =i{I-A-I+B)=UB-A), and the second-degree term can be written as &(A + B)-C}P 2 (sin0) 21.079 +i(fi-^)7 > |(sin^)cos2o>. 54. II the body itself and the distribution of mass in it are symmetrical about the z-axis, then I n = \fhx 2 , which is equivalent to the moment of inertia about the yz-plane, is obviously the same wherever we take the y-axis; the same applies to 7 22 = ^ my 2 . We could interchange the x- and y-axes without effect on 7" and 7 22 , and we conclude that in this symmetrical case p 1 _ 722 If, in addition, the z-axis is a principal axis, then it is evident from Equations 21.077 that A = B. The moment of inertia is the same about any axis in the xy-plane, and any pair of perpendicular axes in the ncy-plane are principal axes of inertia. In this case, we have also from Equations 21.078 C 20 =A-C C 22 = 0, and the second-degree term in the potential (mul- tiplied by r 3 ) reduces further to the single zonal harmonic 21.080 (A-C)P* (sin<*>). 168 Mathematical Geodesy which is the form generally used for the attraction of planets on their satellites when the satellites are distant enough for the planet to be considered ro- tationally symmetrical and for the higher order terms in the potential to be neglected. These as- sumptions are, of course, too drastic in the case of near-Earth satellites and for the general expres- sion of the Earth's gravitational field to the degree of accuracy now attainable. ROTATION OF THE EARTH 55. It can be shown that the rotation of a rigid body is stable about a principal axis of greatest moment of inertia: if the motion is slightly disturbed, the axis of rotation will describe a cone about the principal axis. 7 The same is true of an elastic body, except that the period of disturbed oscillation will be different. 8 In the case of the Earth, the period of this oscillation (the Eulerian free nutation or the Chandler wobble) can be computed theoretically at about 14 months; this oscillation is confirmed by measurements of variation of latitude over long periods. The amplitude, which has similarly been observed, depends on the nature and duration of the disturbance and on damping effects, but does not seem to exceed one- or two-tenths of a second of arc. In addition, there are small annual variations of about the same magnitude caused by shifts of mass resulting, for example, from seasonal weather changes. 9 The conclusion seems to be that the instantaneous axis of rotation coincides with a principal axis (of greatest inertia) to within a few tenths of a second of arc. It may eventually be possible to provide worthwhile corrections for this variation from data provided by the International Polar Motion Service (prior to 1962, known as the International Latitude Service), but meanwhile the effect seems to be negligible for our present pur- poses. We have seen that the whole group of terms of the same degree in the potential is invariant under rotations of the coordinate system; the only effect of such errors in orientation (these errors are, in any case, small) is to change the magnitude of some terms at the expense of others of the same degree. 56. We can easily show that the center of mass 7 Routh (Dover ed. of 1955), The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies, original 6th ed. of 1905, 86-130 (especially § 155, 101-102). 8 Jeffr ys (1959), The Earth; Its Origin, History, and Physical Constitution, 4th ed., 211-229 (especially § 7.04, 216-218). !l For a complete discussion of this entire question, see Munk and MacDonald (1960), The Rotation of the Earth, A Geophysical Discussion. must lie on the axis of rotation of a freely rotating body. If we take the axis of rotation as z-axis, then we have seen in §20-10 that the Cartesian com- ponents of centrifugal force on a particle of mass m at (x, y, z) for uniform angular velocity u> would be (md> 2 x, mary, 0). Because the rotation is free, there is no force acting on the axis to balance any resultant of these centrif- ugal forces, and we must therefore have mx- Vmy- summed over all masses. We find from Equation 21.061 that the center of mass must lie on the axis of rotation. 57. In § 19-13, we considered means of setting up a coordinate system whose z-axis is parallel to the axis of rotation. This can be done, and it is of fundamental importance that it should be done, although all major survey systems are not oriented in this way as yet. Satellite triangulation using stellar photography automatically ensures and preserves such an orientation for a worldwide coordinate system, but we have no geometrical means of setting up a coordinate system whose z-axis coincides with the axis of rotation. If the z-axis is parallel to the axis of rotation but does not coincide with it, then the first-degree terms are not absent in the harmonic series for the potential because the center of mass does not lie on the z-axis and is not therefore the origin of coordinates. We have seen, nevertheless, in § 21-48 that if the z-axis is reasonably close to the axis of rotation, then it is also a principal axis of inertia within the limits of the Chandler wobble and seasonal variations. In accordance with §21-51. therefore the Cm- and S^i-terms must be omitted from the potential series even though first-degree terms are present. 58. If the z-axis coincides with the axis of rotation, then all three first-degree terms and the C 2 r and Sai-terms must be omitted from the potential series. Conversely, if we set up a series in which these terms are omitted, then § 21-41 and § 21-47 together with the dynamical considerations in § 21-54, allow us to assert that the z-axis and the axis of rotation coincide. If we use this form of the potential series in the equations of motion of s satellite, we must ensure that the tracking stations are located in the same coordinate system. If the coordinates of the tracking stations are in the "parallel" system of § 21-57, then we must include origin corrections in the observation equations which, provided the observations are sufficient!) The Potential in Spherical Harmonics 169 widespread, will accordingly determine the position of the center of mass in the tracking system. In short, angular measurements, which must include astronomical observations, will enable us to set up the z-axis parallel to the axis of rotation. Coin- cidence of the two axes, through location of the center of mass, can be assured only by global measures of gravity or potential. 59. We have seen in Equation 20.08 that the total potential or geopotential is obtained by adding -id> 2 (* 2 +y 2 ) to the attraction potential, that is, to MG r to allow for the centrifugal force on points attached to the rotating Earth. In deriving this result, we assumed that we have x — y=Q on the axis of rotation — or, in other words, that the z-axis coincides with the axis of rotation — so that the rotation term in the form — %r 2 (3 cos-7-l)/f 3 = 2^mr 2 P-Acosy)lf :i so that the third term in the series of Equation 21.082 is -KMoP'V/G; proceeding in this manner, we may verify that Equation 21.082 can be rewritten in the form Vp=V + (V s ) p s + HVst)op s p t + • • ■ 21.085 +^(F. s .,... ( „))„pV. . . p l ">+. . .. But this is simply the Taylor expansion of the potential function over a distance r in the direction OP. Equation 21.085 is convergent within the same domain as the equivalent series in Equation 21.082, and we conclude that the potential can be expressed by means of a convergent Taylor series within the sphere of convergence specified in connection with figure 21. Expression in Spherical riarnionics 63. Returning to Equation 21.082 and using the Addition theorem Equation 21.034 for the Legendre functions, we find as before that the potential can be expressed in the form V * n — ^=2] X r "P n ( sm ){[Cnm J cos mco " n = m = 21.086 + [Snm] sin mco} , provided that [£<«>] =2 7^7777 P »(si"& [C ]=2% _ ( f +1) /" " m !| n'(sin<£) cos mco [S,„„] = 22 _ ( f +n 7 I 7,; P;,"(sin0) sin mai in which the summations are carried out over all particles m(tf), to) in the attracting body. To dis- tinguish these coefficients from those obtained in Equations 21.037, we have enclosed the coefficients of Equations 21.086 and 21.087 in brackets. 64. If a is a constant, such as the radius of the sphere of convergence, we can divide Equation 21.086 by a", provided that we multiply the C"s and S's by a". This device ensures the convergence of the series in Equations 21.087 and of the cor- responding integrals in the case of continuous dis- tributions. The coefficients can be normalized in accordance with Equations 21.038. 65. To express the Cs and S's in terms of the coefficients of the Taylor series (Equation 21.085), that is, in terms of successive differentials of the potential at the origin, we use Equation 21.083 and convert the differentials of (1/r) to spherical harmonics as explained in §21-22. For example, using Equation 21.032, we have -(Vu2hlG = ^mair)u2 — 2^4 ] 2 fU s ^ n 4>) sinw — iP|(sin(/>) sin 3to\ 21.088 =3[5 3 i]-90[5 33 ]. The zonal coefficients can be found ai once Irom Equation 21.027 as (F333 . . . (« ))o/G = ^m(l/r ) 33 3 (»i = 2 7^717 ™ P '' (sin ^ 21.089 r ( " + 1) (n + m)! 21.087 = (-)"« ![C,..]. 66. Proceeding in this way, we find that -VolG=[C 00 ] ~(V 1 )olG=-[C n ] - (V 2 ) /G=- [S„] 21.090 - (K 3 )o/C=- [Co] -(F,,)o/C=-[Co]+6[Ca] -(F 22 ) /G=-[Co]-6[C 2 ] -(V: is )olG = 2[C i0 ] -(F 12 )o/G = 6[S 22 ] -(Fi3)o/Cf=3[C] 21.091 -(F M ) /G = 3[&i] The Potential in Spherical Harmonics 171 21.092 -(^ii)o/G=9[C 31 ]-90[C 3 3] -(^ 2 i)o/G = 3[C 31 ]+90[C 33 ] -( V Xil ) /G =- 12 [C:„] -(*m)o/G=3[S 31 ]-90[S 33 ] -(^ 22 )o/G = 9[S 31 ]+90[S 33 ] -(V 332 ) /G = -12[S- n ] -(Fn 3 )o/G=3[C.,o]-30[C 32 ] -(F 223 )o/G=3[C 30 ]+30[C 32 ] -(r 333 ) /G = -6[C 30 ] -(F, 23 )o/G=-30[S 32 ]. Only five of the second-order terms and seven of the third-order terms are independent because of the relations provided by the Laplace equation and its derivatives (Pn)o+(T 22 )o + (^33)0|=0 Tnr)o+(T 22r )o+(JW)o = (r=l,2,3). 67. The reverse formulas can be found by ex- pressing the C's and S's as (necessarily harmonic) homogeneous polynomials and by substituting the homogeneous polynomial as f(x, y, z) in Hobson's formula (Equation 21.031) in which all the La- placian terms will be zero. For example, we have [Cm] = eV 2 ~ Piisin (f> ) cos 2o> = |V- sin cos 2 (j) (cos 2 a> — sin 2 w ) =^f 7 {xH-yH) = -^2™{(l/rln 3 -(l/r>22 3 } = + . T (2n)! 2"-'n!(« + m)! X Ijdkr^+W , , [n — m )(n — m— 1) , ? (n-m) — tl^n-m-2) 2(2/1 — 1) + . . . . After expanding (jt + iy)"\ we can substitute for each polynomial m the corresponding term derived from Hobson's formula (Equation 21.031), that is, (1 ,(F m ...(p) . . /£, 1 -3. . .(2 n _i)^ 1 »---«' Jffl2 ---w> 333 and finally separate the real and imaginary parts. 68. Powers of f cancel in relating the two sets of constants, just as they did in relating the Ps with the C's and S's in the formulas for the potential at distant points. We cannot, however, relate the con- stants in the two formulas for the potential at near and distant points, even if both series were con- vergent over the same region, because the fs ap- pear in different places and vary during summation over the entire mass. A comparison of Equations 21.037 and 21.087, in which the C's and S's have different meanings, will make this statement clear. Invariance 69. It is evident from the tensor form of Equation 21.085 that each group of terms ^(V st . . . ooJbpV • • • p M is invariant under rotation of the coordinate axes, provided the origin does not change. If the origin changes and the point P remains within the new sphere of convergence, then the new Taylor expan- sion from the new origin remains convergent. The values of (V s i . . . („)) at the new origin and the position vector of P will become different tensors so that the group of terms is no longer invariant on change of origin. However, we can show, almost exactly as in the section on invariance of the poten- tial at distant points (§21-35), that the sum of all terms in the new series remains the same, provided the term of infinite order in the original series is 172 Mathematical Geodesy zero, which must be so if the series is convergent. The situation on invariance is accordingly exactly the same as for the potential at distant points. ANALYTIC CONTINUATION 70. We have seen that Equation 21.017 or its spherical harmonic equivalent may be divergent at points inside a sphere centered on the origin, which just encloses all the matter, and so may not represent the potential at most points on the Earth's surface. This would mean that values of the co- efficients Cnm, etc., obtained from observations on artificial satellites — the must convenient and ac- curate method for evaluating at least the lower harmonics — could not properly be used in conjunc- tion with observations on the ground. We might expect to overcome the difficulty by using Equation 21.017 to evaluate all the successive differentials of the potential at a point P outside the sphere of convergence of Equation 21.017 (fig. 22). These Figure 22. successive differentials of the potential are then substituted in the Taylor series (Equation 21.085), which we have seen is the same by groups of terms of the same degree as the series derived from the law of Newtonian attraction. This latter series is convergent within a sphere centered on P which just touches the attracting body atS, and the equiv- alent Taylor series is accordingly convergent within the same sphere PS. The whole process is equivalent to the standard operation of analytic continuation within this sphere. 71. Symbolically, the process leads to the follow- ing expression for the potential at a point 7" (fig. 22), within the sphere of convergence PS, whose posi- tion vector PT relative to P as origin is p p , _^ = y f .LL^'V...(„>(T) G »lTo^6 m! " ! \rJstu...(„) P9 r...(m) 21.093 x P »p«p' . . . p cn). In this formula, inertia tensors are taken with respect to the origin O from which Equation 21.017 for the potential at P was evaluated; all the (m + n) derivatives of (1/r) refer to changes in (l/OP) for virtual displacements of P or for changes in the coordinates off, with O fixed. 72. Equation 21.093 can be written as an infinite matrix M(llr) -/*(l/r), +H st (llr) s t • . . M{llr) pP P -Pair) sp p" + !/*'( l/r),, p p" . . . W0-lr) Pq pPp 9 -y s {llr) spg p p p g +H st (Hr) stp gpPpi . . . 21.094 in which the inertia tensor of zero order is the total mass M. The first row summed represents the po- tential at P. The second row summed is the first derivative of the potential at P contracted with the fixed bounded vector p 1 ', and so on. The fact that the series in Equation 21.093 is convergent implies that the matrix is convergent if the rows are summed first. On the other hand, if we sum the columns first (and this process is not necessarily valid), then it can be shown that the final result would be 21.095 Vt G )" n'. \OT Ltu (») in which the derivatives are now evaluated at T. But this last Equation 21.095 is the same as Equa- tion 21.017 evaluated at T; if Equation 21.095 cor- rectly represents the potential at 7\ then Equation 21.017 must be convergent at T even though T lies inside the sphere of convergence of Equation 21.017 (fig. 19). The convergence of Equation 21.017 at T accordingly depends on whether interchanging the order of summation in Equation 21.093 is valid. The necessary and sufficient conditions for the interchange to be valid do not yet appear to have been established rigorously, but the question may be considered in general terms by taking Equation 21.093 for a point K on PT (fig. 22), which lies out- side the sphere of convergence of Equation 21.017. In that case, Equation 21.017 certainly represents the potential at K, and the summation interchange in Equation 21.093 is valid at K. But the two con- The Potential in Spherical Harmonics 173 tinuation series (Equation 21.093) for K and for T — both of which are convergent — must also have the same properties of absolute and uniform con- vergence because the coefficients of the vectors are the potential at P and derivatives of the potential at P, and are therefore the same for both series. The only difference between the two series is the magnitude, not the direction, of the contracting vector p 1 '; this does not affect the convergence of either series. Accordingly, if the necessary and suf- ficient conditions for the summation interchange depend solely on convergence properties, then these conditions would seem to be satisfied at T as well as at K. However, this demonstration is a long way from a formal proof, and there is another factor which we shall now consider. 73. Proof of convergence of Equation 21.093, obtained in § 21-62, depends on absence of matter within the sphere PS. This proof would not neces- sarily be valid if there is an alternative distribution of matter, which is nearer to P than the actual distribution and gives the same potential and deriv- atives of the potential at P as the actual distribution. According to Kellogg, 1 " there is always such an alternative distribution which could invalidate the whole process of analytical continuation in this case. The question has been considered from a different angle by Moritz, " who concludes that the series is divergent, but his demonstration is also a long way from a formal proof. More research is needed on this controversial question of con- vergence, which cannot yet be considered as definitely settled. In particular, it may be that the series at points on the topographic surface, although divergent, can be truncated at a certain number of terms to give a better answer than a formally convergent series would give for the same number of terms. THE POTENTIAL AT INTERNAL POINTS 74. For some purposes, it is desirable to have formulas for the potential at points inside the Earth, developed from the same geometrical definition of a Newtonian potential, although the physical meaning of the result may be doubted. We have no means of inserting a test particle or of making any measurements at such points; therefore, we have 10 Kellogg (1929), Foundations of Potential Theory, 197. " Moritz (1961), "L'ber die Konvergenz der Kugelfunktion- sentwieklung fur das Aussenraumpotential an der Erdober- flache," Osterreichischen Zeitschrift fur Vermessungswesen, v. 49, no. 1, 1-5. no experimental verification of the law of Newtonian attraction so close to the attracting matter. 75. We draw a sphere, centered on the origin O, which passes through the point P (fig. 23) where we Figure 23. require the potential. There will be matter outside this sphere, where we shall add a subscript E, as well as inside, where we shall add a subscript /. The contribution to the potential at P arising from the internal matter can be expressed as a convergent series in the form of Equation 21.035 in which the summations or integrations in the Cs and S's are carried out over all the internal matter. We write the resulting coefficients as (Cn(l)/, (C,nn)l, (Siiiii)l- The contribution to the potential at P arising from the external matter can be expressed as a con- vergent series in the form of Equation 21.086 in which the summations or integrations in the C's and S's are now carried out over all the external matter. We write the resulting coefficients as |_C„i>J/.;, [ChioJa', LSh/mJ/i in which the brackets indicate that Equations 21.087 are to be used. The potential at P is then V x » — — = V V P'H (sin (f>) {Cnm cos mo) + S n m sin mu>} " 11 = », = where we have 21.096 C = (C nm )il^' +i) + r n [C ],• and a similar formula for S„,„- 76. We can rewrite this last formula as C Hm = (C„„ i ) /+£ /^" +,) -(C„ w ) JS /H" +1) +r''[C„ m ] £ 21.097 174 Mathematical Geodesy in which (C n m)i+E are the result of summing or inte- grating over the entire mass in Equations 21.037. The (C n m)i+E are also the coefficients of a spherical harmonic series which would be obtained by obser- vation on distant points such as artificial satellites. We may consider the remaining terms in Equation 21.097, that is, 21.098 -(Cnm)El^ l ^+r n [Cnm} E , as a correction '- which must be added to the first term, obtained from satellites, in order to give the corresponding coefficient in the potential at the internal point. If we proceed in this manner, we do not need to know or to assume the mass distribution or densities at points inside the sphere passing through P in figure 23. 77. In deriving the preceding formulas, we have assumed that both series for the internal and ex- ternal contributions to the potential at P are con- vergent actually on their respective spheres of convergence, an assumption which is not neces- sarily true. Moreover, the contribution to the potential at P becomes infinite for masses infinitesi- mally close to P, and we have no right to add the corresponding elementary series to the others. We can overcome this difficulty by the usual device of supposing that a thin spherical shell of matter of radii (r+ e) and (r— e) is removed. We can then show that the contribution of the removed matter to (C„w)/^" + " and r"[C n m] for all n is negligible when e is reduced indefinitely. ALTERNATIVE EXPRESSION OF THE EXTERNAL POTENTIAL 78. We have seen that all the harmonics of the external potential can be obtained by repeated differentiation of the primitive (1/r), that is, the potential of degree zero among the resulting har- monics. Another form of the primitive potential is, however, indicated in a formula by Hobson 1:i m/2 21.099 Pg'(sin ) = (-)" d" r I r + z '-' Equivalent formulas for the correction have been obtained in 1966 by A. H. Cook (see Cook (1967) "The Determination of the External Gravity Field of the Earth From Observations of Artificial Satellites," The Geophysical Journal of the Royal Astronomical Society, v. 13, 297-312) and by F. Foster Morrison (Validity of the Expansion for the Potential Near the Surface of the Earth, paper not yet published). The latter paper was read at the 6th Western National Meeting of the American Geo- physical Union, Los Angeles, Calif., September 7, 1966. 13 Hobson, op. cit. supra note 2, 106-107. Note, however, that Hobson's f,'," is ( — y» times our P'», which is the more usual convention. which degenerates to Equation 21.027 for m = and has the advantage that it can generate all the required harmonics by differentiation with respect to z only. We may note that any function of the longitude at is constant under differentiation with respect to z because tan a) = y/x, so that we have P J!'(sin ) /cos moi] f('H 21.100 sin mxo - )" d" — m)\ dz" ' f r— z] "'/Vcos m(o\ r + z\ \sin mw) This last formula shows that the Newtonian attrac- tion potential, which we have seen is expressible as a sum of the Legendre functions on the left (Equation 21.035), is equally well expressed by a sum of the derivatives on the right. The resulting series will converge in the same way as the spherical harmonic series; the two series are in fact equiva- lent term-by-term. 79. At this stage, we introduce the spherical isometric latitude i//, defined by the following ex- pressions which are easily shown to be equivalent e^= cosh i//+ sinh t//= sec + tan = tan (rn- + !(/>) 1 + sin /l + sin (/)^ !/ - cos 4> 21.101 1 — sin

m(ij/-(cu) in which we can change the sign of o» independently of the latitude functions to have also Pll (sin ) fin + l) (-)" d" (n-m)\ dz" -»l(i|i+/aj) 21.104 80. The appearance of the complex variable {ifj+io)) in these equations suggests an analogy with the theory of orthomorphic or conformal map projections, which arises in the following way. The Laplace equation d 2 V s " dx r dx s * ■g rs r* s v k =o The Potential in Spherical Harmonics 175 in spherical polar coordinates can easily be written from formulas in Chapter 18 by substituting (Ri + h)= (R-2 + h) = r. The metric tensor is given by Equations 18.03, etc., the Christoffel symbols are given by Equations 18.34, etc., and the final equation expands to sec 2 (ft d 2 V , 1 d 2 V , d 2 V tan ± §V + 2 dV _ r 2 d

^ (cos (ft^r) +^-0 _d_ d or 21.105 d 2 V | d 2 V di// 2 dor = 0. Equation 21.105 is a well-known equation, satisfied by either coordinate (x, y) of any conformal pro- jection of the sphere or by the complex variable (x + iy). The general solution of Equation 21.105 — that is, all solutions of the Laplace equation not containing r — is well known to be 21.106 V=f(i}j+ia>)+g(ilj-iw) in which /, g are arbitrary functions. Instead of 1/r, we could take 21.107 V = j{f(^ + ie"" 17 = x — iy= r cos (f>e~' w . The following relations are easily verified, r 2 =^ + z 2 21.108 21.109 0+iu e Uo = (£/t7) ,/2 e*=(r+z)/(£ri) 1 l 2 e^""= (r+z)lri = tjl(r — z) e*- i »=(r+z)l£=ril(r-z) iLL — i 2L d£~ 2 r dr - _! £ 3tj 1 21.110 — = dz r d(ih+ico) 1 ,, . , — -i- i = — p-{ili+iw) . dg 2r d(i//-ico)_ 1 ,,,,_„. dt; 2r d(ifj + iu>)_ 1 , _ dr) 2r dr) 2r d(\jj + i(o) 1 . d( (// — i to ) _ 1 dz 21.111 21.112 2— =— -i — d£ dx dy d _ d d dr) dx dy 82. The metric in these coordinates is ds 2 = dx 2 +df + dz 2 = (dx + idy) {dx — idy) + dz 2 21.113 =d^dr}-Tdz 2 so that the only nonzero components of the metric tensor are *» -i gss - 1 ; \g\ r i2 = 2 ; ^=1. All Christoffel symbols are zero, and the Laplacian is accordingly 21.114 A = 4 d 2 d^_ d{dr) + dz 2 which shows that any function of £ only, or of rj only, is harmonic. This property introduces some simplification in the use of Hobson's formula (Equa- 176 Mathematical Geodesy tion 21.031) in these variables because any function of £ or 77 can be treated as a constant on the right- hand side of this equation. A corresponding sim- plification is also introduced into Equations 21.025 and 21.026, which are tensor equations true in these or in any other coordinates, because there are only two nonzero components of the metric tensor. Using Equations 21.111, etc., we can also show that \p, a> and any function of (t/> + ia>) or (1// — i(o) are harmonic, as is clear from Equation 21.106. 83. Using the relations in § 21-81, we can obtain the following formulas 21.115 2 j- (7 e-"^-^ = -^ (± e -<'"-w-->) 21.116 2 JL (- p-mUH-iw))=— (- e -(m.+XM+iu) d£ V J dz\r d (I 21 117 2— (— e~ m( ^~ im A : =~ [— e _(! " ""'' '"" 21 118 2— (- e - m{ '*' +io ' ) ) = — — ( -e "" ""'' " ! dr) \r / dz \r which enable us to differentiate Equations 21.103 and 21.104 with respect to £ and 17, to switch into a higher differential with respect to z, and then to move into a Legendre polynomial of higher degree. For example, we have d_ f Pi?(sin ) ■„ afl r<» +1 > (— )n+l gn+1 A A „-(m-l)(ii-icu) (n-m)!3z" +1 \r and by rewriting Equation 21.103 for the (n + l)th- degree and (m — l)th-order, we have also An+2.) (-) n+1 ^«+l -(m-ll(i|/-!tu) {n-m + 2)\dz" +1 \r therefore, together with three other similarly derived equations, we have finally 2 i_ \pi^r& ein dt; [ r+" +1) (n — m + 2){n — m+1) . +2) r e'"" J) 21.119 3_ f Pg(sin <£) _„„,.,] /^(sin ^) , m+lW d£ I r ( " +1) i ,-(?i+2) 3 [ P',?(sin 0) ^„_ 21.121 2)(n m + 1 ) ^ e " j(m— l)(u a [ P;;'(sin 4>) . 21.122 P;;W(sin 0) & ^n+2) Formulas corresponding to Equations 21.119 and 21.122 have been given by Bateman, 15 and the other two formulas can be obtained from them by chang- ing the signs of a» and y. If we separate real and imaginary parts, it will be found that Equations 21.119 and 21.121 are equivalent as are Equations 21.120 and 21.122 also. By differentiating Equations 21.103 and 21.104 with respect to z, we have in much the same way d fP;,"(sin <$>) , , l v Pg +1 (sin-M) /cos mw r<" +1) \sin mat There are (n — m) axes coincident with the axis of 2, and the remaining waxes are equally spaced in the .vy-plane at intervals of Trim. If A r , B r , C r are as usual the (x, y, z) coordinate axes, then the axes of n"(sin (/)) ,-(« + !> cos mco are (a) Q>C*C T . . . 0"-' n) and (b) (modd) Ai{A k cos (trim) + B k sin (tt I M)} (m — 1 )77 A (m) cos m , n , , . (m— 1)77 + £<'"> sin- — y, or in -) (m even) \A J cos h/?* sin - — [ [ Zm Zm) \A k cosl^ + B* sin ^4 zm zm] /4 (m) cos 21.125 The axes of (2m -l)7r 2m n . . . (2m -1)77 + # >' 1 (n — rn)l (— )«2 m - 1 (m even: sin mco) ~^ ^77^ (_)(m-2)/2 (n-m)\ 21.127 ' 7 Ibid., 132-135. An interesting derivation is also given by -Hilbert and Courant (Interscience ed. of 1953), Methods of Mathematical Physics, original ed. of 1924, v. I, 510-521. The axes of the zonal harmonic P„(sin ) can be seen from Equation 21.027 to be all C'\ and the scalar is ( — )"/"!• 86. The determination of the n poles or axes of a general harmonic of the nth-degree with arbi- trary coefficients C,,m, S mn for the Legendre har- monics is a matter of considerable difficulty, and the authorities seem to be content with proving the existence of a unique solution if all the poles are to be real. 18 The standard method converts the spherical harmonic to a homogeneous polynomial f n (x, y, z), as explained in §21-27 and §21-29. The polynomial, which is, of course, a harmonic function, is then substituted in Equation 21.031 to give - V H r (2n + 1) f4aZ, T7j» Tz)(z) = Mx, y, z)' 1 • 3 . . . (2n-iy"\dx' df dzj\r 21.128 If f(n-2)(x, y, z) is an arbitrary homogeneous polynomial of degree (n — 2), we note that (x 2 + y*+z 2 )f (n - 2) (x,y,z) can be added to the right-hand side of Equation 21.128 without affecting the left side because the resulting additional term on the left would be in the form of /<«- d d d \dx 2 dy 2 dzT -\dx dy dz/\r, which is zero by virtue of the Laplace equation The next step is to factorize 21.129 fn (x, y, z) + (x 2 + y 2 + z 2 )f Ul ., ) (x, y, z) into (atx + biy + ciz)(a 2 x-\- b-ry+c-iz) . . . , in which case the left-hand side of Equation 21.128 can be put into the form d , d d (i\— + bi— + ci— dx dy dz CD' Mi. pq- (n) ' Hobson, op. cit. supra note 2, 135-136. 306-962 0-69— 13 178 Mathematical Geodesy where («i, 61, Ci) are proportional to the Cartesian components of the axis LP, etc. The scalar C is the product of all the moduli (a 2 + b\ + c?) 1 ' 2 . The difficulty lies in factorizing Equation 21.129 when each separate term in the expression has an arbitrary coefficient. 87. It is sometimes better to work in terms of the inertia tensors, which should be easier to break down into vectors. For example, we can see at once from Equation 21.062 that the single axis of the first-degree harmonics is in the direction of the center of mass (distant r from the origin), and the scalar is Mr n . In regard to the second- degree harmonics, we can, without any loss of gen- erality, take the principal axes of inertia as co- ordinate axes (A r , B r , C r ), in which case we have seen in Equations 21.071 and 21.077 that the in- ertia tensor can be written as /"« = (I-A)ApAi+ (I-B)B»B" + (I-OCpC" where A, B, C are the principal moments of in- ertia and I — i(A + 5 + C). The total second-degree potential is then given by U" q {Hr) m =h{{I-A)AvA* 21.130 +(/-fl)5^ + (/-C)C'0}(l/r) / „ / ; this result is unaffected if we add any multiple of AiA«+B"B>i + 0>C» to the inertia tensor because (APA* + Bi>Bi + C»C«) ( \jr) pq = by virtue of the Laplace equation. We can use this fact to eliminate one term from Equation 21.130, but if the remainder is to be split into real factors, the two remaining terms must be opposite in sign. If C > B > A, we accordingly subtract = |(/ - B) (A»A« + B»Bi + C"C") (l/r) pq from Equation 21.130 and have iP'"(\lr) lltl = U(B-A]A"A"-(C-B)aO}(llr) l)(l . 21.131 This last equation factorizes to H(B-A)^Ap+(C-Byi 2 0>}{(B-A) l i 2 Ao ~{C-B)^C}{\lr) liq so that the two axes are parallel to 21.132 (B - A) 1 ' 2 Ap ± (C - BV' 2 Ck , which can be verified from results previously given- such as Equations 21.023. 21.073, and 21.066. This result (Equation 21.132) does not require the principal axes of inertia to be the coordinate axes. The expression for the potential in Equation 21.131 is an invariant which has the same value in any coordinate system having the same origin. If the attracting body is symmetrical about the C- axis, we have seen in §21-53 that A = B, so that both axes coincide with C p . 88. If the mass distribution is known, then all components of all the inertia tensors can be cal- culated, but only in this sense is there any depend- ence between inertia tensors of different order. The only known "recursion" formula connecting the inertia tensors is a differential relation for change of origin obtainable from Equation 21.057, the symmetric form of which is Equation 21.058. We could, nevertheless, use the methods of § 21-87 to find the axes of the higher order tensors. Pro- ceeding as in §21—46, we might look for three preferred orthogonal directions, which would not necessarily be the principal axes of inertia related to the second-order tensor but would contract the tensor to zero as in Equation 21.069. The expres- sion of the potential in terms of these preferred vectors, and of certain components associated with these preferred vectors, corresponding to Equation 21.130, would then contain fewer terms: these terms could be still further reduced by adding multiples of differentials of the Laplace equa- tion, such as (A'A* X' + B r B* X< + CO X') ( 1/r) ,,., = in which X' is an arbitrary vector, until finally the result can be split into linear factors, of which there would be n in the case of the nth-order tensor. We can be assured that such a result exists, if only we can find it, and the result con- taining real axes would be unique. Further re- search is needed on this question, which might also result in more knowledge of the nature and properties of the inertia tensors. 89. An apparent advantage of expressing the potential in the polar form of Equation 21.124 instead of in Legendre harmonics is that we ob- tain expressions of the same form by differen- tiation. For example, the component of the grav- itational force, arising from the potential in Equa- tion 21.124, in the direction of a fixed unit vector A"' is CD'M"N> . . . y . . . (1/r),,,,,.... „....<„+,), 21.133 which is evidently a harmonic of degree (n + 1) The Potential in Spherical Harmonics 179 with the same scalar C and the same axes as the corresponding nth-degree term in the potential plus the additional axis k"'. This facility is, how- ever, mainly of theoretical use as indicating the nature of the harmonics in the gravitational force because of the difficulty in locating the poles of the general harmonics in the potential. Much the same theoretical advantage is obtained by using the inertial form of the potential in Equation 21.017, as we have done in investigating invariance and analytic continuation. For practical purposes, however, we require formulas in Legendre har- monics at least for the first differentials. REPRESENTATION OF GRAVITY 90. Attempts, which have been made to express gravity (g) in Legendre harmonics, have not met with much success because g is not a harmonic function. Like most other functions, g can be expressed over a sphere in surface spherical harmonics of the geocentric latitude and longitude. For that matter. g can be expressed in spherical harmonics of the latitude and longitude of the normal to any surface, as we can see at once if we consider the spherical representation of the surface in § 13-16. For ex- ample, we can express g over an equipotential surface in terms of spherical harmonics of the astronomical latitude and longitude. We cannot express g as a series of solid harmonic functions of any sort. However, we can express the component of the gravitational force in any fixed direction, such as a Cartesian coordinate axis, in solid har- monics; and if we do so in three fixed directions, we shall have expressions which give us the direction as well as the magnitude of the gravitational force. 91. Addition and subtraction of Equations 21.119 and 21.120 and use of Equations 21.112, followed by separation into real and imaginary parts, give d \ P,"'(sin ) ( cos mco dy Pi," ( sin (f) ) / cos mco sin mco dx r ( n + i ) (n sin mco m + 2)(n-m+ 1 )P» 1 ^ ( sin cf>) cos (m — l)a) sin {m — \)co P»' + V(sin(/)) /cos (m+l)co r ( " + 2) \ sin (m+l)co 21.134 (n-m + 2) (n-m+ 1 )P»»-f ( sin sin (m — \)co cos (m — I) co P'n+x ( sin ) are the astronomical longitude and latitude, we have 21.135 we can write finally g cos (ft cos co ^ ^ "v ^'ViUin » = (» m=o X (C{n+ i). in cos mco 21.136 +S(n + \),m sin mco) where C(„ + i).fi = 2 n{n +l)C,n C(H + n, i = — C„o + \ n (n — 1 )C, r > S(n+i), i=in(n — l)S /)2 G(n+1), m 2 Gn, (m—l) + 2 (n — m + 1) (n — m)C„,( m+ i) S(n+1), m ~~ 2 ■J», (m-i) + | {n — m + l)(n — m)Sn,(in + \) (m = 2,3, . . . (n-1 G(n + i ), „ : S(n + 1 ), « " G(i, + i), (h + i) : S(n + 1), (» + l) : 21.137 2 Gn, (w— l) : 2 ill, («-l) 2 ^ n , n ■ — i S 2 "J II . II • 180 Mathematical Geodesy In deriving this result, the Legendre functions on the right of Equations 21.134 for m = contain (n + l)(n + 2)P;U(sin), which must be transformed to -/>,', + , (sin (ft) in accordance with the usual formula. Alternatively, we could obtain the term m — in Equations 21.134 by direct differentiation as d_ / P„(sin (ft) \_ ? ^» + i(sin (ft) cos w dx \ r ( " + 1 » / r ( " + 2) 92. In the same wav, we have g cos (ft sin qj ^ ^ (S, 1 P'^i (sin 0) X(C(n+l), m COS fflft) 21.138 where 'J;/,(lli-l) + i(ra — m+l)(« — m)S„, (m+1 , 0( n + 1 ) , m zd n, ( hi — l ) — Un — m+1) (n — m)C„,(m+i) (i»=2,3, . . . (n-1)) m« + i), « —~$Sn, (n-i) •3(ll+l), II ~ 2"C(»-1) C(ll+1), («+l)— 2S/1, „ 5( ii + 1 » , < h + 1 1 = jC,i , „ . 21.139 93. Derived in the same way from Equation 21.123, the third component of the gravitational force is d ( V\_g sin (ft Bz\ G G * » + ' P»' + ,(sin + S lim sin mxo) 11 = m = <> can be found. Rotating Field 94. If the field is rotating about the z-axis with angular velocity d) and if ((ft, o>) are to refer to the direction of the total gravitational force, then instead of (-V/G) in Equations 21.135 and 21.140, we should have -VIG+'2U-(x- + y-)IG. To the right-hand side of Equation 21.136 we should add 21.142 ,<«+n in Equations 21.137, because these har- monics can be derived from the general formula for C(«+i), ml we must remember that the order of a harmonic cannot exceed its degree so that the term containing C„, (m+i) must be omitted if (m + 1 ) exceeds n. In any case, the coefficient t(n — m+ l)(n — m) becomes zero for m = n or m = n+ 1. We shall find that similar con- siderations apply to the second differentials. 96. The second Cartesian differentials of the potential are, of course, harmonic functions which can be written, for example, in the form a 2 (V\ ^ '!£* P|;' +2 (sin » cos mat + o(/,+2), »i sin mw). OX \W H _ ffl=0 T If this result is obtained by differentiating Equation 21.136, then Equations 21.137 tell us that t>(H+L'i, m = 2M«+i >, (»i-i)"r 2(ti tn + 2) (n Tn~r l)Li(n+i), (m+\)i and substituting Equations 21.137 for the coefficients of the first differential with respect tm, we have for m > 2, €(„+■>), m = \C„, (»i-2) — i(n — m + 2)(n —m+ 1)C„,„ + Un -m + 2)(n-m+l){ —%C,im + \(n - m) (n — m — 1 )C„, (m+i)} = \C n , (»t-2) — i(/i — m+ 1) (ra — /n+ 2)C„„, 2 1 . 144 + i(n - m— 1) (n — m) {n— m+ 1) (n — m+ 2)C„, (MI+2 ). If m were 1 or 2, we should have substituted instead the zonal or first-order harmonics given earlier in Equa- tions 21.137. A complete list of all the harmonics in all six second differentials are given as follows: i)x- \ G CiH+t), o = - Un + 1) {n + 2 )C,„> -f i(n — l)n(n + 1) (n + 2 )C H2 C(„ +2) . i=-|n(n + DC„,+i(n -2) (n - 1 )n(n + l)C„ :i §(*+»). i = — in(n + 1 JS„, + i(n - 2) (n - 1 )n(n + 1 )S, a Cu,+2), -2 = hCno-kn (n- 1 )C„ 2 + }(/i -3) (n-2) (n- 1 )nC,n S(„+2), 2 = — i»(n— 1 )S„2 + t(» — 3) (n— 2) (n — 1 )nS„ 4 C , ( M+ 2), m = ^C« > o«-2) — i(ft — w+l)(n— 7n+2)C» w +i(7i— m— l)(n — m)(n— m + l){n — m + 2)C„, (,„+•>) S(«+2), m=4S„,(,„-2i — 2('? — m + l)(« — m+2)S„ m +|(n — m — l)(n—m)(^ — m + l)(/? — m + 2)S„. (,„+.>) 21.145 (m>2) ay- V 6' C(» +a) , o = -i(w + 1) (n + 2)C„„ - 2-U* - 1 )«(n + 1) (n+ 2)C H2 C(„ +2 ), i=-in(n+l)C„i-i(n-2)(n-l)«(n+l)C H:i S( H+ 2), i = -in(n + l)S, n ~i(n - 2) {n -l)n\n + l)S„ :i C (/(+ 2), 2 =-iC„ -in(n - 1 )C„-> - £(/i -3) (ra-2) (n- 1 )raC, M S(»+2), 2=— \n{n — 1 )Sn2 — !(rc — 3) (n — 2) (n — 1 )nS„.j C(,, + i).ni = — \C„, („,-■>)— h{n—m-\-\)(n — m-\-2)C, im —\(n — m — \ )(n— m)(n— m+1 ){n — m+2)C„,o»+2> S(»+2), w=— iS H ,(m-2) — 2(" — m + l)(/i— m + 2)S„,„— 7) 21.146 (m>2) 182 dz- Mathematical Geodesy 21.147 C( W+ 2),o= (« + 1)(" + 2)G„ ( > C(„+-i), i — /i(77 + l)C„i S(„+2), i = 77(7! + l)S /n C'( / , + 2),2 = n(n —l)C, r > S(„+-2),-> = n(n — 1)S„ 2 C( W+ 2),»i= (« — m + l)(n — m + 2)C„, S( H+ 2), »/= (/i — m+ 1) (n — m + 2)5,,^ (m>2) -^— (-^): C 0(+ 2),o = i(Ai-l)n(n+l)(ra + 2)S H 2 d%dy \ G/ C ( „ + 2), , =-ire(n + 1 )S„, +i(re - 2) (n- 1 )n(n + 1 )S„ :! S(„+a), i =-i«(n + 1 )G„, -\{n-2) (n - 1 )n(n + 1 )C„ :i C( H+ 2), 2 = {(/J — 3) (n — 2) (n — 1 )«S„ 4 S( M+ 2), 2 = iCo — i(« — 3) (ti — 2) (n — l)nC«j C(« + 2), m= — iS„, ( M ,-2) + i(n — 772 — 1 ) (n — m) (/i — m + 1 ) (n — m + 2)S„, {m+z) S(,i+2), m — iC„, (»,_2) — \(n —m — l)(n — m)(n — m+l)(n — m + 2 ]C„, ( , H+L >, 21.148 (77! > 2) dvd~ \ _ r) : C( H+ 2).o =— i«(n+ l)(7i + 2)S„i G(„+2), i= — 2(7? — 1)77(77 + 1)S„2 S(« +S ), , = (n + 1 )G,„> + 2-(/i - 1 Mu + 1 )C»2 C(«+2), 2 = — 2""S„i — i(n — 2) (n — DtjShu S(H+2), 2 =277C„i +?(7?— 2) (7! — 1 )nC„:i C( H+ 2) lW = — 1(77 — 777 + 2 ]S„, ( H( -i) — z(« — 77i) (7; — 777+ 1)(« — 771 + 2)S„, ( HH -i) 5(h+2), Hl=i(77 — 777 + 2)C„, (,„-|) + i(n — 777) (ll ~ 777 + 1 ) (77 — 777 + 2 )C„. („,+ 1 ) 21.149 {m>2 ) The Potential in Spherical Harmonics 183 d 2 ( V dzdx \ G C( W+ 2), o =—ln(n + 1 ) (/i + 2)C„, C(»+2), i = (n+ l)C„(i — 5(n — 1 )h(h + 1)C„> S(„+2), i =— i(n— 1 )«(« + l)S„-j C( H+2 ), 2 = i/iC , „i — A(« — 2){n — l)nC„3 S(h+2), 2=inSwi — i(u —2) (n— l)nS„ 3 C(h+2), m = l(« — "' + 2 )C n ,(m- 1) — i(n — m) (n — m + 1 )(n — m + 2)C», (Wl+l) Soi+2), m=2(n — m +2)S h ,(hj-i)— i(/i — m)(n — m + l)(n — m + 2)S Hl ( H ,+i) 21.150 (m>2). The Laplace equation r).V-U;/ rV\67 r)z-V67 is satisfied by each harmonic of the same degree and order; also, the mixed derivatives d 2 /clxdy or 3 2 /dydx give the same result, although the first differential is not the same in both cases. 97. Second differentials of the geopotential W are given by d 2 (W\ d 2 (V O) dx 2 \G) Bx 2 \G) C dy 2 \G) dy 2 \Gj G from Equation 20.08. There is no difference between the other second differentials of W and V. 98. We have finally the six Cartesian components of the Marussi tensor JF ;> , which can be contracted with the base vectors A.'', fx r , v r of the equipotential surfaces to give us the six curvature parameters of the field, as in Equations 12.162. The Cartesian components of the base vectors are given by Equations 12.008 in which are the latitude and longitude of the line of force, obtainable together with gravity g from Equations 21.136, 21.138, and 21.140. To avoid confusion with the geocentric latitude and longitude in the spherical harmonics, we shall overbar the latitude and longitude of the line of force — that is, the astronomical latitude and longitude — from Equations 12.008. For example, the median curvature k> is given by §L(E d\-\ G ■ u , - 3 2 / sin-m cos- a) : d.X- \ ■21 • , - ^ ( W sin-0 sin- w— 77 /~»r»c 2 2 2 d2 ( W \ co *+J?\g) , . ,7 . - - d 2 (W\ — z sin* 1 ® sin to cos 10 — v Bxdy \ G J + 9c" A 1 ■ ~ d ' Z ( W \ + z sin0 cos cpsin o> — — — aydz \G / 21.152 +2 sincj) cos $cos oj dzdx \G 184 Mathematical Geodesy with similar equations which can be written at once for the other parameters. DETERMINATION OF THE POTENTIAL IN SPHERICAL HARMONICS 99. The advantage of using Equations 21.136, 21.138,_and 21.140 for g cos (f> cos o>, g cos cf> sin w, g sin (/>, compared with expressions for g, is that these equations are spherical harmonic series in the usual form and are linear in the C n m, S nm of the potential. Accordingly, we may use these three equations as linear observation equations to determine the C nm -, S nm to the limit of computer capacity from sufficient and widespread measure- ments of g, (f>, u>; time will be saved in the com- putation of the coefficients of the Cnm, S„ m from the positions of the observing stations. In these equa- tion, (f> and o) are astronomical latitude and longi- tude, but it would not usually be necessary to make astronomical measurements at every gravity station in an intensive local survey; it would be sufficient to apply regional deflections to geodetic values. The lower harmonics could not be deter- mined in this way from regional surveys, but are already well determined from satellites. The satellite values should be substituted in the equa- tions, leaving the higher harmonics to be determined from regional surveys. 100. The same considerations apply to Equation 21.152 and to similar expressions for the other curvature parameters. These equations are also linear in the coefficients Cnm, S„ m of the potential and could be used as observation equations in conjunction with Equations 21.136, 21.138, and 21.140. The curvature parameters, other than the vertical gradient of gravity, can already be meas- ured to a high degree of accuracy and might be of value in the determination of the higher har- monics in local or regional surveys. This has not yet been done, and further research is required to explore the practical possibilities. MAGNETIC ANALOGY 101. If we take a small magnet QQ' (fig. 24) of pole strength p situated inside the Earth, then in accordance with the usual geophysical convention, the negative pole will be at Q nearest the north and the positive direction of the axis (unit vector L s ) will be QQ' . The magnetic potential at an external point P, writing fjb = p XQQ' for the mag- Figure 24. nitude of the magnetic moment, will be QQ' V r ' l ^)s LS 21.153 in the limit when QQ' — > 0, and the magnet becomes a dipole. In this expression, c is a constant depend- ing on the units employed. The differentiation of (1//) refers to displacement of Q relative to a fixed origin at P so that the gradient of r is in the direction PQ. The magnetic potential at P can then be written as CfJL T cp -7 r s L s = f cos y. 102. In deriving this formula, we have assumed unit permeability of the medium between the dipole and P. We are not proposing to determine the actual external magnetic field of a dipole buried in the Earth; all we want to do is to set up a mathematical model analogous to the gravitational field, and in doing so we can make any stated assumptions, such as a completely permeable medium. The reason for this assumption is that we shall later super- impose the fields of dipoles in different locations, and the analogy would break down if the per- meability changed. 103. Instead of the dipole, we shall now suppose that we have a particle of mass m at Q. The gravita- tional potential at P will be {—Gm/r), and the component of force at P in a direction parallel to QQ' will be -{-Gmlr) s L s in which the gradient of r is now in the direction QP because the differentiation must be carried out by displacement of P relative to a fixed origin at Q. The component of force at P parallel to QQ' is accordingly Gm T — — i"sL : r- Gm cos y, The Potential in Spherical Harmonics 185 which is exactly the same as the potential of the dipole if we make cp = Gm. Subject to this relation and to the assumption of unit permeability, the component of the gravitational force in a given direction is the same as the potential of a dipole situated at the mass point and oriented in the same direction. 104. If we set up dipoles at all other mass points with the same proportion of mass to magnitude of magnetic moment and with the same orientation, the total magnetic potential at P will be the same as the component in the same direction of the total gravitational force exerted by the whole body at P. Moreover, the same conclusion will evidently apply to a continuous dipole distribution and to a continuous mass distribution, provided the dipoles are oriented in the same direction. Finally, we could set up at each mass point a cluster of three dipoles of equal moment, oriented in the direction of the coordinate axes, and so obtain all three components of the gravitational force. The scalar magnetic potential of such an arrangement could not, however, represent the vector gravita- tional force field, and there would be no physical correspondence. 105. Nevertheless, the correspondence between magnetic potential and the component of gravita- tional force in a fixed direction of magnetization is established, and a similar correspondence clearly exists between successive differentials of these scalar quantities in fixed directions — such as the coordinate axes. Thus, components of magnetic force correspond generally to second differentials of the gravitational potential so that magnetometer and torsion balance measurements correspond. This is not to say that the actual magnetic field of the Earth can be used to derive the gravitational field, or vice versa, but merely that methods applied to the one field can often be applied to the other. Torsion balance interpretation formulas are, for example, used in the calculation of magnetic anomalies. 1 " We might also expect that frequencies in harmonic analysis of the magnetic field would generally be one higher than the harmonics of the gravitational field in relation to the noise level, although the amplitudes might differ widely. Multipole Representation 106. We note that the magnetic potential in Equation 21.153 of a dipole situated at the origin ,!l Heiland (1940), Geophysical Exploration, 393. is proportional to the first-degree terms in the polar form of the gravitational potential (Equation 21.124). We shall now show that the higher degree terms can be represented by multipoles at the origin. We reverse the direction of the axis of the original dipole at Q in figure 24, transfer it to figure 25, and add another dipole at R in the direction of the unit vector M'. This second dipole has the same mag- netic moment in magnitude (jx) and direction (L") as the original dipole at Q. The magnetic potential at P, arising from the whole arrangement, is then - cfx( 1 IrjJL s + c/x( II r )sL s . We now define a quantity v — pXQR in much the same way as we have defined p = pXQQ\ and suppose that v remains finite (because p. in- creases) when QR is decreased indefinitely. The limiting arrangement is known as a quadrupole Figure 25. of moment v, and its potential at p is T *{GK l s =■ M-l L°M> If L\ M' are the axes and (cv) is the scalar of the second harmonics of the gravitational potential (-V/G), then the latter is represented by the po- tential of the quadrupole. In the same way. we can set up another quadrupole at a point S in the di- rection QS = N". The limiting potential of this arrangement, when QS decreases indefinitely while v X QS remains finite, will be proportional to (l/r),,„LW/V", that is, to the third harmonics of the gravitational field if L\ M', N" are chosen as the axes of these harmonics, and so on. 107. The multipole analogy is mostly of theoret- ical use for indicating possible applications of electromagnetic methods in the gravitational field, and vice versa. For example. Maxwell introduced 186 Mathematical Geodesy 1 it has suggested ional field in the ine found his theory of use, not only in itself, but also 21.017. his theory of poles in connection with electro- because it has suggested representation of the magnetic problems, but we have, nevertheless. gravitational field in the inertial form of Equation CHAPTER 22 The Potential in Spheroidal Harmonics THE COORDINATE SYSTEM 1. In this chapter, we shall first develop a special coordinate system; the /V-surfaces of this system are all oblate spheroids formed by rotating a family of confocal ellipses about their common minor axis, which we shall choose as the Cartesian C r -axis. Later, we shall obtain by standard methods a general solution in these coordinates of the Laplace equation which can represent a general attraction potential, and we shall investigate the corresponding mass distribution. 2. There are currently two main gravimetric uses of this spheroidal coordinate system: The expression of the potential in spheroidal coordinates has less restrictive properties of formal convergence than the corresponding expression in spherical harmonics, and leads also to an exact formulation of the standard gravitational field to be considered in Chapter 23. The coordinate system itself and the properties of the meridian ellipse on which it is based, nevertheless, have other uses, and the system will be considered in more detail than is necessary for the immediate gravimetric purposes. THE MERIDIAN ELLIPSE 3. Any meridian plane containing the axis of rotation cuts each spheroid of the family in an ellipse as shown in figure 26. We begin by collecting, without proof, some well-known properties of this ellipse. We denote the equatorial radius or semi- major axis CA — CA' by a, and the polar radius or semiminor axis CP = -CP' by b. The two foci S, S' are located on the major axis at CS — CS' — ae where e is the eccentricity of the ellipse. If is any point on the ellipse, then we have 22.01 SO + S'0 = 2a so that S'P = SP — a. It is usual in classical geodesy to define a complementary eccentricity as b/a or (1 — e 2 ) 1/2 , and yet another eccentricity as ae/b. Instead, we shall introduce the auxiliary angle a = S'PC, in which case the three eccentricities become sin a, cos a, and tan a, respectively. 4. A circle on A' A as diameter is known as the auxiliary circle. We can consider the ellipse as formed from this circle by shortening all ordinates parallel to the minor axis in the ratio b/a so that we have 22.02 ON/0'N=bla = cosa. The tangents to the ellipse and to the auxiliary circle meet on the major axis at T. The angle OCT is known as the reduced latitude u. 5. The normal OGH to the ellipse at O makes an angle (f> with the major axis, which is evidently the latitude of the spheroidal normal as usually defined throughout this book. We shall call this latitude the spheroidal latitude and shall use the same symbol for it as we used in Chapters 12 and 21 187 188 Mathematical Geodesy Figure 26. for different but similar quantities. The context serves to avoid confusion. The spheroidal longitude is, in the same way, denoted by o» and is the angle between the meridian plane of figure 26 and the meridian plane through an origin such as Green- wich, with the same conventions as in § 12-7. We have not yet identified the spheroid with an equi- potential surface, in which case the spheroidal latitude would be also the latitude of the line of force at points on the spheroid: or, with the base surface of the coordinate system of § 18-23 used for the description of geodetic positions, in which case the spheroidal latitude would be the geodetic latitude of points in space. For the present, we are dealing with the spheroid solely in its ordinary mathematical sense as an ellipsoid of revolution. 6. It can be shown that the normal bisects the angle SOS' , and the half-angle (3 is given by 22.03 or sin (3 = sin a sin (j> tan /3 = tan a sin u. From the fact that the tangents at O, 0' intersect at T on the major axis, we infer that 22.04 tan u = cos a tan $, leading to other formulas connecting the reduced and spheroidal latitudes as follows, sin u — cos a sec (3 sin 4> 22.05 = cos a sin (f>l (I — sin' a sin- (j)) 11 ' 1 cos u = sec j8 cos 22.06 = cos 0/(1- sin- a sin- (fr) 1 ' 1 22.07 (1 — sin 2 a sin 2 )(l — sin-'a cos 2 u) = cos 2 a 22.08 (1 — sin 2 a cos 2 u) cos a sec /3. 7. In this chapter, we shall denote the principal radii of curvature of the spheroid by p (the radius of curvature of the plane elliptic meridian) and v (the principal radius of curvature perpendicular to the meridian). We found in §12-49 that the radius of curvature of the parallel of latitude is — 1 / ( A: i sec ), which in this case is v cos (/> and is The Potential in Spheroidal Harmonics 189 evidently OM in figure 26, for any surface of revolution. Consequently, we have v = OH in figure 26 and 22.09 ^cos = a sec (3 = a sec a/(l + tan 2 a cos 2 ) 1/2 = a sec a{\ — sin 2 a cos 2 «) 1/2 22.10 =a-l{a- cos 2 + b 1 sin 2 (/>) l/2 . If dm is an element of length of the meridian ellipse, then by projecting small corresponding arcs of the auxiliary circle (a, du) and of the ellipse [dm) on the major axis, we have sin cf) dm — p sin $ d deb p — a cos 2 a sec' 5 /3 = a sec a/(l + tan 2 a cos 2 ) :,/2 = a cos 2 al{l — sin 2 a sin 2 <£) :!/2 = a sec a{\ — sin 2 a cos 2 u) 3/2 , together with the following differential relations which are often useful, 22.13 dfi/dcb — sin a cos u 22.14 d/3/du — tan a cos 2 /3 cos u 22.15 d(\n p)/d(b — 3 sin a tan a sin u cos u 22.16 d(v cos 4>)/d({) — — p sin 22.17 cfti; sin (f>)ld<^ — p sec 2 a cos 22.18 dv/d(f>= {v-p) tan 0. The last equation is equivalent to the sole Codazzi equation of the spheroid as derived in Equation 18.22. 8. The principal curvatures k\, k> are —1/v and — 1/p so that the curvature invariants are 22.19 K= ll(pv) 2H = -(Hp+llv). 9. In the case of the Kepler ellipse used in or- bital geometry, the origin of rectangular coordi- nates (<7i, q>) is taken as a focus S (fig. 26), and the angle TSO is known as the true anomaly f. The reduced latitude u is known as the eccentric anom- aly E. By relating rectangular coordinates in the two systems — origin S and origin C — we have at once q x = OS sin/= b sin E = a{\ — e 2 ) 1/2 sin E q> — OS cosf— (a cos E — ae) — a (cos E — e); 22.20 by squaring and adding, we have r=OS = a(l-ecosE). From the last two equations, we have (1-e 2 ) [1 + e cosf) so that 22.21 r=OS = a(l-ecos£) = (1 — e cos E) a(l-e-) 1 + ecos/V These equations are sometimes useful in branches of geodesy other than satellite geodesy. 10. The three-dimensional Cartesian coordinates of — considered as a point on the meridian ellipse whose longitude is w — with respect to the usual axes are x = CN cos (o — a cos u cos oj — (ae ) cosec a cos u cos at y= CN sin to — a cos u sin w — (ae) cosec a cos u sin a> z = ON=(b/a)0'N=b sin u = (ae) cot a sin u, 22.22 from which we may obtain the radius vector CO as r=(ae)(cos 2 u + cot 2 a) ,/2 = (ae )(cos u + i cot a ) 1/2 (cos u — i cot a ) ,/2 . 22.23 The tangent of the geocentric latitude OCS is given by 22.24 U 2 + y 2 ) 1/2 = cos a tan u. SPHEROIDAL COORDINATES 11. It is evident that a is a constant over the one spheroid we have been considering, and would be 190 Mathematical Geodesy a different constant over any other spheroid. We now consider a family of confocal spheroids for which CS — CS' — ae (fig. 26) is the same for all, so that (ae) is an absolute constant in space. It is then clear from Equations 22.22 that, instead of (x, y, z), we could equally well take (a>, u, a) as space coordinates, in which case the confocal spheroids will be the constant a coordinate surfaces. The other two space coordinates, which as usual we shall also use as surface coordinates on the spheroids, will be the longitude and reduced latitude with reference to the particular spheroid passing through the point in space under consideration. 12. By differentiating Equations 22.22, we find after some manipulation that the metric of the space in the coordinates (w, «, a) is ds 2 — dx 2 + dy 2 + dz 2 — (a- cos- u)d(i)-+ (v 2 cos- a)du 2 22.25 +(v- cot- a) da 2 : the only nonzero components of the associated metric tensor are normal, we have g u — 1/ (a 2 cos 2 u) ; l/{v 2 cos 2 a) ; 22.26 g 33 = 1/(V 2 cot 2 a). The coordinate system is accordingly triply orthogo- nal, and the surface coordinates (w, a) are constant along the spheroidal normals. Consequently, the coordinate system is a normal system with a spheroidal base, and all formulas of Chapter 15 apply with N— a and with the spheroidal latitude (/> converted to the reduced latitude u by means of the formulas given in the last section. We can, however, retain the spheroidal latitude and the principal radii of curvature, etc., as functions, which are defined in relation to the spheroid passing through a point in space, as long as we remember that they are now functions of the two variables (u, a). 13. We shall require the differentials of some of the spheroidal quantities — in particular a and v — along the normals before we can substitute in the formulas of Chapter 15. The basic gradient Equa- tion 15.01 is now 22.27 a r — nv r where n has its usual geometric significance and v r is the unit outward-drawn normal to the spheroids, not to be confused with the gradient, which we shall not require, of the radius of curvature v. If ds is an element of length along the outward-drawn da = nds, while differentiations along the normal and with respect to a are related by d_ = l 3_. da n ds 22.28 From Equations 15.03 and 22.25, we have at once 22.29 tan a We have taken the negative sign for rc. which appears in the metric only as 1/n 2 , in order to make the positive direction of the a-coordinate outward in spite of the fact that a decreases numerically outward, and so we preserve the right-handed system used throughout this book in the order (co, u, a)= (1, 2, 3). This device enables us to use all formulas in Chapter 15 as they stand. 14. By differentiating sin a— (ae) /a with, of course, (ae) constant, we have 3 (In o)_ _ _ d(ln a)._l 22.30 da --— cot a ; ds in which a is the semimajor axis of the coordinate spheroid, not to be confused wiih the determinant of the surface metric. By differentiating Equations 22.03 and 22.04 with u constant and simplifying, we have also 22.31 22.32 d P a ■ a. — = sec a cos p sin

cos (/>; then by differentiating other relations in the last section, we have 22.33 22.34 22.35 d In (p cos (/>) _ d In (a cos u) 5a da cot a d In v da '-— cot a-\- tan a sin 2 (f> — - — "=— cot a — 2 tan a + 3 tan a sin 2 d>. da By differentiating the metric tensor in accordance with Equation 15.13, we have the components of the second fundamental form of the spheroids as 22.36 ba/3 — (— v cos 2 cos (/> rj ! , = tan a cos- (/> r| 2 = (a 2 tan aVt' 2 d Inn 1 3 In n a- I 3 d 3 = — = — cot a r-7- oa n ds n\p v) n „.. i) In n . . 1 ^2 = : = sin a tan a sin (p cos cp. v sin a cos a (Hi 22.38 The remaining space symbols, which are the same as the surface symbols, are obtained directly from the surface metric as r, 1 ., =— tan u If, = sec a sin cf) cos (f> rj = sm a tan a sin cos (}) = — d In « r) In v ihi <)ii 22.39 16. The components of the surface tensor \njap are required in many of the formulas of Chapter 15 to compute variation along the normals. We can easily obtain these components either from the Codazzi Equation 15.25 or by direct covariant dif- ferentiation, using the values of the Christoffel symbols given in Equations 22.38 and 22.39, as n (1/n )n =— tan 2 a sin 2 (/> cos 2 n(l//i)i2 = n(lln)>-2 — tan 2 a cos 2c/>(l — sin 2 a sin 2 4>) 22.40 =tan 2 a cos 2 /3 cos 2(/>. THE POTENTIAL IN SPHEROIDAL COORDINATES 17. We can readily expand the Laplacian of a scalar V, &V=g r 'Vrs, in spheroidal coordinates either by using Equation 3.18 or by expanding the covariant derivative and using values of the Christoffel symbols given in Equations 22.38 and 22.39. The result in either case is (v 2 cos 2 (f>)AV = ^ J 1 d' z V dco 2 + sec 2 a cos 2 <$> sec u — \ cos u du dV du 22.41 + tan 2 a cos 2 (f> d 2 V da 2 in which we must make AF=0, if V \s to be harmonic and so to represent a Newtonian potential. For rea- sons which will become apparent later, we change the independent variables in the resulting partial differential equation to 22.42 so that we have p — sin u q — i cot a 8 d — = cos u — d u dp — = — i cosec^ a — ; doc dq the differential equation becomes then r ._d 2 V , 2 d f n .,. dV\ U — — ^+sec- a cos 2 — \ (1— p 2 ) —\ d(o 2 dp [ dp) — sec 2 a cos 2 4> — \ (\ — q 2 ) — dq dq We propose to obtain solutions analogous to the expression for the attraction potential in spherical harmonics, that is, in the so-called "normal" form 22.43 V=ilPQ in which H, P, Q are, respectively, functions of d \ , .. ,, dP 0= n^ + p -dlA {l ' ir) ^P 22.44 sec 2 a cos 2 d> d f . , ,. dQ Q--T q Y x - q) T q 192 Mathematical Geodesy The last two terms in this equation are independent of longitude co and the first term is a function of (o only, so that if the equation is to hold for all values of co, we must have 22.45 1 am, ft dco 2 m in which m is an arbitrary constant. If A and B are constants of integration, the general solution of Equation 22.45 is 22.46 Q, = A cos ma) + B sin raw. Combining Equations 22.44 and 22.45, we have = m 2 cos 2 a sec 2 = m 2 ( 1 — sin 2 a cos 2 u ) sec 2 u 1 d j n 2 . dP\ 1 d , (l-p 2 ) {\-q 2 Y using Equations 22.06, 22.07, and 22.42. The vari- ables in this last equation are now completely separated, and we can write m Pd P [ [i p ' dpi (1-p 2 ) m" 1. Consequently, it is advisable to include Qn(q) = Qn(i cot a) in our solution only when we have cot a > 1, that is, when we have b > ae for the coordinate spheroid through the point under consideration. 2 For the same reason, we cannot include (?'"(sin u) if we require u to be zero. For the external potential (at great distances from the Cartesian origin), we take D = E = in the general solution, Equations 22.47, to give the potential in the form y «. n — -~= ^ ^ Qn(i cot a)P^(sin u)(A„ m cos mw h=0 m=0 + B„ m sin mw) 22.50 in which we have amalgamated the constants in Equations 22.47 with those in Equation 22.46, and we have included the gravitational constant G. Equation 22.50 corresponds to Equation 21.035, which we know to be sufficiently general. On the other hand, if we require an expression for the internal potential which has to be valid at and near 2 No matter how we express Q',','{i cot a), we cannot include it in the potential if cot a is small. If we differentiate this function in the direction of the normal to a coordinate spheroid and use Equations 22.28 and 22.29, we have dQ',?{i cot a) ds n + 1 Q',','(i cot a) i{n — m + 1) tan a Qn+Ai cot a) which becomes infinite for cot a = 0, that is, for points on the limiting "spheroid" formed by rotating the interfocal line. The function will not therefore serve as part of a Newtonian potential in such a region. The Potential in Spheroidal Harmonics 193 the Cartesian origin, we must make D = F=Q in Equations 22.47 to give y * n -77= V V TO cot a)P!f'(sin u)([A„ m ] cos moj n=o m=0 r , + Ld„,„J sin mw), 22.51 which corresponds to Equation 21.086. All quanti- ties in these last two equations are dimensionless except V, A nm - B nm \ we conclude that the dimensions of Anm, B„ m are the same as those of VjG, that is, L W. 18. The first three Legendre functions of the second kind in our notation are Qu(i cot a) — — ia Q\ (i cot a) — a cot a— I Q>{i cot a) = 2i(a + 3a cot- a — 3 cot a); 22.52 the remainder can be found from the recursion formula (n+l)Q H+l -(2n+l)i cot aQ n + nQ„-i = 0. 22.53 To derive the associated functions, we use Ferrers' definition, even though the argument is imaginary, so that we have n,„/- s ,„ d'"Q„(i cot a) C? 'U cot a) = cosec'" a — — — — — d(i cot a)'" 22.54 19. In most of the literature on spheroidal har- monics, the third coordinate is r\ where we have sinh rj = cot a with other relations which can easily be derived from Equation 21.101. This alternative, however, loses the advantage of the simple geometrical interpretation of a given by figure 26. THE MASS DISTRIBUTION 20. To relate the A n m, B„„, in the general formula for the potential to the mass distribution, we require an expression in spheroidal coordinates for the elementary potential at (w, u, a) arising from a single particle of mass m at (w, u, a): in short, we require an expression for the reciprocal of the distance between the two points. We shall deal with the case illustrated by figure 19, Chapter 21, to find the ex- ternal potential when the origin is inside the body. and for this purpose Equation 22.50 is appropriate. The case illustrated by figure 21, Chapter 21, can be dealt with similarly by using Equation 22.51. 21. The reciprocal of the distance (1/ro) between two points in spheroidal coordinates is itself a potential function, and must therefore be expres- sible in the form of Equation 22.50 in which the constants A>,m, B nm will be functions of the coordi- nates of the overbarred point. Moreover, if we inter- change the overbars, we can expect the formula to change to the form of Equation 22.51. By taking a temporary origin for longitude at the barred point, we see that the longitude term must take the form Anm cos mio) — a>) + Bum sin m (&> — a»): and because the field is symmetrical in longitude so that l/r () does not change if the signs of both w and d) are changed, the B„ m must be zero. These considerations are satisfied by the form 1 ro 2 2 Qn(i cot a)P„'(i cot a)P|!'(sin u)P»'(sin u) X A nm cos m (a> — a>). and in fact the final formula, due to Heine, is —=i y (2 re +i) Q„(i cot a)P„(i cot a)P „(sin a) (n — m)\ X P„(sin a) + 22(-)" \(n + m)l X Q'»(i cot a)P%(i cot a)P^"(sin u 22.55 xP{;'(sin u)cos m(a) — a>) A rigorous proof is given by Hobson.' 5 22. If we multiply Equation 22.55 by the mass m of the particle at the overbarred point and sum over the whole mass of the attracting body, we find that the constants in Equation 22.50 are given by . „ i(2n + 1) _ _ . _. _ A m) =2_ l ~ ~~ m "nU cot a)r„(sin u) A„, Bn, 22.56 2 ae 2i(2n + l) n — m)l\- n + m)\ iP'"(i cot a) xP;;'(sin u) cos rnw sin mw '■' Hobson, op. cit. supra note 1. 430. Hobson's conventions for the associated Legendre functions are different, but make no difference in this case. Hobson omits the overall factor i necessary to give real values of r». 306-962 0-69— 14 194 Mathematical Geodesy in which to can be any integer between unity and n inclusive, and (n — to)! is to be interpreted as unity if we have m — n. These equations correspond to Equations 21.037 in spherical harmonics. As in § 21—15, we can replace m by pdv where p is the density at the overbarred point and dv is an element of volume. The summation is then replaced, in the case of a continuous distribution of matter, by a volume integral taken over the whole attracting body. CONVERGENCE 23. The series in Equation 22.55 converges 4 if we have a > a. which implies that the overbarred mass point must lie within the coordinate spheroid passing through the unbarred point (o>, //, a) where the potential is sought. If this condition is to apply to all mass points, then the point where the potential is sought must lie outside the coordinate spheroid which just encloses all the matter. Moreover, the spheroid enclosing all the matter must have b > ae so that we have cot a > 1, if the external potential is to be expressed by Equation 22.50. In that case, all the individual particle series can be multiplied by the particle mass and added term-by-term, and the resulting series for the total potential will con- verge. As in §21-11, we cannot say, however, that convergence of all the individual particle series is necessary, although it is certainly sufficient. 24. The sphere of convergence of §21-11 and figure 19, Chapter 21, is accordingly replaced by a spheroid of convergence if we express the potential in spheroidal coordinates, and the conditions are otherwise exactly the same. In the case of the actual Earth, it is possible to choose a coordinate spheroid which just encloses all the matter and is generally much nearer to the topographic surface than any sphere that also encloses all the matter. Accordingly, we can say that the expression of the potential of the Earth in spheroidal harmonics can be made certainly convergent much nearer to the topographic surface than the potential expressed in spherical harmonics. RELATIONS BETWEEN SPHERICAL AND SPHEROIDAL COEFFICIENTS 25. For the same mass distribution, Equations 21.035 and 22.50 for the potential in spherical and spheroidal harmonics, respectively, must give the same answer at all points in space where both series are convergent. Accordingly, there must be some 4 Ibid., 430. relation between the Cnm, S, im and the A nm , B nm . To obtain this relation, we make use of a formula, due to Blades, 5 which, in our present notation and with a slight modification arising from changing the sign z and therefore of u also, is 1 f n p t x cos t + y sin t — iz \ /cos mt\ , 277 J-rr " \ ae ) \sin mt) - (n - "0! , )n+m p m{i cot a)P ,n {s[n B , /cos rm> (n+m)\ sin mw 22.57 The corresponding formula 6 for the spherical har- monics of the geocentric latitude (cp) and longitude (to), as used throughout Chapter 21, is J_ 2tt (x cos t + y sin t — ii 'cos mt\ dt vsin mt) n< - /"»-»V«P»<(sin ) f COS mW \ )'. \sin m, r) and (oj, //, a) represent the same point where a mass m is situated. We can expand x cos t + v sin t iz Pn in Equation 22.57 in powers of (x cos t + y sin t — iz), and substitute Equation 22.58. The result is mul- tiplied by m, summed over the whole attracting body, and Equations 21.037 and 22.56 for the co- efficients Cnm, S nm and Anm, B nm are substituted. The final result after some simplification is An% B,n, 1-3-5 (2/i + D (7! + to)! m)(n — m — 1) 1 + + - 2-(2n-l) (a — m)(n — m — l)(n- Jnn t>(«-2), m) S(;,-2), m) 2)(n-m-B) 1 (ae)"- 2 -4(2/i — l)(2/i ■J(n-4), m i 3) 22.59 5 Whittaker and Watson (reprint of 1963), A Course of Modern Analysis, 4th ed. of 1927. 403. A simple proof of the formula with the usual difference in conventions is given by Hobson, op. cit. supra note 1, 423. 6 Whittaker and Watson, op. cit. supra note 5, 392. In this case, a change in the sign of z is not required. The Potential in Spheroidal Harmonics 195 The same formula gives the zonal coefficients A n0 in terms of C n0 , C( n -z),o . . . simply by making m = 0. 26. The reverse formula for C« m , S nm in terms of A nm , B nm is obtained by expanding (x cos t + y sin t — iz\" in Equation 22.58 in terms of the Legendre functions (x cos t + y sin / — iz\ P„ Pn- (l(> x cos t + y sin t — iz etc., for substitution in Equation 22.57. The result after simplification is „„+„+,, /C,„A = (ae)»+Hn-m)l 1-3-5 (2ra + l) (n + m)l (An 2n + \ (n + m- -2)1 2 (n — m- -2)! ] (2n + l)(2n-l) (n + m- -4)! 2-4 (n — m- -4)! (2n + l)(2n-l)(2n-3) (n + m- -6)! 2-4-6 (n — m- -6)! (n — m) ! \Z? n ,„ A(n-2), in B(„-2), m A(„-4), m o(n-4), m ^(n-6), m t>(„-6), in 22.60 The same formula gives the zonal coefficients C„ ( > simply by making m — Q. Equations 22.59 and 22.60 enable us, for example, to transform rapidly an expression for the potential in an area where this expression is certainly con- vergent. The corresponding expression in spheroidal harmonics is then certainly convergent almost to the topographic surface. 27. We are now able to relate the spheroidal Anm, Bnm to components of the inertia tensors by means of formulas given for the C nm , S nm in § 21-28 through § 21-34, and, in particular, to the total mass, to the center of mass, and to the moments of inertia of the attracting body. Zero-Order Inertia Tensor 28. From Equations 21.016, 21.037, and 22.60, we have the inertia tensor of zero order as the total mass M where 22.61 M = C (m = -i(ae)A m , which shows that the coefficient Aqo is imaginary. 29. The leading term (m = 0, n — 0) in Equation 22.50 for the potential is 22.62 A oQo(i cot a) = — iaA 00 = Ma/(ae) , which is not the same as the leading term in the spherical harmonic expression (M/r). However, the two terms become nearly the same at great dis- tances from the attracting body where the co- ordinate spheroids become nearly spheres of radius /' and a ~ (ae)/r. First-Order Inertia Tensor 30. From Equations 21.061, 21.037, and 22.60, the Cartesian coordinates of the center of mass are I'IM=(CnlM,SnlM,CiolM) 22.63 ^hiae)^Maer%-Haer%). Consequently, if the origin of spheroidal coordi- nates—that is, the common center of the coordinate spheroids — is at the center of mass, then we have A n = Bn=Aio = 0, and all the first-degree harmonics are absent from the expression of the potential in spheroidal har- monics. Conversely, if these harmonics are missing, the origin is at the center of mass as shown in § 21-42. Second-Order Inertia Tensor 31. From Equations 21.043 and 22.60, we have C2o=I 33 -i(I n +r z2 )=ii(ae) 3 (iA i0 + A o) C 21 =/«=f (ae) 3 ^2i S 21 = / 23 =t(ae) 3 fi 2] C 22 = i (/"-/'--) =-%i(aey i A 22 S 22 =4/ 12 =-f i(ae) 3 £ 22 22.64 from which we can draw much the same conclu- sions as in § 21—52 and § 21-53. For example, if the z-axis — the minor axis of the coordinate sphe- roids—is a principal axis of inertia, then we have A 2l = B n = 0. If all three Cartesian axes are principal axes, then, in addition, we have B 22 = 0; certain relations between the three principal moments of inertia are then given by Equations 21.078 and 22.64. If the distribution of mass is 196 Mathematical Geodesy symmetrical about thez-axis, then we have A- 12 = 0. The same conclusions, as were drawn in § 21-57 and § 21—58 regarding the omission of certain terms in the spherical harmonic expression of the potential, apply similarly to the expression in spheroidal harmonics. THE POTENTIAL AT NEAR AND INTERNAL POINTS 32. If the point P at which the potential is re- quired is nearer to the origin of coordinates than any point of the attracting body, we consider that P is at the overbarred point in Equation 22.55 and the particle of mass m is situated at the unbarred point. In that case, the series will remain con- vergent because a is still greater than a. We must also take the potential in the form of Equa- tion 22.51, in which we now suppose that the coordinates (w, u, a) are overbarred. Proceeding as in § 22-22, we then find that IAo] = X -fhQ„(i cot a)P„(sin u) ae [Bum] ^ 2t(2re+i; ae !\2 (n + m)l m 22.65 X(?i?(i cot a)P%(sm u) cos moj sin moj in which we have finally overbarred the mass point to correspond with Equations 22.56, so that Equation 22.51 may be used as it stands for the potential. As usual, the summations are taken over the whole mass of the attracting body and can be replaced by volume integrals in the case of continuous density distributions. The equations for the potential and the mass distribution cor- respond to Equations 21.086 and 21.087 in spheri- cal harmonics. We may note that the only differ- ence from the formulas for the potential at distant points in Equations 22.56 consists of an inter- change between Legendre functions of the first and second kinds in much the same way as the corresponding difference in spherical harmonics consists of an interchange between r" and l/r" +1 . 33. The spheroidal harmonics P','!(i cot a )/>',!' (sin u) cos mw sin met) in Equation 22.51, for the potential at points near the origin, can be transformed to spherical har- monics in the form ^cos mco\ sin mwj r"P;?(sin (/>) by the method of § 22-25, without assuming that the mass distribution remains the same. This transformation illustrates the fact that the po- tential given by either Equation 22.50 or 22.51, or by either of the corresponding series in spheri- cal harmonics, does not uniquely settle the mass distribution; the same external or internal poten- tial can arise from a variety of mass distributions. To settle the mass distribution, we require knowl- edge of the potential at all points throughout space which cannot be provided by a single series divergent in some areas. 34. In the same way, it must be possible to trans- form the spheroidal harmonics ^ , . , r. • , /cos mu>\ Q'S (i cot a)PX (sin u) . \sin fflto/ in Equation 22.50, for the potential at distant points, to spherical harmonics in the form 1 p;:'(sin) cos ma> sin mw We have so far achieved this transformation only by assuming the same mass distribution. It is more difficult to effect the transformation without mak- ing any assumption about the mass distribution, although Jeffery 7 has given a formula correspond- ing to Blades" Equation 22.57 which could be used for the purpose. However, there is no need in any current geodetic application to suppose that the mass distribution changes during the transforma- tion. DIFFERENTIAL FORM OF THE POTENTIAL 35. It is evident from Equation 22.41 that a is a harmonic function. Also, we have seen in § 22-29 that a behaves like (ae)/r at great distances and is accordingly proportional to the Newtonian po- tential of some finite mass distribution. Accord- ingly, we infer from § 21-6 and § 21-7 that 22.66 V n—0 («) (a), st on 7 Whittaker and Watson, op. cit. supra note 5, 403, and Hob- son, op. cit. supra note 1. 424. The Potential in Spheroidal Harmonics 197 represents the Newtonian potential of a general mass distribution, provided the Js are arbitrary and are constant under covariant differentiation. Equation 22.66 corresponds to Maxwell's form of the potential as expressed in Equation 21.017. 36. We have seen in § 21-27 that the sum of all terms of the same degree is the same whether the potential is expressed in general spherical harmonics or in Maxwells form. For example, the three first differentials of (1/r) are the first- degree spherical harmonics. There is no such simple relation between even the first differen- tials of (a) and first-degree spheroidal harmonics. For example, it can be shown that we have ( ae ) i)a\i)z = £ (2n + 1 )Q„ ( i cot a )P„ ( sin // ) 22.67 (n odd only). which is a spheroidal harmonic although not solely of the first degree. The other differentials must similarly be expressible in spheroidal harmonics because they are harmonic functions, but the ex- pressions become progressively more complicated. Equation 22.66 is not therefore of much use for deriving spheroidal harmonic properties of the potential, but the expression of the potential in this compact form can be of use in theoretical investigations. CHAPTER 23 The Standard Gravity Field FIELD MODELS 1. To facilitate calculation of directions and distances between widely separated points in the gravity field, we require a mathematical model of the field which shall be near enough to the actual field for us to form first-order or linear observa- tion equations. In much the same way, it is useful to have a model or standard field in which it is easy to compute the potential and derivatives of the potential, so that we can confine our attention to the small departures from the model encoun- tered in actual measurement. Departures from the mathematical model are known as anomalies, disturbances, and deflections; the smaller we can make these departures, without sacrificing the simplicity and regularity of the mathematical model, the better. The model field is often called the nor- mal field in the literature; however, the word "standard" describes the situation at least as well, and the word "normal" is already overworked in a book which also deals with the differential geom- etry of families of surfaces whose normals define a vector field. Standard gravity is usually denoted by y in the literature: we shall use this convention later when actual gravity appears in the same formulas. In this chapter, we shall be dealing en- tirely with standard gravity; we shall use the ordi- nary symbol "g" to avoid confusion with the curvature parameters yj, y>, which appear later in the chapter. Standard potential is usually denoted by U in the literature, either for the at- traction potential or the geopotential, and we shall follow this convention in later chapters when the actual potential is used in the same formula: in this chapter, there is no ambiguity in continuing to use V or W for the potential of the particular standard field. SYMMETRICAL MODELS 2. An obvious simplification of the general ex- pression for the potential would be to suppose that the field is symmetrical about the z-axis of rotation, thereby making all the tesseral harmonics anoma- lous. In that case, the model potential is independent of longitude and is given in spherical harmonics as 9 , ni r = y Co/MsiiKft) jd^ (*- + /') G ,^o r" + 1 G in which, as always in spherical harmonic expres- sions in this book, (f) is the geocentric latitude or latitude of the radius vector. Moreover, the longitude of the line of force in the model (w in Equations 21.136 and 21.138) is the same as the geocentric longitude. There will be no zonal harmonics in Equations 21.136 and 21.138, and the only tesseral harmonics will be of the first order so that Equations 21.136 and 21.138, corrected for rotation, reduce to the single equation g cos (t>= ]£ ——-P^-iisin <)>) — a> 2 rP\(sm (/>). n=0 r 23.02 There will be no tesseral harmonics in Equation 21.140, which is unaffected by rotation and becomes 23.03 g sin $= f (n+ ^f C "° P H+1 (sin *). n=0 r As noted in §21-93, Equations 21.136, 21.138. and 199 200 Mathematical Geodesy 21.140 give the outward components of the gravita- tional force, which is, nevertheless, positive inward. We have accordingly introduced an overall change of sign in Equations 23.02 and 23.03 in order that g may be positive. 3. Alternative gravity formulas can be given in terms of (<£ — $), which might be termed the deflec- tion in latitude with respect to a central field, by combining the last two equations, \llcr sonic manipulation involving well-known properties of the Legendre functions, we have # sin (<£-<£) =-|^P>(sin) 23.04 + 6) 2 r sin (f> cos gcos (-(£)= 2) ^i P„(sin) 23.05 — G) 2 r cos 2 0. At any point where the direction of the line of force is radial, the first of these equations is zero and the second gives g direct. For example, this situation would occur at the poles in a symmetrical field. In that case, the first equation is identically zero be- cause we have P/,(1) = 0, and the second could be obtained by radial differentiation of the potential in Equation 23.01. 4. A further simplification would be to make the model also symmetrical about the equatorial plane, in which case the C n o in Equations 23.01, 23.02, 23.03, 23.04, and 23.05 would become zero for n odd. After omitting all the tesseral harmonics and the zonal harmonics of odd degree, we might as well omit all the zonal harmonics beyond the fourth or even the second degree. However, the most con- venient coordinate system for geometrical purposes is a (a», , h) system with a spheroidal base. We must be able to relate the geometric and gravimetric systems and, the simplest way of doing this is to make the base spheroid of the geometrical system the same as an equipotential surface of the gravi- metric system. We shall accordingly investigate this type of model next, instead of an arbitrarily trun- cated spherical harmonic model. field is chosen as nearly as possible to fit the geoid, that is, the actual equipotential surface nearest to Mean Sea Level. The minor axis of the spheroid is oriented parallel to the axis of rotation of the Earth. Ideally, the minor axis should coincide with the axis of rotation, and the center of the spheroid should coincide with the center of mass of the Earth so as to provide also a unique worldwide geometric reference system, as discussed in § 21-57 and § 21-58. 6. The model field rotates with the same angular velocity o> as the actual Earth. Whenever we need to consider the mass distribution which gives rise to the model or standard potential, we suppose that the total mass in the model is the same as the total mass M of the actual Earth, although the mass can- not, of course, be distributed in the same way. 7. The problem of developing such a model field would already have been solved if the field were static. We have seen in § 22-33 that the sphe- roidal coordinate a is proportional to a Newtonian potential and is constant over each of the coordi- nate spheroids, one of which can be chosen to approximate the geoid. The potential would be given by Equation 22.62 as 23.06 V = Ma G ae " and gravity by Equations 22.28 and 22.29 as GM tan a 23.07 {ae)v so that gv would be constant over any one equi- potential surface. 8. However, we are not concerned with a static field: the case we have to consider is a field rotating with uniform angular velocity d». In that case, it is still possible to arrange for the geopotential to be constant over one of the coordinate surfaces of a spheroidal coordinate system. The other co- ordinate spheroids will not, however, be equipoten- tial surfaces. THE SPHEROIDAL MODEL 5. The model most often used for gravimetric purposes consists of an axially symmetrical field in which one equipotential surface is an oblate sphe- roid. This spheroidal equipotential of the model THE STANDARD POTENTIAL IN SPHEROIDAL HARMONICS 9. The geopotential W in a field rotating with constant angular velocity G> is obtained from Equa- tions 20.08 and 22.22 as The Standard Gravity Field 201 -W = -V+\6i 2 {x 2 + y 2 ) =— V +\ o) 2 a 2 cos 2 u 23.08 =-J / +iw-a 2 -iw 2 a 2 / 3 2 (sinw), assuming that the axis of rotation coincides with the minor axes of the coordinate spheroids. If we assume also that the standard or model field is axially symmetric so that the potential is inde- pendent of longitude, then we have m = in Equa- tion 22.50 for the attraction potential V, and we have — W= V GA n oQn(i cot a)P„{sin u) + ^w 2 a 2 n = 23.09 -iw 2 a 2 P 2 (sinu). The geopotential on one particular coordinate spheroid, for which we have a = ao, a — ao, is — Wi) = GA {m Q (i cot oco) + i 6)-al + GAioQi(i cot aii)fi(sin u ) + {GA-zoQ-Ai cot an)— ia) 2 a7 ( }P2(sin u) + GA M Q 3 (i cot aoJ/^sin u ) + . . . ; if the geopotential is to be a constant over this spheroid for all values of u, we must have — Wi) = GA iW Q»{i cot ao) + i&) 2 ag ,4,o = GA 2 oQz{i cota ) = iw 2 a'5 23.10 ^„ = (n>2). The first of these results, combined with Equation 22.62, gives the potential on the base spheroid (a = ao) in terms of the dimensions of that spheroid and the total mass M as 23.11 -r = GMao/(a (( sina„) + iw 2 a 2 . It is usually supposed that the total mass is con- tained within the base spheroid, so that the potential on and outside the base spheroid may be repre- sented by a convergent series in Equation 23.09, in accordance with §22-23. If this series was not convergent, we could not have proceeded beyond Equation 23.09. 10. From Equations 23.10, we have also GA 2 o = ioD 2 a 2 ) IQ2(i cot a ) 23.12 iiG) 2 a'l using Equations 22.52. The coefficient iA 2 o can accordingly be computed definitely for a particular base spheroid. Finally, the geopotential can be written in the spheroidal coordinates (u, a) as -W=GMa/(ae)+GA 20 Q 2 (i cot a)P 2 (sin u) 23.13 +{Wa 2 -Wa 2 P 2 ( sin «)} in which (ae)=ao sin «o is an absolute constant, while we have 23.14 a = ao sin ao cosec a. The term within braces in Equation 23.13 arises from rotation, and is equally well expressed by W(x 2 + f). We may note that if we have d> = 0, the potential is the same as we obtained for the static case in Equation 23.06. THE STANDARD POTENTIAL IN SPHERICAL HARMONICS 11. For the same mass distribution, whatever that may be, the spherical and spheroidal coefficients are related by Equation 22.60, which in the sym- metrical case (m = 0) takes the form {ae \" +1 nl 3-5 .. . (2n + i; ' . 2n + 1 23.15 A( n - 2 ), . (2/i + l)(2n-l) , ~x — : A( n . 2-4 o + Because the only nonzero spheroidal coefficients are A 2 $ and A 00 , the Co are zero if n is odd, as we should expect from the equatorial symmetry of the model. We can rewrite the last equation, after considering the terms of lowest degree in the A's, as 23.16 Cn (_)(»/2)+l( Qe )n+l (n+l) iAc ni A no U + 3)J in which n is to have only even values. To reflect this restriction to even values, we may write (2/i — 2) for n so that C(2n- {~)"{ae) 2n - 1 (2n~2), Q- (2/7-1) iA oa + (2n-2)iA 20 (2/i + V 23.17 3 cot a — a ( 1 + 3 cot 2 a ) in which the range of n is from unity to infinity. The geopotential, including the rotation term, can 202 Mathematical Geodesy now be written in the form G W * (2n+ l)iA 00 +(2n-2)iA 20 H = l (2n-l)(2n + l) i)+- ^ in which $ is the geocentric latitude. For substitu- tion in this equation, we have A20 from Equation 23.12 and 23.19 A 00 = iMI(ae) from Equation 22.61. 12. We can check this result' by considering the potential along the z-axis where figure 26, Chapter 22, shows that we have cot a = zl(ae) and sin u = 1 . In that case, we have from Equation 23.13 -77 = — tan" 1 — +^20 "•^ 8 " (a 2 cos 2 (/) + 6 2 sin 2 (/)) ,/2 in which is the latitude of the normal to the coordinate spheroid; a, b are the semiaxes of the coordinate spheroid; and g e , g p are the values of gravity on the equator and at the poles of the coordinate spheroid. The formula, due originally to Somigliana, gives the component of gravity normal to the coordinate spheroid in latitude 4>. If the coordinate spheroid is the base equipotential sur- face of the standard field, then g„ is the total force of gravity, but the formula is of more general application and gives one component of gravity normal to the coordinate spheroid at any point in space. Clairaut's Formula 15. With some manipulation and use of Equation 23.12, we may rewrite L in Equations 23.21 as 3L (or cot a){a 2 Q>{i cot a u ) — af t Q>{i cot a)} 23.25 Q- 2 (i cot a„) ia> 2 a$Qi(i cos a) Q 2 (i cot a ) From Equations 23.22 and 23.23, we have also 23.26 g p g e _ tan a 3L a 2 X 2 ; b a The Standard Gravity Field 203 on the base (equipotential) spheroid (a = au), this is gp g e _ (o-Qi{i cot go) 23.27 (i cot a{))Q-2{i cot «<>) which, if we use the expansion in Equation 22.48, reduces to it b .&-J«»(i. tan L aor 2^ tan 4 «„ + .). 23.28 If we omit the small terms in tan a», this equation reduces to the classical Clairaut equation. It should be noted that Equation 23.28 applies only on the equipotential spheroid. The corresponding equation at other points in space is Equation 23.26 with Equation 23.25. Pizetti's Formula 16. From the metric in spheroidal coordinates in Equation 22.25. an element of area dS of a coordi- nate spheroid is dS = av cos a cos u du d 2 v, using Equation 20.15. In this equation, M is the total mass contained within the coordinate spheroid over whose surface we have integrated. The last two members of Equation 23.32 are compatible there- fore if, and only if, the coordinate spheroid we have been considering contains all the mass. In that case, we can rewrite Equation 23.32 as 23.33 h — — 4-nGp— 2ar a b in which p is the average density obtained by di- viding the total mass by the volume of the coordi- nate spheroid. This result is due to Pizetti; it holds true, not only for the equipotential spheroid, but also for any coordinate spheroid enclosing all the mass. General Remarks on Gravity 17. If we know the dimensions of the equipoten- tial spheroid, then g e and g,, are directly related by the Clairaut Equation 23.28; we need to know only one of these quantities, for example, g e . The Somigliana Equation 23.24 then gives us gravity at any point of the equipotential spheroid in terms of this one constant g e , which is directly related to the average density or total mass by means of the Pizetti Equation 23.33. Provided d> is known, the three Equations 23.28, 23.24, and 23.33 accordingly allow us to express gravity on the equipotential spheroid — of known a and b— in terms of a single constant: either the total mass, or gravity on the equator, or gravity at any point. Several approxi- mate formulas are given in the literature, all of which can be derived from these three exact equa- tions; the degree of approximation involved appears in the derivation. The exact equations are not, however, more difficult to compute. Some formulas are given in terms of the flattening of the spheroid, f= (a — b)/a; of the gravitational flattening, (g,, — ge)lg e : and of the ratio of centrifugal force on the equator to standard gravity on the equator, q = a) 2 alg e . For example, the Somigliana Equation 23.24 can easily be expanded in the form 23.34 g =g e ( \+B 2 sin 2 (b + B, sin 2 2 + . . .). If we omit tan 4 a and higher powers in Equation 204 Mathematical Geodesy 23.28, the coefficients become B,=-f+U-Hqf+^q 2 23.35 R^ip-kqf, omitting P, qp, and higher powers. 18. The international gravity formula, adopted in 1930, is in the form of Equation 23.34. However, g e and B 2 do not have their theoretical values for a spheroid of given dimensions, enclosing a given mass, but were obtained empirically from gravity measurements. The # 4 -term was obtained theo- retically, but is not in line with modern ideas of the flattening. The 1966 international values of the constants were g e = 978.049 cm./sec. 2 B 2 = 0.0052884 23.36 Bi = - 0.0000059. Values recommended by the International Associa- tion of Geodesy in 1967, and also by the Interna- tional Astronomical Union, are g e = 978.031 cm./sec. 2 B 2 = 0.0053024 B 4 = - 0.0000059. 19. In the same way, there are many classical formulas expressing the coefficients of the second- and fourth-spherical harmonics of the standard potential in terms of e or / and q (which usually appears as m) to various degrees of accuracy; the usual line of development is to add second- and third-order terms to Clairaut's first-order result. However, it is as easy, if not easier, to compute the coefficients of the second, fourth, or any harmonic from the exact formula given as Equation 23.18. STANDARD GRAVITY IN SPACE 20. Because the standard geopotential is inde- pendent of longitude, there is no component of gravity in the direction of the parallels of latitude of the coordinate spheroids. To find the meridian component, we note that the metric in Equation 22.25 gives an element of length along the meridian of the coordinate spheroid as (i> cos a)du. Accordingly, the magnitude of the northward com- ponent of gravity in the direction of the meridian is 1 dW gin ■ v cos a du 3 sin u cos u v cos a [GA 20 Q 2 (i cot a) — 3ora 2 ], 23.37 which is zero (as it should be) on the equipotential spheroid, where the terms in brackets become zero by Equation 23.12. The meridian component is also zero at the poles and on the equators of the coordi- nate spheroids where u is \tt or zero. 21. At other points in space, we can combine normal gravity g„, given by the Somigliana Equa- tion 23.24, with the meridian component g,„, given by Equation 23.37, to give both the magnitude and direction of the total gravitational force. If (f) is the latitude of the spheroidal normal, then the latitude of the line of force is 23.38 (j>- tarn- 1 (g m /g n ); and the magnitude of the total force is 23.39 g =( g 2 l + g 2yi2_ STANDARD GRAVITY IN SPHERICAL HARMONICS 22. It is necessary for some purposes and con- venient for others to have standard gravity ex- pressed in spherical harmonics. Because the field is axially symmetric, Equations 23.02 and 23.03 apply; and because the field is equatorially sym- metric, the rc-odd terms are zero. Accordingly, we write (2n — 2) for n and substitute for the Cs from Equation 23.17 to obtain t A rt ,„ (2n + \)iA {M +(2n-2)iA 2 o g cos d>= 2 (A-)" {2 n-l)(2n + l) 23.40 x M = l (ae) 2 "- 1 P|„_i(sin (f)) — a)-r cos (f> t v> r< \» (2n + 1)L4qq+ (2n — 2)/, 4 20 **n*=2G(-) ^^ 23.41 x %, P 2 „-i(sin0) in which (/> is the geocentric latitude and $ is the latitude of the line of force. 23. From Equations 23.04 and 23.05, we have similarly The Standard Gravity Field 205 B**U>-ti-\GH n + (2»-l)(2fc+l) (ae) 2 " -1 X .,„ P,' w -a(8iP0) 23.42 gcos ($ — ) = ^ G( 23.43 + a) 1 r sin ^> cos <£ (2n + 1 )i^oo + (2n - 2)i^ 2 o H = 1 (ae) 2 "" 1 (2/i+l) X P>„-2 (sin 0) — a; 2 r cos 2 $. The difference in the two latitudes ( — (/>) can be considered the "deflection" in latitude of the standard field relative to a centrally symmetrical gravitational field. There is, of course, no cor- responding deflection in longitude. 24. On the equator of the base spheroid, for ex- ample, we have <£ = (/> = 0, and either Equation 23.40 or 23.43 reduces to g e = -w 2 a-^ G (2n + 1 Moo + (2n — 2)iA 2n {ae ) (2/1+1) a 23.44 1-3-5 . . . (2re-3) 2-4-6 . (2re-2) CURVATURES OF THE FIELD 25. The curvature parameters of the field can be evaluated in any coordinate system from Equations 12.162 by contracting the Marussi tensor, in this case Wrs where W is the standard geopotential. However, evaluation of the parameters in spheroidal coordinates is not simple, even though the potential in spheroidal coordinates includes only two terms. The covariant derivatives of the potential are found by successive differentiation of Equation 23.13 and by use of the Christoffel symbols in Equations 22.38 and 22.39. The covariant derivatives are then contracted with the base vectors of the equipotential surfaces, which can be found from the matrix equation 1 [X.' 1 = 1 cos ( — (/>) —sin ((f) — (f>) v / \0 sin ((/) — (/>) cos (4> — (j)) 23.45 T/(i>cos(/>) l/(v cos a) where 4> is the latitude of the line of force obtained from Equations 23.40 and 23.41, $ is the latitude of the normal to the coordinate spheroid, and the other quantities v, a also refer to the coordinate spheroid. Curvatures in Spherical Harmonics 26. An alternative method is to evaluate the Marussi invariants of Equations 12.162 in the fixed Cartesian system, as explained in general in § 21-95 through § 21-98. Components of the tensor N r s or W rs become ordinary differentials of the geopoten- tial with respect to Cartesian coordinates; these components are easily obtained from Equations 21.145 through 21.150. 27. In the case we are considering, the only nonzero harmonic coefficients in the attraction potential V are C n o, and the only nonzero harmonic coefficients in the first differentials are obtained from Equations 21.137, 21.139, and 21.141 as _B_IV" dx\G M„ + i), 1 — CnO 8 23.46 By \ G I ' A (Y. dz\G ■J(«+ I ), 1 — Gnl) M«+l),() — (/l + l)C)iO- The only nonzero harmonic coefficients in the second differentials are given by Equations 21.145 through 21.150 as C(w+2), o = — 2 (n+ l)(n+2)C„ C( n + 2), 2 = 2 CjK) C(n+2),0 = — !(n+l)(/l+2)C ft o C(n + 2), -2 = — 2 Cm) C(" + 2),..= (n + l)(n + 2)C„ B- Bx 1 (I b- dy* [l B- Bz- (i B- ( V c BxBy B- ByBz \ G a 2 IV dzdx \G 23.47 0(n + 2). 2 — 2 C»(> S( n +2), i — (n + 1 )C„o C( n +2), i — (n+ 1 )C„n- 206 Mathematical Geodesy 28. Allowing for the rotation term in the geo- potential from Equations 21.151 and substituting in Equations 21.152 for -h sin 2 0) C = — (ti+ 1) sin 20 23.49 Z)=isin 2 0, and the Co are given by Equation 23.16. To re- fleet the fact that the only nonzero values of C,,» occur when n is even, we can substitute {2n — 2) for n in Equation 23.48 and in the coefficients B, C, and can use Equation 23.17 for the C(2n-2),o- As always in spherical harmonic expressions, is the geocentric latitude in Equation 23.48 and is the latitude of the line of force, computed, together with g, from Equations 23.40 and 23.41. 29. The remaining nonzero parameters are given by formulas similar to Equation 23.48, but with the following coefficients for gki: A = co 2 B = -i(n+l)(n + 2) c=o 23.50 D=-i for gy 2 : A =i&> 2 sin 20 fl = -f(n+l)(n + 2) sin 20 C = — (n + 1) cos 20 23.51 Z) = isin20 for & ds A = — ft) 2 COS 2 fi = -(n+l)(n + 2)(sin 2 0-|cos 2 0) C = — (n + 1) sin 20 23.52 Z) = -icos 2 0. From Equation 12.021, the curvature of the lines of force is y% because we have y\ = in this case. Equations 23.52, giving the variation in gravity along the lines of force, provide a rigorous form of the "free air" or height correction to the value of gravity on the equipotential spheroid. At points close to the equipotential spheroid, the correction would be given with sufficient accuracy by Equation 20.17, which in this case reduces to dg /l . 1 23.53 ds 2ft, 2 at points near the spheroidal equipotential; the cor- rection is often still further simplified by taking p and i' as a mean radius of the Earth. 30. As a check, we find that the law of gravity in the form of Equation 20.17, that is, dg/ds -g(ki + k 2 )=-2a) 2 , is satisfied by each harmonic in Equation 23.48 formed for each of the appropriate parameters. The remaining parameters ti, yi are found to be zero, as they should be in the symmetrical field we are considering. We can also use these results to determine gravity at points where the curvature is known. For example, on the equator of the equipotential spheroid, h\ —— 1/a, and we then have -SL = ^ + G f _^1L { _i (n+ l)(„ + 2)P n + 2 (0) -hPU-AO)} .•- + G £ <-) (n + 2)/2 L/,(i . . I'd "5 (n+1) 2-4-6 n with n even. Because n is even, we can rewrite this equation with (2n — 2) instead of n as ge_-,. r ^ t .„ C (2 „- 2) ,„ 1-3-5. ■■ .(2/i-D ---»HCJH ^^ X 2-4-6.. .(2n-2)' which agrees with Equation 23.44 if C(2n-2), o is substituted from Equation 23.17. 31. Another interesting comparison arises from the fact that — 1/ki for any surface of revolution is the length of the normal intercepted by the axis of rotation. Consequently, — (1/Ai)cos0 for the equipotential surface is the perpendicular distance between the point under consideration and the axis of rotation; this distance isrcos0 where is the geocentric latitude. Substitution in Equation 23.48 for gk] and use of Equations 23.50 give gcos(f) = — &r rcos0 + G y^ 1 T [2-(n + l)(n + 2)cos0/ > , l + 2 (sin0) n = ' + 2 cos P'n+2 (sin 0)] = -w 2 rcos0 + G ]T -^PLt (sin0). The Standard Gravity Field 207 applying some well-known properties of the Leg- endre functions. If we write (2n — 2) for n and sub- stitute for C(2n-2),o from Equation 23.17, this last result becomes Equation 23.40. 32. This method of determining the curvature parameters is rigorous and can be applied at any distance from the equipotential spheroid to any required degree of accuracy. However, for many purposes, it will be sufficient to use first-order formulas close to the spheroidal equipotential; we shall now investigate this third method, due originally to Marussi. Curvatures in the Neighborhood of the Equipotential Spheroid 33. The curvature parameters at points in the neighborhood of the equipotential spheroid can also be found by Taylor expansions along the normals or isozenithals. 1 For example, in this sym- metrical case, we have from Equations 12.075 and 12.143, with N or JFas the standard geopotential. d fl\ , , dbu , d(l/g) 1 dW\kJ ^ ^dN tan — — — dW ^ ddj 23.54 dKUg) 1 which can be evaluated on the base spheroid from Equation 23.24 or 23.34. The expansions to a first order along the isozenithals follow from the initial spheroidal values of p, v. 34. We can expand along the normals by using Equation 14.32, which in this case becomes 'See also Marussi (1950), "Sulla variazione con l'altezza dei raggi di curvatura principali nella teoria di Somigliana," Bol- lettino di Geodesia e Scienze Affini, v. 9, 3-9. Marussi does not use the physical convention for the sign of the potential. ~ 8 dN ds „„ d(lng) d d(f> d d(f> ' 35. The variation along the normal of the other nonzero parameter yo can most easily be found by direct substitution in Equation 20.25; in this case, that equation becomes, for points on the equipoten- tial spheroid, 23.56 p d + 4d)-y..» ll v where 4> is the latitude of the normal to the spheroid. THE GRAVITY FIELD IN GEODETIC COORDINATES 36. If we are given the position of a point in geodetic coordinates (oj, 0, h) and if we require the potential or its derivatives at the point, the simplest procedure is to convert the geodetic to geocentric coordinates and then to use spherical harmonic expressions for the potential or its deriva- tives. The geodetic and Cartesian coordinates are related by Equations 18.59, which in our present notation become x= ( v + h ) cos 4> cos a> y— (v-\- h) cos <$> sin a» 23.57 z— {v cos- a-\-h) sin where v, a refer to the base spheroid of the geo- detic system. The geodetic and geocentric longi- tudes are the same; if $, r are the geocentric latitude and radius vector, we have 23.58 r cos = ( i' + h ) cos d) r sin (/>= (v cos- a + h) sin $. The same procedure applies to both a general field and the standard field; the only difference is in the formulas for the potential and its deriva- tives, whether for those formulas given in Chapter 21 or for those given in this chapter. CHAPTER 24 Atmospheric Refraction GENERAL REMARKS 1. Almost all geodetic measurements of direction and distance are necessarily made through the Earth's atmosphere, which refracts the line of observation into a complicated space curve. The universal practice is to remove the effect of refrac- tion by applying corrections to the observations, the effect of which is to replace the curved line of observation by the straight chord joining the end points of the line. In following this procedure, we shall begin with a rigorous treatment, which may become necessary in future developments, and then introduce progressive approximations that are justified by our present inability to measure com- pletely the refractive index and its gradient, even at the two end points. Atmospheric refraction is particularly important in the three-dimensional methods used throughout this book, although no method of reducing the observations can overcome uncertainty in the refraction; three-dimensional methods are no better and no worse in this respect than any other. Ac- cordingly, we shall treat the subject fully and, in addition to the rigorous theoretical treatment, we shall give some account of the empirical methods in current use. 2. The geometrical corrections depend on the curvature and torsion of the refracted ray, which in turn depend on the first and second covariant derivatives of the index of refraction. The first approximation will accordingly be to choose a geodetic model atmosphere, which, in most cases, allows us to ignore the torsion of the ray and fixes the direction of the gradient of the index, leaving us with the problem of measuring the magnitude of the gradient. The index of refraction itself can be found from measurements of temperature, pressure, and humidity, but in the present state-of-the-art some further assumptions are necessary to establish the magnitude of the gradient of the index. However, the meteorologists may before long be able to supply, in addition to such field measurements as may be possible, a sufficiently accurate model of the actual atmosphere at the time and in the lo- cality of the observations. In that case, the method of reduction may switch to numerical integration of the rigorous equations of the ray. Moritz, 1 for ex- ample, proposes a direct solution of the eikonal Equation 24.05. In addition, programs are well advanced to measure by two-wavelength techniques the total effect of refraction over the observed line at the time of observation; the theoretical basis of these methods also will be examined in this chapter. THE LAWS OF REFRACTION 3. The basic physical law for studying the propa- gation of light or other electromagnetic waves in a refracting medium is known as Fermafs principle which states that light, for example, will follow that path between two fixed points involving the least traveltime t. Moreover, the refractive index jjl of the medium is related to the velocity v of light in the medium by the equation 24.01 p = c/v 1 Moritz (1967), "Application of the Conformal Theory of Refraction," Proceedings of the International Symposium Figure of the Earth and Refraction, Vienna, Austria, March 14-17, 1967. 323-334. 209 306-962 0-69— 15 210 Mathematical Geodesy in which c is the constant velocity of light in a vacuum. Accordingly, if ds is an element of length along the path, we have 24.02 pds = (c/v)ds = cdt. The optical path length or eikonal is defined as (ct) and denoted by S so that we have 24.03 ct- pds. This integral has to be a minimum along the actual path, compared with any other path joining two fixed terminals. The integral in Equation 24.03 is taken along the actual path. 4. We may also consider a family of light rays emitted in all directions from a point source at the same instant. After a given time t, the fight will arrive at a surface known as a geometrical wave front; for different values of t, we shall have a family of surfaces S = ct= constant. The integral pds will have the same value over the actual path between the source and a given S-surface. 5. We suppose that the medium is isotropic, but not necessarily homogeneous, so that p is a point function, having a definite value at each point of the space considered. In that case, we can transform the space conformally to a curved space with scale factor p as in § 10-19. Because of the minimum principle in Equation 24.03, the rays become geo- desies of the curved space and S becomes the length of any of these geodesies between the source and the transform of the S-surface. The geometrical wave fronts accordingly transform to geodesic parallels, and the rays are normal to the wave fronts in both the transformed and untransformed spaces because of the conformal properties of the trans- formation. As in § 10-20, we can say that the basic gradient equation 24.04 S r = pi. holds true in the untransformed space. In this equation, /,• is the unit tangent to a light ray, or the unit normal to the wave front. Equation 24.04 is fundamental in geometrical optics, and can be reconciled with wave theory even though it has been derived geometrically. Born and Wolf, for example, derive the equation for short wavelengths both from the Maxwell equations and from the electro- magnetic wave equations, and then use the equation to prove Fermat's principle.- The expression of the space in Equation 24.04 by means of a single scalar S and the direction of its gradient, which we have seen in Chapter 12 can be made the basis of a gen- eral coordinate system, is equivalent to Fermat's principle and to other physical laws based on a simi- lar minimum or variational principle, simply by giving the symbols an appropriate connotation. 6. Contraction of Equation 24.04 with g rs S s =pg rs l s gives 24.05 S7S=g rs SrS s =(JL 2 . This equation is generally known as the eikonal equation. 7. Instead of a point source, we could equally well have considered a family of rays perpendicular to any given surface, whose transform could initiate a family of geodesic parallels in the curved con- formal space. In either case, the gradient Equation 24.04 holds true, and we have already developed completely the geometry of the rays and of the wave fronts in Chapters 12, 13, and 14. To use any of the results in these chapters, all we need do is to change the notation from (N, n, v r ) to (S, p, l r ). 8. In particular, Equation 12.020 tells us at once that the principal normal to a ray is an S-surface vector, the principal curvature of the ray is the arc rate of change of (In p) in the direction of the princi- pal normal, and there is no change of (In p) in the direction of the binormal. These results agree with § 10-15. If the principal normal, binormal, and curvature of the ray are m'\ n'\ x~ we have 24.06 (In p) r m r = x (In p),-n r =0. DIFFERENTIAL EQUATION OF THE REFRACTED RAY 9. We can eliminate the scalar S from Equation 24.04 by covariant differentiation along the ray. Using the fact that S rs = Ssr-, we have (pi,) s l s = Srsl* = Ssrl s = ( pi* ) rl S = fJLr + pU s . The last term is zero by Equation 3.19 because /.s is a unit vector, so that the intrinsic derivative of (pi,) along the ray is 8(pl r ) 24.07 55 Pr - Born and Wolf (1964), Principles of Optics; Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 2drev.ed., 110-115, 128. Atmospheric Refraction 211 in which p r is the gradient of the refractive index. This equation is equivalent to either Equation 24.03 or 24.04. For example, if we expand the intrinsic derivative, we have (/X.s/ S )/, ■ + pl rs l s=: Pr- This equation contains the same information as Equations 24.06, obtained by transforming the basic gradient Equation 24.04 which we have seen is equivalent to Fermat's principle. THE SPHERICALLY SYMMETRICAL MEDIUM 10. An important particular solution of Equa- tion 24.07 is obtained by considering the variation along the ray of the vector product e rst (ixl s )p t in which p t is the position vector. We have 8 (e'-y/sp, ) _.,,, §( pi,) 8s ■=£' 8s p, + e rst pl s p tk l« 24.08 = e rst fi s p t +e rst filslt, if we remember that p^- is the metric tensor g^-- The last term is the vector product of two parallel vectors and is therefore zero. For the same reason, the preceding term also is zero if p s is parallel to the position vector, that is, if p. is a function of the radius vector r only. But the left side of the equa- tion is a tensor, all of whose components are now shown to be zero. We can say therefore that if p is a function of r only, we have 8(e rst pl s pt) 24.09 85 = in Cartesian coordinates. The Christoffel symbols are zero, and the equation can be immediately integrated along the ray to show that 24.10 pe rst l s pt=(p,r sin p)q r is constant along the ray where /3 is the angle be- tween the ray and the radius vector whose length is r. The vector q r is a unit vector perpendicular to both /, and p r so that q r is perpendicular to the plane — containing /, and p, — which passes through the origin. Because q' is a constant vector, this plane must remain fixed, and the ray must be a plane curve lying wholly within the plane con- taining the source, the origin, and the initial direc- tion of the ray. The origin must be the center of symmetry for p, which could not otherwise be a function of r only: the refractive index p can, however, be any continuous function of r. The con- stant p -surfaces are spheres centered on the origin. It is clear also from Equation 24.10 that along any ray in this medium we have 24.11 pr sin ft — constant. GEOMETRY OF FLAT CURVES 11. A refracted ray in the actual atmosphere will approximate to a straight line, and is best treated as a Taylor expansion from one end. Quantities at the other end of the line will be denoted by over- bars. If F is any continuous, differentiable scalar and if s is the arc length of the ray from the unbarred end, then the Taylor expansion is 1 ds 24.12 ds 2 d^F ds 3 + k°-^r)s* + If the unit tangent, normal, and binormal of the ray are / r , m r , n r , if the curvature and torsion of the ray are x, T, and if we use the Frenet Equations 4.06, we have dF/ds = F r l r d 2 Flds 2 =(F r l r )sl s = F rs l r l s + xF r m r d 3 Flds 3 = F rst l r l s l> + 2> X Frsm r l s 24.13 +(dxlds)F r m r + xFr(rn r -xl r ). These successive differentials are evaluated at the unbarred end of the line. 12. Next, we suppose that F is any one of the Cartesian coordinates (x, y, z). All components of the tensors F rs , F rs t, etc., are then zero in Car- tesian coordinates, and are therefore zero in any coordinate system. The invariant F,m r becomes, for example x r m r = A r m r , which is the x-component of the vector m r . If p r is the position vector, Equations 24.12 and 24.13 in Cartesian coordinates become p r = p r + sl r + $s 2 xm r 24.14 +ss 3 {-x 2 / r +(dx/ds)m r + XTn r }+ - . ., which as a vector equation is true at the unbarred point in any coordinates if we consider a parallel to p r through the unbarred point. Also, the equation is true for any curve which is sufficiently flat (X, t small) for the Taylor series to be convergent. It should be noted that x< t, (d\/ds) and all the 212 Mathematical Geodesy vectors except p r have their values at the unbarred point. 13. The difference of the two position vectors p r and p r is the chord vector whose magnitude and unit vector will be denoted by (5), k r so that we have (p>- P nis={(s)is}k>- = l r (i-ix z s 2 ) + ni r {i X s + : k(dxlds)s 2 } 24.15 +n'-(i X ™ 2 ), correct to a second order in the small quantities X, t. Taking the modulus of this last vector equation to the same degree of accuracy, we have { (s)/s} 2 = 1 -i X V + ix 2 * 2 = 1 -tVx 2 * 2 24.16 (s)ls=l-iix 2 s 2 so that the correction to the arc length s to obtain the chord length (5) is 24.17 . _J_ V ' 2 S 3 24 A A • 14. The angle 8 between the chord and the principal normal is given by 24.18 cos 8 = is{x + iT^s} omitting second derivatives of the curvature, if X3 is the curvature at a point one-third the way along the ray. Equation 24.17 shows that the factor (s)/s can be dropped without affecting the result to a second order. The simple "one-third" rule in this subject seems to have been introduced first by de Graaff- Hunter. Equation 24.14 shows clearly that it holds true, for both plane and twisted curves, as far as the third-order terms in s 3 in the Taylor expansion along the line; this is the highest degree of accuracy we can attain without introducing second derivatives of the curvature. The validity of the Taylor expan- sion depends on the existence of successive deriva- tives of x- For example, Equation 24.18 would not give the correct answer at a point well outside the effective atmosphere (x = 0; dxlds — 0) over a line extending to the surface of the Earth. 15. Dufour 3 obtains a different formula for the angle (Itt — 8) between the chord and the tangent at the starting point of a plane curve. In our present notation, his formula is :1 Dufour (1952), "Etude Generale de la Correction Angulaire Finie (Reduction a la Corde) Pour une Courbe Quelconque Tracee sur le Plan ou sur la Sphere," Bulletin Geodesique, new series, no. 25, 359-374. 24.19 $77-8 = ^1 {(s)-s} X ds. The integral is taken over the whole curve from the unbarred to the overbarred end, and the chord length (5) is considered a constant during the integration. It will be shown in § 25-16 that, if F is any scalar, the expansion 24.20 (F-F) = U(F' + F') + ^sHF"-F") is correct to a fourth order where the superscripts refer to successive derivatives of F with respect to the arc length s. If we take F as the indefinite integral (5) {(5) — 5} xds and substitute in Equation 24.20, Dufour's formula becomes Tr-d=hx+^s 2 \iz + i_J^x,x^_x_ \ ds (s) (5 To compare this with Equation 24.18, we shall have to introduce the approximations dX cos 8 = 2 tt — 8 when Dufour's formula becomes the same as Equation 24.18 with the factor (s)/s dropped. The approximations involve only terms of the third order in s :! . Accordingly, we may say that to a sec- ond order the two formulas are equivalent, and either may be used as more convenient. There is no reason to suppose that Dufour's formula is any more accurate than the simple "one-third" rule, which holds true for twisted as well as plane curves. 16. The angle e between the chord and the bi- normal is given by 24.21 {{s)/s} cos e = ix T52 in which the factor {(s)/s} can be ignored. But sx and st are of the same order as the radian measure of the angles swept out in the whole course of the ray by the normal and the binormal, respectively; in the case of a flat curve, sx and st are small quantities so that 8 and e must be nearly 90°. Compared with cos 8, which is a first-order quantity, cos e is a second-order quantity and can often be ignored. Atmospheric Refraction 213 ARC-TO-CHORD CORRECTIONS 17. The basic physical law of refraction as ex- pressed by Equations 24.06 does not directly introduce the torsion, which must, nevertheless, be expressible in terms of derivatives of (In fx) and in the direction of the ray if the law is to be sufficient to settle the course of the ray. To investigate this matter further, we set up a (w, , N) coordinate system in which we have /V=ln /x, as we can do if the field is to be defined uniquely at all points by (In /x) and its derivatives. In this system, the bi- nomial n' to any ray must be an /V-surface vector to satisfy Equations 24.06, and is therefore per- pendicular to both the ray /' and to the /V-surface normal v'\ Any other vector perpendicular to the binomial, such as the principal normal m'\ must accordingly lie in the plane of/' and v r . If a, B are the azimuth and zenith distance of the ray in this coordinate system, the azimuth and zenith distance of the principal normal will be a, B + ^tt; we can write then /' = V sin a sin B + fx' cos a sin B + v r cos B m 1 — k 1 sin a cos B + /x r cos a cos B — v r sin B 24.22 n r =— \ r cos a+(x r sin a in which the base vectors A.' , fx'\ v v have their usual significance in a (w, $, N) coordinate system. 18. To determine the torsion of the ray. we con- tract the third of the Frenet equations in Equations 4.06 with v r to obtain t= n rs P r l s cosec B = — Vrsi r l s cosec )6, using Equation 3.20 = (k> — ki ) sin a cos a — ti( cos 1 ' a — sin 2 a) + (yi cos a — y-> sin a) cot B 24.23 in which the parameters have their usual signifi- cance in a (oi, <£, IV) system, and we have substi- tuted Equations 24.22, 12.016, 12.046, etc. We may note from Equation 12.050 that the first two terms in Equation 24.23 are the geodesic torsion of the fi= constant surface in the azimuth of the ray. In the third term, (— yi cos a + y 2 sin a) is the com- ponent—in the direction of the binomial — of the vector curvature of the normal to the /x = constant surface. We can therefore rewrite Equation 24.23 in the alternative form 24.24 T — t— VrsU'v* cot B. The basic gradient equation of the coordinate system is 24.25 (In fx) r = — qvr in which q, corresponding to n in a general (o>, , , In /x ) system, we can combine Equations 24.15 and 24.22 to give { (5 )/s } sin {a ) sin (fB ) = A sin a sin /3 + B sin a cos fi — C cos a { (s )js } cos (a ) sin (/3 ) = A cos a sin /3 + B cos a cos fB + C sin a 24.28 { (s )ls } cos (B) = A cos p - B sin in which s =M* + *(f7M =i * 24.29 (s)/s=l-^x'V- where Xs is th e curvature at a point one-third the way along the ray. An alternative expression ob- tained from the first two equations of Equations 24.28 is 24.30 tan{(a)-a}=- . — - — , A sin B + B cos )8 which gives us a direct arc-to-chord correction for azimuth. If we can neglect C, there is no azimuth 214 Mathematical Geodesy correction, and the arc-to-chord correction for zenith distance would be given by 24.31 {{s)ls} sin {(/3)-j3} = fi. Otherwise, we can compute the zenith distance correction from the third equation of Equations 24.28. /i-surfaces in this system are straight so that the zenith directions are unrefracted. We do not as yet make any assumptions about the magnitude q of the gradient of (In p). The only assumption we have made so far relates to the direction of the gra- dient of (In p), which will not usually differ from an exact gravitational model by more than a few min- utes of arc. THE GEODETIC MODEL ATMOSPHERE 21. Investigation of the form of the refracted ray could be carried without any difficulty to terms of higher order on the same lines, but we should then be involved with higher differentials of the curvature and torsion which we have no hope of measuring. Moreover, the (In p ) pattern may change rapidly with time. For these reasons, we require an atmospheric model, which makes as few assumptions as possible and leaves room for such measurements as we can make, such as measurements of temperature, pressure, and humidity at the two ends of the line. 22. One possible assumption is that the model atmosphere is in static equilibrium, which might be approximately so in settled weather conditions during the afternoon. This would mean that the isopycnics — or surfaces of equal density — which are nearly the same as the surfaces of equal re- fractive index, are gravitational equipotentials: the gradient of (In p) is accordingly in the direc- tion of the astronomical nadir. In that case, v r in Equation 24.25 is the unit normal to the equi- potential surfaces: the torsion of the ray can be calculated from Equation 24.24, which now con- tains nothing but gravitational parameters and the astronomical azimuth and zenith distance of the ray. Equations 24.28 and 24.29 give arc-to- chord corrections as corrections to the observed astronomical azimuth and zenith distance of the ray, provided that we also assume or can measure q, the magnitude of the gradient of (In p), for sub- stitution in Equation 24.26. 23. However, the present state of measurement of the gravitational parameters and of (/ hardly justifies the use of an exact gravitational model, which itself rests on the unreal assumption of static equilibrium. We shall accordingly use a simpler model in which the isopycnics are A-surfaces in the geodetic (w, cf>, h) coordinate system, so that the gradient of (In p) is everywhere in the direction of the geodetic nadir. The normals to the ARC-TO-CHORD CORRECTIONS- GEODETIC MODEL 24. If the refractive index is to be considered constant over the geodetic h -surfaces, then it follows from Equation 24.24 that the torsion t of the refracted ray is simply the geodesic torsion of the /(-surface through the point under considera- tion in the geodetic azimuth a of the ray. From Equation 18.19, we have 24.32 _ (p — v) sin a cos a (p + h)(v + h) in which we have written p, v for the principal curvatures of the base spheroid in the meridian and parallel directions, respectively. We shall know, or can infer, the curvature \ of the ray from measurements to be described later, and we shall also know, at least roughly, the length s of the ray. We can therefore compute the quantities A, B, C, (s)/s from Equations 24.29. Next, we compute the arc-to-chord azimuth correction from Equation 24.30, using the zenith distance /3 of the ray at the end where we are correcting the azimuth. If the azimuth correction is significant, we determine the zenith distance of the chord from the third equation of Equations 24.28. If the azimuth correc- tion is not significant, and it will seldom be sig- nificant, we obtain the arc-to-chord zenith distance correction from Equation 24.31. Finally, the geo- metrical arc-to-chord distance correction is obtained from Equation 24.17 as i ., ., To the degree of accuracy we are working, this last correction should be the same if computed from simultaneous observations for the curvature at either end of the line; if not, the two corrections can be meaned. This is not to say, however, that the curvature x ' s assumed the same at both ends. 25. The system can be simplified if we make some further assumptions. For example, we can assume that t can be neglected compared with \- this assumption can easily be justified for a whole Atmospheric Refraction 215 series of observations by rough computation of t from Equation 24.32 compared with an average value of x- The ray is then a plane curve; there is no azimuth correction. Equation 24.21 shows that the chord is perpendicular to the binormal. The arc-to-chord correction for zenith distance A/3, known as the angle of refraction, is obtained from Equation 24.31, dropping the factor (s)/s, as 24.33 A/3=isx£ where X3 ' s the curvature at a point one-third the way along the ray. 26. If we make the further assumption that x is the same at all points of the ray, then the angle of refraction is the same at both ends of the ray and is given by 24.34 W = is X : as we should expect, because the ray is now a circular arc and sx is the angle in radian measure, subtended at the center of the circle by the ray. The ratio of x> assumed constant over the ray, to a mean curvature of the Earth, expressed as the reciprocal of a constant radius /?, is defined as the coefficient of refraction 4 / so that we have 24.35 /=x*. However, this notion is merely a matter of nomencla- ture and does not introduce any new approximation. If x is constant over the ray, so is /. Indeed, it is frequently assumed that /is a constant for all rays at a given time, and even for all rays at all times within the afternoon period of minimum refraction. For optical wavelengths,/ varies at different times and places between about 0.10 and 0.15; for micro- waves, /is more likely to be 0.25, depending much more on the humidity. 27. If we merely assume that x (° r /) is constant along a particular line and that simultaneous meas- urements are made of zenith distances /3, /3, then Equation 25.39 shows that, to a fourth order, the error in height difference arising from refraction is T2S-x(sin/3 — sin/3) — -pV s :i x cos/3(/c sin/3 + x) 24.36 where k is the normal curvature of the /i-surface in the azimuth of the line, and we have anticipated 4 In the literature of surveying, the coefficient of refraction is usually denoted by k, which in this book is mainly used for the normal curvature of a surface. Sometimes the coefficient is defined as 1 aR. from Equations 24.40 that 24.37 dplds=(k sin /3 + x). For most rays between terrestrial stations, /3 is nearly 90° and the effect of refraction in Equation 24.36 can be entirely neglected. In fact, the assump- tion of uniform curvature, combined with simul- taneous reciprocal measurement of zenith distances at the two ends of the line, provides surprisingly accurate results, especially during the afternoon period of minimum refraction. We can, moreover, use the reciprocal observations of /3 and /3 to determine x (or/). We have /3 - j8 - s(dpids) = s ( k sin /3 + x) • If we assume that /3 is nearly 90° and write k = —l/R where R is a mean radius of the Earth, we have p-p=(slR)(f-l) 24.38 /=1 +(«/«) 08-/3) in which /3, /3 are, of course, in radian measure. The sum of the measured vertical angles at the two ends of the line (elevations positive) equals (/3 — /3). 28. For the methods of adjustment to be de- scribed in later chapters, only arc-to-chord correc- tions (as described in this chapter) should be applied, together with the velocity correction in electronic distance measurement (considered in the following section). We do not require any other corrections, such as "reductions to sea level" or to any supposed equivalent curve on the base spheroid. VELOCITY CORRECTION 29. All electronic distance measurement sys- tems in current use measure the time taken by either light waves or microwaves to travel in air over the distance to be measured and back. If t is one-half this measured time, then we have from Equation 24.03 ct fxds in which c is the constant velocity of propagation in a vacuum and the integral is taken over the actual path from the emitting point P to the distant point P. 30. If we return to Equation 24.20 and substi- tute for F the indefinite integral ixds , 216 Mathematical Geodesy then (F — F) is the definite integral fxas . Also, we have F' = jx, F' ~= fx and F" = ~—= ix(\n p ),l '* = /x(ln p ) r v r cos (3 as = — fx(\n (x)rm' cot /8 = — /xx cot /3 in which we have used Equations 24.22 and 24.06 and the fact that there is no change in the refrac- tive index in the A.'- and jx' -directions, which lie in the constant h -surface and therefore in the con- stant p-surface of the geodetic model. In the last equation, /3 is the geodetic zenith distance of the refracted ray and x is the curvature of the ray. We have finally 24.39 ct = i s(ji + fi) + A s 2 ((xx cot /3 - fix cot /3). This equation is very easily solved for s from a pre- liminary value, obtained by dividing ct by the mean index. The preliminary value is then used to eval- uate the second-order term, which under various disguises is usually known as the "second velocity correction" or as the "velocity component of the curvature correction." 31. For example, we have cos /3— v r l r - sin /3(dp/ds ) = v rs l r l s + v r l r J s 24.40 =-£sin 2 /3-xsin/3 in which k is the normal curvature of the /(-surface, so that to a first order, we have (cot /3 — cot (3 ) = — 5 cosec 2 /3 (d/3/ds ) — — s (k cosec fi + x cosec 2 /3 ). If, in evaluating this small correction, we consider that we have /x — fx—l and that x is constant along the fine, then the velocity correction, considered as an additive correction to the preliminary value of 5, is Y2 s 3 x (k cosec /3 + x cosec 2 /3 ) . If there is no considerable difference in height over the fine, cosec fi is nearly unity; if we confuse — \jk with a mean radius R of the Earth and write /= xR for the coefficient of refraction, the cor- rection becomes finally which is the form given by Saastamoinen 5 or Hijpcke. 6 The correction can further be combined with the geometrical curvature correction (Equa- tion 24.17) as 24.41A Agi/(2-/). Saastamoinen 7 combines this result with a further chord correction to sea level, which we do not require. 32. In microwave measurements, it is usual to assume /= 0.25. For precise Geodimeter measure- ments, Saastamoinen recommends evaluating the coefficient of refraction by Equations 24.38 from reciprocal zenith distances or from vertical angles measured at the same time as the Geodimeter observations. But if any such special measurements are to be made, the reciprocal zenith distances /3, 180° — /3 can enter the precise Equation 24.39 without any of the mass assumptions made in deriving Equations 24.40 or 24.41. For substitution in the precise formula, fx and jx will be known from temperature and pressure measurements at the two ends. The end curvatures are obtained by differ- entiation, as we shall see, and will depend on lapse rates of temperature and humidity. THE EQUATION OF STATE 33. We have next to consider how the refrac- tive index and its gradient may be measured or otherwise determined in order that the curvature of the ray may be deduced and substituted in formulas for the arc-to-chord and velocity correc- tions. This is always necessary in the case of electronic distance measurements, and may be necessary in the case of zenith distance measure- ments when there is a considerable difference in height over the line and the assumption of con- stant curvature no longer holds. For this purpose, we shall require certain physical laws affecting the behavior of gases: these laws are collected here for easy reference. 34. At low pressures p and high temperatures T on the Kelvin (°K.) or absolute scale, p and T are related to the density p of a gas by the perfect, 24.41 -w^i/(l-/), 5 Saastamoinen (1964), "Curvature Correction in Electronic Distance Measurements," Bulletin Geodesique, new series, no. 73, 265-269. 6 Hiipcke (1964), Uber die Bahnkriimmung Elektromagnetischer Wellen und Ihren Einfluss auf die Streckenmessungen," Zeitschrift fur Vermessungswesen, no. 89, 183—200. 7 Saastamoinen, loc. cit. supra note 5. Atmospheric Refraction 217 or ideal, gas law as 24.42 P = cpT in which c is a constant for a particular gas. 35. Another required physical law, which is very nearly in agreement with the latest ideas on the subject if p, is .nearly unity, is that the refrac- tivity, defined as (p — 1), is proportional to the density p of the medium for a particular wavelength of radiation so that we have 24.43 (p — l)/p = constant, although the value of the "constant'" will depend to some extent on the wavelength, as we shall see later. 36. Because of the difficulty of measuring den- sities in the field, we need to replace p by other quantities, such as the pressure and temperature, which can more easily be measured. If there is to be no vertical movement of the atmosphere, and this, of course, is an assumption, then the change in pressure (dp) between the top and bot- tom of a column of air of unit cross-sectional area must equal the weight of air in the column so that we have 24.44 dp — — pgdh in which dh is the height of the column and g is gravity. Equations for Moist Air 37. We shall see later that dry air behaves very nearly as a perfect gas, whose equation of state is Equation 24.42, in which the gas constant is 24.45 c= 2.8704 X 10 H c.g.s. units. 8 If we suppose that moist air also behaves as a per- fect gas and that both dry and moist air obey Dalton's law of partial pressures, then the equation of state for moist air !) is 24.46 , V-^fU in which p, T, p are the pressure, absolute tem- perature, and density of the moist air, and where c is the gas constant for dry air, e = 0.62197 is the ratio 10 of the molecular weight of water vapor to that of dry air, and 8 Smithsonian Institution (1951), Smithsonian Meteorological Tables, 6th rev. ed., 280. 9 Ibid., 295. 10 Ibid., 332. r is the mixing ratio (grams of water vapor per gram of dry air). 38. In terms of the vapor pressure " e, we have e _ r p r+e' Equation 24.46, after we eliminate r, becomes cpT cpT 2447 P ~ 1 - ( 1 - e) (e/p) " 1 - 0.37803(e/p) ' 39. If it is necessary to consider departures of moist air, and of the dry air in a mixture, from a perfect gas, the Smithsonian Meteorological Tables 12 provide the necessary modifications to the formulas and tables, but this refinement is not necessary in current geodetic practice. Integration of Equation of State 40. Combining Equations 24.44 and 24.47, we have 24.48 dp_ {l-0.37803(elp)}gdh p cT which gives, on approximate integration between limits denoted by subscripts 1 to 2, In ft) = J^-{1- 0.37803 (e/p) m }(A2-M 24.49 where the subscript m refers to a mean value over the interval. For routine use, the natural logarithm is converted to base 10 by means of the relation log,,, (— ) = 0.43429 In (^A- 41. Equation 24.49 is usually known as the hyp- sometric formula, which is normally used to obtain a difference in height from simultaneous measure- ments of p, e, and T at the two ends of a line and by substitution of a mean of the two end values for (elp)m and T m . Mean gravity g m is obtained sufficiently and accurately by application of the free-air height correction to the international gravity formula at an initially estimated halfway point. Saastamoinen l:! proposes the use of the "Ibid., 347. vl Ibid., 295-317. 1:1 Saastamoinen (1965). "On the Determination of the Refrac- tive Index of Electromagnetic Waves in Mountainous Terrain." Survey Review, no. 135, 11-13. 218 Mathematical Geodesy hypsometric formula to obtain, from a large known difference of height in mountainous country, a value of T m , which he claims is better than the mean of measured temperatures at the two ends. This value of T m is used to obtain a mean value of the index of refraction over the line from Equations 24.54 and 24.57 for use in the reduction of precise Geodimeter measurements. In the case of optical wavelengths, uncertainty in e and (e/p),,, is of little importance. 42. An alternative integral of Equation 24.48 in terms of the temperature lapse rate l = -dT/dh is frequently useful in the form In (^) = {1 - 0.37803 ( e/p ) m } ^~ In & 24.50 A knowledge of the actual mean lapse rate is of particular importance in the case of lines covering a considerable difference in altitude, and thus a considerable range of temperature and pressure. 14 This tormula should give a better answer than (Tz-TJIh, even when the difference in height h is accurately known. INDEX OF REFRACTION -OPTICAL WAVELENGTHS 43. The refractivity for a gas, defined as (/a— 1), depends not only on the density and composition of the gas, but also on the wavelength of the light. The dependence on the wavelength A is expressed by an experimentally determined dispersion for- mula, which is naturally subject to continual minor improvements, such as those recently sum- marized by Edlen. 15 The formula adopted by the International Association of Geodesy in 1960 is in the Cauchy form of {(A S - 1 ) x 10 7 = 2876.04 + 16.288A" 2 + 0. 136A 4 24.51 in which k is the wavelength in microns (10 _t! meters) of monochromatic light in a vacuum. The constants in the formula are due to Barrell and Sears. 16 The formula applies to "standard air" at a temperature of °C, with a pressure of 760 mm., Hg., and with a carbon dioxide content of 0.03 per- cent. More recent determinations, which are usu- ally expressed in terms of the wave number (de- fined as the reciprocal of the wavelength in microns) and in a different form, suggest that the Barrell and Sears result is correct to better than one part in 10 7 . 44. The Barrell and Sears formula gives the re- fractivity in terms of the wavelength of monochro- matic light. However, the measurement of dis- tances in such instruments as the Geodimeter implies the use of a group of waves of slightly different wavelengths, which have slightly differ- ent velocities of propagation in a refracting medium. In such cases, it is appropriate to use a group velocity, compounded from the individual waves, or, what is equivalent, a group index of refraction given by the formula 1T 24.52 so that we have dfJLs (llc,-1) x 10 7 = 2876.04+ (3 x 16.288R - 24.53 +(5x0.136)\- 4 from Equation 24.51. This formula should be used in preference to Equation 24.51, even for lasers in a refracting medium. If a true monochromatic source ever becomes available, the formula should still be used if the light is modulated. 45. Measurements are not, of course, made in a "standard atmosphere,"' and we have to allow for the effect of different temperatures and pressures and for a different composition of the air. particu- larly the inclusion of water vapor. The formula lx 14 See, for example, Rainsford (1955), "Trigonometric Heights and Refraction," Empire Survey Revieiv, no. 98, 164—177. 15 Edlen (1966), "The Refractive Index of Air," Metrologia, v. 2, 71-80. 16 Barrell and Sears (1939), "The Refraction and Dispersion of Air for the Visible Spectrum," Philosophical Transactions of the Royal Society of London, Series A, v. 238. 1-64. The con- stants in the international formula, Equation 24.51, have been derived by substituting t = 0°C. and p=760 mm., in Barrell and Sears' Equation (7.7), 52. 17 See Born and Wolf, op. cit. supra note 2, 19-21. '"International Association of Geodesy (1963). Report ofl.A.G. Special Study Group No. 19 on Electronic Distance Measurement 1960-1963, 2-3. This is the second report of the SSG19: the first report, delivered at the 1960 Xllth General Assembly in Helsinki, Finland, has been incorporated in the proceedings of that Assembly (Secretariat of the International Association of Geodesy (1962), Travaux de I'Association Internationale de Geodesie, Tome 21, 62-64). The finally adopted formulas are given in Resolution 9 of the 1963 XHIth General Assembly in Berkeley. Calif, (see Secretary General of the International Union of Geodesy and Geophysics (1965). Comptes Rendus de la XIII e Assemblee Generates de I'U.G.G.I., 159-160). Atmospheric Refraction 219 adopted by the International Association of Geodesy in 1960 is a slightly simplified version of a formula due to Barrell and Sears iy and is 24.54 _ 1)= i^(M (fi-D 55 X 10V (l + at) \760/ (l + at) where p.— actual refractive index, pa — group refractive index calculated from Equation 24.53, t = temperature of the air in °C, p — total atmospheric pressure in mm., Hg., e = partial pressure of water vapor content in mm., Hg., a — temperature coefficient of refractivity of air (or the coefficient of thermal expansion), (0.003661). 46. The full 1938 Barrell and Sears formula in the notation and units of the International Association of Geodesy formula, Equations 24.51 and 24.54, is (p - 1 ) 10 6 = [ 0.378125 + 0.0021414X- 2 + 0.00001793X- 4 ] p{l+ (1.049 -0.0157f)p x lO" 6 } X' (l + at) 24.55 - [0.0624 -0.000680A- 2 ] X l + at) Barrell and Sears themselves suggest simpli- fication of the vapor pressure (last) term to the form given in the international formula, except that their recommended value of the constant is 55.6 instead of 55. The constants in the disper- sion formula — the content of the first brackets — become the same as in the international formula for p= 760mm., t = 0, and e — 0. After making this adjustment, the international formula drops the term (1.049 — 0.0157t)p X 10" 6 , which indicates a slight departure of dry air from the ideal gas law Equation 24.42. Dry air in the international for- mula, as thus modified, obeys the ideal gas law, provided Equation 24.43 takes the form 24.56 (p-l)_ C(PG-1) p 760a ' which, together with Equation 24.53, exhibits the dependence on the wavelength of the "con- stant" in Equation 24.43. In deriving this result, we have used the fact that 24.57 (l + at)=aT where T is in °K. According to the latest determi- nation, °K. equal °C. plus 273.16, which exactly fit this last formula if a = 0.003661. Equation 24.57 can be said to define the Kelvin scale. INDEX OF REFRACTION - MICROWAVES 47. Although a number of slightly simpler for- mulas have been extensively used for the refrac- tion of radio waves employed in such instruments as the Tellurometer, the formula adopted by the International Association of Geodesy in 1960 is due to Essen and Froome 20 and is 103.49 (p-1) X10 6 = — ^— (p-e)+- 86.26 5748 \ 1 + ^^J e 24.58 where T— temperature in °K. (°C. plus 273.16), p= atmospheric pressure in mm., Hg., e— partial pressure of water vapor in mm., Hg. 48. In place of the first term on the right, the original Essen and Froome formula contains the two terms 103.49 177.4 —f Pl+ m P-2 in which p\, p> are, respectively, the partial pres- sures of dry air and carbon dioxide, so that the international formula assumes no carbon dioxide content. In view of the very small proportion of car- bon dioxide generally present in the atmosphere, Essen and Froome themselves consider that the effect can be neglected. We can therefore make p-i zero, and substitute (p — e) for p\ if p is the total measured pressure. 49. Interestingly, the first term in the Essen and Froome formula is simply an expression of the ideal gas law for dry air, if the density is assumed to be proportional to (p — 1) and if the electrically deter- mined experimental value of 288.15 X 10~ H is sub- stituted for (p — 1) at 0° C, with a pressure of one atmosphere. The formula also reflects the fact that water vapor behaves as an ideal gas at any one tem- perature. The effect of water vapor on the refraction of microwaves is much greater than on the refraction 19 Barrell and Sears, op. cit. supra note 16, Equation (7.7), 52. 20 Essen and Froome (1951), "The Refractive Indices and Di- electric Constants of Air and its Principal Constituents at 24,000 Mc/s," Proceedings of the Physical Society of London, Series B, v. 64, 862-875. 220 Mathematical Geodesy of optical waves, and the vapor pressure needs to be measured about as accurately as the total pressure. 50. Although short by radio standards, the micro- waves in common use for distance measurement are much longer than optical wavelengths. The effect of different wavelengths in a dispersion formula, which, on theoretical grounds, could not differ much from Equation 24.51, would be very small: there is no sensible effect of group velocity. It is interesting to note that the dispersion Equation 24.51 for a wavelength of 1.25 cm., which is too long to have any appreciable effect, gives (p — 1)= 0.00028760 for standard air. The experimentally determined figure of Essen and Froome is 0.00028815. Essen and Froome performed their measurements at a fre- quency of 24,000 Mc/s. [MHzJ (wavelength 1.25 cm.), and they estimated that their results held true for all wavelengths above 7 mm. MEASUREMENT OF REFRACTIVE INDEX 51. Determination of p from Equation 24.54 or 24.58 depends on measurements of temperature, pressure, and humidity, which can normally be made only at the two ends of the line. Sufficiently accurate measures of pressure and humidity can be made without difficulty, even if this is not always done in current practice. Humidity is usually ob- tained from wet-and-dry-bulb temperatures, from which the vapor pressure can be derived from for- mulas and tables given in the Smithsonian Meteoro- logical Tables? 1 Sometimes data may be in the form of relative humidity, considered equal to 100eM, where e w is the saturation vapor pressure over water at the dry-bulb temperature. 22 52. The accurate measurement of air tempera- ture requires fairly elaborate precautions 2:i which are not always employed. Angus-Leppan 24 has found that radiation intensity, as measured by "black-bulb" thermometers placed on the ground at the observing station, is a more reliable indicator 21 Smithsoniam Institution, op. cit. supra note 8, 365-369. 22 Ibid., 350-359. 23 See Angus-Leppan (1961), "A Study of Refraction in the Lower Atmosphere," Empire Survey Review, no. 120. 62-69; no. 121, 107-119; and no. 122, 166-177. In addition to experi- mental results, these three papers provide a useful summary of the subject. 24 Ibid. of refraction than air temperature measured by ordinary thermometers: but more work is required before the appropriate modifications can be made to Equations 24.54 and 24.58, which in their present form require the actual air temperature. CURVATURE 53. The curvature of the ray is found by simply differentiating Equation 24.54 or 24.58 and using Equation 24.26. For example, if we differentiate Equation 24.54 with respect to geodetic height h in the geodetic model atmosphere, we have sin (3 dp p dh sin /3 p(l + at) {p — l)a dt p<; — 1 dp dh 760 dh 24.59 + 55X10- dh with a similar equation for microwaves from Equa- tion 24.58. We can substitute 24.60 dp dh pg- 91. from Equations 24.44 and 24.42 on the assumption that the moist air behaves as a perfect gas. The determination of curvature then depends on a knowl- edge of the lapse rates — dt/dh, —de/dh, which we shall consider more fully in the next section. Approximate Formula — Optical Waves 54. An approximate formula for curvature of optical paths, based on the assumption that we have e = 0, is often used. In that case. Equations 24.59 and 24.60 reduce to 24.61 l/n-DsinjS pT dh c in which T is the absolute temperature; here, we have used Equations 24.57 and 24.54 for e = 0. In most cases, we shall already have computed p., and there will be no need to introduce any more approximations. Bomford 25 derives a formula in this form without using the international Equation 24.54 for p. However, we have seen in § 24-46 that the international formula for e = is simply an expression with appropriate constants of the perfect 25 Bomford (1962), Geodesy, 2d ed.. 212. Atmospheric Refraction 221 gas law, which Bomford does use, so that Bomford's result must be equivalent to Equation 24.61 with some further assumptions. We can combine Equa- tions 24.61 and 24.54 for e = as 24.62 (fi(;— 1) sin /3 p 760/jLa T 2 dZ + 8 dh c which should be equivalent to Bomford's 24.63 X=16-5 /' dT dh ' 0.0334 where p is in millibars, T is in °K., h is in meters, and X is in seconds per meter. To reconcile the two formulas, we can use the definition 1 millibar = 0.750062 mm., Hg. (standard). 2 * Bomford makes the additional assumption that /3 = 90° and also makes reasonable assumptions for fie, /x, and g. Such assumptions are not necessary if we use Equation 24.61 in which c is the gas con- stant for dry air (2.8704 X 10 K in c.g.s. units). We can use any realistic value for g, such as the inter- national formula with a free-air height correction. 55. The question arises whether neglect of humidity in these approximate formulas is justified or whether it has been too readily assumed that, because humidity has little effect on the refractive index for optical waves, the effect is equally small on the curvature — that is, on the first differential of the index. As an example, we take e as the satura- tion vapor pressure over water at 15° C, which is about 13 mm., Hg.; p is 760 mm., Hg.; and (jjl ( , • — 1) is 0.00028. The humidity term in Equation 24.54 is then about 0.25 percent of the pressure term and can certainly be neglected. For substitution in the curvature Equation 24.59, we calculate de/dh from Equation 24.64 as — 1/210 mm. of pressure per meter of height. The omitted humidity term within the brackets of Equation 24.61 is then 55X1Q-9 210 760T 0.00028// which is about (-0.25 X 10 3 )°K. per meter. If the temperature lapse rate is 0.0055 °K. per meter, which is an average figure, the omitted humidity term is accordingly equivalent to an error of about 5 percent in the temperature lapse rate. At present, we are unlikely to know the lapse rate within 5 per- cent, but humidity may become more significant in the future. LAPSE RATES 56. We have seen that the curvature and thus the arc-to-chord and velocity corrections depend on the vertical gradients of temperature and vapor pressure. Sufficiently representative values of these quantities cannot at present be obtained by direct measurement near the ground. We shall now fill in the present state of our knowledge of these quantities. Humidity 57. All that seems to be known at present about the lapse rate of vapor pressure (—de/dh) is an empirical formula by Hann, 27 determined in 1915 as e/c„=10- /! / 6300 where e is the vapor pressure at a height of h- meters above sea level and e () is the vapor pressure at sea level. Differentiating logarithmically, we have 24.64 de_ dh 6300 x 0.43429 26 Smithsonian Institution, op. cit. supra note 8, 13. Temperature 58. Some idea of the possible values of the tem- perature lapse rate can be obtained from Equation 24.63. If we have dT/dh = - 0.0334 °C. per meter, then the ray is straight. If dT/dh has an even greater negative value than that figure, the ray will curve upward; we know from the common observation of mirage conditions, which do not by any means occur only in deserts, that this condition is possible close to superheated ground. Also, we know that tempera- ture inversions are frequent, especially on clear nights, and, in that case, dT/dh would be positive and the ray would be very strongly curved. We can- not expect to obtain accurate results by assuming that dT/dh is constant at all times and at all places. Accordingly, we shall consider first whether the lapse rate can be assumed constant at certain times, such as the afternoon period of minimum refraction. The Adiabatic Lapse Rate 59. Much consideration has been given to the lapse rate associated with the adiabatic expansion of air. 28 The theory assumes that a given volume of 27 Ibid., 204. 28 For example, de Graaff-Hunter (1913). "Formulae for Atmos- pheric Refraction and Their Application to Terrestrial Refraction and Geodesy," Survey of India Professional Paper No. 14, 1-1 14. 222 Mathematical Geodesy air is heated mainly by radiation from the ground, not by direct solar radiation, and then rises, without acquiring or losing any more heat, to an equilibrium height where its temperature is settled by the out- side pressure. We may expect the process to be complete during the afternoon, that is, around the period of observed minimum refraction. 60. Absolute temperature (T) and pressure (p) in an adiabatic expansion of dry air are related to some initial temperature (T ) and pressure (po) by the equation 29 24.65 TJT= (pn/p) 5 where 8 is the ratio of the gas constant for dry air to the specific heat of dry air at constant pressure. We can take 8 as 2/7. If we differentiate this equation logarithmically with respect to height and use Equa- tions 24.44 and 24.42, we have }_dT T dh ' z\dp p dh Tc so that the adiabatic lapse rate, in a very suitable form for substitution in Equation 24.61, is given by dT = _ 2 g dh 7 c Using the value for g/c in Equation 24.63, the lapse rate (—dT/dh) is very nearly 0.01° C. per meter. Unfortunately, this is almost double what is usually found during the period of minimum refraction from reciprocal vertical angle measurements. The reason for this condition may be that there is some delay in reaching adiabatic equilibrium, if indeed it is ever reached. Also, the adiabatic assumption may be invalidated by acquisition of latent heat through condensation. Whatever the reason, this value of the lapse rate is no longer used. Other Constant Lapse Rates 61. Either of the two standard atmospheres in common use 30 employs a lapse rate of 0.0065° C. per meter, which seems too high to fit geodetic observations at minimum refraction. For such pur- poses, a rate of 0.0055° C. is usually assumed, but there are considerable departures from this figure at different seasons and heights, especially near the ground. Attempts to measure the local lapse rate at a ground station by taking temperatures over a known height difference have seldom given 29 Smithsonian Institution, op. cit. supra note 8, 308. 30 Ibid., 265-268. satisfactory results, partly because of the difficulty of accurately measuring the small difference of air temperatures without elaborate precautions and partly because such local measurements would not be sufficiently representative of the air actually traversed by the ray. Recent Work 62. Because the assumption of a constant tem- perature lapse rate appears too drastic, even at restricted times, various attempts have been made to find formulas, other than the simple adiabatic formula, for the variation of temperature with height (h) and time {t). One such formula, 31 based on eddy conductivity K of the atmosphere, is T=T Q -lh+Ae~ bh sin (qt-bh) b 2 = ql(2K) in which / is a mean lapse rate and e~ bh is an ex- ponential damping factor. Other formulas contain- ing more harmonics have been proposed on much the same basis. Unfortunately, the eddy con- ductivity K, which was expected to be constant, is known now to be even more erratic than the lapse rate. 63. More recently, Levallois and de Masson d'Autume 32 have proposed a formula in the form T= To ~ lh + e-» h f{t - hIV) + (b(t) in which V is a velocity of upward transfer of heat; these geodesists obtained the form of the periodic functions and the constants from large numbers of meteorological observations covering a consid- erable range of altitude and time. 64. Angus-Leppan 33 finds that formulas of the same type fit the observations within a restricted range of heights in other localities, but the con- stants vary considerably with locality; the practical use of the method seems to be restricted until more work has been done in a particular area. 65. Meanwhile, the more developed meteor- ological services could no doubt provide a reason- able estimate of lapse rate at given heights and times within a particular locality — for example, by 31 Sutton (1949), Atmospheric Turbulence, 1st ed., 33, 40. 32 Levallois and de Masson d'Autume (1953), "Ltude sur la Refraction Geodesique et le Nivellement Barometrique," lnstitut Geographique National, 1-112. 33 Angus-Leppan, loc. cit. supra note 23. Atmospheric Refraction 223 interpolation from radiosonde records — particularly where the observed rays cover a considerable range in height; undoubtedly, such facilities will increase. For rays between points at about the same height, it is very doubtful if our present knowledge has progressed much beyond the simple assumption of constant curvature over the path, and the deter- mination of that curvature from reciprocal vertical angles. The lapse rate, used in conjunction with Equation 24.61, is mainly required for rays covering a considerable difference of height when the assumption of constant curvature is no longer valid; but for that purpose, we require representative values of the lapse rate at the two ends of the ray. If this information is not available, then the simple assumption of constant curvature would have to be made also in this case. ASTRONOMICAL REFRACTION 66. We have so far considered terrestrial ob- servations where measurements of temperature, pressure, and humidity can be made at the ends of the line and used to sample the actual refractive index. The gradient of the index, necessary to establish the curvature of the ray, cannot in the present state-of-the-art be directly measured, and we have to rely on more-or-less plausible atmos- pherical models to provide the necessary lapse rates. In the case of observations to stars or satel- lites, the ray passes through the effective atmos- phere, and it becomes necessary to develop a more complete atmospheric model based on measure- ments at one end only of the ray. 67. In all investigations of astronomical re- fraction so far made, a spherically symmetrical model is assumed leading to Equation 24.11; the various investigations which have been made differ only in the assumed radial variation of the index of refraction, or of density, or of temperature, and in the methods used for the further integration of Equation 24.11. A good historical summary is given by Newcomb. 34 The latest investigation, using a discontinuous radial variation of temperature in line with modern meteorology, is due to Garfinkel, 35 but even so the values of the atmospheric param- 34 Newcomb (Dover ed. of 1960), A Compendium of Spherical Astronomy With its Applications to the Determination and Re- duction of Positions of the Fixed Stars, original ed. of 1906, 173-224. 35 Garfinkel (1944), "An Investigation in the Theory of As- tronomical Refraction ," The Astronomical Journal, v. 50, 169-179. Also, Garfinkel (1967), "Astronomical Refraction in a Polytropic Atmosphere," The Astronomical Journal, v. 72, 235-254. eters are subject to continual revision as more data become available. 68. In the spherically symmetrical case, we have from Equation 24.11 24.66 sin /3 p {) r () sin /3n where /3 is the angle between the ray and the radius vector, and the zero subscripts refer to the ground station. Because the gradient of (In p) is radial and because the ray is a plane curve, the curvature is 24.67 x = (ln/x)rOT r ={V(ln/u,)} 1 / 2 sin /3; we have the variation of refraction along the ray as 24.68 3(ln fi)lds = tin p),l r = - {V(ln /a)} 1 ' 2 cos j8. The total angle of refraction is Jx*— J tan £ d(\n p) ds ds 24.69 ■FA p' p r sin /So -1/2 , N) coordinate system are given by l r = (dx s ldx r % = ax, + by,- + cz r — \ r ( — a sin (o + b cos co) + fJL r ( ~ a sin (f) cos co — b sin cf> sin co + c cos c/>) -\-v,\ + a cos 4> cos &>+6 cos sin co+c sin c/>), 25.05 using Equations 12.009. If the azimuth and zenith distance of the line in relation to the /V-surface normal are a, (B, we then have from Equation 12.007 sin a sin /3 = — a sin oj + b cos co cos a sin /3 = — a sin cos (o — b sin c/> sin co + c cos cos co + b cos c/> sin co 25.06 +csin<£ in which a, 6, c can be considered as constants of integration. Only two of these equations are inde- pendent because /, is a unit vector and a 2 +b- + c' 1= l. We obtain an identity by squaring and adding the three equations. 7. The (oj, , N) components of the unit tangent are now given by substitution of the appropriate components of the base vectors V. \,. etc., from Equations 12.037. 12.041, etc., in Equation 25.05. We have (sec (f>)li =— (ki/K) sin a sin /3 + (ti/K) cos a sin /3 1-2= (ti/K) sin a sin /3 — {k\/K) cos a sin fi ls= — sec cp sin a sin p 25.07 and a(i/n . cos j8 , — cos a sin tH (cos c/))/ 1 =— &i sin a sin /3 — 1\ cos a sin /3 + yi cos /3 l 2 = — t] sin a sin /3 — A'j cos a sin /3 + y-2 cos (3 25.08 / :! =nco Si 8 in which, of course, a, /3 have the values given by Equations 25.06. An alternative expression for the third covariant component is obtained from Equa- tions 12.097 as 25.09 l 3 =(Un) sec/3 cos & in which f3 is the zenith distance of the isozenithal k r and cos d"=/,A'. 8. In terms of the Q-matrix of Equation 19.26 we can rewrite Equations 25.06 in the form {sin a sin jS, cos a sin /3, cos fi} = Q{a, b, c}, 25.10 The Line of Observation 229 which implies that Q 7 {sin a sin/3, cos a sin/3, cos/3} = { sin a sin /3 — sin (/> sin a) cos a sin /3 + cos (/> sin a; cos /3 25.12 c = cos (/> cos a sin /3 + sin cb cos /3. THE LINE IN GEODETIC COORDINATES 9. Using Equations 25.03, we can also write {5 sin a sin /3, s cos a sin B, s cos /3} 25.13 =Q{(x-x), (y-y), (z-z)}, which enables us to calculate azimuth, zenith dis- tance, and length of the line if we are given the latitude and longitude of one end of the line and the Cartesian coordinates of both ends. This equation holds true in any (a>, all refer to the same system. We cannot, however, proceed further unless we know the relationship between the Cartesian and (w, (b, N) coordinates, that is, unless we specify the particular (o>, (b- N) system. The simplest results will be ob- tained if we can express (x, y, 2) directly in terms of (oj, (/>, N) because, in that case. Equation 25.13 would lead to closed formulas for {s, a, B) in terms of (at, (b, N). To provide such formulas, we should integrate Equations 12.009, having first substituted the components of the base vectors from Equations 12.041, etc., and this would hardly be possible in the case of a general (w, c/>, N) system. However, reference to Equation 17.64 shows that we can do so in a (oj, , h) system, provided the equation of the base surface is expressible in the Gaussian form of Equation 6.03. We could then rewrite Equation 17.64 in Cartesian coordinates as X = *o(0), (/>) + /* cos

, cb) + h cos 4> sin a> 25.14 2 = zo(o), (/>) + /* sin (/>, substitute in Equation 25.13, and so obtain closed formulas for s, a, fi. In Equations 25.14, Xo, yo,zo are the Cartesian coordinates at the foot of the normal to the base surface (/i = 0), and are functions of {(1), (b) only. 10. Greater simplicity can be achieved if we use a symmetrical (a>, (b, h) system, as discussed in Chapter 18, leading to the Cartesian Equations 18.28 and 18.30. Still greater simplicity results from the use of a spheroid as base surface because Equation 18.30 is then integrable and the Cartesian coordinates are given explicitly by Equations 18.59. To avoid confusion with the overbars, which in this chapter we shall reserve for quantities at the far end of the line, we rewrite Equations 18.59 in terms of v — the principal radius of curvature of the base spheroid perpendicular to the meridian — as x — (i>-\~h) cos 4> cos oj y—(v + h) cos (b sin oj 2 = (e-j^+/i)sin ) in these formulas refer to the straight normals to the base spheroid. It is apparent from the first two equations of Equations 25.15, or from Equations 18.28 in the case of a more general symmetrical system, that v is also the length of the normal, intercepted between the base surface and the z-axis of symmetry. We shall in the future refer to the (o». + h] (P + h 230 Mathematical Geodesy in which a, ft have the values given by Equations 25.06. These results agree with Equations 18.14. Reverse Problem 12. Substitution of Equations 25.15 in Equation 25.13 gives {s sin a sin ft, 5 cos a sin ft, s cos ft} = Q{x,y,z}-Q{x,y,z} = (i> + h )Q{cos cos a>, cos sin w, sin Q{0, 0, 1} — (f + h )Q{cos (j> cos w, cos 4> shi w - s i n sin — v sin (f>){0, cos , sin $}. 25.19 In particular, we notice that we have s sin a sin ft = (v+ A)sin cr sin 6c* — {v + h)cos $ sin (a) — at), and from Equation 25.18, we have 5 sin of sin j8 = (i> + h)sin a sin a* = (i> + h) cos 4> sin (a» — a>) so that 25.20 (i> + A)cos sin a sin /3 has the same value at the two ends of the line, and is therefore constant along the line. It is of some interest to compare this last equation with the gen- eralized Clairaut Equation 18.51 for geodesies on the A-surfaces, which in this notation is (v + h) cos , h, a, (3 and we re- quire d>, (/>, h, d, jS — which is usually known as the "direct" problem — then we can rewrite Equa- tion 25.13 as {x, y, z} = {x, y, 2} + Q T {s sin a sin ft, s cos a sin ft, s cos ft} = {x, y, 2} +s{a, b, c} , 25.21 which enables us to compute x , y, z directly. From Equations 25.15, we then have at once 25.22 tanw = y/i, but some process of iteration is necessary to deter- mine (/>, h from {v+ h)cos (/> = (.?- + y-yl' 1 and 25.23 (e-^ + /i)sin = z, starting with an approximate value tan 4> = zl{e-(x- + y 2 ) il -}. Azimuth and zenith distance then follow from Equa tion 19.27. Chovitz ' has shown that iteration wil not always converge if we have e 2 ^ 2, but this case does not arise in the present context. 15. Alternatively, we can use one of the differ 1 Chovitz (1967), On the Use of Iterative Procedures in Geodetu Applications (unpublished manuscript). The paper was reac at the 48th Annual Meeting of the American Geophysical Union Washington, D.C., April 17, 1967. The Line of Observation 231 ential methods developed in the next two chapters. From Equations 27.19 and 27.20, we have, for example, {(v + h) cos 4> da), (p + h)d(j>,dh} 25.24 =A{ds, sdfiu, — s sin (3 dao} in which the overbars refer to the far end of the line and ds, rf/3o, dan are corrections to length, zenith distance, and azimuth at the near end of the line associated with changes doj, d(f>, dh in the coordi- nates of the far end. The matrix A is obtained from the azimuth a and from the zenith distance /3 of the line produced at the far end as (sin a sin /} sin a cos /3 —cos a\ cos a sin /3 cos a cos (3 sin a • cos jQ -sin/3 / 25.25 To apply this method, we start with assumed ap- proximate coordinates a>, (/>, h at the far end, and compute s, a, /3, a, /3 from Equations 25.18 and 25.19. We are given "observed" values of 5, a, f3, and we substitute observed minus computed values as ds, dao, d[3o in Equation 25.24, which directly gives corrections dco, d<$>, dh to the initial approxi- mate values. The whole process is then repeated as necessary to obtain results of the desired accuracy. TAYLOR EXPANSION ALONG THE LINE 16. Subject to the usual conditions of differen- tiability and convergence, which we shall assume are satisfied by intuitive physical considerations in the cases we are going to discuss, or at least are justified by results, we can expand a scalar function of position F along a line of finite length s as (F-F) = sF'+h 2 F"+h :i F' 25.26 + ^js 4 F"" in which the overbar refers to the value of the function at the far end of the line, and the super- scripts mean successive derivatives with respect to 5. Quantities without overbars, F and its succes- sive derivatives, are supposed to be evaluated at the near end of the line. If the derivatives are measured in the same sense of the line at the far end, that is, in the direction of the line produced, then the corre- sponding expansion from the far end of the line is obtained by interchanging overbars and changing the sign of s as (F-F)=sF'-h 2 F"+h*F'"-^s A F"" + 25.27 In the mean, we have (F-F)=h(F'+F')+h 2 (F"-F") +^s s (F'"+F'") 25.28 +^snF""-F"")+ .... The derivatives can be considered as functions of position, defined at all points along the line, and can similarly be expanded as (F'-F')=sF"+h 2 F'"+h :i F""+ . . . = sF"-h 2 F'"+h :i F""~ . . . so that we have = s(F"-F")+h 2 (F"' + F"') 25.29 +h :i (F""-F"")+ . . .; while by direct expansion, as in Equation 25.28, we have 0=(F"-F")+h(F"'+F"') 25.30 +h 2 (F""-F"")+ .... Multiplying Equations 25.29 and 25.30 by —is and T2S 2 , respectively, and adding to Equation 25.28, we can eliminate the third- and fourth-order terms and can say that 25.31 (F-F)=h(F'+F')+-ks 2 (F"-F") is correct to a fourth order. We could, of course, have eliminated the second-order term instead of the third to the same degree of accuracy, but did not do so because it will usually be possible to measure the second order, but not the third-order terms. Also, we could have eliminated the third- and fifth-order terms instead of the third and fourth, but this would have no effect on the second-order term. We could continue the process by adding equations similar to Equations 25.29 and 25.30, starting with fourth-order terms, and so could eliminate more terms of still higher order, but this also would have no effect on the second-order terms. We may conclude that Equation 25.31 gives us the best possible second-order approximation in cases where we have values of the derivatives at both ends of the line. EXPANSION OF THE GRAVITATIONAL POTENTIAL 17. For the sake of greater generality, we shall assume in this case that the line is curved and that its binormal is an equipotential surface vector. In the case of a refracted ray, this relation is equivalent to the assumption that the isopycnics are level 232 Mathematical Geodesy equipotential surfaces. The principal normal (m r ) to the curved line then lies in the plane of normal section, as shown in figure 28, where we have also Figure 28. shown a unit equipotential surface vector q r in the same azimuth as the line l r . We then have l r = q r sin (3 + v r cos /3 m r = q r cos (3 — v r sin f3. The first differential of the potential along the line is dlV/ds = N r l r = nv r l r = n cos (3 in which we identify N with the geopotential and n with gravity. The second differential is d 2 Nlds 2 =(N r l>y s = N rs l r l s + N r l r sl S — (n s v r + nv rs )l r l s + x np rm r in which \ is the principal curvature of the line so that we have (lln)d 2 N/ds 2 = (\n n) s l s cos j8 + v rs (q r sin (3) (q s sin (3 + V s cos /3) = — k sin 2 /3 — x sin j3 + 2 (In n) s q s sin /3 cos /3 25.32 + (In n) s v s cos 2 /3 where /r is the normal curvature of the equipoten- tial surface in the azimuth of the line and where the zenith distance (3 of the line is measured from the plumbline to the refracted ray. While k is always negative, 2 \ is usually positive, so that curvature and refraction have opposite effects in the deter- mination of the second-order terms of the potential. The final Taylor expansion is (N - IV)ln = scos(3 + h 2 {- k sin 2 (3 - x sin /3 + 2(ln n) s q s sin (3 cos f3 25.33 +(ln n) s ^cos 2 /3}. 18. We have seen in § 20-31 that (In n) s v*, the vertical gradient of gravity, is not at present meas- urable to a high degree of accuracy. However, from Equation 20.17, we have 25.34 (In n) s v s = 2H-2fo 2 ln, which shows that the vertical gradient is of the same order as the normal curvatures of the equi- potential surface. The zenith distance f3 will nor- mally be near ^77, so that the last term on the right of Equation 25.32 will usually be small compared with the first term. Even so, we should require a knowledge of the vertical gradient in order to deter- mine k from torsion balance measurements. Subject to these considerations, everything in Equation 25.33, except the refraction curvatures, can be measured at both ends of the line; by substitution in Equation 25.31, we can determine either the difference in potential or a relation between the refraction curvatures. It is noteworthy that the effect of refraction cancels if n\ sin (3 is the same at both ends of the line. Because n sin (3 is usually nearly the same at the two ends, this fact means that to a fourth order in the expansion of the poten- tial, the effect of refraction depends solely on the difference in curvature of the ray at the two ends. 19. The difference of potential to a first order is 25.35 N-N = h(n cos (3 + h cos /3). If gravity n at the two ends is assumed to be the same, then we have N-N 25.36 is(cos f3 + cos (3); this equation is the difference in "height" which would be obtained by the ordinary surveying process of calculating "trigonometric heights" from re- ciprocal vertical angles measured from the plumb- line. This process accordingly gives heights related to the first-order difference of potential, comparable with results which would be obtained from spiril 2 Otherwise, two adjacent plumblines could intersect in air at points which would have double values of astronomic latitude and longitude. This is contrary to experience. The Line of Observation 233 leveling, within the limits of approximation and of the observing procedure. This first-order result is unaffected by refraction, which is a second-order effect, provided that /3, J3 refer to the same ray, that is, to observations taken at the same time. EXPANSION OF GEODETIC HEIGHTS 20. Equations 25.32 and 25.33, before the intro- duction of Equation 25.34, hold true in any (o>, (/>, N) system. In the geodetic (&>, , h) system, we have n — \ so that dh/ds = cos /3 25.37 ^^-Asin^-xsin^ in which /3 is now the zenith distance of the refracted line from the geodetic (spheroidal) normal and k is the normal curvature of the /?-surface in the azimuth of the line. From Equation 18.18, we have 25.38 , _ sin 2 a cos 2 a (v + h) (p + h) 21. Including the effect of refraction, the differ- ence in geodetic heights is given by Equation 25.31 as h — h—\s{ cos /8 + cos /3 ) + 1 1 2S 2 (A : sin 2 i 8 + x sin fi 25.39 -k sin 2 £-x sin/3), correct to a fourth order. It should be noted that j8, /3 must be measured simultaneously because the changing refraction would alter the curvature of the line between observations; the formula has been derived on the assumption that /3, /3 refer to a single state of the line. In accordance with the convention adopted throughout this book, /3 is the zenith distance of the line produced. The observed zenith distance at the overbarred end will be (180° — /3). 22. As in the case of the potential, we find that the effect of refraction cancels if x sin /3 is the same at the two ends of the line. Apart from the effects of refraction, the formula obtained from Equation 25.31 for the difference in geodetic heights is ex- tremely accurate. For example, over a line 80 kilometers long in the worst azimuth, the error in height is no more than 3 mm. in 2,700 meters, that is, about one part in a million, compared with exact calculation from formulas given earlier in this chap- ter. The second-order terms in this example amount to 138 mm. EXPANSION OF LATITUDE AND LONGITUDE 23. Geodetic latitude and longitude may be expanded along the line in much the same way as geodetic heights. For example, the expansion of longitude in radian measure to a second-order along a straight line is 25.40 a> — co = si' + h-ojr.j'l* in which we have 25.41 and /' da) ds ' sin a sin fi (v + h)cos iwj r i*=-mj r i s _ sin a sin /3(sin cos a sin ^3— cos (/> cos /3 ) 25.42 (v + h)' 1 cos 2 4> using values of the Christoffel symbols given in Equations 18.34 and 18.35. Calculation of the ter- minal coordinates in this way seldom is justified in comparison with the exact methods given in § 25-14 and § 25-15, but the first-order expansions are sometimes useful to give preliminary values. Equations 25.41 and 25.42 are, of course, evaluated at the unbarred end of the line. ASTRO-GEODETIC LEVELING 24. In this section, we shall enclose quantities related to the astronomical system in parentheses, while quantities not in parentheses are related to the geodetic system. Quantities at the far end of a line, whose unit vector is /'' and length is s, will as usual be overbarred. In § 19-23, we defined the vector deflection as A' = (v r ) —v r and showed that, to a first order, this definition is equivalent to the classical first-order notions of deflection. The component of deflection in the di- rection l r is accordingly A = A'7 r =(cos/3) -cos/3. At the far end of the line, the component is A = A'7,.= (cos /3) -cos /3 so that, using Equations 25.36 and 25.39, we have £s(A + A)=M(cos)3) + (cos/3)} — is{ cos /3 + cos /3} = (l/n){N-N}-{h-h} 25.43 = rise in "trigonometric heights'" minus the rise in geodetic height along the line, all to a first order. 234 Mathematical Geodesy 25. The components of deflection A are obtained to a first order from Equation 19.42 as 25.44 A = A r /, = (cos (/> 8a>) sin a sin /3 + (§) cos a sin (3 in which So», for example, is the astronomic minus the geodetic longitude. The astronomic coordinate is directly measured, and the geodetic coordinate is carried forward by calculation along the sides of a triangulation or traverse. Starting from known or assumed values of (N/n — h), we can accordingly derive values of (N/n — h) at all other points. If N is the potential relative to the potential of the geoid, then N/n is roughly the depth of the geoid below the observing station, and (N/n — h) is roughly the local separation of geoid and spheroid. The approxima- tions involved are, however, equivalent to the assumption that the deflections are the same at a point on the topographic surface as the deflections would be if measurable at a point "vertically" below on the geoid or spheroid. This assumption would require the actual plumblines to have the same curvature as the geodetic normals. At points not much above the geoid (or spheroid) in gravita- tionally undisturbed country, the approximation might be justified; but in other circumstances, the accumulation of error could be serious, and the results should be accepted with reservation until such time as they can be checked by other methods. DEFLECTIONS BY TORSION BALANCE MEASUREMENTS 26. We have seen that deflections of the vertical, relating the normals to the third coordinate surfaces of two (a», , TV) systems, usually the astronomic system and a geodetic (o», (/>, h) system with a spheroidal base, can be obtained by direct astro- nomical measurement of latitude and longitude (or azimuth) and by comparison with the geodetic coordinates extended from an origin by triangulation or traverse. The results are, of course, affected by accumulation of error in the triangulation or trav- erse. Relative deflections can also be obtained, or at least interpolated, from measurement of zenith distances, but the results in this case may be vitiated by uncertainty in atmospheric refraction. The two methods may be combined in the adjustment of a space network, as we shall see in Chapter 26. 27. A third method is to integrate gravity anom- alies over large areas surrounding the point where the deflection is required. For accurate results, gravity measurements should be made over the entire globe; even so, the results would be vitiated by smoothing the actual measures of gravity. 28. A fourth method, which we shall now con- sider, uses the torsion balance as originally pro- posed by Eotviis — the inventor of the balance — and since used by Mueller and a few others. Some of the disadvantages of this method, mentioned in §20-34, restrict its use to rather flat terrain where the deflections are of least interest and where simplified formulas are justifiable. With a view to the possibility of a more extended future use of the instrument or of a much improved modern gravity sensor, we shall consider the basic theory rigorously so that the nature of any approximations made may be fully understood. 29. We shall adopt exactly the same notation as in the spherical figure 15. Chapter 19. The nor- mals to the equipotential surfaces at the two observ- ing stations P, P will be v r and V; the fixed vector /'", represented in the spherical diagramby Q, will be the unit vector of the straight line PP. In addi- tion, we shall require a unit vector m r normal to the plane containing v T and /'. that is, perpendicular to the plane of normal section at P. This vector m r is shown in the spherical figure 29 as the pole of the great circle PQ. In the same way, the unit vector m r is perpendicular to the plane of normal section_ at P and is shown as the pole of the great circle PQ in figure 29. The angle between the two planes of normal section is shown as A in figure 29. All other quantities shown in figure 15, Chapter 19, will be required and are connected by Equations 19.01 through 19.18. 30. We shall now consider the integral 25.45 Vrs(m' cosec f$ — m r cosec /3)l s ds along the line PP. The reason for considering an integral in this form will appear later. In this integral, v T is the normal to the equipotential surface at the current point and ds is an element of length of PP so that we have I s ds = dx s - The vector in parentheses is evidently constan during the integration, so that we have the value of the integral as /; {v r (m r cosec /3 — m r cosec f3)} s dx li 25.46 = [i>,(m r cosec /3 — m r cosec /?)]',' = sin 6 cosec /3 + sin 8 cosec (3 = 2 sin K The Line of Observation 235 (P)v T (Q)l 1 Figure 29. where 0, are as shown in figure 29. 31. The next step is to obtain an approximate value of the integral in Equation 25.45 — in terms of the gravitational parameters at the two ends of the line — by evaluating the integrand at P and P and by meaning the results on the assumption that the integrand varies uniformly along the line. Less approximate methods of integration, such as the use of Equation 25.31 with F equal to the indefinite integral of Equation 25.45, would require measure- ment of the gradients of the gravitational parameters which is not at present possible. For our immediate purpose, it will be easier to evaluate the integrand in the alternative form (cosec (3 cosec (i)i> rs € rpq {v l) + v, / )l Q l s , obtained by using the relations e rp %/ v = (sin/8)m r c rpQ l„v q = (sin j8 )m r . The value of the integrand at P is accordingly (cosec /3 cosec P)v r te rpq (vp + Vp)l q l l in which v„ is taken as translated to P by parallel displacement whose components are accordingly given by Equation 19.19, so that we have (vp + Vp) =(cos $ sin 6o>)A ; , + (sin cr cos a*)/jLp + (1 + cos cr)vp = (sin cr sin a*)K p + (sin a cos a*)fip + (1 + cos cr)vp, 236 Mathematical Geodesy using Equation 19.09. We have also l q = \ q sin a sin /J + \x q cos a sin /3 + I'q cos /3. Some labor may be saved by evaluating e™(v p + v p )lq first and ignoring terms in v T because we have VrsV r = 0. The curvature parameters are introduced from Equations 12.016, 12.046, and 12.047 as h = — v rt \ r k* k-i = — PrsfJi' IX s ti=— V rs k r IJL s — — V rs /A r \ S 25.47 y2 = v, s ix r v s ; we have finally for the value of the integrand at P, Ip=+ ki sin /3 cosec f3 {sin a cos a(l + cos cr) — sin a cot /3 sin cr cos a*} — A'2 sin /3 cosec fl {sin a cos a(l + cos cr) — cos a cot /3 sin cr sin a*} + fi sin /3 cosec /3 {(cos 2 a — sin 2 a) ( 1 + cos cr) — cot /3 sin cr cos (a + a*)} — y i cos /3 cosec /3 { cos a ( 1 + cos cr ) — cot /3 sin cr cos a*} + y2 cos /3 cosec (5 {sin a( 1 + cos cr) 25.48 — cot /3 sin a sin a*}. The value of the integrand at P is very easily ob- tained by interchanging overbars in this formula and changing the sign of cr so that we have Ip = Jti sin /3 cosec fi {sin a cos «(1 + cos cr) + sin a cot /3 sin cr cos a*}, etc. From Equation 25.46, we then have 25.49 s(I P + Ip)=4. sin K where 5 is the length PP of the line. The sole as- sumption made in the derivation of this result is that the integrand varies uniformly along the line. Otherwise, all the formulas are exact and, in addi- tion to the five parameters, require five of the seven observable quantities (/>, c/>, 8&>, a, (3, a, (3 from which all other required quantities can be calcu- lated from Equations 19.01, etc., in accordance with §19-12. 32. In practice, /3 will be somewhere near 90° and sin cr will be small so that the second terms within the braces of Equation 25.48 will be very small compared with the first terms, and we may usually write //■ — sin fi cosec /8 (1 + cos a) X { (A'i — k 2 ) sin a cos a + t\ (cos 2 a— sin 2 a) 25.50 — yi cos a cot /3 + y-i sin a cot /3} - In this form, the curvature parameters (A"i — k-z), t\, yi, y-i may be obtained from torsion balance measurements. In fact, reference to Equation 20.36 will show that the expression within braces could be obtained by a single torsion balance reading if it were possible to set the line joining the weights in the azimuth and zenith distance of PP. Similar results for measurements at P are obtained by interchanging overbars in Equation 25.50, if we remember that, in accordance with our usual convention, a, /3 refer to the line PP produced through P, and not to the back direction PP. 33. A further approximation may often be made in cases where /3 and J3 are nearly \-n by writing Ir — ( 1 + cos cr ) { ( A'i — k-z ) sin a cos a 25.51 + Mcos 2 a — sin 2 a)}- Moreover, in these approximate formulas, it will usually be sufficient to evaluate cv, /3, cr, s from geodetic coordinates without making astronomical observations. From Equation 12.050, we can see that //• in this last result is directly proportional to the geodesic torsion of the equipotential surface in the azimuth of the line. 34. If the equipotential surfaces were spheres, then PP and the normals at P and P would be coplanar, so that /'', v'\ v r in figure 29 would lie on the same great circle and A. would be zero. The magnitude of A, obtained from Equation 25.49, is accordingly an indication of the departure of the field from spherical symmetry. 35. So far, we have been working entirely in astronomical coordinates, but the formulas apply equally well in any other (w, (/>, N) coordinate sys- tem, provided we substitute appropriate values of the curvature parameters. We shall normally work in the geodetic (o>, in geo- detic coordinates; by substitution in Equation 25.49, we have a geodetic value for A. which we shall write as k(,. We shall write k\ for the value of X, obtained in the astronomical system from torsion balance measures of the gravitational parameters. The geodetic value k (; is the angle between two planes, one containing the line PP and the geodetic normal at P and the other containing PP and the geodetic normal at P; we have, of course, assumed through- out that the positions of P and P remain fixed in space, whatever coordinate system is used to describe these positions. 36. We can now obtain a first-order relation between (k A — ka) and the deflections at P and P. For this purpose, we consider changes d(f>, doo in the latitude and longitude of P in the triangle CPQ of figure 15, Chapter 19. We have seen in § 19-4 that d$>, d(o can arise from either a change in the coordinate system or a change in the posi- tion of P. In this case, we consider that dcf), dio arise from a change in the coordinate system, with P and P fixed, so that the vector l r (Q) as well as the axis of rotation C' are fixed. In the spherical triangle PCQ, we have sin Q — cos 4> sin oc cosec CQ. Logarithmic differentiation of this equation with CQ fixed gives cot Q dQ — — tan <$> d+ cot a da — c?a>(cot a sin cf> — cot a cos a cot /3 cos cf>) + d(f)(cos a cot/3 — tan 06) on substitution of Equations 19.29. Division by cot Q — cosec a sin ft tan <£ — cot a cos /3 gives us finally dQ— (cos a cosec /3 cos 4>)da> — (sin a cosec (3)d(f>. If we start with geodetic coordinates ax. . (;, the changes dco, dxf> to the astronomical system cu 1 , (/m are (oj.i — o»c), (4>a~4><:) an d dQ= (cm 1 — coo) cos a cosec /3 cos cf> — ((f) a — 4><; ) sin a cosec /3 , with a similar equation dQ — (riU — air;) cos a cosec ft cos (f> — (4>a — <$><:) sin a cosec /S arising from a change in the coordinates of P. In this equation, <5, /3 refer as usual to PP pro- duced, and (o>,4 — wr,), {4>a~4 > g) are the deflections at P. The difference is dQ-dQ=Q A -Q G - {Qa-Qg) = (Qa-~Qa)-{Q<;-Q<,) = k A ~ kr, so that we have finally (A..4 — ka) = — cos a. cosec /3 cos (goi — (be) + sin a cosec f3 (4>a — 4>a) + cos a cosec (3 cos c/> (om — &>(,) 25.52 — sin a cosec /3 (0i — 4>r,) . This single relation, which is in the nature of an observation equation, does not, of course, deter- mine the four deflections at both ends of the line. Observations at the three vertices of a triangle would give us three equations connecting six un- known deflections. A fourth point would add two more equations and two extra unknowns, while a crossed quadrilateral would provide six equations for eight unknowns. In theory, a strong network would eventually provide sufficient and even redundant equations to determine the deflections at all points. Nevertheless, the main application of the method is likely to be the interpolation of de- flections between known values, which could be substituted in the observation equations, such as Equation 25.52, before solution. CHAPTER 26 Internal Adjustment of Networks GENERAL REMARKS 1. In this chapter, we shall consider the forma- tion of differential observation equations for most of the usual systems of geodetic measurement, including, in some cases, the derivation of finite for- mulas that may be necessary to provide computed values. Differentiation of such formulas leads to the observation equations. Instrumentation and observation procedures will be considered only to the very limited extent necessary to understand the nature of the resulting measurements insofar as this affects the formation of the observation equations. We shall not deal with the formation and solution of normal equations from the obser- vation equations; these matters are not peculiar to geodesy and are best studied in the standard litera- ture. The old distinction between adjustment by observation and condition equations is ignored; any fixed condition can always be turned into an obser- vation equation by differentiation and given a very large weight in the solution. The order in which various systems are treated and the amount of space devoted to each system have nothing to do with relative importance, but have been decided partly by history and mainly by simplicity and continuity of explanation. Line-crossing techniques are given last, for example, because they introduce a mini- mum principle not required in any of the other systems. Lunar methods are discussed after stellar triangulation and satellite triangulation, not because lunar methods are later and more sophisticated, but simply because they require less explana- tion in that order. In every case, only enough detail is given to provide a full understanding of the method in the general context of this book. Satellite triangulation, for example, which grows in sophistication every week, will eventually need to be presented in a separate book when the rate of growth slows down enough for a detailed de- scription to remain in date long enough to justify publication in print. THE TRIANGLE IN SPACE 2. If we are given the geodetic coordinates (a), , h) of a Point 1 (fig. 30) and have also the Figure 30. geodetic azimuth, zenith distance, and distance («i2, )3i2, 5i2>^>f Point 2 from Point 1, we can com- pute (a), <£, h, «i2, /3i2) at Point 2 from formulas given in this chapter. If we are also given (ai 3 , /3 J3 ) 239 240 Mathematical Geodesy at Point 1 and (a^s, ^23) at Point 2, then the posi- tion of Point 3 can be found by intersection, but in order to compute the position of Point 3, we have to solve the triangle 123 for the two sides (513, 52.5). We can always do this by computing the angles 312 and 123 from azimuths and zenith distances, by deducing the third angle 231, and then by applying the rule of sines. For example, we have cos 312 = cos /3i2 cos /3i3 26.01 + sin /3i2 sin j8m cos (ai2 — CK13). We are here dealing with computations in geodetic coordinates; the as and j8s are referred to the geodetic or spheroidal normal and are assumed to be free of error. Later in this chapter, we shall relate the geodetic quantities to .actual measurements, necessarily referred to the astronomical zenith and subjected to atmospheric refraction and observa- tional error; but for the present, we are merely discussing operations in the geodetic coordinate system on the assumption that we are given a con- sistent set of quantities in that system. In that case, the two lines in space, 13 and 23, will intersect in a unique Point 3, whose position will be the same whether it is computed from Point 1 or from Point 2. 3. Alternatively, we obtain direct expressions for the sides (513, 523) in a convenient matrix form. The basic vector equation of the triangle is 26.02 5,2/(2 +523^ 3 -5l3^ 3 = 0, which expresses the condition that there shall be no change in the Cartesian coordinates of the Point 1 on proceeding around the triangle. As a relation between vectors, Equation 26.02 is true in any coordinates, provided that parallels to the vectors are considered at a single point in space. If we substitute the Cartesian components of the three vectors in Equation 26.02 and use Equation 25.11, we have Si2Q T {sin ttio sin /3i2, cos a V 2 sin /3y>* cos ySi 2 } + 52sO r {sin «23 sin ^23, cos 0:23 sin /S23, cos ^23} — 5i3Q r {sin a vi sin /3 K 5, cos a yi sin /3 K ), cos /3k?} = 26.03 in which overbarred quantities refer to Point 2 and all other quantities refer to Point 1. We thus have two independent equations to determine 5 )3 and 523. We can eliminate S13 and so directly determine S23 if we first premultiply Equation 26.03 by Q and then by (cos a )3 . —sin 0:13. 0), which gives us S12 (cos «i3, — sin «i3, 0) X {sin a v > sin /812. cos «i2 sin /3i2. cos fivi) — ~ "523 (cos ai3, — sin 0:13. 0) X QQ r {sin a 2 3 sin /3 2 .3. cos a 2 3 sin /3 2 3, cos ^23} ■ 26.04 This entire operation is equivalent to contraction of Equation 26.02 with m,, a unit /(-surface vector at Point 1 perpendicular to /J3, so that we have m,—K r cos CK13 — /jl, sin an. In the same way, if we premultiply Equation 26.03 by Q and then by (cos 0:23, —sin 0:23, 0),we have 5i2 (COS «23, ~ ~ Sin «23, 0) X QQ r {sin a\2 sin /3i2, cos a V i sin /612, cos /3il>} = 5i3 (COS «23, —Sin CX23. 0) x QQ T { sin a, 3 sin 1813. cos a i3 sin (3 v .l cos ^8,3}. 26.05 The matrix QQ T is given by Equation 19.25, with auxiliary angles as in Equations 19.01, etc., and contains only latitudes and longitudes of Points 2 (overbarred) and 1. 4. The triangle can also be solved by the differ- ential method given in § 2.5-15 from initial approxi- mate values of the geodetic coordinates of Point 3. but in this case, the correction to length ds would be unknown. Thus, for the line 13. we have three equations connecting four unknowns: three cor- rections to t he coordinates of Point 3 and one correction to the length 13. The line 23 adds three equations and only one more unknown: the correction to the length 23. Accordingly, we have six equations connecting five unknowns, and the problem is soluble with a complete check if the data are consistent. If the data are incon- sistent or refer to a different coordinate system, we must treat the triangle as part of a network by methods described in the following sections. VARIATION OF POSITION 5. We shall now consider first-order changes in the components of the straight-line unit vector /' arising from changes dx' , dx' in the coordinates of the two ends of the line. If we suppose that we Internal Adjustment of Networks 241 are working in Cartesian coordinates, then we have from Equation 25.04 26.06 sl r = x r -x'\ which can he differentiated as 26.07 sd{l r ) + l r ds = dx' -dx r - We know from Equation 3.19 that the differential of a unit vector, which remains a unit vector after the change is a small vector perpendicular to the original vector so that d(l') is perpendicular to /''; if we contract Equation 26.07 with /, (or /,, which has the same Cartesian components at the far end of the line), we have 26.08 ds = l,dx' — l r dx' giving us the change in the length of the line aris- ing from dx r and dx' . But dx'\ dx' are small vec- tors at the two ends of the line, and this last equa- tion is accordingly an invariant equation which is true in any coordinates, even though we derived the equation in Cartesian coordinates. We can substitute the changes in any coordinates for dx'\ dx'\ provided that we substitute the covariant com- ponents of / r , /r in the same coordinate system. 6. Elimination of ds between Equations 26.07 and 26.08 gives sd(l') = dx r - dx' - ( l s dx s - l s dx s ) I' = {dx s -dx s )(8 r s -l r l s ) 26.09 = (dx s — dx s )(m s m r -\-n s n r ) , using Equation 2.07 and denoting by m'\ n' any perpendicular vectors which form a right-handed orthogonal triad with /'". If m. s , h s are parallel vec- tors at the overbarred end of the line, we can re- write this last equation as sd(l') — (m s dx s — m s dx s )m r + (h s dx s — n s dx s )n'\ 26.10 which again shows that d(l') is perpendicular to l r because it is in the plane of m r and n r . Moreover, Equation 26.10 is a vector equation with invariant coefficients, holding true in any coordinate system. VARIATION OF POSITION IN GEODETIC COORDINATES 7. If the azimuth and zenith distance of /'" are a, /3, we have from Equation 12.007 /' = A'' sin a sin f3 + /jl'' cos a sin /3 + v r cos /3. 26.11 Differentiation of this equation for changes in a;, (f>, a, f3 and use of Equations 12.008 or 12.014, etc., give d(l') = (fx r sin d(o — p' cos dw) sin a sin (3 — (X r sin da>+ u'dcj)) cos a sin /3 + (V cos (f> dco + fx'dcf)) cos (3 + m'dfi — n' sin (3 da 26.12 in which we have written m' — A' sin a cos /3 + /u/' cos a cos /3 — v r sin (3 n'= — A' cos a + jx' sin a 26.13 so that the azimuth and zenith distance of m r are (a, T7T + /3) and of n r are (§7r+ a, 277"). It is evident that (/', m'\ n') form a right-handed orthogonal triad and that m'\ n r can accordingly be used in Equation 26.10. Also, it must be possible to express d(l r ) in Equation 26.12 completely in terms of m'\ n 1 ' because d(l r ), being perpendicular to /', must lie in the plane of m 1 and n r . Indeed, we find after some manipulation d(l') = m'{efy3+cos <£ sin a Joj+cos a d} + n'{— sin (3 da+(sin sin /3 — cos 26.15 s sin {3 da = — h.^dx- + n s dx s + s(sin sin /3 — cos <$> cos a cos (3)da> 26.16 +5 sin a cos (3 drf>, giving the changes in azimuth and zenith distance at the unbarred end of the line that arise from changes of da), d(f), dh and dco, d(f), dh at the two ends, provided that we use the (a>, , $, N) system. In the geodetic system, substitution of -ki = ll(v + h), -k- z = l/(p + h), f,=0, n=\ in Equations 19.31 and 19.32 and use of Equa- tions 19.34 give /sec /> + A) 0\ R=(S r )-> = ll(p + h) \ 1/ I (v + h) cos 0\ 26.19 S= (p + h) \ 1/ so that we may write m, m-2 m 3 {v + h) cos<£ ' (p + A) = {sin a cos j8,cos a cos /3, — sin /6} ft-i {v + h) cost/)' (p + h)" 1 26.20 = { — cos a, sin a, 0} and m i m 2 (v + h) cos' (p + ^)' = QQ 7 '{ sin a cos /3,cos a cos /3, — sin f3} n 2 -, "3 l(^+/?)cos0 (p + h) = QQ 7 '{— cos a, sin a, 0}- 26.21 The matrix QQ r , set forth in Equation 19.25. depends solely on the terminal latitudes and longitudes. 9. Some checks may be applied at this stage. Because the right-hand sides of Equations 26.21 consist of orthogonal matrices and a unit vector, we can premultiply each side by its transpose and obtain + 7^ (v + h) 2 cos- (/> (p + h) ; + m% = 1 , 26.22 with a similar equation for the components of h r , together with comparable equations without the overbars for the components of m r and n r . In Equations 26.15 and 26.16. we may note that an alteration in the origin of longitudes could have no effect on these equations because of the longitudinal symmetry of the geodetic coordinate system. The longitude terms must accordingly reduce to some multiple of (da> — d(o), or, in other words, the co- efficients of da) and do) must be equal in Equations 26.15 and 26.16. Extracting the da), d sin a h\ =ni + s (sin 4> sin /3 — cos (f> cos a cos fi), 26.23 which can be verified algebraically from Equations 19.27. 25.18, and 25.21. For reasons which will appear in the next section, we do not, however, use these relations to simplify Equations 26.15 and 26.16 at this time. OBSERVATION EQUATIONS IN GEODETIC COORDINATES Horizontal and Vertical Angles 10. We start with approximate geodetic posi- tions (a>, , ft), computed roughly from formulas given in Chapter 25. In the case of a triangulation, we may first have to compute the unmeasured side- lengths from Equations 26.04 and 26.05. If the posi- tion of a point is computed from more than one other point, the mean can be accepted. The approx- imate coordinates are then used to compute accurately s, a, /3, a, /3 from Equations 25.18 and and 25.19, and thus to compute the components of the vectors m r , n r , m r , n r from Equations 26.20 and 26.21. If we could measure geodetic azimuths and zenith distances, Equations 26.15 and 26.16 would become the observation equations by substituting "observed minus computed" values for da and d/3. Internal Adjustment of Networks and could be solved in the usual way to provide corrections dx s , dx" to the initial geodetic coordi- nates. However, observations for azimuth and zenith distance are necessarily made in relation to the physical plumbline or astronomical zenith, and we must, in addition, correct the geodetic a, /3 by adding Equations 19.29 to effect a transformation to the astronomical system. In Equations 19.29, Sou, 8(f) will accordingly be the astronomical minus the corrected geodetic coordinates, with longitude positive eastward and latitude positive northward as in figure 12. Chapter 12. In Equations 26.15 and 26.16, dco, d(f) will be the corrected minus the initially computed geodetic coordinates. Conse- quently, (8co+dw), {8(f> + d) will be the astro- nomical minus the initially computed geodetic coordinates. 11. Two further corrections are necessary. If no astronomical azimuth has been measured, an initial direction for the astronomical meridian must be assumed, and we must add a station correction Aa to the assumed astronomical azimuth (or sub- tract Aa from the calculated azimuth). To reduce the observed zenith distance to the straight line on which Equation 26.15 is based, we must also add the angle of refraction A/3 to the observed zenith distance (or subtract A/3 from the calculated zenith distance). 12. Application of Equations 19.29 and the cor- rections Aa, A/3 to Equations 26.15 and 26.16 give us the following observation equations, (Observed Minus Computed) Zenith Distance = — A/3 + fh\d6)ls + rh-zdfyls + m 3 dn/s — mida>ls — models — m^dh/s — (do) + 8u>) cos c/> sin a— (d(f> + 8(f)) cos a 26.24 (Observed Minus Computed) Azimuth = — Aa — h\do) (cosec fl)ls — n 2 d(f) (cosec /3)/s ■— n 3 dh (cosec /3)/s + nido) (cosec /3)/s + n 2 d(f) (cosec /3)/s + n a dh (cosec /3)/s + (da) + 8w)(sin 4> — cos cos a cot /3) + (d(f) + 8(f)) sin a cot /3. 26.25 243 Reverse Equations 13. If measurements have been made at the other end of the line, as will almost always be the case, we must form observation equations for the reverse direction for which the vectors m r , n r , m r , n r are not the same. The same equations, nevertheless, hold true if we remember that the initial azimuth and zenith distance are now (180° + a) and (180° — /3) and will already have been computed. If we retain the same overbarred notation for what is now the initial point, the matrix QQ T remains unaltered; the vector components at the new initial point are given by Equations 26.20 as m. in-. m 3 (v + h) cos (f)' {p + h — { sin a cos )8, cos a cos /3, — sin 0} Til TT» "3 (v + h) cos (/>' (p + h 26.26 = {cos a, — sin a, 0} , while the components at the new far point are given by Equations 26.21 as J ni\ m 2 ] \{v+h) cos0' (p + h)' m ' A \ = QQ r {sin a cos /3, cos a. cos /3, —sin fi} I Tl\ Hi n 3 (v+ h) cos 0' (p + h) ~QQ T { cos a, — sin a, 0} 26.27 in which we may substitute for QQ T the transpose of the original matrix QQ T . The advantage of pro- ceeding in this manner is that the vectors for the reverse direction are easy to compute and refer to the same points as for the forward direction. We must make the same substitutions for azimuth and zenith distance in the remaining terms and remem- ber that the initial point is now overbarred. The full observation equations for the reverse direction are then (Observed Minus Computed) Zenith Distance = — A/3 — fnidw/s — m>d(f)l s — m 3 dhls + m\do)ls + m 2 d(f>ls + m 3 dhls + (dti) + 8a)) cos 4> sin a + (d($> + 8(f)) cos a 26.28 244 Mathematical Geodesy (Observed Minus Computed) Azimuth = — Aa + fiida) (cosec ft)/s + n 2 d4> (cosec ft ) Is + fizdh (cosec ft) Is — riida) (cosec ft)ls — n 2 d^> (cosec ft) Is — nzdh (cosec ft) Is + (do + 80) (sin — cos (f> cos a cot ft) + {d. General Considerations Affecting the Angular Equations 14. If an astronomical longitude has been meas- ured, then (do + 8o), which is the astronomical minus the initial approximate geodetic longitude, is known. The corresponding terms in the observa- tion equations can be computed and added to the absolute terms. This procedure does not ignore the possibility of random error in the measured astro- nomical longitude, which would appear in the residuals. If an astronomical longitude has not been measured, it may be advisable to assume one from the general values of deflections in the area. In that case, the corresponding terms in the observa- tion equations can be computed with the assumed value and added to the absolute terms. We should, however, retain terms — do\ cos (f> sin a and dcui (sin — cos 4> cos a cot ft) in which d(x)\ is a correction to the assumed astronomical longitude to be found from the solution. Exactly the same pro- cedure should be followed for the (d(}) + 8(l)) terms. 15. Apart from numerical considerations, no reason exists why (do + 80), (d(f)-\- 8(f>) should not be considered as independent unknowns and evaluated by the solution, even though the terms contain the independent unknowns do, d(f). Such a combination of unknowns does not invalidate any principle of least squares. 1 We can finally deter- mine the deflections 8o, 8(f) by subtracting do, d in the /3-equations are all small so that these terms could be omitted in a first solution. The main function of the ft- equations, controlled by frequent astronomical observations, is to interpolate deflections and to Internal Adjustment of Networks 245 determine dh, dh. The coefficients of these un- knowns in the a-equations are, however, small so that fairly large errors in these coefficients would have little effect on the determination of du>, d(j>, da>, d(\> from the a-equations. The one exception is the term (du) + 8w) sin c/> in the azimuth equa- tion; it can be inferred from Equation 19.30 that uncertainty in this term would mainly affect the determination of Aa. 20. In the case of lines radiating from the origin, da), d(f), dh are all zero, and astronomical longitude, latitude, and azimuth should be measured so that the 8fe», 5(f>, Aa terms can be evaluated. (If astro- nomical values are accepted as the initial geodetic elements, then 8a>, 8(f), Aa would all be zero.) The effect of this procedure on at least two lines will be to ensure proper orientation of the geodetic coordinate system by satisfying Equations 19.29. The inclusion of frequent astronomical observa- tions will similarly preserve orientation of the geodetic system. Lengths 21. The observation equation in geodetic coordi- nates for a measured distance between stations is given at once by substituting Equations 25.17 and 25.20 in Equation 26.08 as (Observed Minus Computed) Distance = (v+ h ) cos (j) sin a sin fi(da) — dot) + (p + h) cos a sin f3 d(f> + cos /3 dh — (p + h) cos a sin ft dcf) — cos /3 dh . 26.30 The equation should be divided by a constant of the same order as the average side-length in the network so that the equation may have roughly the same dimensions as the a- and /3-equations. The length and angular equations may, of course, be weighted differently if there is reason to do so. Present (1968) experience suggests that electronic distance measurements are generally of about the same order of accuracy as the best angular measure- ments and that relative weighting is unnecessary. 22. The only correction required in electronic distance measurements is for refraction; the cor- rection reduces the actual measurement to the straight air-line distance between the two stations. Spirit Levels 23. If the two ends of the line are connected by spirit levels, it is possible to construct a first-order observation equation to reflect the measurement and to include the equation in the general adjust- ment of the network. The observing procedure virtually frees the spirit leveling from the effects of atmospheric refraction and makes the inclusion of such observation equations in the network ad- justment of considerable value. 24. The right-hand side of the /3-Equations 26.24 and 26.28 without the refraction terms A/3, A/3 can be considered as a correction to the computed straight-line geodetic zenith distance, arising from changes in the end coordinates and from the change from the geodetic to the astronomical zenith. Consequently, the "observed" zenith dis- tance in these equations, apart from observational error, is the zenith distance of the straight line measured from the astronomical zenith. To a first order. Equation 25.36 relates these "observed" zenith distances (/3), (j3) to the rise h t in spirit levels from the unbarred to the barred end of the line, except that the zenith distance at the barred point in Equation 25.36 refers to the line produced. In our present notation, Equation 25.36 becomes fc,=£s{cos (/3)+cos [180°- (£)]} "*M(j8)-G8)} because the two zenith distances are nearly 90°. If we subtract Equation 26.24 for the forward direc- tion from Equation 26.28 for the reverse direction, the left-hand side of the resulting equation will be 'computed zenith\ /computed zenith\ 2/(, + distance at \unbarred end distance at \barred end 26.31 The computed zenith distance at the barred end will be 180° — /3 where /3 will have been computed from the initial approximate geodetic coordinates by Equation 25.19. The right-hand side of the final observation equation is similarly derived as Equa- tion 26.28 minus Equation 26.24 without the A/3, A/3 terms. 25. We have used only a first-order formula for the difference in potential (or spirit levels), whereas the spirit levels will usually have been measured to a high degree of accuracy which we have not used. However, the effect on all the unknowns except dh, dh will be small, and we should expect to evaluate 246 Mathematical Geodesy dh, dh within the limits only of the first-order as- sumption and not to the degree of accuracy of pre- cise spirit leveling. If the required data are available, we could first remove the second-order terms from the measured spirit levels by using Equations 25.33, in which the x _ term should be omitted because, in this case, we are expanding the potential along the straight line. In many cases, it would be sufficient to use geodetic curvatures and standard gravity in the evaluation of the second-order terms. If allow- ance is made for the second-order terms in this way, the adjustment should provide better values of the geodetic heights. Initial Values 26. Before the observation equations can be formed, we must have approximate values for the geodetic coordinates of all points in the network. These approximate values can be obtained from formulas given in Chapter 25. Alternatively, we can start with very rough positions, obtained from maps or triangulation charts, and solve some of the observation equations themselves for corrections to the initial positions. For this purpose, we could ignore minor terms, such as deflections in the azimuth equations, and substitute as best we can for the angle of refraction and for deflections in the zenith-distance eauations. OBSERVATION EQUATIONS IN CARTESIAN COORDINATES 27. The invariant terms in the observation Equa- tions 26.24, 26.25, and 26.30 and the equation for spirit levels, that is, m,dx'\ m,dx r , h r dx r , n r dx r , l r dx r , l r dx r , can, of course, be evaluated in any coordinate sys- tem; all that is needed is to evaluate the compo- nents of the vectors in the proposed system. In this case, the unknowns dx r , dx r will be corrections to the end coordinates, not in the geodetic system (o>, 4>, h), but in the system which has been used to evaluate the vector components. If, for example, we evaluate these invariant terms in Cartesian co- ordinates, the parallel vectors /,, m r , h r will have the same components as /,, m r , n r , and the invariant terms in Equation 26.24, for example, become mi(dx — dx)ls, m,2(dy — dy)ls, m 3 (dz — dz)/s. This does not mean, however, that we have reduced the number of unknowns which will appear in dif- ferent combinations in the observation equations for adjacent lines. Nevertheless, with the exception of the length Equation 26.08, we cannot express in Cartesian coordinates all the observation equations so far discussed because azimuth, zenith distance, and the astronomical latitude and longitude cannot be expressed simply and solely in Cartesian co- ordinates. We have to find the geodetic coordinates of points, even to express the Cartesian components of the vectors, and we have finally to convert the Cartesian results to the geodetic system. Thus, the only overall advantage of working in Cartesian coordinates in the cases so far considered seems to be that the more elementary Cartesian system is easier to understand than a curvilinear system. This conclusion applies only to observation equa- tions so far discussed for use in connection with horizontal and vertical angular measurements, distances, levels, and astronomical measures. Other forms of measurement, as we shall see, may indicate different coordinate systems. 28. When required, the Cartesian components of the vectors /,-, m r , n, are very easily found from Equations 12.013 and 19.22, that is, from 26.32 [A r , B,, C,} = Q r {\,, n,, v,}. The Cartesian components (a, b, c) of /, in azimuth a, zenith distance /3 are found by contracting this last equation with /' as {a, b, c} = Q'{sin a sin f3, cos a sin /3, cos /3}, 26.33 which agrees with Equations 25.11 and 25.12. The Cartesian components of m, in azimuth a, zenith distance (z7r + j8) are given by 26.34 Q'{sin a cos f3, cos a cos j8, —sin /3}: the Cartesian components of n, in azimuth (|ir + a), zenith distance \tt are given by 26.35 Q T {— cos a, sin a, 0}. FLARE TRIANGULATION 29. So far, we have considered only observations made at intervisible ground stations, whereas observations from ground stations that are not intervisible to elevated beacons that cannot be occupied have received much attention in recent years. The object is to increase the distance be- tween ground stations so as to provide a more open network quickly or to bridge wide water gaps. One such system, used, for example, by W. E. Browne Internal Adjustment of Networks 247 to bridge the Straits of Florida, is to make simul- taneous observations from ground stations to parachute flares dropped from aircraft. 30. Whenever observations to the flare consist of horizontal and vertical angular measurements in relation to the astronomical zenith, the observation Equations 26.24 and 26.25 can be used as given. In this case, we start with an approximate position for the flare as well as for the ground stations which we are required to fix; we form two observation equations for each line, containing corrections to the coordinates of the flare and of the ground station as well as the astronomical and refraction correc- tions. There are, of course, no corrections to the coordinates of known ground stations in these equa- tions. For simultaneous 2 observation of one position of one flare from three known ground stations and one unknown ground station, we have, for example, eight equations between six unknowns, assuming that full astronomical observations have been made at all ground stations and that valid corrections have been made for refraction. Theoretically, we have enough equations to fix the unknown station, and the equations might prove to be sufficient in practice if the stations covered a considerable range in altitude. However, it is usual to observe several flares in widely separated positions from the same ground stations, including several unknown stations, and also to make several observations to the same flare as it falls, treating the position of the flare as unknown for every such additional observa- tion. In this way, we can form enough observation equations to dispense with astronomical measures, if necessary. The determination of geodetic heights from vertical angles is weaker than from reciprocal observations between ground stations because only one end of each line can be occupied. However, if the flares are dropped roughly midway between the known and unknown ground stations, residual errors of refraction tend to cancel as between the heights of ground stations, although the (unwanted) flare heights are seriously affected. Additional ob- servation equations can, of course, be formed and used in the adjustment for observations between such ground stations as may be intervisible. STELLAR TRIANGULATION 31. Simultaneous photography from two or more ground stations of a luminous beacon — a rocket 2 The observations are synchronized by radio signals. In some systems, the circle readings are photographed by small cameras operated by the radio signals, and all the observer need do is to keep the flare continuously intersected. flash or a flare dropped from an aircraft — against a background of stars was first proposed and ex- tensively used by Vaisala in Finland about 1946. In principle, the direction to the beacon is inter- polated by measurements on the photographic plate from the known right ascensions and declinations of the background stars; a single photograph can be considered an observation for the right ascen- sion and declination of the beacon. A simultaneous observation from a second ground station gives the orientation of the plane containing the beacon and the two ground stations. Two such planes for two different positions of the beacon intersect in the line joining the two ground stations, whose direction is accordingly determined in the right ascension- declination system. This direction is, of course, "absolute" in the sense that it does not depend on the local direction of the plumbline, which would be the case if horizontal and vertical angles were measured. 32. To develop the theory in more detail, we shall use the same Earth-fixed, right-handed orthogonal triad of unit Cartesian vectors A r , B r , C r as set up in § 12-8 and § 12-10 to define latitude and longi- tude. As usual, C r is parallel to the Earth's axis of rotation, the plane A r , C determines the origin of longitude or hour angle, and B' completes the right- handed triad A r , B r , C r . We define the declination (D) of a unit space vector L r as the angle between L r and the plane A r , B' — positive north. This defini- tion follows the usual astronomical convention. The origin-hour angle (H) of the vector L r is defined as the angle between the planes A r , C and //, C — positive in the direction of a positive right-handed rotation about C r , that is, positive eastward from A r toward B r . This definition reverses the sign of the usual astronomical convention for the hour angle, which is positive westward, but enables us to adhere to normal mathematical conventions as used throughout this book, and to relate declination and origin-hour angle directly to latitude and longi- tude (which also is positive east). It will then be apparent that Equations 12.003, 12.004, and 12.005 hold equally well for declination D and origin-hour angle H in relation to the vector L'\ which can accordingly be expressed as Z/ = (cos D cos H)A r + (cos D sin H)B r + (sin D)C r . 26.36 The origin-hour angle H is the right ascension of the direction L r minus the local sidereal time at the origin, both expressed in angular measure. The local sidereal time at the origin is the Greenwich sidereal time plus the astronomical longitude of 248 Mathematical Geodesy the origin relative to Greenwich, measured as always positive eastward. 33. The right ascension and declination from a ground station to the beacon at a given time, obtained from plate measurements, give H, D for substitution in Equation 26.36. (We shall defer a description of the process until we deal with the modern techniques of satellite triangulation in §26-43.) A simultaneous observation from another ground station provides a similar equation for the unit vector L 1 from the second station to the beacon. We know that the unit vector joining the ground stations is coplanar with V and L'\ a fact which gives us one relation between the hour angle and declination of the line joining the ground stations. Repetition of the whole process from the same two ground stations to another position of the beacon will then determine //, D for the line joining the ground stations. 34. The situation is illustrated by the spherical diagram, figure 31, in which the unit vectors to the Figure 31. beacon are represented by L'\ L' and a unit vector perpendicular to the plane of L'\ V is G"\ the pole of the great circle L'D . The unit vector of the line joining the ground stations is shown as G'\ neces- sarily on the great circle L'L' because the three vectors are coplanar. The hour angle and declina- tion of G" are obtained from the triangle Z/-pole-()' in figure 31, and then Equation 26.36 gives Q '= A '(sin H cos a — sin D cos H sin a) + B'{ — cos H cos a — sin D sin H sin a) + C 1 (cos D sin a) 26.37 in which the quantities within parentheses are the Cartesian components of G" and must therefore have the same values at all points of the great circle L'L'G '. We can, of course, compute a from the ele- ments //, D, H, D of figure 31 so that these com- ponents (/, m, n) are known. If (//), {D) are the origin-hour angle and declination of G'\ the line joining the ground stations, we then have sin (//) cos (a)— sin (D) cos (//) sin (a) = I — cos (H) cos (a)— sin (D) sin (//) sin (a) = m 26.38 cos (D) sin (a) = n, two equations of which are independent. From the second position of the beacon, when {a) becomes (a*), we have similarly two independent equations connecting (//), (D). and (a*) which are easily solved to determine (//), (D), (a), and {a*). The difference (a) — (a*) is the angle between the planes containing the ground stations and one angle each of the beacon positions. The magnitude ol this "angle of cut" is a measure of the geometrical accuracy of the result. 35. We could compute (//), (D) in this manner for each pair of simultaneous observations to the beacon and use the results as observed values in a system of observation equations. However, (H) and (D) are a long way from the actual obser- vations, which are measures of rectangular coordi- nates on the photographic plates. Moreover, it would be difficult to ensure a proper weighting of such derived "observations," especially when simultaneous observations are made from more than two ground stations or when it is difficult to select pairs of observations with a good "angle of cut" without using the same observation twice. For these reasons, it will usually be better to form observation equations for each observed direction to the beacon; we shall now do this. Observation Equations for Directions 36. If H, D are the origin-hour angle and declina- tion to the beacon — unit vector // — we define two auxiliary unit vectors M r (origin-hour angle //, declination D — ^tt) and /V' (origin-hour angle H + ^tt, declination zero) as M' = (sin D cos H)A' +(sin D sin H)B r -(cos D)C r 26.39 26.40 /V = - (sin H)A r + (cos H)B r . The triad L r , M r , /V r is right-handed in that order. Because the Cartesian vectors A r , B'\ C r are fixed. Internal Adjustment of Networks 249 we can then differentiate Equation 26.36 to have 26.41 d(L r ) = -M r dD + (cos D)N r dH. Substituting in Equation 26.10, where the triad / r , m r , n r is any right-handed unit orthogonal set, and equating coefficients of M r , N r give us the observation equations 26.42 sdD = -M s dx s + M s dx s 26.43 (5 cos D)dH = N s dx s -N s dx s in which dx s , dx s are, respectively, corrections to initial approximate coordinates of the ground station and of the beacon, whose declination and origin-hour angle are D, H in the direction from the unbarred (ground) to the overbarred (beacon) end of the line. The components of the auxiliary vectors M s , N s (and of the parallel vectors M s , N s at the beacon end of the line) are computed from Equations 26.39 and 26.40 in the same coordinate system as dx s , dx s . On the left-hand side of the observation equations, 5 is the length of the line computed from the approximate end coordinates and dD, for example, is the measured declination minus the declination computed from the approximate end coordinates. 37. The observation equations must necessarily contain corrections to the initial approximate posi- tion of the beacon, which we do not usually require. These unwanted corrections can, however, be elimi- nated at some suitable stage — either before or during the formation of the normal equations. Time Correction 38. In relating the photographic image of the beacon to the stars, we are in effect observing the right ascension of the beacon in the system of the star catalog. The observed origin-hour angle is then obtained by subtracting, from the right ascension of the satellite, the local sidereal time at the origin of the instant when the beacon was photographed, while the computed origin-hour angle is obtained from the approximate end coordinates of the line. If, however, we do not know the precise local sidereal time of the observation, then we must assume an approximate value to, which must be corrected to (to + dt). This assumption amounts to adding a time correction dt, expressed in radian measure like dx/s, to the right-hand side of Equa- tion 26.43 and to evaluating this extra unknown together with the corrections to the end coordinates of the line in the solution of the observation equa- tions. The time correction dt would, however, be the same for all stations engaged in the simul- taneous observation of the beacon because it is a correction to the assumed local sidereal time of the observation at the origin, which is common to all stations. This correction could also be the clock correction to one particular (master) station clock used to define / () . An additional clock correction would have to be included for every other station clock which has not been synchronized to the master clock. Observation Equations in Cartesian Coordinates 39. In this case, there is evidently some ad- vantage of working in Cartesian coordinates. From Equation 26.36, we have at once the difference in Cartesian coordinates of the two ends of the line referred to the axes A 1 ', B' , C'\ 26.44 x — x — s cos D cos H y — y—s cos D sin H z — z = s sin D. The Cartesian components of the auxiliary vec- tors from Equations 26.39 and 26.40 are M r = M r = (sin D cos H, sin D sin H- cos D) N r = N r = (- sin H, cos H, 0) 26.45 so that the observation equations become (Observed Minus Computed) Declination = — sin D cos H (dx-dx)ls — sin D sin H (dy-dy)ls + cos D (dz~— dz)ls 26.46 (Observed Minus Computed) Origin-Hour Angle = — sec D sin H {dx — dx)/s + sec D cos H (dy— dy)/s. 26.47 Equations 26.44 are used to obtain computed values of s, H, D from initial approximate values of the end Cartesian coordinates. Either these values of H, D or the observed values may be used for the coefficients on the right of the observation equations. 40. Approximate positions of the ground stations will usually be more accurately known than the position of the beacon. In that case, we could form 250 Mathematical Geodesy Equations 26.44 for the other ground station, solve with Equations 26.44 to determine the two dis- tances to the beacon, and thus obtain the approxi- mate position of the beacon from the observed values of hour angles and declinations. Once we have decided on the approximate positions, we must, of course, use them in Equations 26.44 to obtain accurate "computed" values of D and H. A similar procedure can be followed if the observation equations are formed in geodetic or geocentric coordinates. Observation Equations in Other Coordinate Systems 41. The observation equations can be solved to give corrections to geodetic coordinates instead of Cartesian coordinates by substituting the geodetic components of the auxiliary vectors in the observa- tion Equations 26.42 and 26.43. From Equation 26.39, we have M x = (sin D cos //, sin D sin //, -cos D){A S , B s , C s }, which, by using Equation 19.35, can be written as (M, , M 2 , M 3 ) = (sin D cos H, sin D sin H, — cos D) 26.48 x Q'S, an equation holding true in any (a>, 4>, N) coordinate system, provided the appropriate S-matrix is used from Equation 19.32. In geodetic coordinates, the S-matrix is / (v + h) cos (f> 0\ (p + h) , V 1/ while Q' is obtained from Equation 19.26. Ex- pansion of Equation 26.48 then gives M\= (v + h) cos cf> sinD sin (H — cu) M 2 = - (p + h ) {sin sin D cos (H-oj) + cos (f> cos D} Mx — cos 4> sin D cos (H — , M s ) = (sin D cos //, sin D sin H, - cos D) 26.50 X Q'S; this equation expands to give the same result as Equations 26.49, provided v, p, h, , a) are over- barred. This fact means simply that the approxi- mate values of these five quantities at the beacon must be substituted in Equations 26.49. In the same way, we have 26.51 (N u N 2 ,N s ) = (-smH,cosH,0)Q T S, which expands to N\ = (v + h) cos (f> cos (H — o») A^2 = (p + h ) sin , oj at the beacon in Equations 26.52. 42. The observation equations can also be written in geocentric coordinates, which are more closely related to the observed right ascensions and dec- linations. In that case, the A r -surfaces are spheres of radius r centered on the origin, and latitude and longitude refer not to the astronomical or geodetic zenith, but to the radius vector. Because Equations 26.48 and 26.51 hold .true in any (o>, (f), N) system, we have merely to substitute r for (v+h) or (p + h) in Equations 26.49 and 26.52 and interpret (a>, ) as the geocentric longitude and latitude. If com- putation is to be done directly from Equations 26.48, etc., in matrix form, the appropriate S-matrix is now r cos r 1 and Q 7 is given by Equation 19.26 for the geocen- tric latitude and longitude. SATELLITE TRI ANGULATION - DIRECTIONS 43. Although there are other means of fixing positions by observations on near-Earth artificial satellites, we shall understand the term "satellite triangulation'" to mean stellar triangulation using the satellite as a beacon, which either emits flashes on command or reflects sunlight. In the latter case of a passive balloonlike satellite, accurate timing is necessary and can be obtained to ensure that observations from two or more ground camera stations are automatically synchronized. If the same "instantaneous" flash or series of flashes is observed by several ground stations, the event still has to be timed, but less accurate timing is neces- sary because the stars, which are required to de- termine the orientation of the camera, move more Internal Adjustment of Networks 251 slowly than the satellite; synchronization of the observation to the satellite is achieved by the flashes themselves. Although some tracking cameras are sidereally mounted so that the stars appear as point images while a continuously illuminated satellite appears as a trail, we shall consider only the case of rigidly fixed cameras so that both the stars and a continuously illuminated satellite form trails on the photographic plate. The trails are "chopped" by shutter closures at accurately recorded times when the image of the star or satellite is con- sidered to be at the break in the trail. Before the operation, the camera is set in altitude and azimuth from predicted orbital data so that the satellite trail will pass close to the center of the plate. The star trails are chopped in a distinctive manner be- fore and after the satellite pass, and between passes. Measurement of the plate coordinates of the breaks in the star trails determines the orientation of the camera and of the photographic plate as well as a number of calibration parameters, and gives assur- ance that the camera has not moved between star calibrations. Finally, the known orientation and calibration enable the direction to the satellite to be computed from plate coordinates of breaks in the satellite trail. Variations in procedure do not seriously affect the method of reducing the ob- servations now to be given in barest outline. For example, the only difference in the case of a flash- ing satellite is that measurements are made to point images and not to trail breaks. There are considerably fewer images to measure with a flash- ing satellite; whereas, a continuously illuminated satellite can be chopped all the way across the plate until it ceases to be illuminated by the Sun. Choice of Coortlinate Systems 44. The first step is to determine the direction of the camera axis in a specified Cartesian coordinate system. There are three main possibilities: (a) An inertial system whose z-axis is parallel to the axis of rotation of the Earth and whose xz-plane defines the origin of right ascensions. (b) An Earth-fixed system, as used so far through- out this book, whose z-axis is parallel to the axis of rotation of the Earth and whose xz-plane defines the origin of astronomical longitudes. The base vectors in this system in our usual notation are A'\ B'\ C'\ as defined in Chapter 12; we shall denote coordi- nates in this system by (X, Y, Z). The relation between this system and the inertial system are described in § 26-32. (c) A "local" (X, Y, Z) system in which the geo- centric latitude and longitude of the Z-direction are (, to) and the origin in the (X , Y, Z) Earth- fixed system is {X», F<>, Z»). The P-direction will be northward in the plane containing the Z-direction and the axis of rotation. The ^-direction will be eastward in accordance with our normal right- handed conventions. The coordinate axes are accordingly (A.'', /x' , v r ) in a spherically symmetric (to, 4>, N) system; by contracting Equation 12.013 with a position vector from the new origin, we have ('X— Xo\ /—sin to —cos to Y — Yo 1 = 1 cos to —sin oi z-zj \ o oi /l X sin cf) — cos (/> \0 cos 4> sin ) are approximately the geocentric coordinates of the camera station in the (X, Y, Z) system. However, both (Xo, Yo, Z ) and (to, (/>) are independent. We can consider that (to, (/>) are two fixed parameters whose values are chosen to be approximately the geocentric coordinates of the camera station in order to facilitate the application of corrections for astronomical refraction. As we have seen in Chapter 24, these fixed parameters are presently based on a spherically symmetric model atmosphere. Indeed, this coordinate system is introduced solely for the purpose of evaluating and applying refraction corrections. 45. The direction of the camera axis and the orien- tation of the plate can be determined in any of these three systems, provided the star directions are transformed to the same system. If we use the iner- tial system, updated places derived from the star catalogs can be used after correction for precession, nutation, annual and diurnal aberration, and astro- nomical refraction; the camera orientation will be in terms of right ascension and declination, as also will be the direction to the satellite. In this case, we shall have to transform to an Earth-fixed system before combining results from different stations at different times. 46. If we use the Earth-fixed system for camera calibration, we shall have to convert right ascen- sions to origin-hour angles by subtracting the local sidereal time at the origin, which means that we must be in a position to apply clock corrections 252 Mathematical Geodesy before we start, although the observation equations could be modified to include time corrections similar to those given in § 26-38. Alternatively, we could set up a temporary Earth-fixed system with an ap- proximate sidereal time and apply a final correction by means of a longitude rotation of the coordinate system into the definitive Earth-fixed system. The final correction could be applied in conjunction with rotations for polar movement, which will be dis- cussed in § 26-62. The camera orientation and direction to the satellite will be determined in terms of declination and origin-hour angle. There will be no need to transform to any other system for the network adjustment. 47. If we use the "local" system, we must, in addition, transform the declinations D and origin- hour angles H of the stars to azimuths a and zenith distances /3 in the local system by means of the relation sin a sin p\ /cos D cos H 1 cos a sin/3 =N T lcosD sin// cos /8 / V sin Z) which is easily obtained from Equation 26.53. We shall then obtain the camera orientation and direc- tion to the satellite in terms of azimuth and zenith distance, and shall transform to the fixed-Earth system for the network adjustment by means of the inverse of Equation 26.54. As explained in § 26-44(c), we must use the same approximate values of the geocentric coordinates (oj, (/>) of the camera station in the matrix N for both direct and inverse transformations. 48. The local system is perhaps most often used for the star calibration. As a means of wider illus- tration, we shall, nevertheless, start with the Earth- fixed system and transform only the updated places of the stars to the local system in order to evaluate and to apply refraction corrections, while still determining the camera orientation and direction to the satellite in the Earth-fixed system. This system would require less modification if a different method of refraction correction is introduced later; but in any case, once any of the systems is fully understood, it is a simple matter to derive the equa- tions for any other system. The Basic Photogrammetric Equations 49. If H c , D c are the origin-hour angle and declina- tion of the camera axis, it will be clear from Equation 12.012 that gives the Cartesian coordinates of the point (X, Y, Z) in a system whose z-axis is the camera axis and whose y-axis lies in the plane of the z-axis and the original C r -axis. On the photographic plate in the Northern Hemisphere, the y-axis joins the principal point (where the camera axis cuts the plate) to the photographic image of the celestial North Pole. The x- or y-axes, from which measurements are made on the plate, are, however, given by fixed fiducial marks in the camera; to effect a final rota- tion to this plate system, we introduce a positive rotation k, known as the swing, about the camera axis by premultiplication with the matrix COS K sin k & - sin k COS K 1 Also, we change the X, Y, Z origin to the camera station (Xq, Fo, Z ) by replacing the vector {X, Y, Z} with {(X-Xo), (Y-Y ), (Z-Z )}. In the result, we shall have coordinates of the original object point (X, Y, Z) in the new system, and we have next to find the corresponding coordinates of the photo- graphic image. If A is the distance from the camera to the object point and if d is the distance from the internal perspective center to the photographic image of the point, then we must reduce the transformed coordinates of the object point in the ratio d/A to obtain the coordinates of the image point. Finally, we can change the origin of plate coordinates so that coordinates relative to the camera axis become {(x — x ), (y— yo),/} where (x , yo) are the plate coordinates of the principal point in the new system. In an undistorted perspec- tive, the camera axis — supposedly perpendicular to the plate — cuts the plate in the principal point. The principal distance f is the length of the per- pendicular between the internal perspective center and the principal point. The final transformation is expressed as 2655 (yH^-z, Internal Adjustment of Networks 253 where the rotation matrix is given by (cos k sin k 0\ — sin k cos k 1/ /l \ /- sin // ( cos H c ON X sin /), cos D c 1 1 — cos H c — sin //, \0 - cos D c sin D c / \ 1/ 26.56 and dV {x-x ) 2 +(y-y ) 2 +P (X-X Q ) 2 +(Y-Y ) 2 +(Z-Z V This equation gives (x, y, /) in the same general sense as (X, Y, Z) for small rotations so that x, y are considered to be measured on a positive print covering the object space. If we measure coordi- nates on the original negative — emulsion side up — in relation to the same fiducial mark as the positive ^-direction, then we would measure (x, —y) and should change the sign of y before insertion in Equa- tion 26.53. This rule, of course, assumes normal right-handed coordinate conventions. 50. An alternative form of Equation 26.55 is useful in the present application. If we write the expanded rotation matrix as /win m\i /»i:s 26.57 M= mi X nin m 23 yn-.u rn-M m.33 and eliminate d/A, we have x — Xp _ m u (X — Xu) + m vl (Y— F») + m v .\{Z — Z») / ~ mn(X-X n ) + m-AY- ft) + rmdZ-Z a ) y- y„ = m- n (X - X n )+ m 21 (Y -F„) + m T ,(Z - Z„) / m- n (X-Xo) + m S 2(Y-Yo) + m ss (Z-Zo) ' 26.58 Equations 26.58 are equivalent to the original equa- tions, only two of which are independent, because the scale factor d/A. has the effect of reducing the vectors in Equation 26.55 to unit vectors, while all the rotation matrices are orthogonal. 51. Equation 26.55 or Equations 26.58 are usually known as the projective equations of photogram- metry or as the conditions of collinearity. In deriv- ing these_equations as coordinate transformations, we have, indeed, assumed collinearity of the image and object points and of the perspective center, so that either set of equations represents an undis- torted perspective. Many different conventions for the rotation angles and coordinate systems are used in photogrammetric literature, including some left- handed systems, but the formulas can be reconciled with the normal mathematical conventions used throughout this book by reversing the signs of some coordinates and the directions of some rotations; in whatever order the rotations are made, the final matrix M, connecting the same two coordinate systems, must be the same. Calibration 52. The process of obtaining the orientation of the camera and certain camera constants is very similar to the method of camera calibration from stars described, with a full bibliography, in the Manual of Photogrammetry. 3 53. If the object photographed is a star of declina- tion D and origin-hour angle //, we can write f X-XJ\ /rcosDcosH ! - Y = r cos D sin H Z — ZJ \ r sin D in which r is very large, but is cancelled by A. Equation 26.55 for stars is accordingly /cos D cos H X Xo 1 :>%..:;.*b ' , \ y-y„ j=M I cos D !-!.. 7 / / \ sin with the alternative form from Equations 26.58 of x — xn _ m.\\ cos D cos //+;»i 2 cos D sin H+niy.j sin D f m.n cos D cos H+HI32 cos D sin H+m :ia sin D y — y _ m>\ cos D cos H+m^z cos D sin H+m^ sin D f m :u cos D cos H+m.12 cos D sin H-\-m-. a sin D 26.60 54. Theoretically, these equations are soluble for k, D c , H c , x , y , / from three stars, but even then the solution would not be simple because the equa- tions are not linear in the unknowns. In practice, we require the use of more than three stars to achieve precise results, and we shall have to in- crease the number of unknowns to allow for the fact that we are not dealing with an undistorted perspec- 3 American Society of Photogrammetry (1966), Manual of Photogrammetry, 3d ed., v. 1, 180-194. 254 Mathematical Geodesy tive. Accordingly, the next step is to form differential observation equations in the usual way by partial differentiation of Equations 26.60 with respect to all six unknowns and the measured (x, y) . In the result, dx, for example, will be the measured x minus the computed x, obtained by substituting preliminary values of the unknowns in the first equation of Equations 26.60. We are then able to solve a large number of such observation equations, formed for a large number of stars and appropriately weighted, by the usual least-square processes to provide corrections a?k, dD c , etc., to the preliminary values of the unknowns. 55. We can also add other unknown parameters to the observation equations before solution by expressing their effect on the measured {x, y). For example, considerations of symmetry indicate that the radial lens distortion can be expressed as Ar= for* + *,/* + for 7 in which Ar is the outward displacement from a true perspective position, r is the radius vector from the principal point, and k\, fo, k 3 are unknown param- eters to be derived from the calibration. From similar triangles, we have A* Ay (x — xo) (y — yo) r k^ + kz^ + k^r 6 . The component A.x of the distortion must, for example, be subtracted from the measured x to give the value of x which would be measured on an un- distorted photograph. But the observed x in the observation equation is measured on a true undis- torted perspective. Accordingly, if we insert the actual measured x in the observation equation, we must subtract A* = (x -xo) ( Ai/ 2 + A- 2 r» + for 6 ) from the (observed minus computed) x in the origi- nal equation, and similarly must do the same for y. This relation adds three unknowns to each observa- tion equation. 56. In addition to the use of three parameters to determine the radial lens distortion, it is usual in current practice to introduce parameters to allow for (a) nonperpendicularity of the coordinate axes and other sources of error in the plate-measuring instrument; (b) difference in scale in the x- and y-directions arising from emulsion creep, which is equivalent to the determination of two principal distances; and (c) lens deviation or decentering, nonradial dis- tortion (involving five extra parameters), and cor- rection for nonperpendicularity of the optical axis and the plate. Residual Atmospheric Refraction 57. We have not yet included any correction to the apparent places of the stars for astronomical refraction. One possibility is to convert the apparent places from hour angle and declination to approxi- mate azimuth and zenith distance by Equation 26.54, using approximate values $, w of the geo- centric latitude and longitude of the camera station. The zenith distances are then corrected for refrac- tion from tables, and the corrected star positions are converted back to hour angle and declination, using the same values o/$, oj, before insertion into Equations 26.60 where H, D would then be held constant during differentiation. Errors in the assumed values of <£, o», affecting only the refrac- tion correction through the corresponding error in zenith distance, are of little consequence. However, it is usual in current practice to determine residual refraction parameters — in much the same way as lens distortion and other parameters — from the solution of the calibration observation equations. For this purpose, we simply combine Equations 26.54 and 26.59 to give x — Xa 1 sin a sin /3 2\ y~y» =MN cos a sin/3 / ] \ cos/3 in which the rotation matrix is now "ii "12 n^ \\ \ n lx n-2-2 "23 | n 31 n :vl n 33 , whose components are obtained from Equations 26.56 and 26.53 and contain k, D c , H c , oj, <\>. Equa- tions 26.60 become x — x _ n n sin a sin fi + n v2 cos a sin /3 + "i;s cos /3 / n 3 \ sin a sin /3 + ";s2 cos a sin fi + n-.w cos ft y — y»_ n-2\ sin a sin /3 + "22 cos a sin fi + nij cos /3 / "si sin a sin (3 + " ;i 2 cos a sin (5 -f n A3 cos /3 26.62 In forming the observation equations by differen- tiation, we hold co, 0, a fixed and equate d/3 to the expression for the refraction correction. If we insert apparent zenith distances into Equations 26.62 to derive the computed (x, y), d/3 will be the (true minus apparent) zenith distance, which is the same Internal Adjustment of Networks 255 as the normal convention for refraction. The sign of dft is of little importance, however, because a wrong sign would simply result in reversed signs for the parameters in the refraction equation, which we shall now consider. 58. The expression for astronomical refraction, introduced by Hellmut Schmid, is T*iW[Ki tani0+K 2 tan :! $0 + £:, tarrHfl + k tan 7 £0] where the refraction is in seconds of arc, and r=*/273.16, t = observed temperature at camera station in °C, r=/y(76or 2 ), Po = observed pressure at camera station in mm., Hg., tan = O.1147618r'/ 2 tan 0, j8 = apparent zenith distance. The formula is to some extent empirical, but does follow Garfinkers theory (§ 24-67) by using a modi- fied zenith distance in the classical expansion in powers of the tangent. In fact, the formula fits Garfinkel's model very accurately for zenith distances of less than 75°. In using the formula for satellite triangulation, the four parameters K\, Kz, K.u Ka are determined at each calibration from the observation equations. Direction to Satellite 59. The camera calibration provides data for correcting the (x, y) coordinates of each satellite image through the now-known parameters for vari- ous distortions, etc., listed in § 26-55 and § 26-56, and for atmospheric refraction. Also, a correction should be applied for differential aberration, that is, for the traveltime of light to the camera station from the satellite in relation to the stars. In addition, the parallax correction (Equation 24.72) for differential refraction has to be applied in a sense opposite to the astronomical refraction. Finally, a correction is applied for phase angle, arising from unsymmetrical illumination of a passive satellite by the Sun. 60. In current (1968) U.S. Coast and Geodetic Survey practice on the worldwide satellite triangu- lation, the next step is to reduce all the satellite images, which are exposed (or chopped) at equal intervals of time on each pass, to a single equivalent or "fictitious" image. Other organizations naturally use somewhat different procedures, especially when there are fewer images, but the principles are much the same. The reduction to a single "ficti- tious" image is done by fitting the corrected x- and y-coordinates of the images separately to poly- nomial functions of time, usually of the fifth order. A time is then selected for all simultaneous obser- vations of the satellite involving two or more plates, so that the satellite image at that time would have been formed as near as possible to the center of each plate. The actual (x, y) of the satellite at this selected time is then computed from the polynomial, after applying clock corrections and after adding the time that light takes to travel from the satellite to the camera. The result is equivalent to a single meaned position of the satellite, simultaneously observed from two or more ground stations, at a given time. Net Adjustment 61. The origin-hour angle H and declination D of the satellite at this mean position may now be computed from the inverse of Equation 26.59, that is, (cos D cos H\ /(x—Xo)ld\ cos D sin // =M' (y-y„)ld\ sin/) / \ fid ) where we have d 2 = {x — x») z -\- (y— yo) 2 +/ 2 , using, of course, the calibrated values of k, H c , D c , Xo, yo, f. If the difference in scale for x and y is signifi- cant, (x — Xu) and {y — y») could first be corrected to a mean /. 62. We have so far worked in coordinate systems, oriented with respect to the actual "instantaneous" pole or rotation at the time of observation. If polar movements, as discussed in § 21-55, are found to be significant, the coordinate system could be changed at this stage by applying the appropriate rotation matrices to the left-hand side of Equation 26.63, or by applying the transpose of these rotation matrices to the right-hand side, with consequent modification of M T . 63. Explicit formulas in terms of the components of the original matrix M of Equation 26.57 are tan// = tan D = sin HX = cos//X 26.64 m,->(x — go) + m>>(y — y<») + m-.viif) m n (x—xo) + m2i(y—yo) + m 3 i(j) m v . i (x — x i )) + m>i(y — yu) + m-.^f) m vi (x — Xo) + m-zziy — y») + m^if) miaU- ■*(>) + m-2-Ay — y») + m M {f) m u (x — Xt)) + m-2i(y — y») + m M (f) 256 Mathematical Geodes 64. Observation equations for the network can now be formed exactly as, for example, in §26-36 and combined with duly weighted equations for such measured distances, etc., as are to be included in the adjustment of the network. 65. An alternative method is to form observation equations for the net adjustment by differentiating Equations 26.58 with respect to x, y, X, F, Z, Xo, Fi>, Zo, holding all other quantities fixed by the calibration and thus deriving corrections to initial approximate positions of the satellite (X, F, Z) and of the ground station (AH, F», Z»). Corrections to the satellite position are not, of course, required for triangulation purposes and can be eliminated during the solution. 66. The choice between the two methods does not involve any question of principle; we are entitled to consider that (H, D) are observed as much as the averaged (x, y). A decision will, no doubt, rest on what programs are available; if, for example, pro- grams designed primarily for photogrammetric purposes are available, the choice will probably fall on the second method. SATELLITE TRIANGULATION - DISTANCES Observation Equations 67. If electronic distance measurements are made to the satellite from ground stations, the observation Equation 26.08 can be used in any coordinates; we have simply to assume initial coordinates for the satellite and for the ground station, and then use these coordinates to compute components of the unit vector joining the two ends of the line in the same system. In Cartesian coordinates, for example, if quantities at the satellite are overbarred, we have (Observed Minus Computed) Distance = (dx — dx) (x — x)/s+ (dy — dy) (y — y)/s + (dz — dz) (z — z)/s. 26.65 68. To derive the observation equations in any (oj, 0^ ! ip + h) 1/ 69. In geocentric coordinates, which are oftei the most suitable in dealing with satellites, we havi x = r cos (f) cos co y= r cos $ sin a> z = r sin c/>: the S-matrix is given by (r cos 4) 0\ > ! I r 0|- ly 70. In some systems, such as SECOR, only dis tance measurements are made and must be mad simultaneously from a number of ground stations For example, if simultaneous measurements ar made from four ground stations and the position of three ground stations are known, we have onl four observation equations and six unknowns, observations are made from the same ground sti tions to another, widely different position of th satellite, we add four equations and only thre unknowns — that is, corrections to the secon position of the satellite — so that the problem c fixing the position of the unknown ground statio becomes soluble from three satellite positions. I practice, many observations are made over a Ion period to many satellite positions. Net Adjustment 71. Observation equations for distances can b combined with observation equations for directions only if simultaneous measurements are made to th same position of the satellite. However, it is pos sible to form normal equations separately for th direction and distance measurements. The two set of normal equations could be appropriately weighte* and solved together, but the extent to which such Internal Adjustment of Networks 257 combined adjustment could be done without vitiat- ing the more accurate measurements would re- quire statistical study in each case. 72. If distances are measured in conjunction with directions, the same coordinate system would have to be used for both sets of observation equations. If distances are measured separately, there would be some advantage in using the simpler Cartesian form of the observation equations. Simultaneous meas- urement of both distance and direction would give the complete vector to the satellite and, by sub- traction, would give the complete vector between ground stations, which could be treated as an obser- vation without deriving corrections to the satellite position. Each such ground vector would provide two observation equations for direction and one observation equation for distance. LUNAR OBSERVATIONS 73. We can fix positions on the Earth by photo- graphing the Moon against a background of stars, in much the same way as by photography of any other Earth satellite. One difficulty is that the stars and the Moon require different exposures, but this difficulty has been successfully overcome by the Markowitz moon-camera, designed for and widely used during and following the International Geo- physical Year 1957-58. The camera is equatorially mounted to hold the exposure of the stellar back- ground. Moonlight is reduced by a parallel-plate filter which can be rotated, in much the same way as the parallel-plate micrometer of a precise surveying level, to hold the photographic image of the Moon fixed in relation to the stars. The time of an observation is considered to occur when the rotating filter introduces no relative displacement between the Moon and the stars. Another difficulty arises from irregularities in the Moon's limb; these irregu- larities have always limited the accuracy of geodetic observations, such as the determination of longitude from lunar occultations of stars. Improved knowl- edge of the topography of the Moon may before long enable us to correct these irregularities; meanwhile, the Markowitz system reduces the effect of these irregularities by obtaining the right ascension (or hour angle) and declination of the Moon's center from photographic measurement of a large number of stars. Apart from the fact that the Moon costs nothing to launch, a considerable advantage of the system is that the elements of the Moon's orbit are accurately known; this advantage makes simul- taneous observation unnecessary, although the observation must be accurately timed. Photography of the Moon in at least two different positions from the same station will fix the position of that station in all three coordinates, relative to the center of mass of the Earth. At this time (1968), the results will be of lower accuracy than those obtainable from artificial satellites, but this fact may not be always true. 74. The Ephemeris'* gives the right ascension ol the Moon. We shall reduce the right ascension to origin-hour angle by subtracting the local sidereal time at the origin, which may, of course, be one of the points we propose to fix. The origin-hour angle will be the longitude a» in a geocentric system whose zero of longitude is the plane parallel to the axis of rotation of the Earth and parallel to the astro- nomical zenith at the origin. The listed declination of the Moon will be the same as its geocentric lati- tude (£. In addition, the Ephemeris gives us the parallax tt of the Moon's center; the parallax being related to the radius vector r from the center of mass of the Earth by the formula 26.68 r=etcosec7r in which (/ is an assumed equatorial radius of the Earth. 75. As usual, we start with initial approximate values of (w, $, r) for the Moon at the time of observation and also of (w, , r) for the ground station in the same geocentric (spherical polar) coordinate system. The origin of longitudes co is the plane containing the axis of rotation and the astronomical zenith at the station selected as origin. The approximate values (oj, (/>, r) are used to find computed values H, D, s of the origin-hour angle, declination, and length of the line joining the ground station to the Moon's center from the equations 5 cos D cos H = r cos cos aj — r cos

s cos D sin H—r cos (j) sin a> — r cos $ sin a> 26.69 s sin D — r sin $ — r sin dls - M-idr/s + M x du)ls + M-zd(f>ls + Msdr/s 26.70 (Observed Minus Computed) Right Ascension (or Origin-Hour Angle) of the Same Line = Nt sec Ddw/s + N; sec Dd^/s + N-.i sec Ddr/s — N] sec Ddoj/s — N 2 sec Dd/s — N :i sec Ddr/s. 26.71 It is assumed that the local sidereal time at the origin for the instant of observation is accurately known so that in this case there is no need to in- clude a time correction in the second observation equation. 76. If the position of the Moon really were ac- curately known at the instant of observation, we could put dco, d(\>, dr equal to zero in these equa- tions. Unfortunately, we cannot be sure that the Universal Time of the observation is exactly the same as the Ephemeris Time used as an argument in the Lunar Ephemeris; there is a difference, which varies slowly, between the two times. The simplest way to overcome the difficulty is to envisage a correction dt to the time of observation, to find doi/dt, etc., from the tabular differences, and to replace da), etc., in the observation equations by (d(bldt)dt. This procedure reduces the number of unknowns by two. The four unknowns dt, da), dxj), dr can be determined from observations to two widely separated positions of the Moon, provided dt can be taken as the same for both. In practice, the Moon will be photographed in many positions from several ground stations; it may also be pos- sible to derive corrections to the orbital elements, or to the position of the Moon's center, by expressing dxij, etc., in terms of these elements. 77. Another difficulty arises from the indefinite- ness of the constant d in Equation 26.68. However, we can take d as unity, thereby reducing the scale of the whole model so that we finally determine the radius vector of a ground station as rjd. We must also start with an approximate value of rjd, then s in Equations 26.69 becomes s/d. A measured ter- restrial distance between ground stations would then serve to scale the model and to determine the constant d. LINE-CROSSING TECHNIQUES 78. As a final example of the formation of differ- ential observation equations, we shall consider such systems as hiran where slant radar ranges are measured from two ground stations (Si, S->) to an aircraft (A ) flying a straight-and-level course across the line joining the two ground stations, which are usually not intervisible. Continuous measurements are made during the crossing; the minimum sum of the two ranges, corrected for refraction, is used to determine the distance between the two ground stations. It is assumed in the usual method of reduction that the minimum position occurs when the plane SiAS-z is vertical at A. The limitations of this assumption can be seen at once by considering the aircraft course as tangential to a prolate spheroid whose foci are Si, S>. The sum {S\P-\- PS->) is the same for any point P on this spheroid and is less than for any point Q on the straight aircraft course external to the spheroid, so that the minimum posi- tion occurs at the point of contact of the course with the spheroid. The usual assumption is accord- ingly justified only if (a) the aircraft course is per- pendicular to S1S2 (this situation is usually not the case), or (b) the aircraft crosses in the midway position. The problem can, however, be solved simply and rigorously in three dimensions with- out making any such assumptions. 79. We shall denote values of quantities at the aircraft position A by an overbar, and at the ground station S-> by a double overbar. The coordinates of Si, A, So are then x'\ x'\ x' . Unit vectors in the directions S\A, AS-> are p 1 ', q'\ and the unit aircraft course vector is a'\ Parallel vectors at the three points are denoted by appropriate overbars; for example, parallels to the course vector at Si, A, S->. respectively, are a' , a' , a'. The slant ranges S\A, AS->, corrected for refraction, are u, v. 80. Equation 26.08 for the variation of the two slant ranges is then 26.72 26.73 du = p r dx r — p r dx r dv = q r dx r — q r dx r . Internal Adjustment of Networks 259 To establish the minimum position, we first assume that Si, S2 are fixed and that the aircraft alone moves by dx r , while dx r , dx' are zero. At the minimum posi- tion, we have also du-\~dv = so that the minimum condition is 26.74 p r dx r — q r dx r But dx r is proportional to the contravariant course vector a r , which reduces the minimum condition to 26.75 p r (i r — Qra r — cos P = cos Q where P and Q are the angles that the aircraft course makes with S\A and AS 2 , respectively. The equality of these angles is the correct minimum condition. 81. Next, we suppose that the aircraft course a r remains fixed, and we seek corrections dx r , dx r , dx r to initial approximate positions of Si, A, S2. The correction positions of the three points must satisfy the minimum condition, Equation 26.75. The changes in cos P, cos Q arising from dx r , dx r , dx r are given by u X {final (cos P) minus initial (cos P)} — ud(cos P) — ud(a r p r ) = ua r d(p r ) = a,(dx r — dx r )— (cos P)du where we have used Equation 26.07 and v X {final (cos Q) minus initial (cos Q)} = vd(cos Q) = vd{a r q r ) — va r d(q r ) = a r {dx r — dx r ) — (cos Q)dv. Subtraction of these two equations, after equating the final values of cos P and cos Q to satisfy the minimum condition, gives initial (cos Q) minus initial (cos P) u \u V 26.76 1 = _ d u n , dv _ a r dx r cos P-\ cos (J v a v in which cos P, cos Q are initial values computed from the initial approximate coordinates and du (or dv) is the observed minus the computed value of u (or v). The P, Q, u, v and the components of the vectors need to be accurately computed even though the aircraft course is only roughly known. Equations 26.76, 26.72, and 26.73 can be used either as con- dition equations or as observation equations in conjunction with any other measurements which may have been made to connect Si, S2. If one end of the line is fixed, for example Si, then we have dx r — 0, and the equations are somewhat simplified. 82. Although the equations are true in any co- ordinate system, provided components of the vectors in the same system are used, it will be usual to work in geodetic coordinates. Azimuths, zenith distances, and distances between the initial approx- imate positions of the ground stations and the air- craft are computed from Equations 25.18. We can then expand Equations 26.72 and 26.73 exactly as in Equation 26.30. If the azimuth of the level air- craft course is A, then we have ci\ = (v + h) cos 4> sin A a2 — {p + h) cos A a 3 = 0, and components of the parallel vectors a r , a r are found as often before from Equation 19.39. Lastly, if a, /8 are the azimuth and zenith distance from Si to A and if a, (5 refer as usual to the same direc- tion at A in the same sense, then we have cos P — p r a r — sin A sin a sin /3 + cos A cos a sin /3 26.77 =sin/8 cos (A — a), with a similar equation for cos Q. 83. If the only measurements connecting the ground stations are aircraft crossings, it will be impossible to determine corrections to geodetic heights, and the terms containing dh, dh, dh must be dropped. In a simple trilateration, for example, where a third ground point is to be fixed from two known points, there would be only the three Equa- tions 26.72, 26.73, and 26.76 for each of the two sides; these six equations could do no more than determine corrections to the latitudes and longi- tudes of the third point and of the two aircraft posi- tions. Even though we do not require the aircraft positions, corrections to them must, of course, be left in the equations. The result is not very sensitive to height changes, but the omission of the dh -terms must to some extent affect the determination of latitude and longitude; this omission must be ac- counted as a weakness of the method. 84. If the initial approximate positions are within 15 seconds of the truth, and this degree of approxi- mation can usually be arranged by rough spherical computation and by placing the aircraft along the line in simple proportion to the measured ranges, then a single solution provides results correct to about 2 feet. In a test case of a single trilateral. 260 Mathematical Geodesy deliberately rough initial values of latitude and solution gave results within 0.025 second of correc longitude of the unknown ground station proved values. Movements of the aircraft are not very sensi to be, respectively, 8 minutes and 5 minutes adrift, tive. Equation 26.76 is soon satisfied; when tha and the initial aircraft position was 3 minutes adrift. situation occurs, the corrections to the aircraf The first solution averaged about 14 seconds adrift, position have the same coefficients in the remaining the largest difference being 47 seconds in the longi- equations because of Equation 26.74, and thus cai tude of the unknown ground station. The second be eliminated. CHAPTER 27 External Adjustment of Networks CHANGE OF SPHEROID 1. If we retain the same origin and the same Cartesian vectors, it is evident that the (x, y, z) coordinates of all points in space will be unchanged. To derive the changes in geodetic coordinates result- ing from changes da, de in the major axis and in the eccentricity of the base spheroid, we will need to differentiate Equations 25.15 for dx= dy= dz = 0. Because we have tan (d — y/x, there will be no change in longitude, and so we will need to differentiate only (* 2 + y 2 ) 1/2 = (v + h) cos 4> and 27.01 z = (e 2 v + h) sin(/>, with x, y, z constant. In the last equation, e is the complementary eccentricity given by e 2 = 1 — e 2 . 2. From Equations 18.55 and 18.54, we have dv/da = v/a dv/de = {ele 2 )p sin 2 $; from Equations 22.16, 22.17, and 22.18, we have d(v cos 4>)/d(}> — — p sin (/> d{v sin ()))lr)

/e 2 dv/d(f) — (v — p) tan cj>. Differentiation of Equations 27.01 then gives (p + h) sin dh 27.02 = (via) cos (f) da + (e/e 2 )p sin 2 cos de and 0= {e 2 (v — p) tan (f> sin (f) + e 2 v cos (f) + h cos (f>}d dh + (v/a)e 2 sin da + {ep sin 3 4> — 2ev sin (f>}de. The last equation, with the help of Equations 18.54 and 18.55, simplifies to (p + h) cos d(f> + sin 4> dh =—{v/a)e 2 sincf) da + (e/e 2 ) (p cos 2 +ve 2 ) sin (/> de; 27.03 Equations 27.02 and 27.03 are readily solved to give finally dw = (p + h )d(f> = {e 2 v/a) sin cos da + (e/e 2 )(p + ve 2 ) sin cos $ de 27.04 dh = -(a/v)da + ev sin 2 c/> de. If preferred, we can include the flattening f=(a-b)la=(l-e) instead of the eccentricity by using the relation df= (e/e)de. CHANGE OF ORIGIN 3. Next, we introduce a change (dXo, dY^, dZ,)) in the Cartesian origin, involving a corresponding translation of the base spheroid in the geodetic coordinate system. The effect will be the same if we keep the Cartesian origin and the spheroid fixed and if we alter the Cartesian coordinates of each point in space by ( — dXo, — dY , —dZ n ). The cor- responding changes in the geodetic coordinates could then be found by differentiating Equations 25.15, with a, e fixed, and by solving the resulting three equations for dco, d<$>, dh. However, we can in this case obtain directly the changes in geodetic coordinates from results already given. 261 262 Mathematical Geodesy 4. We have, for example, dco dco dco aa> = — —— oAq — - — diQ — — dZ ax ay oz = -A 1 dX -B 1 dY -C 1 dZ< ) where A 1 , B 1 , C 1 are the 1-components of the Car- tesian vectors in geodetic coordinates; similarly, we have dcf> = - A 2 dX - B 2 dY n - C 2 dZ dh = - A 3 dX - B*dY - C 3 dZ . Using the notation of Equation 19.36, we then have {dco, deb, dh}=-(Q T R) T {dX , dY , dZo] 27.05 = -R T Q{dX , dY , dZ }, an equation which would be true in any (a>, cf>, N) system, with the appropriate values of the matrices from Equations 19.26 and 19.31. In geodetic co- ordinates, we have R r =R l/{(v + h)cosc})} '.)\ llip + h) 0|; 1> expansion of Equation 27.05 gives (v + h) cos = (sin cos co)dX + (sin cos co)dXo 27.06 — (cos sinco)dY — (sin cf>)dZo. If there is both a change of spheroid and a change of origin, these first-order results should be added to Equations 27.04. We may write Equations 27.06 in matrix form as {(v + h) cos cj) dco, (p + h)dcb, dh} 27.07 = -Q{dX , dY Q , dZ } where Q is given by Equation 19.26. If the new spheroid is to be parallel to the old at the origin, then we have dco = dcb — in the observation Equa- tion 27.07 for the origin. CHANGES OF CARTESIAN AXES 5. It has been assumed throughout this book that all (co, (/>, N) systems — in particular, the astronom- ical and geodetic systems — share the same Car- tesian axes, which in the geodetic applications are physically related to the axis of rotation of the Earth and to the astronomical meridian at a fixed datum or origin. The conditions to ensure common Cartesian axes at the origin of a survey have been investigated in § 19-13 through § 19-15; this situation will be continuously preserved if frequent astronomical observations are made throughout a network which has been adjusted by using the observation equations developed in § 26-12. If this procedure has been followed, there should be no need to consider reorientation of the Cartesian axes. Unfortunately, many surveys of considerable extent have not followed this rigorous procedure. At most, a few Laplace azimuths have been used in the adjustment; as we have seen in § 19-14 and § 19-15, this procedure is not sufficient to ensure correct orientation. Whenever it becomes necessary to join two such surveys or to adjust them into a correctly oriented system, we should include orien- tation parameters to allow for a change of Cartesian axes and for a corresponding change in the orienta- tion of the base spheroid of the geodetic coordinate system. 6. Large rotations of the coordinate axes must be made in a prescribed order to provide unique results, although the many ways of prescribing the order can lead to some confusion. For our purposes, it would be advantageous to adopt the most common definition of Euler's angles because these angles are used in celestial mechanics and in satellite geodesy. 1 Unfortunately, two of the three Eulei angles are indistinguishable in the case of small rotations which concern us in the present appli- cation. On the other hand, small rotations can be made in any order and compounded as vectors so that we have no need to specify the order. Never- theless, we prescribe an order required for large rotations so that the results may be used for othei applications. 7. We begin with one set of Cartesiaji axe (A r , B\ C r ) and derive others (A\ B\ C r ) by right-handed rotations, which are positive if clock- wise, when looking outward from the origin along the positive direction of the axis of rotation, as follows: First, a rotation of a>i about the x-axis, A r , Second, a rotation of C02 about the new y-axis, and Third, a rotation of (03 about the new z-axis. The combined effect of these three independent rotations is described by the following product matrix, which premultiplies the initial vectors (A r , B r , C r ) or the initial Cartesian coordinate vector (x, y, z) to obtain the final vectors {A r , B r , C r ) or (x, y, 2). The product matrix is See Kaula (1966), Theory of Satellite Geodesy, 17-18. External Adjustment of TSetivorks 263 27.08 COS £03 Sin CO3 M = | - sin to3 cos o>3 cos o> 2 — sin oj 2 1 sin co> cos (1)2 cos toi sin o»i — sin wi cos oj] which expands to cos co 2 cos a*) (c° s w i s i n ^3 + sin co, sin co 2 cos co :i ) (sincoi sin (1)3 — cos coi sin co 2 cos a) :i ) 17.09 i —cos co 2 sin 0)3 (cos o>i cos 0*3 — sin a»i sin co 2 sin co 3 ) (sincoi cos CO3 + COS coi sin a> L > sin C03) J sin (1)2 ~ s ' n w i cos u)> cos coi cos 0)2 For small rotations, the expanded matrix reduces to 1 CO3 — co> 2 7J

  • 1 i-o> 3 1 CO, co 2 — COi 1 It may be noted that M is an orthogonal matrix because its three component matrices are orthogo- nal, so that the inverse transformation is given by the transpose M T . The approximate matrix N is not orthogonal, but the inverse will, nevertheless, be given by N r , which is the approximate form of M r . Because of the antisymmetric properties of N, the transpose N r is equivalent to rotations (— coi, — 0)2, — cos) which restore the original situation. 8. Next, we have to find the changes in the geodetic coordinates of a point (co, , h) resulting from these rotations. If the new Cartesian coordi- nates of the point are (x, y, z) , we have {i, j, 2} = N{.r, y, 2} ; the change in Cartesian coordinates is given by {dx, dy, dz} = N{x, y, z} — I{x, y, z} 27.11 =N {x, y, 2} where I is the identity matrix and 27.12 N : W.( -co -C03 0)] C02 — COi But, from Equation 27.07 in which the changes of coordinates of the point are { — dXo, —dY n , —dZo}, we have {(v + h) cosc/)c7co, (p + h)d4>, dh} = Q{dx, dy, dz} 27.13 -QN„{x, y,z}. Expanding and substituting for the Cartesian coordinates from Equations 25.15, we have after some simplification {v + h) cos (f) do>=— a)a{v + h) cos c/> + (coi cos C0 + CO2 sin co) X (e z i> + h) sin c/> (p + /; )d(f) = (0)2 cos co — coi sin co) X (h + a 2 /p) dh = (co 2 cos co — coi sin co) 27.14 X (e 2 v sin c/) cos c/>), which are in a suitable form to add to Equations 27.04 and 27.06 in those cases where there are changes in the shape and size of the base spheroid, in the Cartesian origin, and in the orientation of the Cartesian axes. 9. An interesting alternative way of deriving the same result is to start with the equation {A r ,Br,C r } = N{Ar,Br,Cr} or with the equation { (Ar-Ar), (B-Br), (C,.-C,)} = T$ {A r , B r , C,} = NoCr{X r , fir, V,} from Equations 12.013 and 19.23. If we contract this last equation with a position vector p' and use Equation 12.169, we have {dx, dy, c/z} = N Q r {(sec c/>)(dp/do)), dp/d da>, (p + h)d){dpld, p} in which QN Q T contains only the three rotation angles and the latitude and longitude, while the last vector contains the spheroidal elements. From 264 Mathematical Geodesy Equations 12.170, etc., in geodetic coordinates, we have p — (v + h) cos 2 4> + (e 2 v+ h) sin 2 (f> ={h + a 2 /v) ■ dp/dcf) — — e 2 v sin cos , (03, but this procedure would be even more unsound. We shall accordingly investigate separately the effect of systematic error of scale and orientation within the network by hold- ing the origin fixed, or by holding some central point of the network fixed if there is no origin. We shall choose as parameters (a) a proportional scale change ds/s, where 5 is the straight-line dis- tance of the point under consideration from the origin, and (b) changes da , dft;, in the azimuth and zenith distance at the origin of the straight line to the point under consideration. The parameters dsjs, da , dfto will, of course, be given the same values for all lines radiating from the origin. The straight-line distances, azimuths, and zenith dis- tances of all points from the origin are first computed from Equation 25.18 in which the unbarred point is the origin, and are used in the coefficients of the observation equations that we shall now form. 12. We could simply differentiate Equation 25.18, with the unbarred origin and therefore the matrix Q fixed; we could then obtain three equations con- necting ds/s, da = da , dft — dft with changes da>, d(j>, dh in the geodetic coordinates of the point under consideration. These three equations could then be solved to give da>, d(f>, dh explicitly in terms of the parameters ds/s, dao, dfto. However, we shall find it more instructive to proceed from first principles and to derive the results in matrix form. 13. We make dx r , da), dcf) all zero in Equations 26.07 and 26.14 and obtain l r ( ds/s ) + m r dfto — n r sin ft da — dx r /s. 27.17 In this equation, dx r are corrections to the Car- tesian coordinates of the point under consideration arising from changes da , dfto in the azimuth a and zenith distance ft of l r . The auxiliary vectors m r , n r are defined by Equations 26.13; we must use the Cartesian components of all vectors in Equation 27.17, which is a vector equation only in Cartesian coordinates because of the derivation of Equation 27.17 from Equation 26.06. If we form the orthogonal matrix /sin a sin ft sin a cos ft — cos a A= cos a sin ft cos a cos ft sin a V cos ft — sin ft 27.18 and if we refer to Equations 26.33, 26.34, and 26.35, we find that the matrix of Cartesian components of /'', m r , n r is Q r A where Q is as usual given by Equation 19.26. Ac- cordingly, the left-hand side of Equation 27.17 can be written in matrix form as Q T A{ds/s, dfto, - sin ft da }; from Equation 27.07 in which we must substitute External Adjustment of Networks 265 dx for — dX , etc., the right-hand side is Q T {(l> + h) cos0 da>, (p + h)d, dh}/s so that we have finally {(v + h) cos dco, (p + h)d, dh} 27.19 =sQQ T A{dsls, dp , - sin (3 da }. The overbars in this last equation refer to the point under consideration, while the unbarred quantities and da , dfio refer to the origin. The matrix QQ T is given by Equation 19.25; by using Equation 19.27 for each of the three vectors l r , m r , n'\ we can write 27.20 QO r A=A where A is the same matrix as A but formed from the azimuth and zenith distance at the overbarred point, that is, at the point under consideration. If O is the origin, we must as usual form this matrix by using the azimuth and zenith distance at P of the line OP produced, which could have been computed just as easily from Equation 25.19 as the azimuth and zenith distance at O. The parameters da , df3 still refer to the origin, but once the parameters are known, they can be substituted in the same equation to give the changes in coordinates of other points which have not been used in the adjustment, regardless of their actual meaning. We can ac- cordingly drop the overbars and rewrite Equation 27.19 as {(v + h) cos da, (p + h)d, dh} 27.21 = sA{ds/s, dp ,-sin ft da }, provided we form the matrix A from the azimuth a and zenith distance /3 at the point P under considera- tion of the line OP produced. In this final form, the equation is suitable for combining with the equa- tions for changes in the geodetic coordinate system. EXTENSION TO ASTRONOMICAL COORDINATES 14. Most of the preceding analysis applies equally well to a general (o>, (f>, N) system, including the astronomical system in which N is the geopotential, provided we use the more general R-matrices given in Equation 19.31. The derivation of Equation 27.05, for example, shows that for changes (dx, dy, dz) in the Cartesian coordinates of a point, we have 27.22 {dw, d, dN} = R T Q{dx, dy, dz} where ■A'i sec 4> —ti sec (/> y r sec (/>\ K' ■ I -t, -k 2 y 2 n and is given by Equation 19.26. Equation 27.22 gives, for example, the changes in (a», 4>, N) co- ordinates for an origin shift of (dXo, dY n , dZo) by simply substituting dx=—dXo, etc. 15. The change in (a>, , N) coordinates, arising from operation of the rotation matrix N (Equation 27.12) on the Cartesian axes, is obtained from Equa- tions 27.11 as 27.23 [dot, d, dN} = R r QN {*, y, z}. To apply this result, we must know the Cartesian coordinates of the point; in the case of a general (0, at, N) system, there are no such integral for- mulas as Equations 25.15. However, the Cartesian coordinates appear only in the coefficients of the first-order rotations a>i, w 2 , w :i , and approximate values would suffice. 16. The result of changes in the scale and orienta- tion of the network corresponding to Equation 27.21 is similarly given by {da>, d, dN}=sR T A{ds/s, d/3„, - sin /3 da }. 27.24 To apply this equation in the astronomical system, we need to know the length s and the astronomical azimuth and zenith distance at the point P under consideration of the line OP produced, where is the origin. Approximate values, such as geodetic values, would suffice, corrected if possible for the deflection at P. 17. There is, of course, no corresponding equation to reflect changes da, de in the base-spheroid parameters, which arise solely from the special choice of a (w, (f), h) system. We make such a spe- cial choice in the case of a general (a>, cf>, N) sys- tem by identifying N, for example, with the geo- potential, which settles all the components of the R-matrices at their actual physical values. To apply the system, we accordingly need values of gravity and of the curvature parameters at all points of the network. ADJUSTMENT PROCEDURE 18. The total change in the geodetic coordinates (d(o, d(f), dh) arising from application of the four sets of parameters (da, de), (dX , dY , dZ ), (a>i, (o>, w.i), and (ds/s, dfi a , da ) in Equations 27.04. 27.06, 27.14, and 27.21 can be obtained by adding these equations. This procedure assumes that the parameters are independent and that second-order effects can be either neglected or removed by some 266 Mathematical Geodesy process of iteration, although in some cases, the parameters, especially the rotations, will be strongly correlated (see §27-27). In the result, we have three equations for each point containing 11 parameters. In most cases, the two spheroids will be known so that the {da, de) terms can be computed and re- moved from the equations. For the remainder of this section, we shall assume that this procedure has been followed and that we are left with three equations for each point containing nine parameters. 19. We shall consider the case of two adjacent networks which are to be adjusted into sympathy through common points. If one network (overbarred) is held fixed, we substitute d(o — 0) — (o, etc., for the difference in coordinates of each com- mon point and solve for the parameters to correct the unbarred system. All the coefficient matrices are computed in the unbarred system. We need at least three common points for a solution; if there are more points, the equations can be treated as observation equations and appropriately weighted in a least-squares adjustment. For a stable solution, the common points must, of course, be widely separated. 20. If neither network is to be held fixed, we sup- pose that the final values of the coordinates will be oj*, etc. We can then form equations in each net- work, whose absolute terms are a>* — o> and oj* — to, and subtract these equations in pairs to eliminate o)*. We are left with three equations for each com- mon point containing 18 independent parameters, and we shall need at least six common points. An extension of the same procedure would enable us to connect several networks. 21. We have supposed that all three geodetic coordinates of each common point are known in both adjacent systems. Unfortunately, geodetic heights will seldom be known. Vertical angles, con- trolled by frequent astronomical observations as proposed in Chapter 26, have not been measured in several major triangulations because of the (excessive) fear of the effects of atmospheric re- fraction and in the expectation that the stations would be connected by lines of spirit levels. How- ever, for economic reasons, spirit leveling has for the most part been confined to roads, and triangula- tion stations sited on hills still have no accurate heights. Where accurate vertical angles have been measured, there are usually too few astronomical ob- servations to provide adequate geodetic heights, and the vertical angles have been reduced as indicated in §25-19 to provide a first-order approximation to spirit levels. 22. If spirit levels, or an approximation to spirit levels, are available for the common points, the best procedure would be to replace the d/i-equations by c/A-equations, formed as in § 27-14, using geo- detic values of (o», ) in the coefficients and the best possible values of the gravitational parameters in the R-matrix. An additional parameter may be required to allow for difference of level datums in the networks. The unknown parameters are other- wise the same in the o?/V-equations, which can accordingly be used in conjunction with the dw, d(f> geodetic equations. 23. If no adequate heights are available in any defined system for the common points, no valid adjustment is possible; the points of the network are, in fact, located in three-dimensional space, and we cannot expect to achieve a rational answer by arbitrarily stripping a dimension, even though such procedures have been common in classical geodesy. The most we can do is to drop the dh- equations and to solve for the unknown parameters from the dw, d(f) equations only, using the best available values for h in the coefficients. In that case, we should need at least 50 percent more com- mon points, and even so we could not expect to derive valid values for some of the parameters. For example, dX t) , dY», dZ f) would probably be ficti- tious because we should not have taken any definite steps toward positioning the spheroids. It would be better to defer the adjustment altogether until adequate observations have been made. FIGURE OF THE EARTH 24. In modern language, the old problem of deter- mining a "Figure of the Earth" becomes the prob- lem of finding a geodetic coordinate system which best fits the astronomical system. The problem is very easily solved if we substitute the astronomical minus the geodetic longitude (or latitude) for do) (or d(j)) in the observation equations of this chapter and retain the parameters da, de of Equations 27.04. All the points used in the adjustment should be in the same network, although the network may have been formed by joining adjacent networks as pro- posed in the last section. In addition to da, de, other parameters may be included in the adjust- ment, depending on the kind of network we are using. External Adjustment of Networks 267 25. The parameters {dXo, dY», dZ») should nor- mally be included in order to locate the origin of the final system as near as possible at the center of figure; we should be unable to locate the origin at the center of mass by any geometrical adjustment. However, if we know the positions of some of the stations of the network in relation to the center of mass, whether by lunar observations as described in Chapter 26 or by dynamic observations to arti- ficial satellites, we can relate the origin derived from the geometrical adjustment to the center of mass and so can shift the origin to the center of mass. For example, if the geocentric coordinates of a point in relation to the center of mass as origin are (w, $, r) and if the geodetic coordinates of the same point are (oj, (f>, h), then dx= (v-\- h) cos (/> cos a) — r cos cos 6j dy— (t> + h) cos (/> sin oj — r cos $ sin o» 27.25 dz= (e 2 v + h) sin — r sin $ give the coordinates of the center of mass from the origin of the geodetic system. Mean values of the shift from a number of points should ultimately provide a close result. As always, we require the geocentric and geodetic systems to share the same Cartesian axes. If we know dx, dy, dz from Equa- tions 27.25, then we can substitute dx=dX», etc., in the observation equations and so can derive a geodetic system whose origin is the center of mass. 26. The Cartesian rotations (oji, oh, oj.s) should be included in the observation equations if we have any reason to suspect the initial orientation of the network. These rotations should not be included in a passive satellite triangulation network where every line, apart from observational error, has been correctly oriented. 27. Scale and orientation parameters (dsjs, dfio, dao) could be included, but would be confused with da and (ct>i, a>>, ai.-s) unless the network is of great extent. These parameters should not be included in a worldwide satellite triangulation network which has been closed and internally adjusted, but should be included in the adjustment of an existing triangu- lation to satellite control. 28. We have so far considered only the observa- tion equations for latitude and longitude in deter- mining a Figure of the Earth. The question arises whether we also can include equations for the third dimension. If we know the geopotential at points of the network, whether by spirit leveling or by other means, we can find a point whose geodetic coordi- nates are (a>, $, h) where the standard potential has the same value. If h is the geodetic height of the network point, we could write (h — h) for dh in an observational equation. Inclusion of this equation in the adjustment would result in values of the parameters which would minimize (h—h) as well as the astronomical minus the geodetic latitude and longitude. The adjustment would thus bring the standard gravitational field into closer accord with actuality. 29. There are, of course, certain advantages in adopting a geodetic system close to the astronomical system. It is convenient to confuse the two systems within allowable limits of error for such purposes as small-scale mapping; it is essential that first- order transformations between the two systems should be sufficiently accurate for even the most refined geodetic work. However, there are serious practical and economic disadvantages in changing the geodetic system too often. The next justifiable occasion to make the change may well be on com- pletion of the worldwide satellite triangulation. CHAPTER 28 Dynamic Satellite Geodesy GENERAL REMARKS 1. The static use of artificial satellites as elevated beacons has been described in § 26-43 through § 26-72. In addition, it is possible to derive much geodetic information by observing and analyzing the motion of the satellite in orbit. The lower har- monics of the gravitational field can be obtained more accurately from satellites than by any other method, until it becomes possible to cover the entire globe with gravity observations to a high degree of accuracy and with uniform density; corrections to the positions of tracking stations may be obtained in a worldwide reference system, supplementing direct geometrical fixation by satellite triangulation; and the origin of the reference system can be located at the center of mass of the Earth, which is impossible by any other method until gravity surveys are completed over the entire globe. But to obtain all this information, as well as the changing elements of the satellite orbits from a growing number of satellites, necessarily involves some complexity. 2. Methods initially were taken from the astrono- mers—who did not have quite the same problem of a close satellite of an unsymmetrical rotating parent body — with the result that considerable extensions and modifications have been found necessary. As in astronomical calculations, analytical methods, which must, nevertheless, be studied to gain any understanding of the problem, are giving way to numerical and statistical methods, using larger computers on more sophisticated programs. Against this background of rapid development and of grow- ing complexity, the most explanation which can be provided in one chapter is a fairly detailed account of the basic equations and elementary theory, fol- lowed by notes in bare outline on current methods of solution. EQUATIONS OF MOTION - INERTIAL AXES 3. Newton's second law of motion for a particle of mass m is usually expressed in Cartesian coordi- nates as d 2 x _ d(mv,,) 28.001 dt 2 dt ■=F X with two similar equations for the other coordinates y and z. In these equations, the derivatives are with respect to time t, which is assumed to be independ- ent of the space coordinates: v x , F x are, respec- tively, the x-components of the velocity (dx/dt) and of the applied force. If the equations are to hold in the same Cartesian system over some finite region of space, then that space must be flat (§ 5-2). More- over, if the Cartesian equations of motion are to express a law of nature, these equations must be invariant with respect to manmade coordinate transformations; it can be shown that the equations are invariant, provided the mass does not change either with time or with the coordinate system and provided the two sets of Cartesian axes are fixed or, at most, are moving relative to each other with a constant velocity of translation. The equations do not hold true in any coordinate system if one set of axes is accelerating (or rotating) with respect to the other set. 4. A coordinate or reference system which is either fixed or moving with a constant velocity of 269 270 Mathematical Geodesy translation is known as an inertial system; we can say that Equation 28.001 holds true only in such a system. The inertial system which most concerns us in satellite geodesy has the z-axis parallel to, and in the northward direction of, the Earth's axis of ro- tation; the x-axis is parallel to the plane of the Earth's orbit around the Sun — the plane of the ecliptic — in the direction of the vernal equinox. We must also specify a time or epoch and must correct our obser- vations accordingly because the axis of rotation and the ecliptic vary slightly in time, mainly as a result of lunar and planetary perturbations. Even then, we cannot say that we have a true inertial system. A recent description of the astronomical determination of an inertial frame of reference has been given by Wayman. 1 Nevertheless, it has been said that the only valid definition of an inertial system is one which would make Newton's laws true, and because these laws are not true on the cosmic scale, there is no such thing as an inertial system. However, for our present purposes, the concept is a good approxi- mation. We shall also assume that the origin of the inertial system is the center of mass of the Earth, in which case as we have seen in § 21—42 that first- degree harmonics must necessarily be absent from the expression of the Earth's potential in spherical harmonics derived from the Cartesian inertial system. 5. To express Equation 28.001 in a general coordi- nate system x'\ we must first generalize the velocity vector. In (overbarred) Cartesian coordinates, the contravariant velocity vector is dx s /dt, expressing the time-rate of change of each coordinate. By the ordinary transformation law, the velocity vector in any other coordinate system (unbarred) is then 28.002 dx r dx s V s - dx r dx s dx s dt dx dt ds dx' dt ds vl' where v = ds/dt is the linear velocity, ds is the arc element of the path of motion or orbit, and l r is the unit tangent to the path of motion. The covariant velocity vector is obtained by simply lowering the indices in the first and last members of Equation 28.002. In Cartesian coordinates, the velocity vector is also the time-rate of change of the position vector p'\ and we may generalize this statement to 28.003 -u~ vl with a covariant equation obtained by lowering the indices, provided that we now take the intrinsic derivative (§4-1) of the position vector. Because 1 Wayman (1966), "Determination of the Inertial Frame of Reference," The Quarterly Journal of the Royal Astronomical Society, v. 7, 138-156. Equation 28.003 is a tensor equation which is true in Cartesian coordinates, it is true in any coordinate system. 6. The equations of motion can now be general- ized to 28.004 m § 2 p, 8t 2 8v r 8(mvl,) 8t 8t = F r where F, is the applied force vector and (mvl r ) is defined as the momentum vector. Equation 28.004 reduces to Equation 28.001 and to the two similar equations in Cartesian coordinates, as a tensor equation, Equation 28.004 is therefore true in any coordinate system derived from an inertial system. If we consider F, to be the force per unit mass or, alternatively, if we consider that we are dealing with a particle of unit mass, we may drop the m in the last equation and write 28.005 8-p, _ 8vr _ 8 ( vl,- ) 8t 2 _ "o7~ 8t F r , which expresses the acceleration vector. If the applied force is derived from a scalar potential V, we have F r = -V r from the generalization of Equation 20.05; we can write 8 2 p r= 8v, = 8(vl,) 8t 2 ~ 28.006 8t 8t F,=-V r . 7. The tensor Equation 28.006 can be written in yet another form more suitable for expanding the equations of motion in a particular coordinate sys- tem. We have V r = F r 8(vl,) = d(vl r ) 8t dt = d(vl r ) dt _ d(vl r ) dt - n b (vis) cW dt ~ Th v 2 g xq M« [rk, q]vH"l h d(vlr) dt . i (dgrq | dgkq 2 \ dx k dx r dx" J d(vlr) 1 (dgrq | ?>gkq \ dx k dx'' dt dgrq dx k v 2 M k Dynamic Satellite Geodesy 271 on interchanging the dummy indices (k, q) in the last term and by using Equations 28.002, 3.02, and 3.01. Finally, we have ~~_ ir i- d I dx s \ , dgkq dx k dx'i 28.007 -V^Fr^grs-^)-^-^-^, which enables us to write at once the components of force in any coordinate system from the metric alone. 8. For example, suppose that we wish to work in the symmetrical (a», , h) coordinate system of Chapter 18. By direct substitution of the metric given in Equations 18.03 and 18.04, we have dV d do) dt {(/?, + h) 2 cos 2 4>) dV - d t(3 , L\2l\ i d{{Ri + h)- cos- } ... . d{(R-, + h) 2 } .., --* dj ^ dV d ,;, , d{(Ri + h) 2 cos 2 d>} ., -Jh = dt {h] -t Th w " , d{(R., + h)-} :, ~- Th *' 28.008 in which we have adopted the usual convention of denoting differentiations with respect to time by dots, for example, co = dco\dt. Equations 28.008 apply to any choice of axially symmetrical base sur- face whose principal radii of curvature R\, R.2 are functions of the latitude cf> only. To obtain the equa- tions in geodetic coordinates with a spheroidal base, we have only to use the special values of R u Rz given by Equations 18.55 and 18.54. We can also obtain the equations in spherical polar coordinates by choosing a spherical base surface of radius r - R, = R, so that we have (Ri + h) = (R 2 + h) = r, the radius vector of the point under consideration. Expansion of Equations 28.008 in this case give immediately the well-known formulas dV d , 2 , , ., = — (r* cos-

    dt ■\i/ j — TT = T" < r2 )+ rl sm 4> cos relative to the inertial axes A r , B r , C r (C r — C r ). If t is elapsed time since the two sets of axes coincided, we have from Equations 20.10 A r = A r cos cot + B, sin cot 28.010 B r = — A r sin cot + B, cos cot C r =C r . Because ordinary and covariant differentiation are the same in Cartesian coordinates and because A,-, B,, Cr are fixed, we have dA r dt 28.011 >B, : (IB, dt = — coA, dCr dt 0. 10. The position vector of a satellite at (x, y, z) in the moving system is p,=xAr + yBr +zC r ; using Equations 28.011, the absolute velocity vector relative to the inertial axes is C -^ = xAr + yB,+zC,+co(xBr-yAr), the last two terms arising from the motion of the axes and the first three terms giving the apparent velocity vector v r relative to the moving axes. In the same way, the absolute acceleration vector is given by *'%■= (xAr + 9Br + zCr) dt* + 2co(xBr-yAr)-cb 2 (xA r + yB r ). We may equate the absolute acceleration to the applied force vector per unit mass F, by Newton's second law. The first group of terms in parentheses on the right is the apparent acceleration relative to the moving axes; this group can be expressed in tensor form as 8?v/o7 in which v, is the apparent velocity vector. The second group of terms can be written as the vector product 2a>erpqC l 'v Q , if we remember that covariant and contravariant components are the same in rectangular Cartesian coordinates. The third group of terms is the gradient 272 Mathematical Geodesy of — i(L-(x- + y-); this group can be written as — icir (x'- + y-),-, using Equations 12.009. We have finally 8v, 28.012 F r = - ^+2(be ll)q Ci'v"-^a> 2 (x 2 + y 2 ) r . This last equation is a tensor equation which holds true in any coordinate system transformed from the moving A,-, B r , C, system, if we note that (x' 2 + y' 2 ) is the square of the distance of the satellite from the axis of rotation and is therefore an invariant under such transformations. Moreover, the vector C 1 ' symbolizes any axis of rotation and need no longer have anything to do with the coordinate system. It will be seen that Equation 28.012 differs from the Newtonian equations of motion relative to inertial axes by the addition of the last two terms on the right. The first of these terms is known as the Coriolis force (or acceleration); the second is known as the centripetal force. However, these are fictitious forces arising from the motion of the axes, unlike the applied force F r , the idea being that if we "correct" the applied force by the Coriolis and centripetal "forces," the ordinary Newtonian equations of motion apply to the ap- parent velocity. Another way of handling the matter is to forget that the axes are rotating and to use Equation 28.012 instead of Equation 28.005. It may be emphasized that the covariant velocity vector v,—vl r is not the gradient of the scalar velocity v, whether in an inertial or moving coordinate system. The scalar velocity v has so far been defined only as dsjdt along the orbit and can have a gradient only along the orbit. At the end of this chapter, we shall define the scalar velocity of a family of orbits in space, but meanwhile there need be no confusion. 11. If the impressed force F, is derived from a scalar potential V, we have Fr = ~V r from the generalization of Equation 20.05. Also, if W is the geopotential defined in § 20-10, we have W r =V r -W{x 2 + f), from Equation 20.08, so that Equation 28.012 in this case can be written as 28.013 -W l =^r + 2u€ n>q C»v«. ot If the Coriolis force — the last term on the right — could be written as the gradient of a scalar S, then the equations of motion would take the norma Newtonian form with a potential (W+S). However this is generally impossible; the Newtonian equa tions of motion simply do not apply to acceleratinj or moving axes even with a modified potential. 12. The equations of motion in Cartesian coordi nates, referred to the Earth-fixed A r , B, , C, system are easily obtained by contracting Equation 28.011 successively with A'\ B r , and C'\ which are constan vectors in this system so that the x-component o the applied force is, for example, F,=F,A'=A ,. 8 2 p r 8t 2 2a>e rp9 A r CPv" - d) 2 xx r A' 8HA r p r ) 8t 2 dx q — 2d)B q —TT — (xTX d x dy , = —rz — 2o» — a>-x. dt 2 dt The three equations of motion are then x — 2a>y = F x + co 2 x — —'dW\'dx y + 2 (hx = F y + d) 2 y = -dW/dy 28.014 z =F Z =-dWjdz. The last three members of these equations assume that the impressed force can be expressed as th( gradient of a scalar potential — V, in which case W is the geopotential. INERTIAL AXES -FIRST INTEGRALS 13. It is apparent from Equation 28.002 that th magnitude of the velocity vector v r is v because / is a unit vector. Accordingly, we have V'Vy= V 2 , which can be differentiated intrinsically to giv d(v 2 ) 8v r 8v r dt v 8t- + v '-8t- 8v, "o7 ?rsV s 8v^ 8t 8v r 8(g rs V>) 8v r 8t;.v 28.015 2v 8v r "o7' Dynamic Satellite Geodesy 273 remembering that the metric tensor g rs is constant under covariant or intrinsic differentiation. 14. If we take the equations of motion in the form of Equation 28.006 as 28.016 ot ~ Vr ' contract with the contravariant velocity vector v r =dx r /dt, and use Equation 28.015, we have 8v r , d(v' 2 ) 28.017 ^--i-y- dx' v — ' dt ' The total time differential of V is dV _ dV dx r dV _ dx r dV 28.018 -^-j^- d j + Tt~ Vr ~dT Jr Tv if V does not contain the time explicitly (dV/dt = 0), we can write Equation 28.017 as d 28.019 dt (h 2 +V)=0, which shows that 28.020 H* = h z + V is a constant of the motion and provides a first integral of the equations of motion. The potential V, as we have seen in § 20—3, can be considered a form of energy; whereas (it' 2 ), remembering that we are dealing with a particle of unit mass, is the kinetic energy of the particle. We say that (%v 2 + V) represents the total energy of the particle, which is conserved during the motion. The integral (kv 2 + V) is sometimes known in the literature as the vis viva. 15. If, on the other hand, the potential is time- dependent in the sense that its expression contains the time explicitly, then there is generally no simple law of conservation of energy. In that case, if we add Equations 28.017 and 28.018 and integrate, we have 28.021 fa 2 +V CdV J dt dt + constant; the integral on the right cannot generally be evalu- ated unless we can express V completely in terms of the single time variable. Equation 28.021 can, in some cases, be solved by successive approxima- tion; as we shall see in § 28-91, the equation can be given a definite expression in the case of a uniformly rotating, attracting body such as the Earth. 16. The attraction potential of the Earth is not symmetrical about the axis of rotation and will therefore contain tesseral harmonics when ex- pressed in spherical harmonics related to Earth- fixed, but rotating, axes A,-, B r , C,- The longitude oj in the spherical harmonics is related to the inertial longitude o> by the relation 28.022 OJ — OJ — (Ot, if t is the elapsed time since the two sets of axes coincided and if oj is the constant angular velocity of rotation, as we can see at once from figure 16, Chapter 20. To express the potential in spherical harmonics related to the inertial system, as we must do if we are going to use Newtonian equations of motion, we can substitute Equation 28.022, for ex- ample, in Equation 21.035, which would then con- tain the time explicitly. Another way of considering this matter is to note that the field at a fixed point in inertial space will vary with time as the Earth rotates; the potential is time-dependent, whether we express the potential in spherical harmonics or in any other coordinate system derived from the inertial system. We conclude that Equation 28.021, and not Equation 28.020, holds true in this case. 17. Another law which might assist a solution of the equations of motion is the conservation of angu- lar momentum. In figure 32, the origin is at S, the Figure 32. satellite is at O, and the unit tangent /, to the orbit is as shown in the plane of the paper. The line SQ is perpendicular to the tangent; the magnitude of the angular momentum or moment of momentum for unit mass is defined as v(SQ)=vr sin B, which is the magnitude of the vector product 28.023 e rst p s v, = ve rst p s l, = e rst p s pt, known as the angular momentum vector, whose direction is perpendicular to the plane of the paper 306-962 0-69— 19 274 Mathematical Geodesy and toward the reader. If we assume that both the magnitude and direction of this vector must be constant in time to satisfy a conservation law, we have e rst p s vt = constant. Obviously, this law cannot be a universal law of nature because it depends on the origin S of the coordinate system through the position vector p s . To determine the circumstances in which the sup- posed law can apply, we differentiate intrinsically with respect to time, remembering that we have 8psl8t=v s and that the vector product of two paral- lel or identical vectors is zero. The result, using Equation 28.005, is 28.024 e'*'p,F, = 0, which implies that the force — or the gradient of the potential — must be parallel to the position vector. In the case we are considering, this result restricts the potential to the elementary form GM/r; (vr sin (3) is then constant. Also, the orbit must lie entirely in a plane passing through the origin, that is, the plane of the paper in figure 32, because the angular mo- mentum vector is normal to this plane and is a con- stant vector. There is a clear analogy with the situation discussed in § 24-10 for the path of a light ray in a spherically symmetrical refracting medium, an analogy first noted by Newton 2 himself. 18. We shall now indicate briefly that the same situation would occur if the supposed law required the magnitude, but not the direction, of the angular momentum vector to be constant in time. In that case, we have € rsl p s vt€ r pqp p v Q — constant; by intrinsic time differentiation, we have after some manipulation and use of Equations 2.18, 2.19, and 2.21 28.025 PpVqpPv«=PqV p pl>V*. We now set up the usual triad (A. r , Pr, v r ) of parallel, meridian, and normal vectors in a spherical polar (w, c/>, r) system. The gradient vector of the poten- tial, like any other vector, can be expressed in terms of the triad as 28.026 V q = lkq+ mp q + nv q . If the azimuth and zenith distance of the orbit in the (to, , r) system are a, /3, we have v q =vl q =v(X q sin a sin (5 + p q cos a sin r 3~\-v q cos /3) and also p q =rv q . Substitution in Equation 28.025 gives after some manipulation vr 2 sin (3(1 sin a+ m cos a) = 0, which clearly cannot be satisfied for a general orbit (a, (3 arbitrary) unless we have l = m — 0, again requiring V q in Equation 28.026 to be parallel to the position vector. In this case also, angular momentum is conserved only in an elementary potential field GM/r. The orbit is then plane, and 28.027 vr sin (3 = constant is an integral of the equations of motion. MOVING AXES -FIRST INTEGRALS 19. In our present problem, the geopotential W in Equation 28.013 does not contain the time ex- plicitly. For example, the attraction potential in spherical harmonics given by Equation 21.035 con- tains only spherical polar coordinates derived from the Earth-fixed A r , B r , C r system, and the same applies to the expression of the potential in any other coordinates derived from the A r , B r , C r sys- tem. The centripetal part ldi 2 (x 2 + y 2 ) of the geopo- tential contains only rectangular coordinates in the A r , B r , C r system. The total time differential of the geopotential is accordingly dW_ dx? dt Wr dt ' If we contract Equation 28.013 with the velocity vec tor v r = dx'/dt, the Coriolis force is eliminated. Usin^ Equation 28.015 and integrating with respect te time as in § 28-14, we have 28.028 h; 2 + W '= constant - Quoted by Forsyth (Dover ed. of 1960). Calculus of Variations, original ed. of 1926, 256-257. as a first integral even though the geopotentia contains tesseral harmonics. In this expression, i is the magnitude of the apparent velocity relative to the rotating axes fixed in the Earth. We shal consider in § 28-87 and § 28-88 how to transforn this result to the inertial system, and so to obtair a first integral of the inertial equations of motion 20. The rotating system can be considered ar inertial system in which the ordinary Newtoniar equations of motion would apply, if we interpret the impressed force as the gradient of the geopotentia Dynamic Satellite Geodesy 275 plus the Coriolis force. Therefore, we cannot expect angular momentum to be conserved in the rotating system unless the total force, thus com- pounded, is directed toward the origin. As we found from Equation 28.024, we should require in which k is a scalar; this equation cannot possibly be satisfied for general values of W r and v 9 , any more than Equation 28.024 could be satisfied by general values of the potential. Accordingly, no first integral can be derived from a conservative law of angular momentum except in special cases. THE LAGRANGIAN 21. In an inertial system, the space coordinates are independent of time — an essential feature of the Newtonian system — and are therefore independent also of the velocity components. Generally, we can associate any velocity components with any space coordinates, although the two sets of variables will, of course, be related for a particular orbit. Instead of considering our present problem in terms of three space variables and of their variation with time, we can consider the problem in terms of seven independent variables (x, y, z, x, y, z, t), which can be transformed in various ways, and we derive solutions of the equations of motion for particular orbits in the form of relations between these seven variables. A complete solution, for example, would consist of x, y, z as functions of time from which x, y, z could be obtained by differentia- tion or, alternatively, A, y, z as functions of time from which x, y, z could be obtained by integration. 22. Next, we introduce an expression 28.029 L* = Ux 2 + y 2 + z 2 )-V(x, y, z, t), known as the Lagrangian , in which (±, y, z, x, y, z, t) are considered as independent variables. The first term on the right is the kinetic energy, and V is a scalar potential. We then have d /dL* dt \ dx dx dt dx dL* dx using the Cartesian form of Equation 28.006 for the equations of motion in a scalar potential field, plus two similar equations for y and z which are equivalent to the Newtonian equations of motion. These three equations can be put into index form as 28.030 d /dL* dt \dq r dL* dq r " which is a tensor equation only if the coordinates and the transforming factors dq r dq s dq r dq s are independent of time. The equations of motion may be written in this Lagrangian form for the posi- tions and velocities of any number of particles in a general dynamic system. THE CANONICAL EQUATIONS 23. Although a first integral of the inertial equa- tions of motion in the form of Equation 28.020 will not generally exist, there will, nevertheless, be a quantity H* — known as the Hamiltonian — given by 28.031 H* = h> 2 +V. The H* will be constant in time in accordance with Equation 28.020, only if the applied force is the gradient of a scalar, —V, which does not explicitly contain the time. However, in the general circum- stances of our problem, H* can be written in Cartesian coordinates in the form 28.032 H* = i(x 2 + f 2 + z 2 ) + V(x, y, z, t) , containing seven variables. Differentiation with respect to these variables and substitution in the Cartesian form of Equation 28.006 give three sets of equations of the form dx dx dt dH* dx dV dx dx which can be written in index notation as dH^_ = dx^ dH* _ dxr = dt ; 28.033 dx r dx' dt These last sets of equations are evidently equivalent to the Newtonian equations of motion, except that we now have six first-order equations connecting the six variables x r , x r and the time, instead of three second-order equations connecting the x r and the time. The symmetrical first-order form of the equa- tions of motion in Equations 28.033 is known as the canonical form. We shall see later that these equations can be transformed to others having the same form by a suitable change of variables. 24. The canonical form of the equations of motion has been derived from and is equivalent to the iner- tial equations. We can derive a similar canonical form for the equations, referred to moving (Earth- fixed) axes from Equations 28.014, only if the Coriolis force can be expressed as the gradient of a 276 Mathematical Geodesy scalar which could be used to modify the Hamil- tonian. Generally, this is not possible. Otherwise, the most we can do is to transform back to the inertial system (jto, yo, 20) by the relations x = Xo cos Git + yo sin tot y = — xo sin tot + yo cos cot 28.034 z = z in which t is the time since the two sets of axes coincided. We can then write the canonical equa- tions in the variables (xo, yo, Zo, xo, jo, zo, t) and can transform to other canonical variables, as will be explained later. THE KEPLER ELLIPSE 25. If the attracting body were a single particle of mass M situated at the origin of inertial coordinates or, alternatively, a sphere of uniform density and total mass M centered on the origin, then the external potential from Equation 20.01 would be minus (GM)/r, which in this chapter we shall denote as minus fx/r. In that case, the equations of motion of a satellite can be integrated easily and completely from the first integrals already obtained. The potential is not time-dependent; therefore, Equation 28.020 holds true as 28.035 h) 2 -filr=H* with H* the constant energy of the system. The angular momentum is also constant from § 28-17, and we can write 28.036 vrsin(3 = N in which fi (fig. 32) is the zenith distance of the orbit in a spherical polar coordinate system. Also, we know from §28-17 that the orbit is a plane curve. If (r, /) (fig. 33) are polar coordinates in this plane and ds is an element of length of the orbit, then for any orbit, we have dr/ds = cos fi 28.037 rdf/ds=smP; multiplying these equations by the linear velocity v — ds/dt, we have r — v cos /3 28.038 rf = v sin (3 which, substituted in Equations 28.035 and 28.036, give r*f=N 28.039 (r) 2 + (rj) 2 = v 2 = 2(iJLlr+H*). These equations could also have been obtained from Equations 28.009 for motion in a plane by substi- tuting V= — [x\r, = 0, to=f and integrating. 26. Eliminating fin Equations 28.039, we have 2/x N 2 \ 1/2 K+&HM)r which is directly integrable to give r as a function of time. However, we require the equation of the orbit in polar coordinates as a relation between r and/, for which purpose we substitute r dfdt~dfV) df\r) df\r N) The equation can now be integrated as a standard form to give in which / is a constant of integration. Comparing this last equation with Equation 22.21, we see that the orbit is an ellipse, one of whose foci S is at the origin. If / is measured from the nearest point A of the major axis (fig. 26, Chapter 22), known as perigee in this subject, then we have/o=0. The semimajor axis a and eccentricity e of the ellipse are then obtained by comparison with Equation 22.21 and are given by Ml 'N 2 all- a(l-e 2 ) N\ N 2 from which we have 28.040 yV=V/xa(l-e 2 ) Dynamic Satellite Geodesy 277 28.041 H*=-fi/2a; also, from Equation 28.035 we have 28.042 v 2 = fji(---)=^~ \r a) ar where r' is the radius vector to the other focus. 27. The constant TV is customarily expressed in a different way. An element of area swept out by the radius vector to the satellite is ir 2 ^/, so that the first equation of Equations 28.039 expresses the fact that the time-rate of change of this area is constant, which is Kepler's second law. Moreover, if T is the time required to describe the whole orbit from perigee to perigee, the total area of the ellipse is Trab = TTa 2 (l-e 2 ) l l 2 - JO \rfdt ■■ But if n, known as the mean motion, is the mean angular velocity of description of the orbit over a complete revolution, then we have 2tt 28.043 combining the last two equations with Equation 28.040. we have 28.044 = ^ 2 c -3/2 which expresses Kepler's third law. In terms of «, Equation 28.040 becomes 28.045 A^=V ) Lta(l-e 2 ) = na 2 (l-e 2 ) 1 / 2 . 28. Next, we introduce the eccentric anomaly E, which is the same as the reduced latitude u for the meridian ellipse of figure 26, Chapter 22. By differentiating the purely geometric Equation 22.21 along the ellipse {a, e fixed) with respect to time, we have dr . dE ae{l- e 2 ) sin f df r*e sin / N -r-=ae sin E — r =—-—, : — ttt" , = dt dt (1 + ecos/) 2 dt a(l-e 2 ) r 2 28.046 Using Equations 22.20, 28.045, and 22.21, the last two members of Equation 28.046 reduce to 28.047 f=~= n " jTv dt r (1 — e cos E) which integrates to (E — e sin E) = n(t — t () ) where to is a constant of integration equal to the time of passing perigee (£ = 0). The right-hand side of this equation is defined as the mean anomaly M, giving the position of the satellite as if it were moving at the mean angular velocity n about the focus or origin. We have finally 28.048 (E-e sin E)=n(t-t )=M, usually known in the literature as Kepler's equation. 29. We have now completed the dynamical ex- amination of the elliptic orbit, although we also can use any of the purely geometrical relations for an ellipse, as given in § 22-3 through § 22-10, in which case the notation may require some translation. For example, we shall use /3 in this chapter for the "zenith distance" of the orbit, relative to the focal radius vector as the zenith direction, shown in figures 32 and 33. This symbol is the complement of the angle /3, shown in figure 26, Chapter 22, and used in Chapter 22 as the angle between the normal to the ellipse and either focal radius vector. We shall also use a as the azimuth of the orbit in this chapter, whereas a is an elliptic constant in Chapter 22. For example, the second equation of Equations 22.03, translated into our present notation, gives the zenith distance of the orbit in the form 28.049 cot (3 e sin E (l- e 2)l/2 which can also be obtained in the equivalent form «« n~^ n e sin/ re sin/ 28.050 cot0=-— —7,= -r, T\ (1 + e cos/) a(l — e z ) from Equations 28.037 and 28.046 — the two forms being shown to be equivalent from Equations 22.20 and 22.21. Equation 22.21 is repeated for con- venience as 28.051 r= a(l — e cos E) — - a(l-e 2 ) 1 + e cos/) The rectangular coordinates of the satellite in the plane of the Kepler ellipse are repeated from Equa- tions 22.20 as q\ = r cos /= a(cos E — e) 28.JD52 q-i = r sin /=a(l-e 2 )»/ 2 sin E. We have also from Equations 28.038, 28.046, and 28.045 28.053 ^c 1/2 e sin/ _fM ll2 a ll2 e sin E V COS P — a i/2 (1 _ e 2)l/2~ ~y~ which, together with Equation 28.050, gives ,_yV_At 1/2 a 1/2 (l-e 2 ) 1/2 /n 1/2 (l + e cos/) v sin (3 28.054 a l/2 (1 _ e 2)l/ 278 Mathematical Geodesy Other useful formulas, easily verified, are cos/+ e 28.055 cos E = cos /= 1 + e cos/ cos E — e \ — e cos E , ixH\ + 2e cos f+e 2 ) /x(l + ecos£) 28.056 The components of velocity in and perpendicular to the semimajor axis are na sin/ na 2 sin E v cos (f+P) = qi- v sin (f+/3)=q 1 - 28.057 (l- e 2)i/2 r na(e + cos/) _ /?« 2 (1— e 2 ) 1/2 cos E (1-e 2 ) 1 ' 2 r 30. In the centrally symmetric field we are con sidering, the orbital characteristics (a, e, r, v,f, M etc.) will evidently be the same whatever the attitude of the orbital plane in a three-dimensiona coordinate system. Nevertheless, even in this special case, we must define the orbital plane if w( are to locate the satellite in the inertial system or h any system. To do this, we introduce three angula elements H, i, w as shown in the spherical diagram figure 34. The inertial system specified in § 28~ z is shown as X, Y, Z, the origin being lettered S t< agree with figure 33 and to indicate that the origii is a focus of the Kepler ellipse. The point Z is th< North Pole of the axis of rotation; the great circli XY represents the plane of the Equator. Th< satellite is represented at moving in the directioi shown in figure 34; the great circle PAO represent: the orbital plane intersecting the Equator at P which is known as the ascending node for the direc tion of motion shown. The descending node is 180 direction of motion (satellite) Equator (ascending node) Figure 34. Dynamic Satellite Geodesy 279 in longitude away from P, and the line SP is known as the line of nodes. The line SA is the direction to perigee, already defined as the point on the major axis nearest S. The other end of the major axis is known as apogee. The angle Cl = XP is accordingly the right ascension of the ascending node or longi- tude of the ascending node in the inertial system; i is the orbital inclination; and w = PA is known as the argument of perigee, usually denoted in the liter- ature by oj, which, however, is required throughout this book for various forms of longitude. We also use the term longitude in the geodetic sense as measured in the equatorial plane, whereas astrono- mers often measure longitude, wholly or partly, in the plane of the ecliptic. Auxiliary Vectors 31. We shall require certain unit vectors which are shown in both figure 33, representing the plane of the ellipse, and figure 34. The unit radius vector to the satellite is shown as r s = p s /r. The unit tan- gent to the orbit, represented at l r in figure 34, is the direction of the radius Sl r ; the representative point l r must lie in the great circle representing the orbital plane. The angle between l r and r r , shown as j8, is the zenith distance of the orbit relative to a spherical polar system of coordinates. The azimuth of the orbit in the same system is the spherical angle a. Unit vectors t r , m r in the orbital plane perpendicular, respectively, to r r , l r are as shown in both figures 33 and 34. Finally, a unit vector n r perpendicular to the orbital plane, shown in both figures 33 and 34, is used to complete either of the right-handed triads (l r , m r . n r ) or (r r , t' . n r ). In figure 34, n r is the pole of the orbital plane. 32. From figure 33, representing the plane of the orbit, we have 28.058 l k = r* cos /3 + t k sin /3 n k — — r k sin /3 + t k cos f3 . In terms of the usual meridian, parallel, and normal vectors (p k , k k , v k ) of the spherical polar coordinate system, the vectors are easily found to be l k = k k sin a sin /3 + p. k cos a sin fi + v k cos /3 m k = k k sin a cos /3 + p k cos a cos (3 — v k sin /3 n k = — k k cos a + fi, k sin a r* = v k t k = k k sin a + fx k cos a. 28.059 Inertial Cartesian components are (cos (iv+f) cos Cl — sin (w+f) sin Cl cos i\ cos (w+f) sin il+sin (w+f) cos Cl cos i I sin (w+f) sin i J 28.060 /—sin (w+f) cos Cl — cos (w+f) sin Cl cos i\ t k — I — sin ( w +f) sin Cl + cos ( w +f) cos Cl cos i 28.061 28.062 cos (w+f) sin i (sin Cl sin i \ — cos Cl sin i cos i I 28.063 (cos (w+f+fi) cos Cl — sin (w+f+fi) sin Cl cos i N cos (w+f+P) sin Cl+ sin (w+f + ft) cos Cl cos i sin (w+f+fi) sin i t 28.064 nr sin ( w +f+ /3 ) cos Cl — cos ( w +f+ (3) sin Cl cos i\ sin (w+f+ /3) sin Cl+ cos (w+f+ /3) cos Cl cos i J, cos (w+f+fi) sin i J as we can easily verify by expressing the scalar products of each vector with the Cartesian vectors A r , B r , C r in terms of elements of spherical triangles in figure 34. 33. We can obtain alternative formulas in much the same way as we did from § 12-15 by applying the following positive rotations to the inertial (A k , B k , C k ) system: (a) First, Cl about the z-axis, (b) Second, i about the new x-axis, and (c) Third, (w+f) about the new z-axis. 280 The result is 28.065 cos (w+f) sin (w+f) — sin (w+f) cos (w+f) K{A k ,B k ,C k } Mathematical Geodesy \ cos i sin i — sin i cos i/ cos fi sin fl — sin fl cos fl 1 cos (w+f) cos ft — sin («;+/) sin fl cos i cos I —sin (w+f) cos fl — cos (w+f) sin fl cos t —sin sin 17 sin i 28.066 Because the component matrices of K and therefore 28.067 {A k ,B k ,C k }-- 34. By putting /= in Equation 28.065, we obtain a triad of vectors (j r , k r , n r ) in which j r (the %-axis) is the unit radius vector to perigee and k r (the y- axis) is in the orbital plane. We have {f,kr,n r } = K f =o{A r ,B r ,Cr}; contracting this equation with the position vector of the satellite, we have (w+f) sin fl+sin (w+f) cos fl cos i sin i sin (w+f) (w+f) sin fl + cos (w+f) cos Cl cos i sin i cos (w+f) — cos O sin i cos i / K itself are orthogonal matrices, we can also write = K7{r fe , tK n k }. 28.068 { 91,92,0} =K /=0 {x, y, z) where q\, qz are given by Equations 28.052 and x, y, z are the inertial coordinates of the satellite. The reverse equation is 28.069 {x, y, z} = K$ =0 {qu q 2 , 0} , which enables us to express the inertial coordinates in terms of orbital elements. By splitting the third rotation (w+f) in Equation 28.065 into two suc- cessive rotations, we have also cos/ sin/ 0^ 1^ j -sin/ cos/ |.k, „ 1 28.070 =FK /= o 35. In deriving Equation 28.065, if the third ro- tation were (w+f+/3), it is clear from figure 34 that we should arrive at the triad {l k , m k , n k } . By substituting (w+f+f3) for (w+f), we accordingly have 28.071 { IK m k , n k } = K w+f+e { A\ B k \ C k } and a corresponding inverse. Moreover, by applying a fourth rotation of /3 about the z-axis to Equation 28.065, we have 28.072 K w+f+0 : cos p sin /3 ' sin B cos B ) K. h 36. The velocity vector is given by p k = vl k = (v cos B, v sin /S, 0)K{A k , B k , C k }; 28.073 the three Cartesian components of the velocit vector are given by p k (A k , B k , C k ) = (p k A k , p k B k , p k C k ) = (x, y, z) 28.074 =(t> cosiS, v s\n B, 0)K in which we can substitute Equations 28.053 an 28.054 and so can obtain the velocity vector and it components in terms of the orbital elements. W can, of course, transpose the last equation as {x, y, z} = K T {v cos /3, v sin B, 0} = Kf =0 F T {v cos B, v sin B, 0} 28.075 =KJ =0 {i; cos (f+B), v sin (f+B), 0] using the transpose of Equation 28.070. But th last vector in this equation evidently gives the com ponents of the velocity vector vl k in and perpendicu lar to the radius vector to perigee, that is, {v cos (f+B), v sin (f+B), 0} = {vl r j r , vl r k r , vl r n r } = {9i,92,0}, so that finally we have 28.076 {*, y, z}=K$ =0 { qi , q 2 ,Q}. Dynamic Satellite Geodesy 281 Comparison of this result with Equation 28.069 shows that the matrix K/=o can be considered as constant during time differentiation, as we should expect in a Kepler ellipse because the components of the matrix are all constant. This result holds true for the osculating ellipse of a perturbed orbit, as we shall see in § 28-40. 37. The latitude and longitude of the satellite in a spherical polar system, based on the inertial Car- tesian system, are marked as ((/>, &)) in figure 34, and the following spherical relations will often be found useful, 28.077 cos i= cos (/> sin a 28.078 cos (w+f) = cos 4> cos (a> — Cl) sin (w+f) — sin ( ct> — fl) cosec a= sin (f) cosec i 28.079 cos a = tan (/> cot (w+f) = sin — fl) = sin i cos ( ci) — Cl ) = sin i sec cos (w+f). 28.080 PERTURBED ORBITS 38. If the mass M of a heavy particle located at the origin or if the total mass of a homogeneous sphere centered on the origin is the same as the total mass of the actual Earth, then for the sym- metrical potential we have been considering, CM r is the first and largest term in the expansion of the actual potential expressed by Equation 21.035 in spherical harmonics. We can write the actual potential as 28.081 V=n/r+R so that R represents all the terms which must be added to p\r to give the true potential, whether or not R is expressed in spherical harmonics. More- over, R may contain other small gravitational potentials contributed by the Sun and the Moon. The effect of dissipative and discontinuous forces, such as atmospheric drag and solar radiation pres- sure which cannot be expressed as continuous derivatives of a scalar potential, cannot be included in R; separate treatment is required. We have seen that if R were zero, the orbit would be a Kepler ellipse, defined as an unperturbed orbit. Accord- ingly, R is a measure of the departure of the actual perturbed orbit from a Kepler ellipse; we call minus R the perturbing or disturbing potential and call the gradient Ri< the disturbing force. 39. If we are given the position, and the magnitude and direction of the velocity of a satellite at a given time, then it is possible to find a unique Kepler orbit in which the satellite would have the same position and velocity. The position and direction of motion (or direction of the velocity vector) of the satellite, together with the origin of inertial coordi- nates, settle uniquely the plane of a Kepler orbit. Within this plane, we are given the radius vector r, the zenith distance /3 relative to the radius vector as zenith, and the linear velocity v. These three quantities enable us to determine uniquely a, e, and / from Equations 28.042, 28.054, and 28.053, and so to specify a Kepler ellipse in which the satellite would have the same position and velocity, in magni- tude and direction, as in the actual orbit; the true anomaly f applied to the direction of the radius vector settles the direction of the major axis, and a, e settle the size and shape of the ellipse. Another way of considering this matter is to note that the satellite has six degrees of freedom; that is, we can choose arbitrarily three position coordinates and three components of velocity. Having chosen these six quantities, we can find six, and no more than six Kepler elements Cl, i, w,f, a, e which are necessary and sufficient to establish the same instantaneous motion in a Kepler orbit. The Kepler ellipse which satisfies these conditions is known as the osculating ellipse. (However, this is an incorrect description because the two orbits do not have more than two- point contact.) Instead of the true anomaly /, we may choose either the eccentric or the mean anom- aly (E or M) to describe the position of the satel- lite within the osculating ellipse. There are some advantages in choosing the mean anomaly M. It is sometimes stated, although this is not a very realistic approach to the problem, that the satellite would travel in the osculating ellipse if at any time all perturbing forces were removed. 40. We can say that such relations as Equation 28.076 are true for a perturbed orbit (although de- rived for a Kepler orbit), provided osculating ele- ments are used in such equations, because nothing more is involved than the elements and velocity components which are the same for the actual and osculating orbits. The energy (which is —p/2a in the osculating orbit) is not the same for the two orbits; the kinetic energy is the same, but the po- tential energy differs by the perturbing potential. The accelerations are not the same because the components of force are not the same. Accordingly, 282 Mathematical Geodesy the satellite will depart from a plane osculating orbit and will follow a more complicated curve in space under the action of the more complicated forces. Nevertheless, at any subsequent time, we can fit another osculating ellipse to the actual orbit, so that we can describe the actual motion by means of time differentials of the osculating elements rather than by changes in the actual position and velocity of the satellite. In the next section, we derive expressions for the time differentials of the osculating elements, leading to another form of the equations of motion. VARIATION OF THE ELEMENTS 41. We shall suppose that the total force F r per unit mass is composed of a central force — directed toward the origin or focus of the osculating ellipse and of magnitude /x/r 2 — together with a disturbing force R r , so that we have 28.082 Pr + Rr The central force, if acting alone, would maintain the satellite in the Kepler ellipse, although R r may, of course, have a central component in addition. In cases where the disturbing force is the gradient of a scalar, Equation 28.081 differentiated shows that R r is the gradient of the scalar R; but in this section, we shall assume F r and R r to be forces which are not necessarily derived from a scalar potential. 42. The linear acceleration in the direction of the orbit is the component of total force in that direc- tion, giving 28.083 dv dt ■F r lr = -^^ + R r lr. As in Equation 28.003, the velocity vector is p r —vl r ; from Equation 28.015, we have oo noA d[f) n dr &(prP r ) 28.084 -V i =2r- r = — =2p r p r = 2pr{vl r ) at dt ot dr 28.085 -j t = p r p r lr=v cos p. Semimajor Axis 43. To obtain the time differentials of the ele- ments, we differentiate any suitable Kepler equation without holding any of the elements fixed. For example, if we differentiate Equation 28.042 with respect to time, we have n dv 2u,dr.u, da 2v~r= — -V-r + -^ -77; dt r 2 dt dt' Equations 28.083 and 28.085 then give 28.086 As we should expect, if there is no perturbing force, the semimajor axis a remains constant; whereas, in the presence of a perturbing force R r , this last equation gives the rate of change of a between twc successive osculating ellipses. 44. From figure 33 or 34, either of which illus trates the osculating ellipse as well as a Keplei ellipse, we have /r = r r cos p + t r sm £ so that an alternative equation is ^ = ^{e sin /(Rrrn+^f^ 1 (R r r)}, 28.087 using Equations 28.053, 28.054, and 28.045. Angular Momentum 45. As in §28-17, we can write the angulai momentum vector as 28.088 (vr sin /3)rc r = /W= e r ™p p p g in which n r is the unit vector normal to the plan* of the osculating ellipse. Differentiating intrinsically with respect to time, we have "' V nT + N §£ = e™ PpPq + e™p p p q . dt 8t The first term on the right is the vector product o two parallel vectors and is therefore zero. Th< equations of motion can be written in the form Pq = Fq = -^Pq + Rq so that the last equation becomes dN Ar dn r - a7 nr + N w =e^ Pp F q 28.089 ^e™ PpPq +6™ Pp R q = e r P0p p R q Dynamic Satellite Geodesy 283 because the vector product of two parallel vectors again is zero. Because n r is a unit vector, as in Equation 3.19, 8n r /8t must be perpendicular to n r and must therefore be coplanar with r r and f in figure 33 so that we can write 8n r . contracting with r r , we have on Qr-, _i r 8 A r 8t r*" 8t r " 8t 1 8n> n r l r because n r and l r are perpendicular. In deriving this result, we have used the fact that p, and n r are perpendicular so that we have Prrt r =0, and by intrinsic differentiation, we have on r "67 8p r ~o7 Substitution in Equation 28.089 and successive contraction with n r and t r yield 28.090 — = e r Pin r p P R Q =rR q ti, using the formula for a vector product given in Equation 2.24. Also, we have so that 28.091 NQ = e r ""t r p p R (l ^-rR q nRg) dt~~ v I (l_ e 2)l/2 28.094 Zenith Distance 47. Although the zenith distance ($ is not one of the usual six osculating elements, its variation is sometimes useful and is easily found by differen- tiating N= vr sin ^3 and by using Equations 28.090, 28.083, 28.085. and 28.042. We have then 28.095 dt Aisin/3( \ + r(miR q ). In this last equation, (3, of course, varies even in an unperturbed orbit where its variation is given by the first term on the right. Eccentricity 46. We are now able to find the variation of the eccentricity e by differentiating Equation 28.045, that is, N 2 = fxa(l-e 2 ). Substituting Equations 28.090 and 28.086 in the result, we have (k_ l n .^ da ., dN dt~ ^-^{l-e^-^-N-^ 28.092 ={va 2 (l-e*){l«R q )-N r (t F r m r — v 2 x cos y— (/jl sin /3) I r 2 + R r m r F r n r — v 2 x sin y= R r n T in which we have used Equations 28.140 and 28.082 for the disturbing force R r . These equations enable us to find both y and x from the force components. In an unperturbed orbit (/? r = 0), we have y = 0; the radius of curvature is 1_ v 2 r 2 _ r(2-r/a) X /u.sin/3 sin/3 = r cosec /3(1 + cos E) — a cosec /3( 1 — e 2 cos 2 E) in terms of the Kepler elements. Allowing for difference in notation, this last equation is easily verified from Equation 22.12 as the radius of curvature of an ellipse. 79. For the perturbed curvature vector, we can now rewrite Equation 28.136 as v 2 xm, = F r -(F s hlr = (F s m s )m r + (F s n s )n r 28.146 (tL^A + Rsm$ \ m> . + {Rsn s )nr . 80. The torsion (t) of the orbit involves deriva- tives of the force. For example, by taking the covari- ant derivative of Equation 28.142 along the orbit and by using the third equation of the Frenet Equations 4.06, we have 28.147 F rs n 'I s = jF r m r = v' z x T after substituting Equation 28.138. We can also differentiate Equation 28.141 covariantly along the orbit and can substitute Equation 28.139 to give — Tm r = n rs l s cos y—m rs l s sin y — m r (dyjds) in which ds is the arc element (—vdt) of the orbit. Contracting this last equation with m, and using Equations 28.139 and 3.19, we have 28.148 t = ( dyl ds) — n rs m r l s . There are various ways of evaluating the invariant on the right. An interesting method is to use Equa- tions 28.062 and 28.064 and to evaluate in Cartesian coordinates from rls r d ' lr im rs m r r — m 1 —r- dt = — sin (M7+/+/3) sin i cos (w+f+fB) -j- dil dt = -jy(R q n«)cosP by substituting Equations 28.105 and 28.104. Using Equations 28.145 and 28.054, we have finally n rs m r l s = — x sin y cot fi 28.149 r=(dy/ds) + x sin y cot /3. Evaluation of (dy/ds) by differentiating Equation 28.143 or Equations 28.144 and 28.145 along the orbit again introduces derivatives of the force or second derivatives of a perturbing potential. The torsion of an unperturbed orbit (y = 0) is, of course, zero. THE DELAUNAY VARIABLES 81. Instead of the elements (a, e, i), it is some- times convenient to use three new variables G=Vixa(l-e 2 )=N H = V fxa{\ — e 2 )cos i = N cos i, 292 Mathematical Geodes first introduced by Delaunay, and still retain the other three elements (M, w, ft). Unfortunately, every one of these symbols (L, G, //), which are standard in the literature, also means something else, sometimes in the same chapter of the litera- ture. It is also usual in this context to use (/, g, h) instead of (M, w, ft), the better to exhibit their relationship to (L, G, H). To avoid confusion, so far as possible, we have used L*, H* for the La- grangian and Hamiltonian. The Delaunay variables are not used outside this chapter and are unlikely to be confused with other meanings — of G, g, h,' for example — used elsewhere. Time Derivatives 82. The time derivatives of the new variables are easily obtained by direct differentiation and by use of the formulas for da/dt, dejdt, di/dt already given. We can also relate the results to partial derivatives given in § 28-56 through § 28-76, remembering that in those sections F is any scalar defined in relation to the orbit, such as the dis- turbing potential R. We have dL , fi^da , (1*1*20* „ , vR r l' -vR r l r = dt 28.150 a 1 ' 2 dt .1/2 V dM' using Equations 28.086 and 28.120; we have dG dN „ dR -j-T=-r=r.R»<«=-r— , dt dt aw 28.151 -7- = using Equations 28.090 and 28.127; and we have dH .dN „ . .di —j- — cos i —, N sin i —r dt dt dt = (r cos i)R q t q — r sin i cos {w-\-f)R q n q »»" =% using Equations 28.090, 28.104, and 28.125. 83. We can also express the time derivatives of (M, w, ft), already obtained, in terms of partial derivatives of the disturbing potential R with re- spect to the new variables (L, G, //). For this pur- pose, we need partial derivatives of (a, e, i) with respect to (L, G, H). We have a — L 2 l(i 28.153 G 2 = L 2 (l-e : H = G cos i. For partial derivatives with respect to L, we mui have H, G constant (as well as M, w, ft) so that i : constant and 2L(l-e 2 )dL-2eL 2 de = 0, giving, together with the first equation of Equatior 28.153, da 21 be (1-e 2 ) dL fx, dL eL 28.154 in the same way, we have da _ de _ G ~b~G~ ; ~d~G~~eT 2 ' TL = * di _ cot i ~d~G~~G~ 28.155 da 28.156 1 0; ^=0; — = - dG dH G sin i 84. Next, we have by the ordinary chain rul dR_dadR ctedR di_dR dL dL da dL de dL di 2Lr n (l-e 2 )f = (R r r r )+- — r- 1 \- (a cos f)R r r r ix a eL [ 28.157 + [a + 1-e R r t r sin/ j using Equations 28.154, 28.109, and 28.116; b inspection of Equation 28.102, this is 28.158 dR__dM bL~ dt +n - In the same way, we have G dR dG = — ■ jW~(a cos f)R r r r + la + y__-J 2 ) Rrf sin/ r cot i sin (w+f)(R r n r )/G\ dw ~~~dt" 28.159 using Equations 28.116, 28.123, and 28.107. Als» we have dR r sin (w + / )(R r n r ) _ dVL dt ' 28.160 dH G sin i using Equations 28.123 and 28.105. Dynamic Satellite Geodesy 293 Canonical Equations 85. In this section, we have defined R as the scalar whose gradient is the disturbing force /?,•• To ensure correct signs, we integrate Equation 28.082 and obtain 28.161 V=(fi/r)+R where V is the total potential; therefore, R is, for example, the sum of all the terms in the spherical harmonic expansion of the potential given in Equation 21.035 which must be added to /x/r, thus agreeing with Equation 28.081. 86. The Hamiltonian H* (not to be confused with the Delaunay variable //), as defined by Equation 28.031, is 28.162 H* = W 2 -(fJi/r)-R. We can substitute Equation 28.042 for v because the actual velocity is equal to the velocity in the osculating ellipse so that we have 28.163 /■/* = — — 2a -/? = 2L 2 R in which L is the Delaunay variable V fxa. Also, we have dH*_fjL 2 dR dR dL' L 3 dL n BL' whereas, the partial differentials of H* with respect to the other five Delaunay variables are the same as the partials of (—/?). By substitution in Equations 28.150, 28.151, 28.152, 28.158, 28.159, and 28.160 and writing (/, g, h) for (M, w, CL), we have dL dH* dG_ dt dH* dg ' dH dH* dt dl ' dt dh dl dH* dg dH* dh dH* dt dL ' dt dG ' dt dH 28.164 These six equations are in the canonical form of Equations 28.033, with (/, g, h) replacing the Car- tesian coordinates (x, y, z) and with (L, G, H) re- placing the momenta (x, y, z) . The Hamiltonian has the same value, that is, jv 2 + V, whether the Hamil- tonian is expressed in Cartesian or Delaunay variables. FIRST INTEGRALS OF THE EQUATIONS OF MOTION- FURTHER GENERAL CONSIDERATIONS 87. We shall now consider further the Equation 28.028, that is, 28.165 kv 2 +W= constant, which was obtained as a first integral of the equa- tions of motion relative to the uniformly rotating axes A r , B r , C r fixed in the Earth. To avoid con- fusion, we have overbarred all quantities related to this system so that V is the apparent velocity of the satellite relative to axes rotating with constant angular velocity &>, and W=V-W

    2 d 2 , using Equations 28.003 and 28.059. Substituting this result in Equation 28.165, we find that if 2 + V— (a)d)v sin a sin /3 = constant applies in the inertial system. Using the fact that we have d—r cos (/>, together with Equations 28.077 and 28.054, we can also write 28.166 where N as usual is Vpa(\ — e 2 ). This equation must be a first integral of the inertial equations of 294 Mathematical Geodei motion, equivalent in this case of uniform rotation to Equation 28.021. We can also write 28.167 / dV dt dt = coN cos i + constant. Equation 28.166 can be considered as an expression of the law of conservation of energy in this case and will, in the future, be referred to as the energy integral in our particular problem. It will be noted that (N cos i) in the correcting term is the same as the Delaunay variable H. 89. Equation 28.167 leads us to consider the time variation of (TV cos i). Using Equations 28.090 and 28.104, we have d(N cos i) dt 28.168 rR q {t q cos i — n q sin i cos (u>+f)} (r cos (f))R Q ki by substituting Equations 28.059 and 28.077, and the last term of Equation 28.080. But {R q Ki) is the component of disturbing force in the direction of the parallel and is zero only if the resultant disturb- ing force lies in the plane of the meridian, which would require the field to be axially symmetric. Also, (r cos cpjRqk 9 is the moment of this force component about the axis of rotation. We conclude that (N cos i) is the axial component of the angular momentum vector, which can be verified from Equation 28.023, if we take TV cos i = e rst C,p s pt d(N cos i) j t = e rsl C r p s p< = e rst C rPs F t = e rst C r p s R t 28.169 because the central component of force does not contribute to the vector product, and we are there- fore left with the disturbing force Ri. We have finally d{N cos i) dt = re rs( C r r s R<= (r cos )X'/?„ agreeing with Equation 28.168. 90. We can now verify Equation 28.167 and thus Equation 28.166. If the potential is expressed in the form — ^ = X Tfl P'n'( COS )(Cnm COS mCD + Snm sin ni(o) in which co is the geodetic longitude and in the form co = co — cot where co is the inertial longitude and t is elapse time since the inertial and geodetic meridiai coincided, then we have at once dV . dR . '■ CO -r — = ioRq\ 9 (r COS cf>) dV dt -co 3co dco because we have — V= pLJr-r R and therefoi dVldco = — dRldco. From Equation 28.168, we ha^ 28.170 — = dV . d{N cos i) dt <° dt which is equivalent to Equation 28.167. 91. We find therefore that (N cos i) is a constai of the motion, and thus an integral of the equatioi of motion, only if the field is axially symmetric, : which case (N cos i) is clearly an integral of tl equations of motion relative to either the inerti or the rotating axes. In the case of an axially syi metrical field (tesseral harmonics absent), we ha 1 both 28.171 h 2 +V= constant 28.172 N cos i = constant; whereas, in the case of an unsymmetrical field, \ have only 28.173 W+V—toN cos inconstant, or, using Equations 28.169, we have 28.174 iv 2 + V—Gbe rst C,p s pt = constant. 92. An alternative way of looking at the syr metrical field is of some interest. If the disturbii force is axially symmetric, it can be expressed I Rr = AC,+Bp r where A, B are scalars, but not necessarily constant In that case, we can see at once from the vect product in Equation 28.125 that we have dR/dil = [ therefore, we have dH/dt — from Equation 28.L C where H is the Delaunay variable (iV cos i). A cordingly, we have verified that (N cos i) is constant of the motion in an axially symmetric field. A canonical variable, such as Cl = h in th case, which makes the associated variable constant in this way, is said to be ignorable. INTEGRATION OF THE GAUSS EQUATIONS 93. A standard method of solving differenti equations is to find an exact solution in a speci case which is close to the actual problem; f< example, the exact solution of our present proble: Dynamic Satellite Geodesy for the main term —fi/r in the potential is the Kepler ellipse. This exact solution results in a num- ber of arbitrary constants which, in this case, are six Kepler elements from the three second-order equations of motion or the equivalent six first- order equations. We then obtain more general solutions by writing equations for the (small) variations of the constants required to accommodate the difference between the exact and actual prob- lems—in this case, the perturbing potential — and we solve these equations by successive approximation. Astronomers can claim to have invented this pertur- bation method for this particular purpose, but it is now very generally applied to most of the equations of mathematical physics, usually in the form of an integral equation. A clear introduction to the sub- ject, supported by further references, has been given by the Jeffreys. 4 94. The first Gauss Equation 28.086 can be written as fjb da _ dx r _dR L dR_ 2a 2 dt~ r dt dt dt in which dR/dt is the total differential of the dis- turbing potential, containing explicit time. Inte- grating this equation, we have ~2a~ R+ jl^ But the velocity in the actual and osculating orbits is the same so that Equation 28.042 holds true as r la we have also from Equation 28.081 dt — constant. so that V=^+R r \v 2 V ■It-*- J dt constant; or, using Equation 28.167, we have V — ojN cos i — constant. 1„2. iV which is the same as the energy integral Equation 28.166. The first Gauss equation is accordingly equivalent to the energy integral, and will give us no more information. 95. To illustrate the general method of solution, we shall consider Equation 28.105 for the right ascension of the ascending node, perturbed by the 4 Jeffreys and Jeffreys (reprint of 1962), Methods of Mathe- matical Physics, 3d ed. of 1956, 493-495. 295 second zonal harmonic of the gravitational field, as lxC> () P> ( sin <{> ) _ /xC 2 o 28.175 R r 3 2H 5 (3 sin 2 0-1) where C20 has the meaning assigned in Chapter 21. Using Equations 28.059 in a spherical polar coordi- nate system, we first find the invariant R<,n q = sin a dR 3/xCio r 4 sin (f> cos (}> sin a 3/XC20 • / 1 r\ = — — — sin (w+j) sin 1 cos 1 by substituting Equations 28.077 and 28.079 so that we have dCl iixC'o . » > , ~ 28.176 ~ir= Nf 3 sin- (w+J) cos /. To integrate this equation, we must first transform to a single variable of the osculating ellipse, either t or/, or M, or E\ the obvious choice in this case is the true anomaly/. Using the unperturbed relation d_ = dfd_^Nd_ dt dt df r 2 df 28.177 from Equation 28.098 and substituting the last member of Equation 28.051, we have d£l_r^dn df~N dt _3fxC 2 »(l + e cos/) sin 2 (w+f) cos i ywi-e 2 ) { 1 — cos ( 2w + 2/) + e cos / 3C 20 cos I 2a 2 (l-e 2 ) 2 — \e cos (2w + 3f) —\e cos (2w+f)} : 28.178 To obtain a first-order result, already implicit in the use of the unperturbed Equation 28.177, we assume that the elements a, e, w, i are unchanged during the integration; we then integrate over a complete revolution from/ to/o + 2-7T. The resulting first-order change Aifl (not to be confused with the Laplacian) in the nodal longitude is 28.179 A^ 37rC20 cos i a 2 {\-e 2 ) 2 ' Or, adopting an alternative form of the constant C20 whereby we have 28.180 C n0 = -(a e )»J„ 296 Mathematical Geodes in which a e is a mean radius of the Earth, we have 3-n-a 2 cos ij% 28.181 AxO = — — • a* ( 1 — e* ) z By the same process, we have for the other elements A ia = A ie = Aii = drraUi Aiw= 2t"\~ €J \\2 (1 — 4 sin 2 i) gt (1 — e i ) i (dM \ 37ralfo \~dT~ n ) * = a «(l-e»)»/» (1 ~* Si " 2 °- 28.182 The perturbation of the mean anomaly requires some explanation. If carried from perigee to perigee using Equation 28.043, the integral on the left would be the total change in M minus 2tt, on the same assumption as for the other first-order perturbations that a and therefore n are constant during the integration; otherwise, the last equation is not strictly correct. There is little or no effect on the first-order perturbations whether we integrate from perigee to perigee or between ascending nodes, but the distinction does affect and does complicate the second-order perturbations. 96. First-order or linear perturbations Aift, etc., of the elements caused by each higher harmonic can be calculated in the same way; the results can be added to give the final perturbation Aft as a series containing only the first powers, of the gravi- tational constants C nm , S„ m . If we make enough measurements of the perturbations on different satellites so as to introduce different values of the coefficients of the C nm , S n m, we can accordingly solve the resulting equations for some of the lower order C n m, S n m, assuming that the effect of the higher harmonics on satellites, whose perigee heights are large, can be neglected. The process of integration over a complete revolution will remove some of the higher tesseral harmonics; all the tesseral harmonics will be eliminated if observations of the change in the elements are averaged over a complete day. The method has, in fact, been used most extensively to determine a few of the lower zonal harmonics after suitable corrections for lunisolar perturbations, atmospheric drag, and radiation pressure — the last two of which are small in the case of heavy, compact, high-altitude satellites suitable for determination of the gravitational field. 97. It will be found that the coefficients of J2, 7 4 , Je, ■ ■ . are much larger than the coefficients c 73, Js, • • • in the series for Am; and Aft; these pei turbations are accordingly used mostly for the detei mination of the even harmonics. The Ae- an Aj'-perturbations are best used for the determinatio of the odd harmonics, and can also be used for th higher even harmonics y 4 ,i6, . . . . The integratio of Equation 28.178 over a complete revolution ha removed the short-period terms, consisting t constants multiplied by sines or cosines of angle containing multiples of the true anomaly /. Th resulting first-order perturbations — averaged ove a complete revolution, caused by J2, and give in Equations 28.181 and 28.182 — do not contai any periodic terms (because Ai = 0) and are know as secular terms, the effect of which increase steadily with time. From Equations 28.182, w see that the argument of perigee w changes seci larly so that perigee will eventually complet a whole revolution in the orbit. For this reasor terms containing sines and cosines of w, whic appear in the perturbations caused by som higher harmonics, are known as long-period terms 98. First-order perturbation by A of the argumen of perigee, Aim; in Equations 28.182, becomes zer for an inclination given by sin 2 i = | or cos 2 i=] Close to this critical inclination, perigee oscillate instead of precessing secularly. This case ha attracted much attention, but seems to be of im portance in geodetic applications only insofar as th critical inclination slightly limits the use of pertui bation in perigee. 99. The Kepler elements are not very suitabl for orbits having small inclination or eccentricit because then ft, w, M and their perturbations ar not well defined. The difficulty, which has bee encountered in a different context in § 27—6, ma be overcome by using suitable combinations c the elements as variables. 100. As long ago as 1884, Helmert determine J2 from the orbit of the Moon, using a formul comparable with A^ in Equation 28.181, afte allowance for the large perturbation of the Moon' orbit by the Sun. However, the accuracy of th result, which depends on {a e /a) 2 , is much greate from nearer artificial satellites even though th higher harmonics have more effect. Second-Order Perturbations 101. In common with other perturbation method of solving differential equations, integration Dynamic Satellite Geodesy 297 the Gauss equations runs into trouble when one of the perturbing terms, in this case Cm or J 2 , is much larger than the other terms. It is well estab- lished that C 2 o is about one thousand times larger than any of the other C„ m , Snm', second-order perturbations containing C| or 7| have about the same magnitude as first-order perturbations caused by the other C nm , S„ m - Consequently, we cannot hope to obtain values of the other harmonics unless we includes ./f-terms and possibly also such terms as J2J3, etc. To do this, we must not con- tinue to assume that elements occurring in the coefficients of Equation 28.178, for example, are constant during the integration, and we must use a perturbed relation instead of Equation 28.177. The process will be illustrated by the same example as used for the first-order perturbations, that is, perturbation of ft by the second zonal harmonic. 102. We begin with Equation 28.176 and consider necessary modifications arising from the incon- stancy of w. We have d(d£l/dt) da dt~~ idn ~{dt (dCl ~\dt dw 6ju-C 2 q dw sin (w+f) cos (w +f) cos i dw 28.183 in which the term in braces is the same as we have used on the assumption that the elements are con- stant. From Equations 28.175 and 28.059, we have dR R r r r = — dr 3/jlC 2 2r* (3 sin 2 0-1) cosa dR_3/jLC 2 o . , A r f — zt — — -, — sin © cos ffl cos a r a

    o r 4 sin {w+f) cos (w+f) sin 2 i, using Equation 28.079 and the last term of Equa- tions 28.080. Substitution in Equation 28.096 then gives an equation of the form df dt =4u- 3C 20 X 2ea 2 (\-e 2 in which I is a function containing e, sin 2 i, and trigonometric functions of multiples of w and /. To a first order in C20, we can write dl = j^( SC 20 X df N\ 2ea 2 (l-e 2 ), combining this equation with Equation 28.183, we have dn = r^idil) df N[dt\ r 2 6/jlC 2 h N Nr^ r 2 3C 20 X sin {w+f) cos (w+f) cos i dw dCl N2ea 2 (l-e 2 ) [dt in which we have omitted the term containing Cf , leading to a third-order term. Integration around a complete revolution will give, for the first term on the right, J N{dt\ A,ft, already evaluated in Equation 28.181. The second- order perturbation to be added to Aifi will accord- ingly be 6/u,C 2 o sin {w+f) cos {w+f) cos i A,ft = - [ Nr^ + J 2ea 2 (l-e 2 ) N\dt\ dJ - In these integrals, we have to substitute Equation 28.178 and a corresponding equation for dw/df, and then convert to trigonometric functions of multiple angles. During the evaluation of these second-order integrals we can consider the ele- ments constant, just as we did in the evaluation of the first-order integrals to find the first-order per- turbation Aifl, so that the integration follows the same lines as the integration of Equation 28.178. 103. In addition, we have to include terms in Equation 28.183, such as d(dVLldt) de de, to allow for variation in the other elements; each such term would lead to a second-order integral containing, for example, de/df These terms have to be evaluated, even though the first-order pertur- bation Aie taken between limits is zero. 104. First- and second-order perturbations for a number of harmonics have been derived by Merson,"' 5 Merson (1961), "The Motion of a Satellite in an Axi-sym- metric Gravitational Field," Geophysical Journal of the Royal Astronomical Society, v. 4, 17-52. 298 Mathematical Geodes Kozai," Zhongolovitch and Pellinen, 7 and others. The algebraic equations, even those of the final results, are involved: the labor required to obtain these equations must have been immense. It is probable that the method has served its purpose in the evaluation of a few low zonal harmonics and that future developments will be more in the direc- tion of numerical integration. Meanwhile, other attempts have been made to avoid the complexity introduced by the magnitude of J >. INTEGRATION OF THE LAGRANGE EQUATIONS 105. The Lagrange Equations 28.134 require the disturbing potential R to be expressed in terms of the elements (a, e, i, M, w, fl). This expression has been given by Kaula 8 in the form ix a ", " x R» m = ~nTi^l Fnmp(i) 2 G»i»i(e) p=0 q=- x 28.184 XSnmpa(w, M, ft, 6) where R„ m is the harmonic of order m and degree n in the disturbing potential R, and j in re inn n-m even _ Jnm_ n-m odd -1- O n in n - m even _^ nm_ n - m odd 28.185 cos [(n — 2p)w + (n-2p + q)M + m{CL-d)] sin [(n — 2p)w + (n-2p + q)M+m(n-d)]. Also, d is the sidereal time at the origin of longi- tude—for example, Greenwich — in the original expression for the potential in spherical harmonics. The angle (ft — 6) is accordingly the (Greenwich) longitude of the ascending node. The terms F„m P {i) and C/jqie) are known functions, respectively, of the inclination and eccentricity, which appear in the literature of classical astronomy, and have been tab- ulated for a number of harmonics by Kaula. 9 The symbol a e is a mean radius of the Earth, the inclusion of which requires the C nm , Snm of Chapter 21 to be divided by a n e before substitution in Equation 28.185. It is hardly necessary to say that Equations 28.184 6 Kozai (1959). "The Motion of a Close Earth Satellite," The Astronomical Journal, v. 64, 367-377. 7 Zhongolovitch and Pellinen^ (1962), "Mean Elements of Artificial Earth Satellites," Biulleten' Instituta Teoreticheskot Astronomii, v. 8, 381-395. 8 Kaula, op. cit. supra note 3, 37. 9 Ibid., 34-35, 38. and 28.185 are merely indicial equations and have r tensor significance any more than the constants C m Snm- Transformation of the ordinary expression of tr potential in spherical harmonics to the pole of th osculating orbital plane is not difficult; the con plexity arises almost entirely from the use of M i. one of the elements rather than a purely geometric; quantity such as/, but this complexity is necessai if we expect to use the canonical Equations 28.16' 106. Because the Lagrange equations, like th Gauss equations, are linear, we can substitute di ferentials of individual harmonics R nm on the rigl side of these equations and can integrate term-b; term. For example, the contribution to dil/dt of on term Rmnpq in the double summation of Equatio 28.184, substituted in the last equation of Equatior 28.134, is 28.186 dn 1 ^ n e dF HI dt N sin i a"' di {riipqOn npqunmpqi which can be integrated to find the first-order ( linear perturbation on much the same assumptior as are made for the integration of the Gauss equ: dons. In this case, we assume that (a, e, i) are coi stant during the integration and that w—dw/d M = dMjdt, Cl — dflldt are also constant, whic implies that iv, M, Cl have either unperturbed ( average values obtainable from the correspondin Lagrange equations. The only variable in Equatio 28.186 is then Snmpq', we have Jnmpqdt _ C S, imi > q d{(n-2p)w+(n-2p + q)M+m(n-d) J {n-2p)w+(n-2p + q)M + m{Cl-0) O i nil jig {n-2p)iv+ (n-2p + q)M+m(Ct-d) where Smnpq is the integral of Snmpq in Equatio 28.185, with respect to the argument [n—2p)\ + (n-2p + q)M+m{n-0). In this result, = d> i the constant rate of rotation of the Earth. The fine contribution to the change in the element is LAl Lnlll 1 fxa; X sin i a' x2 [dF,impldi)G„pqSin 28.187 ■2p)w + (n-2p + q)M + m(fi-e) In this equation, n is, of course, an index and nc the mean motion. Dynamic Satellite Geodesy 299 107. First-order perturbations are found to be the same as those obtainable from the Gauss equations. The second-order perturbations may be obtained in much the same way, allowing for variation in (a, e, i, iv, M, Cl), but are just as complicated and just as necessary. Resonance 108. If the denominator {n-2p)w + (n-2p + q)M + m((l-d) of equations, such as Equation 28.187, for the first- order perturbations is near zero, the corresponding term in the perturbation will become very large and the first-order theory will break down. These cases are of considerable importance in the orbits of geostationary communications satellites and in the accurate determination of some higher har- monics. One case, considered by Kaula, 1 " occurs when w-rM + (l-0 is nearly zero; this situation can happen for certain terms in the disturbing function SmnpQ of Equation 28.185. Another case arises when the ratio of the mean motion of the satellite (M) to the Earth's rota- tion rate (9) is nearly equal to m/(n —2p + q) because, for such a term, the denominator will con- tain only the small perturbations iv, Cl. The orbit is then said to be commensurable. There is already a large and rapid growing literature on the subject by such authors as Allan, Anderle, Morando, Wagner, and Yionoulis. One of the latest, which gives reason- ably full references to earlier work, is a paper by Gedeon, Douglas, and Palmiter." INTEGRATION OF THE CANONICAL EQUATIONS Contact Transformations 109. In this book, we have so far used only point transformations, either to a different set of coordi- nates or to a point in another space related by one-to-one correspondence. We now briefly con- sider contact transformations, whereby both the coordinates of a point and a vector associated with the point are transformed in such a way that the 10 Ibid., 49-56. "Gedeon, Douglas, and Palmiter (1967), "Resonance Effects on Eccentric Satellite Orbits," The Journal of the Astronomical Sciences, v. XIV, no. 4, 147-157. canonical form of equations connecting the coordi- nates and components is preserved. The transfor- mation from the six independent variables (x, y, z, x, y, z) in Equations 28.033 to the Delaunay variables (L, G, H, I, g, h) in Equations 28.164 is a contact transformation. For our present purposes, we need to consider the position and velocity of a single particle only in three-dimensional space with six independent variables. However, the same methods apply to dynamical systems consisting of any num- ber of particles, each of which will contribute three coordinates and three components of velocity or momenta. The transformed variables may no longer represent position and velocity separately — the Delaunay variables do not — although the trans- formed variables are sufficient to determine position and velocity either directly or by another transforma- tion. Nevertheless, it is usual to call three of the variables coordinates and to call the other three momenta to fix their position in the canonical equa- tions with the correct sign. We shall denote coordi- nates by q r and momenta by p, so that the canonical equations are, as in Equations 28.033, 28.188 . i)H* dH* in which it is assumed that the Hamiltonian H* can be expressed as a function of p r , q r and of the time t . In writing these equations, we have used index notation and can use the summation convention, but the canonical equations are generally not tensor equations because the variables do not transform in the same way. The total time differential of the Hamiltonian is dH* dH* dH* dt Jp~ p ' + W Q ' + ■ i)H* ~~dT in which the first two terms on the right cancel by Equations 28.188. If the Hamiltonian does not contain the time explicitly (dH*/dt = 0), then we have dH*/dt = 0; the Hamiltonian is a constant of the motion and therefore an integral of the equations of motion. 110. A contact transformation to new variables Q r . P> will result in the canonical equations dK* t 9K* 28.189 Qr = dP r Pr dQ r in which the new Hamiltonian K* expressed in terms of P,, Q r need not necessarily have the same value as //*. It can be shown ,2 that the transforma- 12 A fuller treatment of the subject for different forms of the transforming function is given by Goldstein (1950), Classical Mechanics, 237-243. 300 Mathematical Geode tion equations from Equations 28.188 and 28.189 are 28.190 BS „ dS dq r V dP r ' dt in which the transforming function S may be a function of time and is a function of the mixed variables q r , P r , so that we have 28.191 S=f(q r ,P r ,t). There are alternative transformations in which S can be a function of other variables, for example q r , Q r , but this form in Equations 28.190 and 28.191 is the most useful for our present purposes. The Hamiltonian remains unchanged in value, even though expressed in terms of different variables, if the transforming function does not explicitly contain the time. The Hamilton-Jacobi Equation 111. Next, we seek a contact transformation which will make the new Hamiltonian K* zero so that we have H*(q r ,pr, *) + ?7= 0; at or, using the transforming equation p r — dS/dq r , we have 28.192 V oq r ) dt known as the Hamilton-Jacobi equation. If we can solve this last equation for S, the whole problem is solved because the new canonical Equations 28.189 then show that the new P r , Q r - are arbitrary con- stants of the motion a r , /3 r . The transforming func- tion in Equation 28.191 can then be written S=f(q r , a r , t); the transforming Equations 28.190 become dS(q r , a r , t) Pr 28.193 Q r ^f3 r ^ dq r dS(q r , a r , t) c)a r which enable us to express p r and q' as functions of a r , /3 r and t. We can finally choose the arbitrary constants a r , (5 r to fit given values of p r , q r at a given time, that is, to fit the starting conditions in the orbit. The coordinates and momenta p r , q r are then calculable at any later time. 112. In the form of Equation 28.192, the Hamilton- Jacobi equation applies to general dynamical prob- lems containing any number of coordinates q r . F our particular problem of a single particle in thn dimensions, we can use Equation 28.032 for tl Hamiltonian in Cartesian coordinates and write tl equation as V(x,y,z,,)-H'{(^)\( a - oz) J dt [\dxj \dy, or, using Equation 3.13, we have 28.194 V+WS + f- = 0. at But this last equation is a space invariant whk holds true in any space coordinates, provided v can write the potential V in the same coordinat* and provided the space coordinates in S are ind pendent of time — as we are entitled to assume in ai Newtonian system. If the associated metric tensi of the coordinate system is , b\ are arbitrary constants and the constant (ae) of the spheroidal coordinate system — not to be confused with the Kepler elements of any orbit — is also available as an arbitrary constant. For equatorial symmetry, the potential does not change for ± u, and 6, must be zero. The potential is then 6ii cot a Ubo \ib, cot 2 a + sin 2 u i cot a — sin u i cot a + sin u ' which can be expanded to within a scale constant by Heine's Theorem 15 as X ibu V (2n + l)Q n (i cot a)P„(sin u) (n even). 11 = This result is transformed to spherical harmonics — for the same mass distribution, whatever that may be — by Equation 23.15 as (ae (ae) :i n . . , (ae) 5 n . , ion r^— "•>(sind>H — rr - "-j(sin <&)■ \ ir ir ir 1 We may choose the harmonic of zero order to be — fx/r as usual if we make 28.201 &«, = -—: ae then, the potential is -^{l-(f) 2 p,(sin0) + (f)V,(sm0) Also, we can make the second zonal harmonic the same as in the actual potential of the Earth, if we use Equation 28.180 and make 28.202 (ae) 2 = -C>» = + a?J- 2 , which is Vinti's convention, so that finally the potential is -fll- (yj hP* (sin ) a, ^/Msin) + 28.203 It is of interest to note that Equation 28.202 leads to a real spheroidal coordinate system only if Jo (in Vinti's convention) is positive or if C»o (in our convention of Equation 21.035) is negative, but in the case of the actual Earth, this condition is met. 116. The transforming function W* is obtained from the solution of the Hamilton-Jacobi equation 15 Whittaker and Watson (reprint of 1962), A Course of Modern Analysis, 4th ed. of 1927, 321. 302 Mathematical Geode by Vinti in the form of elliptic integrals containing the constant momenta a, •, which appear during the process of separation; W* is differentiated with respect to these constants to provide the trans- formed coordinates Q r from Equations 28.200 also in the form of elliptic integrals. The constants a r , /3' describe the motion completely in much the same way as initial values of the Delaunay variables do for the potential — fx/r, although not as simply. Nor are the constants a,-, fi 1 as easily related to the Kepler elements, although this relation has been accomplished by Izsak 16 and Vinti. 17 The solution completely takes care of the large second zonal harmonic which causes trouble in the integration of the Gauss and Lagrange equations, but first- order perturbation methods are still necessary to evaluate the higher harmonics. The higher zonal harmonics are evaluated as differences from the corresponding harmonics in the Vinti potential. The von Zeipel Transformation 117. Instead of making the transformed Hamil- tonian zero as in § 28-111, the von Zeipel transfor- mation of the canonical equations successively eliminates the time and the Delaunay angular vari- ables from the Hamiltonian. In the final transforma- tion represented by Equations 28.189, for example, the dK*ldQ r are zero; therefore, the final momenta P r are arbitrary constants. Working backward from the now-known P r and Q r , we can, at any rate theoretically, recover the original p r , q r in terms of arbitrary constants which can be related to the starting conditions. The method has been used to account for the lower-order zonal harmonics, and is complicated enough in this favorable symmetrical case where explicit time and one Delaunay coordi- nate are absent in the initial Hamiltonian. (We have seen in Equation 28.172 that, in this axially sym- metrical case, the Delaunay variable H = N cos i is constant in time; therefore, the Hamiltonian in Equations 28.164 cannot contain the Delaunay coordinate h — il.) An outline description, covering only the C 2 o- or y 2 -disturbing potential, is given by 16 Izsak (1960), "A Theory of Satellite Motion About an Oblate Planet. I. A Second-Order Solution of Vinti's Dynamical Prob- lem," Smithsonian Institution Astrophysical Observatory. Re- search in Space Science. Special Report No. 52. 17 Vinti (1961), "The Formulae for an Accurate Intermediary Orbit of an Artificial Satellite," The Astronomical Journal, v. 66, 514-516; and (1966), "Invariant Properties of the Spheroidal Potential of an Oblate Planet," Journal of Research of the National Bureau of Standards, Section B, v. 70, 1-16. Kaula; 18 a fuller description, including the applic tion to other zonal harmonics, is given by Brouwer. DIFFERENTIAL OBSERVATION EQUATIONS - DIRECTION AND RANGE 118. A solution, which is more in line with curre geodetic practice, is to assume an approxima orbit in much the same way as we start with appro: mate positions in a geodetic network adjustmei The "computed" value of an observed quantity then obtained from this approximate model ai enters the observed minus computed side of differential observation equation. On the other si' of the equation are various terms giving the effe on the observed quantity of the application of C( rections to the approximate orbital elements. The terms are broken down into corrections to the gra tational constants assumed in the approximate orb together with a number of parameters in exprt sions for the drag, etc., which the solution is requin to provide. The observation equations at differe times to a number of satellites are then solved 1 least squares for the corrections and parametei The method differs only from a normal geodet adjustment in that the observations are made different times to a moving object, so that the coel cients of the corrections will usually contain t time of observation. The approximate orbit usually a Kepler ellipse perturbed by the larj C20- or Jo-gravitational term; this orbit ensures th corrections to the gravitational constants will 1 uniformly small. 119. The most useful observations to artific: satellites for geodetic purposes consist of: (a) photography of the satellite against a stell background with, for example, the Baker-Nui tracking cameras or the BC— 4 cameras used f satellite triangulation, as described in §26- through § 26-66; (b) ranging by radio; or, optical-distance measut ment to the satellite using lasers; and (c) range-rate measurement by Doppler-trackii systems or by continuous range measurement. Other methods, such as measurement of horizo tal and vertical angles to the satellite at a kno\ time by kinetheodolite, for example, are general less accurate, but the appropriate observation equ 18 Kaula, op. cit. supra note 3, 43-49. 19 Brouwer (1959), "Solution of the Problem of Artific Satellite Theory Without Drag," The Astronomical Journ v. 64, 378-397. Dynamic Satellite Geodesy 303 tions can easily be formed if required by suitable modification of equations given in Chapter 26. 120. The reduction of photographic observations has already been fully treated in §26-43 through § 26-66, although fewer refinements are usually required in orbital analysis. The end result consists of observational equations connecting observed minus computed plate or film measurements — or deduced right ascensions and declinations — with corrections to the Cartesian coordinates of the i satellite and of the ground-tracking station, such as Equations 26.46 and 26.47. For range observations, \ we can use Equation 26.65. Although these equa- tions were drawn up for the Earth-fixed A'\ B' , C' j system, the equations apply equally well in the : inertial A r , fi r , C r system, provided we replace the origin-hour angle H (as defined in § 26-32) by [H + 9) where 9 is the sidereal time at the origin of longitudes in the Earth-fixed system — in other words, if we interpret H as right ascension. The i inertial coordinates of the tracking station change as the Earth rotates and are different for each obser- ! vation. To derive corrections to the Earth-fixed ! coordinates dxo, dyo of the tracking station, we must , replace the inertial coordinates x, y of the tracking station in the modified Equations 26.46, 26.47, and i 26.65 by 28.204 28.205 x\_ /cos — sin 9\ Axo 07 \sin 9 cos 9 / \yoJ and must replace the inertial differentials dx, dy by dx\ _ /cos 9 — sin 0\ /dxu y dy) \sin 9 cos 9 ) \dyo We can replace 9 by d)t in which w is the rotation rate of the Earth and t is elapsed time since the initial meridians of the Earth-fixed and inertial systems coincided. These transformations do not alter the Cartesian origin. Accordingly, if first harmonics are omitted from the expression for the potential used in forming the observation equations, the corrections dxo, dyo to be found by solving the observation equations will give the final coordinates of the tracking station in relation to the center of mass as origin, as we have seen in § 21—42. 121. We are not interested in obtaining correc- tions to initial values of the inertial coordinates {x r or p') of the satellite, which also would be quite different for each observation. Instead, we seek corrections to approximate values of the orbital elements (a, e, i, M, w, H), which would have been used to compute x 1 ' from Equation 28.069 and so to obtain the computed value of the observed direc- tion or range. The orbital elements (other than M) vary much more slowly than the Cartesian coordi- nates, and can be considered constant in first-order observational equations covering observations over considerable periods of time. Accordingly, we re- place dx, for example, by dx = — da + — de + . . . da de and use Equations 28.130 for the dx/da, etc. More- over, the corrections da, etc., to the orbital elements are themselves composed of: (a) Corrections dao to the values assumed for the approximate orbit. If the approximate orbit takes account of certain perturbations, such as C-n) or J>, then the osculating elements of the ap- proximate orbit will vary with time. Integration of the approximate orbit will give values of the ele- ments do at a particular time to- If the time of observation t is very different, we may have to replace the correction dan in the observation equa- tions by dao + (daldt)o(t — t ), using the Gauss equations for (da/3t), etc., at to; that is, we have to substitute a<>, etc., in the Gauss equations. (b) Corrections arising from the Earth's gravi- tational perturbations. These may be obtained from the integrated Lagrange equations, such as Equa- tion 28.187, by partial differentiation with respect to the Cum, S ini ,. The results will contain sine or cosine terms with arguments {n -2p)w + (n -2p + q)M + m(il- 9). Again, the elements will vary with time if the approximate orbit is not a Kepler ellipse (M will do so anyway). However, if the correcting terms are small, we can write the argument as (n - 2p) (ivo + wkt)+ (n -2p + q) (Mo + MAt ) + m{Clo — do + (il-d)kt} where the zero suffix denotes the initial approximate value of the element at to, At is the time of the observation since to, and w, M, fi are considered constant; C->o or J> is now well enough known to be included in the approximate orbit, in which case the remaining terms, such as da dC n , dC„, could then include such higher zonal and tesseral harmonics as can be handled by computer capacity. (c) Other corrections to the elements arising from atmospheric drag, radiation pressure, and lunisolar 304 Mathematical Geode, perturbations. These corrections will be considered very briefly in the following three sections. Drag 122. Correction terms to the Gauss or Lagrange equations for d, e, w< and M have been given by Sterne 20 and applied by Izsak 21 to an atmospheric model in which the density decreases exponen- tially with height. More realistic high-altitude atmospheric models derived from satellite obser- vations, which among other solar effects show a large diurnal bulge toward the Sun with a pro- nounced lag caused by the Earth's rotation, have been described by Jacchia, 22 by King-Hele, 23 and by Priester, Roemer, and Volland. 24 The results can be used to provide first-order corrections, such as Aa = oAf, to the preliminary values a<> of the elements. However, it has been shown by Kaula 25 that the principal effect is a perturbation AM of the mean anomaly which can be described by a few terms of a polynomial in time as AM = p(At) 2 + q(At) s + .... Kaula also obtains the drag perturbations Aa, Ae, Aw, AO in terms of the same coefficients p, q, which can accordingly be considered as parameters or unknowns in the observation equations in much the same way as the atmospheric parameters Ki, K 2 , K 3 , K 4 (§26-58) are determined in the solution of the observation equations in satellite triangulation. Radiation Pressure 123. Solar radiation pressure as a nongravitational force can have a considerable effect on the large, light, balloonlike satellites used for satellite tri- angulation, but, in that case, a correction is required 20 Sterne (1959), "Effect of the Rotation of a Planetary Atmos- phere Upon the Orbit of a Close Satellite," ARS [American Rocket Society] Journal, v. 29, 777-782. 21 Izsak (1960), "Periodic Drag Perturbations of Artificial Satellites," The Astronomical Journal, v. 65, 355—357. 22 Jacchia (1960), "A Variable Atmospheric-Density Model from Satellite Accelerations," Journal of Geophysical Research, v. 65, 2775-2782. 23 King-Hele (1964), Theory of Satellite Orbits in an Atmosphere. 24 Priester, Roemer, and Volland (1967), "The Physical Be- havior of the Upper Atmosphere Deduced from Satellite Drag Data," Space Science Reviews, v. 6, 707-780. 25 Kaula, op. cit. supra note 3, 57-59. only for orbital prediction. The effect is much le: on small, heavy satellites suitable for the dete mination of the gravitational field, but, nevertheles some allowance must usually be made. The satefli is affected only in sunlight; this intermittent effe requires in practice some form of numerical int gration or harmonic analysis designed to incluc discontinuity. The effect can then be integrate over the time covered by a batch of observation and can be applied as a correction to the ao, en, et( adopted as constant for the batch. A comple treatment has been given by, among other Musen 26 and by Walters, Koskela, and Arsenault. Lunisolar Perturbations 124. We have thus far considered only the attra tion exerted by the Earth on the satellite, and v have justifiably assumed that the attraction exertf by the small satellite on the Earth has no effect ( the motion of the Earth. However, if we introduce massive body such as the Sun into the system, tl effect on the motion of the Earth relative to tl satellite is by no means negligible. The force attraction on the Earth is GM s M E jr 2 where M s , I\ are, respectively, the masses of the Sun and tl Earth and where r (fig. 36) is the distance betwet Sun Earth satellite Figure 36. the two bodies. We have assumed, as we can < because of the great distance r, that the Earth ca for this purpose, be represented by a point mass its center of mass (or a uniform sphere centen 26 Musen (1960), "The Influence of the Solar Radiation Pr sure on the Motion of an Artificial Satellite," Journal of & physical Research, v. 65, 1391-1396. 27 Walters, Koskela, and Arsenault (1961), "Solar Radiati Pressure Perturbations," Handbook of Astronautical En. neering, 8-33, 8-34. Dynamic Satellite Geodesy 305 on the center of mass of the Earth). The Earth, along with the origin of the "inertial" coordinate system we have used throughout this book, is now subjected to an acceleration of GM. s /r L> toward the Sun; the coordinate system is no longer inertial, and the Newtonian equations of motion do not apply. However, we can restore the inertial system by applying an equal and opposite acceleration to all bodies in the system without affecting their relative motion. The satellite (of unit mass) is then subject to the following forces or accelerations: (a) the attraction Fa of the Earth, whatever that may be (we do not assume that this force is directed toward the center of mass of the Earth); (b) the attraction of the Sun GMs/r 2 toward the center of mass of the Sun; and (c) an acceleration of GM s /r 2 , parallel to the direc- tion of the Earth from the Sun, required to cancel the acceleration of the coordinate system. The disturbing force on the satellite is the vector sum of (b) and (c), that is, 28.206 c«« m-f which is equivalent to a disturbing potential at the satellite of -GM S as we can see at once by taking the negative gradient of the latter expression at the satellite with p~j fixed. Expansion of the disturbing potential, as in Equa- tion 21.010, gives GMs(,,r — 1 +- cos w r \ r + . . . — P n (COS l//) r" — — COS l// GM S I f- „ —=- (1+-P-2 (COS l//) + The term not containing ijj drops out on differen- tiation of the potential to form the equations of motion; the remaining terms of the order (l/r ! ) are small, even though Ms is large. 125. Perturbation of the satellite by any number of other bodies, such as the Moon, can be handled by adding accelerations in the same way. Because the Newtonian equations of motion are linear, we can achieve the same result by considering the effect of each body in turn. The disturbing potential in each case can be expanded in spherical harmon- ics related to the inertial system, as in Equation 21.035, and so in terms of the orbital elements of the satellite and of the Sun (or Moon) in a double series similar to Equation 28.184. The full expansion has been given by Kaula. 28 The same methods can then be used as for perturbations of the satellite by terms in the Earth's potential. However, it is now more usual to integrate the Cartesian equations of motion numerically for each small perturbation, using Equation 28.206. DIFFERENTIAL OBSERVATION EQUATIONS -RANGE RATE 126. Continuous measurement of range to the satellite provides a measure of range rate. The range rate is also related to the Doppler frequency of signals emitted by the moving satellite and re- ceived by a ground station, after correction for atmospheric refraction and ionospheric refraction by a two-wavelength technique, although the large number of Doppler observations made on even a short orbital arc requires special initial treatment. Accordingly, we need a form of observation equa- tion for the time rate of change of range, which we shall denote as P. If the inertial position vectors to the satellite and to the tracking station are p r , p r , and if the range and unit vector from the tracking station to the satellite are s, u r so that we have 28.207 su then the range rate is the component of relative velocity in the direction u r , giving 28.208 P=(p'-p'>,. Proceeding exactly as in § 26-5 and § 26-6, we find that we have sdP= (dp'-dp')(p,-p,) 28.209 + (p kPr + q k q,)(p k -p k )(dp r - dp>) in which pk, qk are any unit vectors forming a right-handed orthogonal triad (uk, pk, qk) with Uk— for example, the m r , n r of §26-7 and §26-8 evaluated in spherical polar coordinates. The posi- tion and velocity vectors of the satellite p,. p k are given by Equations 28.130 and 28.131 in terms of the orbital elements. Position and velocity vec- tors of the tracking station are easily obtained in terms of Earth-fixed coordinates (.to, yo, Zo) from Equation 28.204 in the form 'cos d)t — sin wt 28.210 p —\ sin wt \ COS d)t 28 Kaula (1962), "Development of the Lunar and Solar Dis- turbing Functions for a Close Satellite," The Astronomical Journal, v. 67, 300-303. 306-962 0-69— 21 306 Mathematical Geodes 28.211 /— sin ojt — cos G)t 0\ p' = a> cos (bt — sin wf J pg characteristic of the family to be the same constar energy ai or the Hamiltonian where t is elapsed time since the inertial and Earth-fixed meridians coincided. 127. Finally, we must express corrections to the position and velocity of the satellite in terms of corrections to the orbital elements as dp>=^da + d ^de+. . . da de dp' = da + - — de + . . . , da de and then substitute Equations 28.130 and 28.131. As in the case of the observation equations for direction and range, da, etc., are then expressed in terms of da^, dC,,,),, dS„,n, etc. Corrections to the position and velocity of the tracking station are expressed in terms of corrections to the Earth-fixed coordinates p$ by differentiating Equations 28.210 and 28.211 with the time fixed. 128. As in the case of the observation equations for direction and range, the omission of first-degree gravitational harmonics will ensure that corrections to the position of tracking stations are derived in an inertial or Earth-fixed system whose origin is located at the center of mass of the Earth. We have seen in § 21-57 that C 2 r and S 2] -harmonics should also be omitted, although for test purposes these harmonics are sometimes included and the results are com- pared with the theoretical zero. The large number of Doppler observations, which can be made on short arcs, makes this form of measurement par- ticularly suitable, and indeed essential, for the determination of the tesseral harmonics. THE VARIATIONAL METHOD 129. We shall now consider a different approach which yields no fresh results, but affords a deeper insight into the whole problem. So far, we have considered only one orbit, and have defined the linear velocity in this orbit alone. Now we consider velocity as a scalar in three dimensions, defined in some way in a domain surrounding the orbit; this we can do by supposing that the orbit is one of a family, all of whose members have some charac- teristic in common. We shall assume a time-inde- pendent potential V and shall choose the common 28.212 fr 2 +V=ai Because the potential V is defined in space, so is 1 we can introduce the space gradient v r of v, which i not to be confused with the velocity vector vi in one of the orbits. 130. We can also transform the orbit space cor formally with scale factor v to a space in which th line element is ds so that we have 28.213 ds = vds: the velocity vector transforms to 28.214 l r =vl r in accordance with Equation 10.13, provided we us the same coordinate system in both spaces. Thi transformation implies that the point correspondin to the satellite is traveling in the overbarred cor formal space with velocity /,., that is, with constan (unit) linear velocity. We might suppose therefor that the line corresponding to the orbit is a geodesi of the conformal space, just as the free path of point moving with uniform velocity in ordinary spac is a straight line. At present, we introduce thi geodesic property as a reasonable hypothesis; late we shall show that it is equivalent to Newton' second law. 131. It is evident that all members of the family c orbits will transform in the same way to a family geodesies, which will cut orthogonally a family geodesic parallel surfaces generated by assignm different (constant) values to a scalar M*, as ii § 10-19 and § 10-20. Given the family of geodesies it is always possible to construct one surface whicl cuts the family orthogonally; the other geodesi' parallels are then constructed by joining points a equal distances along the geodesies from th initial surface. 132. We can now use any of the results in Chapte 10. Corresponding to Equation 10.27, the equations of the orbits can be written in the vector form 28.215 M* = vl r =l,. From this last equation, we have dM* ds ■=M*l' = l so that M* is the distance between geodesii parallel surfaces (M* — constant) measured alonj the geodesies, as we found in § 10-19. We cai Dynamic Satellite Geodesy rewrite Equation 28.213 as M*= [ vds= I /■■///. Because the length of a geodesic is, in general, less than the length of any other neighboring curve join- ing two fixed points, we can say that the value of the integral vds v 2 dt between two fixed points on an orbit is less if taken along the orbit than along any neighboring path between the fixed points. This is the classical princi- ple of least action, which is now seen to be equiva- lent to the geodesic property in the conformal transformation. We can call M*, integrated along a section of the orbit, the action. 133. The acceleration vector is given by intrinsic differentiation of Equation 28.215 as 8(vl r ) ■Ml dx s vM*l s ot rs dt Because M* is a scalar, we have M* s = M* r and also 8(vlr) 8t = vM*l s =v(vl s ),l S = W r + V 2 l sr l s in which the last term is zero by Equation 3.19, so that we have 28.216 8t VV r (h 2 ), V r by using Equation 28.212; this last equation is Newton's second law as expressed in Equation 28.006 (in which, however, v r is the velocity vector and is not the gradient of the linear velocity). Ac- cordingly, the geodesic principle, the principle of least action, and Newton's second law are all three equivalent. 134. Another way of demonstrating the equiva- lence of the principle of least action and Newton's second law is to write = «, + /_* where L* is the Lagrangian, defined in Equation 28.029, and cti is the constant energy of Equation 28.212. The principle of least action is accordingly equivalent to the assertion that (a i +L*)dt 307 is a minimum for the actual orbit, compared with any neighboring curve between the same points. The Euler-Lagrange equations in the calculus of variations, expressing the condition for this integral to be a minimum, are the same as the Lagrangian equations of motion in Equation 28.030, which we have seen are equivalent to the Newtonian equa- tions of motion. 135. Equation 28.216 can also be written as 8(vl r ) ~8T = Vr in which v r is the gradient of v and s is the arc length of the orbit. This last equation is entirely analogous to Equation 24.07 for the path of a light ray in a medium of refractive index /a. Instead of v in the dynamical problem, we write c/v from Equa- tion 24.01 in the optical problem. We are not at present concerned to reconcile these two problems further. 136. From Equation 28.215, we have 28.217 VM* = g rs M?M? = v 2 g rs lrL s = v 2 because l r is a unit vector. This result should be compared with the eikonal Equation 24.05 in the optical analogy. Substituting Equation 28.217 in Equation 28.212, we have this equation is the Hamilton-Jacobi Equation 28.197 for a time-independent potential, if we takeM*, with any of its various meanings, as the transforming function or Hamilton's characteristic function in the Hamilton-Jacobi theory. Moreover, in all cases where we can solve the Hamilton-Jacobi equation for M*, we can differentiate the result in any co- ordinate system, can substitute in Equation 28.215, and can obtain components of the velocity vector in the same system as a complete first integral of the equations of motion. This approach offers a more geometrical alternative to the canonical solution. 137. Unfortunately, there seems to be no obvious way within the framework of Riemannian three- dimensional geometry to extend this conception to time-dependent potentials. For example, in the geodetic case of a uniformly rotating unsymmetrical field, we could replace Equation 28.212 by Equation 28.174 as a means of defining scalar velocity; but this course at once introduces a preferred direction as well, that is, the tangent to the orbit, which takes the problem out of point transformations into con- tact transformations. On the other hand, if we 308 Mathematical Geodes replace Equation 28.212 by Equation 28.028, that not lead to the correct equations of motion referred is, by the first integral of the equations of motion to rotating axes. The fact remains that the whol referred to rotating axes, we are able to define scalar conception of least action as we have defined it i velocity in the rotating space without the introduc- Newtonian, and the equations of motion referrei tion of a preferred direction, but this course does to accelerated (rotating) axes are not Newtonian. CHAPTER 29 Integration of Gravity Anomalies The Poisson-Stokes Approach GENERAL REMARKS 1. In 1849, Stokes produced his classical paper ' on the determination of the potential at points on a nearly spherical surface by integrating values of gravity or gravity anomalies over that surface. Much work has been done by geodesists on the extension and application of Stokes 1 result to such problems as determining the form of the geoid and the deflection of the vertical — with the object of transforming astronomical to geodetic coordinates — which has resulted in a considerable literature where the basic equations are not always proved or critically examined. As in Chapter 28, we shall accordingly concentrate on deriving the basic equations, and shall indicate methods of solution in bare outline only. Some modern applications are based on even earlier work by Poisson who determined the poten- tial in a field external to a sphere from given bound- ary values of the potential over that sphere. A considerable simplification of the subject results from deriving Poisson's and Stokes' integrals by the same method, and we shall therefore approach the subject in this way. SURFACE INTEGRALS OF SPHERICAL HARMONICS 2. We shall require and shall collect here for easy reference some well-known formulas for the inte- grals of products of spherical harmonics over the surface of a sphere of unit radius or over the whole 1 Stokes (1849), "On the Variation of Gravity at the Surface of the Earth," Transactions of the Cambridge Philosophical Society, v. 8, 672-695. solid angle subtended at the origin. If dCl is an ele- ment of solid angle, if d)(Cnm cos mo)-\-Snm sin mio) 29.02 {&'"} =P"'(sin ) cos mQ)fdO.= I [Pjftsin 0) sin mwpdH 2tt (n + m)! / (2/i + l) fa-ro)! 29.08 29.09 (m#0) [P, ( (sin0)] 2 dft: 47T (2/i + l) 4. Another important integral can be obtained more directly than in most of the literature. In figure 37, P is a. fixed point in (geocentric) longitude Figure 37. and latitude (w, t/>), P is a current point at (&>, 0), and the angular distance between the two is given by cos \\f = sin sin t/> + cos 4> cos <$> cos (ft) — to). The expression {u%} is the value at P of a spherical harmonic, defined over the whole solid angle, and we require the value of the integral / {uJ , }P„(cos ip)dCl. Using the expression for {u%} given by Equation 29.02 and the Addition theorem for P„{cos \Ji) in terms of c/>, t/>, etc., and taking ((/>, co) as constant during the integration, we find without difficulty on using Equation 29.08, that we have / 477 _ { u%} P »(cos \p)dCl = - — -— P;; ! (sin d>)(C nm cos mw Zn+ 1 +S nm sin mo>) 29.10 47T 2n+l {<}? where {«{,"}/> is the value of the spherical harmonic at P. This result also holds for m = 0. SERIES EXPANSIONS 5. The summations of some infinite series con taining Legendre functions are required in this subject and are easily obtained from the following welbknown formula, which is often considered to b< the defining equation of the Legendre functions 29.11 (1-2* cos ijj + k 2 )^ „? k " P " {cos ■W; This equation is absolutely and uniformly con vergent if * < 1 (see, for example, § 21-11). Usually * is considered a constant, but because the equatior is true for all values of * < 1, we can consider £ to b< an independent variable — independent, that is, o i// — so that the equation can be differentiated wit! respect to k. 6. If we differentiate Equation 29.11 with respec to *, multiply the result by 2k, and add to thi original equation, we have 29.12 which can be expressed in the equivalent form £ (2n+l)k"- 2 P n (cosip)=j: 2 (l-* 2 ) (1-2* cos »// + * 2 ) 3 ' — 1—3* cos xp Integration of this equation with respect to / between the limits *, 0, using the standard form contained in most tables of integrals, gives IJ^-'/VcosW 1 1-6* cos i// + 3* 2 k k$> 1 ' 2 -3 cos = (1-2A cos ifj + k 2 ). j After expansion and some manipulation, the value ! of the expression within brackets for A = is found to be (5 cos i// — 3 cos i)j In 4) so that we have finally x (2n + 1 ) — k — 5A 2 cos t// — A ( 1 — 6A cos t// + 3A 2 )/cp'/2 29.14 -3A 2 cos i/; In |(1 -A cos t// + ,/2 ). 7. The basic Equation 29.11, considered as a power series in A, can be differentiated term-by- term, and the differentiated series has the same radius of convergence A < 1. The subsequent opera- tions of deriving Equation 29.12 do not affect convergence, and Equation 29.12 is accordingly convergent for A < 1. The series in Equation 29.14 is also convergent for A < 1 because each term is less than the corresponding term in Equation 29.12. 8. The case A = 1 requires some proof, which we shall not consider, of at least conditional con- vergence, except at the point i//=0. Assuming such a degree of convergence, Equation 29.14 reduces to £o (n-\) 29.15 1—5 cos \\i— (2 — 3 cos i//) cosec i»// — 3 cos i// In ( sin \\\j + sin 2 ji// ) . known in gravimetric geodesy as Stokes' function, with the symbol S(i|/). For a particular value of A , which we shall consider in §29-31, the expression in Equation 29.14 is known as Pizzetti's extension of Stokes' function. However, the even more general Equation 29.14 is simply an identity, al- though this identity does have important applications in gravimetry. 9. We shall also require two other expansions which can easily be obtained by integration of Equation 29.11, using the identity (2/7 + 1 = 2 1 (n+1) " (n + 1 These expansions, which are similar to Equations 29.14 and 29.15, are 29.16 * (2n+ 11 S(*. «/») = £ \ ' *»■»/>„ (cos ' ( n + 1 ) 4> I/2 + A — cos i// 11 = 2A 4)1/2 in which 4> is given by Equation 29.13, and 29.17 £0 (»+D cosec ^— In (1 + cosec h\t). 1 — cos i// These modified functions are convergent to the same extent as Equations 29.14 and 29.15. INTRODUCTION OF THE STANDARD FIELD Potential Anomaly 10. If the actual potential is W and if the standard potential — usually the potential of the standard field described in Chapter 23 — is £/, the difference 29.18 T=W-U is known as the potential anomaly. Because the actual and standard fields are supposed to be ro- tating about the same axis with the same angular velocity w. the terms in the geopotentials containing oj cancel; and it is immaterial whether W ', U are both attraction potentials or both geopotentials. In either case, the potential anomaly T is harmonic. By suitable choice of the standard field, T can be made a small quantity more amenable to approxi- mate solutions. In the literature, T is usually called the disturbing potential, which is too easily confused with the disturbing or perturbing potential affecting satellite orbits (§28-38). Curvature and Deflection 11. At a point P in space (fig. 38). the unit normals to the actual and standard equipotential surfaces, respectively, are v r . v r . The unit vector v r in figure 38, which is not necessarily in the same plane as v r , V, is the unit normal to the coordinate surface through P. We shall assume in this section that l> r refers to the geodetic (o», , h) system in which the meridian and parallel vectors will be denoted by jji r ,k r . We shall also assume that the standard field is as described in Chapter 23 and that the equipotential spheroid coincides with the base spheroid of the coordinate system. From Equation 19.42, we then have in the notation of figure 38 &r =v r_=r ^ ( cos ^ ft^r _|_ (§^)^r _ ^r + £ = r 29.19 where A r is the astronomical minus the geodetic deflection vector and 8a» (8$) is the astronomical 312 Mathematical Geodesy (actual field) (standard field) base coordinate surface Figure 38. minus the geodetic longitude (latitude). In this equation, 8w, 8(f) are supposed to be small, and the equation holds true to the first order in these quan- tities. The first-order meridian and parallel compo- nents of deflection are denoted by £, 17, as usual in the literature. 12. We shall call the angle k between v T and y r the standard curvature correction because it arises from the curvature of the standard line of force when the base coordinate surface is a standard equipotential. There are several ways of finding k from the geom- etry of the standard field. One method is to convert the geodetic coordinates of P to geocentric coordi- nates, compute the latitude of the standard line of force (i> r ) at the same time as standard gravity from Equations 23.40 and 23.41, and subtract from the geodetic latitude of P to obtain the curvature cor- rection k. With the sign convention of figure 38, we then have 29.20 V r = l> r cos k — /JL r sin K. The meridian component of standard gravity y, given by g„i in Equation 23.37 for a spheroidal field, is U r jJL r =yv r ji r = — y sin k, which is another way of computing k. If the base coordinate surface is a standard equipotential, k is zero on that surface and is also zero along the axes of symmetry of the standard field. Gradient of the Potential Anomaly 13. By differentiating the scalar Equation 29.18 we have 29.21 T r =W r —U r = gVr — yv r where W ', U are now considered to be geopotentials and g, y are, respectively, actual and standarc gravity. In this equation, we have used Equatior 20.05 and the physical definition of the potentia (§20-3). Equations 29.19 and 29.20 enable us tc express the gradient of the potential anomaly ir terms of the geodetic parallel, meridian, and norma vectors as Tr=(gr))X r + (gt; + y sin K)fI r -\-{g— y cos x)v r . 29.22 This equation is exact within the first-order defini tion of (£, rj). At points not too far removed from tht Earth's surface, the curvature correction k will be no larger than the deflection components, and te the same degree of accuracy we can write 29.23 Tr **(gr))\r+g(g+K)lJLr+(g — y)Vr. Equations 29.22 and 29.23 hold true for any (co, c6, h coordinate system, provided that X r , jx r , v T are th< parallel, meridian, and normal vectors of the systen and provided that the deflections and curvature correction refer to the same system. For example in a spherical polar system, v r is the unit radius vector and Equation 29.19 gives £, 17 as the meridiar and parallel components of the astronomical zenitl in the spherical polar system. The vector X r is the same in the spherical polar system as in the geo detic system, but the meridian vector jx r is not the same; the deflection component 17 is the same, bu (£ + k) is not the same; and if k is ignored, as usua in the literature, then the meridian deflection £ is not the same. Gravity Disturbance 14. We define the gravity disturbance at the poin P as 29.24 g D =g-y. Equation 29.23 then shows that we have 29.25 g D =T r v r =dT/dh; that is, the gravity disturbance is the componen of the gradient of the potential anomaly in the direc tion of the geodetic normal. In the literature, the Integration of Gravity Anomalies — The Poisson-Stokes Approach 313 gravity disturbance is usually denoted by 8g, which, however, might be confused in this book with any increment of g. 15. If the potential anomaly is expressible in spherical harmonics as T= ^ {r» ! }/r" +1 , n , m then the radial component of the gradient is ar = _ (n + i){r»'| dr *-> r n+2 n, m The geodetic normal can be considered as not far removed from the radius vector in the case of the Earth, and the gravity disturbance will, in any case, be a small quantity if the standard field has been chosen close to the actual field. Subject to these approximations, we can combine the last equation with Equation 29.25 and can write 29.26 go^~^ (n + \){T™}lr»+ 2 . Gravity Anomaly 16. The actual potential at P (fig. 38) is W P . From formulas given in Chapter 23 or from tables based on these formulas, we can find a number of points where the standard potential U is equal to Wp, and we choose such a point B in the direction of the geodetic normal v r so that Wp = Ub- The gravity anomaly is then defined as 29.27 gA = gP — JB , and the length BP = l ! is known as the height anomaly. The gravity anomaly g.\ is usually denoted by A^r, which can, however, be confused in this book with a Laplacian. To a first order, we have W P = Tp + Up = T P + U B + (dU/dh) B £,, and because Wp = Un, this equation reduces to Tp^-{dUldh) B l If we ignore the distinction between v r and i> r , which means neglecting the curvature correction, we have 29.28 7V --?«£, usually known as Brans' equation. Very often this equation is approximated further by assuming that 7b = Jp, so that at any point in space we have 29.29 T^-yl 17. Next, we combine Equations 29.24 and 29.25 to give to first-order accuracy (bt\ /ay ■■ gp — y p = g p — y H I gA i = gA + dh/ B dy\ T Bh a Jh We assume further that PB is small and ignore the distinction between P and B in this last equation, which becomes then for any point in space 8T (8 In y\ dh \ dh 29.30 gA Equation 29.30 is usually known in the literature as the "fundamental equation of physical geodesy." All the approximations in this formula are covered by the single assumption that the potential anomaly T is small. One further approximation is often made. If we ignore the centrifugal part of the standard potential, which is then harmonic, and confuse v r with v r (fig. 38), we have from Equation 20.17 d In y i)h 2H where H is the mean curvature of the standard equipotential surface. Moreover, the standard field differs little from a spherically symmetrical field in which we have 2H = — 2/r, so that we can write 29.31 dT 2T dr r We should obtain the same result from Equation 29.30 by assuming that the standard field is static and spherically symmetrical with a potential of minus p,jr. 18. If T is expressed in spherical harmonics as T= X {T','!}lr" +1 n, in where {T"'} is given by Equation 29.02, substitution in Equation 29.31 gives for each harmonic \{g A W l }=- 29.32 (n + \){T»>} , 2{r;;<} (n-l){7^} This formula was first obtained by Stokes, 2 who made equivalent but different assumptions in deriving it. Summing over m, n, we have 29.33 gA (H-i){r»'} 2j r « + 2 2 Stokes, op. cit. supra note 1, 693. 314 Mathematical Geodes' 19. Various interpretations are given to the points P, B in figure 38, subject to the requirement Wp= Ub- For example, P is often a point on the geoid andfi is a point on the standard spheroidal equipotential, an interpretation which implies that the geoid and spheroid must have the same potential. In that case, measurements of gravity made on the topographic surface are reduced to corresponding values at P on the geoid by making various assumptions about crustal densities. 3 The value of standard gravity at B on the spheroid is subtracted to give the gravity anomaly. In Chapter 30, we shall take P as a point on the topographic surface of the Earth, in which case the locus of the point B is a surface named by Hirvonen the telluroid. To compute the gravity anomaly in this case, Wp would have to be measured by spirit leveling. 20. We may wonder why the more complicated gravity anomaly is used in preference to the simpler, and more logically geometrical, gravity disturbance which compares the two fields at the same point in space. One reason is that, in the earlier applica- tions, P is a point on the geoid and tables of standard gravity are required only for points on the equipo- tential spheroid to compute the gravity anomaly. Another reason is that the geodetic height h is initially known only approximately — in fact, one of the objects of the whole exercise is to find h — so that we cannot calculate standard gravity accurately at P. These arguments are less significant today when standard gravity is readily calculable at any point in space and when h can be, and usually is, found by successive approximation. If the gravity disturbance is used, some iterative procedure, starting with approximate values of geodetic heights, would be necessary and would probably require more computation than the use of the gravity anomaly; there is no certainty that the operation would converge, but this has not yet (1968) been fully investigated. Equation 29.30 for the gravity anomaly has the form of one of the boundary condi- tions of classical potential theory, and this fact has probably attracted theoretical investigators. However, Equation 29.25 is a much simpler bound- ary condition and is more accurate. THE SPHERICAL STANDARD FIELD 21. It will be apparent throughout this chapter and Chapter 30 that this branch of geodesy could be simplified by using spherical polar coordinates 3 See also § 29-42. For full details, see Heiskanen and Moritz (1967), Physical Geodesy, 126-159. and a spherically symmetrical standard field in stead of the geodetic system. If all the standan equipotentials are to be spheres, we can eliminat the leading term in the attraction potential from th potential anomaly by suitable choice of constant in the formulas of § 20-24, but we cannot eliminati the centrifugal term. However, we can eliminati both terms by making one equipotential surface j sphere which coincides with the base surface of th spherical polar system (r = R). In that case, th standard geopotential is symmetrical about th z-axis and is given by Equation 23.01 as - U= 2_, T^l 1" 5 «J 2 r 2 - i oo 2 r 2 P 2 {sm 0). n = ' 29.34 If the geopotential is to be constant (C/o) for al values of the spherical polar latitude 4> over th base sphere (r=/?), we must have -£/ =GCoo//?+id) 2 /? 2 29.35 Q=GC 20 /R 3 -i& 2 R 2 , and all other C„ must be zero. For the leading ten to be the same as the leading term in the actu£ potential, we must have Coo equal to M — the tot£ mass of the Earth. The centrifugal terms will canc« in the potential anomaly if the origin of spheric* polar coordinates lies on the axis of rotation. Th first harmonics (absent in the standard potentia will not appear in the potential anomaly if th origin is located at the actual center of gravit) this condition is compatible with cancellation c the centrifugal terms. If the standard geopotentis Uo of the base sphere is to be equal to the actuc geopotential Wo of the geoid, this requirement woul settle the value of R in accordance with the firs equation of Equations 29.35. Standard gravity ani the latitude of the tangent to the standard line o force, and thus the curvature correction, are give: by Equations 23.02 and 23.03 or by Equations 23.0 and 23.05. All the formulas and remarks in the las four subsections (§29-11 through §29-20) apply i we use the elements of this spherical system ii place of the geodetic system, provided we use oni complete system or the other. The disadvantagi of this spherical system, compared with the usi of geodetic coordinates and a spheroidal standan field, is that the anomalies and meridian deflec tions are generally larger, although probably stil within the limits of the usual first-order assumptions there is not the same necessity in this branch o geodesy as there is in satellite geodesy (§28-101 to provide a model field which eliminates most o the second harmonics. We could, of course, choose Integration of Gravity Anomalies — The Poisson-Stokes Approach 315 a sphere as base coordinate surface and a spheroid as standard equipotential, but, in that case, the angle k in figure 38 will be large in mid-latitudes, and Equations 29.25 and 29.30 for the gravity disturb- ance and the gravity anomaly might no longer be sufficiently accurate; nothing would be gained. POISSON'S INTEGRAL 22. We shall now suppose that values of a har- monic function H are given at all points Q (fig. 39) Figure 39. on the surface of a sphere of radius R. We suppose that H is defined at all points outside as well as on the sphere, and we must find the value of the har- monic function at a point P outside the sphere at a distance r from the center of the sphere. The angle between OQ and OP is shown as i// in figure 39, and the distance PQ is shown as /. 23. First, we put k = R/r in Equation 29.12 to give 29.36 (r 2 -R 2 ) / 3 R" £ (2n+l)^ I P„(cos«J0 Next, we suppose that H is expressible as a con- vergent series of spherical harmonics with O as origin so that we have 29.37 ff«=2 J] {H"}IR*+K Multiplying these last two equations, integrating over the whole solid angle, and using Equation 29.10 on the right-hand side give 29.38 (n-R 2 ) H Q dn=2 2 4ttK" {//;;'}, R ,H in which we have interchanged the order of summa- tion and integration on the right, and {//"'}.s is the value of the spherical harmonic {//"'} at S on the radius vector to P (fig. 39). Because P is fixed during the integration, r and the spherical radius R are both constant during the integration. But {//"'} contains only geocentric latitude and longitude, which are the same at P and S, so that we have Moreover, the value of H at P is 71 = 1)1 = 1) so that Equation 29.38 reduces finally to 29.39 H„ = lj^^H which will be the case if both the actual and standai potentials can be so expressed, it follows from Equ tion 29.33 or 29.26 that rg.\ or rgi> is harmonic so th; Equation 29.47 holds true for the gravity anoma or for the gravity disturbance as 29.48 (gA)p = R 2 (r 2 -R 2 4-rrr r , . da I (gA)Q— t P Integration of Gravity Anomalies — The Poisson-Stokes Approach 317 This last equation is usually known as the upward continuation integral because it enables us to cal- culate the gravity anomaly (or gravity disturbance) at any point in space by numerical integration, if we have values of the gravity anomaly (or gravity disturbance) at a large number of points on a sphere. STOKES' INTEGRAL 30. If we multiply Equation 29.15 for the Stokes' function S(i//) by a bounded function, such as the gravity anomaly g.\ in Equation 29.33, we have so/,) g ,=-jj; { ?"*i) p» (cos w 11 = 2 ^ ' x ^ A (P-1HT?} x 2j L r p+2 p=0 m=0 assuming that g.\ is expressible as a convergent j series of spherical harmonics. Integrating this equation over the whole solid angle or the sphere !in figure 39, noting that the terms of the product on (the right are zero unless p — n (Equation 29.05), and using Equation 29.10, we have n J Tm\ SWr)(ft)Q) («i )<,r) (g A ) Q dn. Contracting this equation in turn with the spheric polar parallel and meridian vectors at P (which ai fixed during the surface integration) and usir Equation 29.23 for a spherical polar system, v have gv 4tttJ as sin a(g A )Qcin 29.57 S^ + x)=-^ r J^ c osoc{g A ) Q dn where we have also used Equation 29.56. The: equations, with k assumed to be zero, were ori inally given by Vening-Meinesz; the function dS/di obtained by straight differentiation of the Stoke function, Equation 29.15, is usually known as tl Vening-Meinesz function. However, in the for given by Equations 29.57, the equations hold tn for Pizzetti's extension of the Stokes' function ai also hold true for the integration of gravity di turbances (instead of gravity anomalies), provid( the modified functions S, given by Equation 29. or 29.17, are used. The azimuth a in Equatioi 29.57 refers to the azimuth at P of the directk QP. If we use the azimuth of the direction Pi we must change both the minus signs in Equatioi 29.57 to plus. 37. However, we must realize that Equations 29.i give the first-order meridian and parallel cor ponents of deflection (£, t?) at the point P in relatk to a geocentric spherical polar system; the secor equation does not, as discussed in §29-13, give tl meridian deflection in relation to the geodetic sy tern, and the difference may be quite considerabl Integration of Gravity Anomalies — The Poisson-Stokes Approach 319 Ti> at P. Because r and / (fig. 39) are functions which can be considered as defined at Q in the same way as at P and because Q is fixed for a displacement of P, we have < r ->'-£ 2r S^-S^ which, on substitution of Equation 29.54, becomes (Tr)p = R^ (■ rj2r 3(r 2 -ff 2 ) cos ft 3(r 2 -/?-) sin ft I 4 rip, Contraction with the spherical polar meridian, parallel, and normal (/>) vectors at P (which are constant during the surface integration) and use of Equations 29.23 and 29.56 give gt g-T) 3R(r- — R-) f sin (3 cos a 4tt 3R(r 2 -R 2 ) [ sin (3 sin a (gi>)r 29.58 R_ 47T 4rr 2r 3(r--/? 2 )cosj8 /■ /' T Q dil T Q dfl T Q dfL. It is by no means easy to discern this fact in the maze of approximations usually made in deriving the Vening-Meinesz integrals. The difficulty arises not so much from first-order approximations as from the nature and multiplicity of first-order approximations. 38. The assumption that k is zero in the Vening- Meinesz form of Equations 29.57 is in order if we are dealing solely with points on the base coordinate sphere, which is taken as a standard equipotential. Otherwise k should be computed for the standard field described in §29-21. The usual assumption that k is zero everywhere implies that all the stand- ard equipotential surfaces are spheres, and this interpretation is possible only if the standard field is not rotating. A nonrotating standard field is incompatible with the definition of the potential anomaly and with the harmonic properties of the potential anomaly on which the entire theory is based, although a nonrotating standard field is assumed in deriving the approximate formula for the gravity anomaly in Equation 29.31. 39. We do not obtain deflections in the geodetic system from the Vening-Meinesz integrals by using gravity anomalies computed for a spheroidal standard field, although the effect may well be dis- guised by doing so. It is true that in the derivation of the Stokes and Pizzetti Equations 29.50 and 29.51 from which the Vening-Meinesz equations are ob- tained by differentiation, the gravity anomalies are merely assumed to be any function related to a harmonic function by Equation 29.32. Neverthe- less, as soon as we identify this harmonic function with the potential anomaly, we introduce a stand- ard field; for example, the use of the approximate formula for the gravity anomaly in terms of the po- tential anomaly, Equation 29.31, introduces a spherical nonrotating standard field, which we must use if we are to be consistent. It is better to use the spherically symmetrical rotating standard field described in §29-21, especially in the more accu- rate formulas containing the gravity disturbance, and so retain the harmonic properties of the poten- tial anomaly. Such a standard field lies within legitimate first-order approximation in this branch of geodesy (unlike the potential disturbance of satellite geodesy, discussed in §28-101), and the resulting geocentric deflections can very easily be converted to geodetic deflections. GRAVITY AND DEFLECTION FROM POISSONS INTEGRAL 40. We are now able to differentiate Poisson's integral, Equation 29.39, for the potential anomaly ■ Heiskanen and Moritz, op. cit. supra note 3, 37-39. Tudil. These equations give the geocentric deflections and gravity disturbance, relative to a spherically symmetrical standard field, at any point in space from given values of the potential anomaly over the reference sphere. The only assumption made is that the geocentric deflections are of the first order. The equations do not determine gravity and deflection at points on the reference sphere (r — R) any more than the original Poisson equation determines potential, although the third equation does apply on the sphere, apart from the singularity in the neighborhood of P(l = 0). Heiskanen and Moritz * remove this singularity by an ingenious device which we shall consider next. 41. The third equation of Equations 29.58 applies to any potential function T and the radial component of its gradient. If we apply this equation in space to the potential function —Rjr, which becomes — 1 on the sphere, we have 3 dr R R r 2 R 477- [I 2r 3(r--R-) cos p !■ dft: 320 Mathematical Geodes multiplying this equation by Tp, which remains constant during the surface integration, and adding to the third equation of Equations 29.58, we have m„ 2r 3(r 2 -/?-) cos/3 l :i r 29.59 x {T u -T,.)dil. Next, we make the substitution r = R + h, where h is small, and find that the contribution of the second term in the integrand over a small area in the neighborhood off is approximately 6Rh cos/3 / f)r\ (ttI- \ ^ 6ttcos 2 j8 / ST Z 4 \dl)\R 2 )^ R \dl which becomes zero in the limit when P lies on the sphere. The contribution of the second term is zero everywhere else for r — R, so that when P lies on the sphere we have 8T\ . Tp = & ( (T Q -T P ) ' R 2t7 29.60 dr !■■■ da. The contribution of a small area in the neighborhood of P to the area integral is flection. The difference may, however, arise i part from misinterpretation of the Vening-Meines integrals, as discussed in §29-37. We shall no consider a simpler method of approaching th problem. 9 See Heiskanen and Moritz, loc. cit. supra note 3. 10 Stokes, loc. cit. supra note 1. 11 Zagrebin (1956), "Die Theorie des Regularisierten Geoid; Geoddtischen Instituts, Potsdam, Veroffentlichungen no. 1-129. This is a German translation of an article that origina appeared in a Soviet journal. See (1952), "Teoriia Regulia zirovannogo Geoida," Trudy Instituta Teoreticheskot Astronom v. 1,87-222. 12 Bjerhammar (1962), "On an Explicit Solution of the Gra metric Boundary Value Problem for an Ellipsoidal Surface Reference," The Royal Institute of Technology, Geodesy Di sion, Stockholm, 1-95, and (1966), "On the Determination the Shape of the Geoid and the Shape of the Earth From Ellipsoidal Surface of Reference," Bulletin Geodesique, m series, no. 81, 235-265. Integration of Gravity Anomalies — The Poisson-Stokes Approach 321 44. The simplicity of using a spherical base co- ordinate surface, as in Poisson's and Stokes' integrals, arises not only from the fact that the radius vector — the third coordinate — is constant during the surface integration, but also from the fact that functions of the radius vector in the potential and gravity anomalies depend only on the degree n and not on the order m of the spherical harmonics used to express the potential anomaly. We can obtain some, but not all, of this advantage by integrating over a base spheroid in the spheroidal coordinates of Chapter 22. The potential anomaly can be ex- pressed in spheroidal coordinates (a>, u, a) by Equation 22.50 as -77C=2 2 ™ (i cot a) =-(n + l){r)»} i tan a(n — m + 1 )Q"' + . (i cot ct ) 0%(i cot a) by summation over m, n, we have 0™ (i cot a TO: ^, A i tan q(/t - m + 1 )(?™ + , (t cot a ) , n = o w = 29.63 which is an extension of Equation 29.26, except that the harmonics are now in terms of the reduced latitude u and not the geocentric latitude. We multiply this last equation by the modified Stokes' function, Equation 29.17, =,, * 2rc+l n = n = cosec ^ i//— In (1 +cosec|i/>) where cos ip is now calculated from the reduced latitudes u, u as cos )//=sin a sin « + cos w cos u cos(o> — cu), 29.64 and integrate over the whole solid angle dfl = cos « dudoj. We then have 4tt7V j tan a(n — m+ \)Q' l " +l {i cot a) Q™(i col a) XS(x},){T>»}dn g D v)S(ip)dn=- -flt J h = m = ( 29.65 which is an extension of Stokes' integral for the gravity disturbance. From Equation 22.49, we have Q™ +1 (i cot a) = (-Y"(n + m + l) Q'Jti cot a) (2n + 3)icota 1 + 1 i cot a and the integral on the right side of Equation 29.65 becomes ■2 2 n = o m = 29.66 (-rtan 2 a(n + m+l)(n-m + l) (2rc + 3) xSty){Tf}dn, ignoring the fourth and higher powers of tan a, which is roughly the eccentricity of the base spheroid. This last integral can accordingly be considered as a correction to the Stokes' integral to allow for the gravity disturbance being given over a spheroid instead of over a sphere. The cor- rection is of the second order in the eccentricity. The other difference from the Stokes' integral in Equation 29.65 — the use of the principal radius v instead of a mean spherical radius R — is also a second-order effect, so that the spherical Stokes' integral holds true for integration of gravity dis- turbances over a spheroid to a first order in the eccentricity, provided we use the reduced latitude and, in effect, integrate over the auxiliary sphere. 45. To calculate the correction in Equation 29.66, we need to expand a first approximation to the 306-962 0-69— 22 322 Mathematical Geodes potential anomaly in spheroidal harmonies, con- verted if necessary from spherical harmonics by Equation 22.59. However, in practice, the zero- and first-degree harmonics are usually assumed to be absent in the potential anomaly because the total mass and center of mass are assumed to be the same in the actual and standard fields. We need usually consider only the comparatively large second-degree zonal harmonic for which the correcting term in Equation 29.66 is -(9/7) tan 2 a j {T 2 } (5/3)P,(cos V p y = — 2 — sin 2 a cos 2 u — 2a) 2 p/y to a second order in the eccentricity, using Equa- tions 22.10 and 22.12. It is usual, as in the approxi- mate formula for the gravity anomaly (Equation 29.31), to ignore the centrifugal term when mul- tiplied by the small potential anomaly. However, the magnitude of this term in the last equation is about 1/150, which is about the square of the eccentricity, so that we have no right to ignore this term in working to a second order. Equation 29.63 can now be replaced by ^=-2 2 ("-Dim n = o »i = o y: II + V V sin 2 a cos 2 u{T"'} ii = in = + jr j (2s»wy){r»} _ Y -A i tan a(n-m + j )(?;,"+ i(i cot a) , , ,1 = i) m = o Q%(i cot a) 29.67 When multiplied by the Stokes' function £() = if^/Mcos.//), n — 2 where cos i// is given by Equation 29.64, and ir tegrated over the whole solid angle, this las equation gives an extension of Stokes' integrz similar to Equation 29.65 with three correctin terms on the right which can be evaluated numer cally for the main second-degree zonal harmoni as in § 29—43. In evaluating the centrifugal correct ing term, it is reasonable to assume that (2a) 2 vl~) is constant at its mean value, which is about 1/145 47. Equations 29.65 and the corresponding equ; tion for the gravity anomaly are in the form ( integral equations whose solution gives the potenti; anomaly at a particular point P on the spheroic this form best illustrates the analogy with th Stokes' integrals for the sphere. However, thes integral equations are exactly equivalent to th system of linear equations in Equations 29.63 an 29.67, expressing the gravity disturbance or anomal as an infinite series of spheroidal harmonics c the potential anomaly. We cannot, of course, solv an infinite number of these equations for the coeff cients A,,,,,, B,„„ of Equation 29.61 any more tha we can evaluate the integrals in the integral equ; tions for all points of the spheroid; the most w can do is to integrate numerically the gravity di; turbance or anomaly at a number of discrete point; which represent average conditions in the localit and are well spaced over the spheroid. In much th same way, we can suppose that the potenti; anomaly is sufficiently well represented by a finit number of the coefficients Anm, B„ m in Equatio 29.61, and we can solve Equation 29.63 or 29.6 for these coefficients. The number of coefficient for which we can solve will naturally be bmite by computer capacity; we must, of course, hav at least as many observations for g/> or g.\ as ur known coefficients, preferably many more so the we may solve the observation equations derive from Equation 29.63 or 29.67 by least square; The advantage of this method is that we are nc limited to points on the spheroid; we can substitut all three coordinates of the observation points i the coefficients of the unknown A nm , B„ m in Equ; tions 29.63 and 29.67, and have no need of Pizzet extensions to the integral Equation 29.65 or th corresponding equation for the gravity anomal; The same method can be used in geocentric c< ordinates to solve Equation 29.26 or 29.33 for th Cnm, Snm of the potential anomaly expressed i spherical harmonics. In § 29-49, we shall conside yet another application of this method. Integration of Gravity Anomalies — The Poisson-Stokes Approach 323 BJERHAMMARS METHOD 48. Realizing the essential simplicity of the classical spherical approach, Bjerhammar, 13 in one of the most modern methods, uses a spherical Figure 40. reference surface completely embedded in the Earth (fig. 40). He then uses the upward continua- tion integral in Equation 29.48, that is, 29.68 (gA)l R*(r*-R*) C(g A ) Q \-rrr / :i dn, first, to determine values of the gravity anomaly (^)q on the reference surface from measured values (gA)p on the topographic surface, and second, to determine values in the external field generally from values on the reference surface. The first operation involves inversion or solution of the integral Equation 29.68 and results in quan- tities which can no longer be physically identified with gravity anomalies because of the intervening matter, but do, nevertheless, satisfy the integral equation. The method is applicable without modi- fication to gravity disturbances or to any function derivable from Poisson's integral in the same way as Equation 29.48. Once the gravity anomaly, or preferably the gravity disturbance, has been found at points in the external field, it is a simple matter to compute actual gravity at such points; the result could be compared with, and be used to supplement, 13 Bjerhammar (1962), "Gravity Reduction to a Spherical Surface," Technical Report, The Royal Institute of Technology, Geodesy Division, Stockholm, 1-2, and (1964), "A New Theory of Geodetic Gravity," Kungliga Tekniska Hogskolans Handlingar, Stockholm, no. 243, 3-76. Details are provided in several other publications of The Royal Institute of Technology. gravity determination from satellites. The external potential could be found by harmonic analysis of gravity, followed by integration or by Stokes' integration. 49. The integral Equation 29.68 can be solved approximately by the standard method of trans- formation to a finite number of simultaneous linear equations. 14 For example, we can express (gA)Q approximately over the reference sphere by a sufficient number of spherical harmonics as 29.69 even if the continuities, form n , in function (gA)o. contains simple dis- Expressing Equation 29.36 in the re=0 and substituting in the integral Equation 29.68. we have Ml *Z i2n ^ 1} 2{»v}Pn(co S +) on the topographic surface or in the external field appear also as power series in h, an interpreta- tion which implies that gravity anomalies are con- stant over level plains. But this result is contrary to experience, as members of the Stockholm school themselves realize; variations in the anomalies over the Gangetic Plain in India have, for example, yielded valuable geophysical results, and the same could be said for many large oilfields. However, representation of (gA)a by a power series serves well to illustrate the principles of the method, which can be applied to more sophisticated polynomials. 51. An alternative method of solving the integral Equation 29.68 has been given by Moritz. 16 From Equation 29.40, we have R 2 = R 2 {r 2 -R 2 ) f dQ. r 2 ~ 477T J l :i ' If we multiply this equation by (gA)s — the value of the gravity anomaly at S in figure 39 — which is a constant during the surface integration, and sub- tract from Equation 29.68, we have R 2 — {g A )s = (gA)p- R 2 (r 2 -R 2 ) ( ( gA )Q-(gA)i 4tt/- P da. The factor (R 2 /r 2 ) on the left is near unity and is ignored, and the last equation is then solved for (g.\)s by iteration. For the first approximation, the anomalies (g.-Oa, (g\)s on the sphere in the surface integral are taken to be their observed values on the topographic surface. Second approximations to the values on the sphere — for example, {gA)s on the left — are then obtained from the integral equation and are used in the surface integral to obtain a third approximation. Similar iterative methods have been proposed in various publications of the 16 Heiskanen and Moritz, op. cit. supra note 3, 318. Stockholm school, including Bjerhammar's first paper on the subject, published in 1962. 52. The method is based on Poisson's integral which requires an absence of matter outside the reference sphere if the quantities gA are to be interpreted as gravity anomalies. It is possible ir the case of many mass distributions to remove matter external to the reference sphere and to adc matter inside the reference sphere without effec on the total mass, on the center of mass, or on the external field, but there is no guarantee that this is possible in the case of such an irregular body as the Earth; if it were possible, the (g.-i)y would ther be gravity anomalies of the alternative mass dis tribution in which all the matter external to the reference sphere has been removed. The existence of a solution of the integral Equation 29.68 coulc then certainly be justified on physical grounds We have no need to interpret values of (gA)o. on the reference sphere as gravity anomalies. We coule consider (£a)q simply a function defined on the reference sphere which, when substituted in the integral Equation 29.68, correctly reproduces the observed quantities (g^)/-, but we still have to shov that such a function can be found by solving the integral equation and is expressible to sufficien accuracy by a practicable number of terms. Fo: example, there are considerable fluctuations in the anomalies over the topographic surface which coule lead to even more violent fluctuations of the (^)q requiring a very large number of polynomial terms for adequate representation. Failing justificatioi on physical grounds by an alternative mass dis tribution, the only way of settling the question is by numerical trials on simulated and unfavorably irregular mass distributions. Members of the Stock holm school are still (1968) engaged in such trials but their results so far seem to indicate that the method will be satisfactory when applied to the actual Earth and will be as good as any other method in addition to being much simpler. THE EQUIVALENT SPHERICAL LAYER 53. We know that the external potential of a soli< body can arise from an infinite variety of mas distributions, and the question arises whether w< can replace the actual mass distribution by a coat ing of density ex spread over a given surface withou effect on the external potential. In some moden geodetic applications, such replacement is assumee possible for any surface; we shall consider this Integration of Gravity Anomalies — The Poisson-Stokes Approach 325 generalization in more detail in § 30-55. In this chapter, we shall consider, as an introduction to the more general case, the classical problem of whether, and in what circumstances, the mass distribution of a given external potential can be replaced by a surface density cr spread over a sphere. An obvious advantage would be that we can then obtain any required elements of the external field straight from the surface mass distribution by surface integration. In fact, we can obtain alter- native, but equivalent, forms of all the integral formulas in this chapter, no more and no less, and the alternatives are subject to the same limitations. 54. Any bounded function, including functions with simple discontinuities, can be represented over the surface of a sphere in a series of spherical harmonics; we can accordingly express a surface density cr in geocentric coordinates (w, (j)) as x II cr= V V f*I"(sin 4>) {c lim cos mo) + s n m sin mcj} 11 = lll = () 29.72 where c,,m, $nm are constants. If (IS is an element of area of the spherical surface, the corresponding element of mass is )dS r ui = 2tt r<2) = + 7r/2' = R" + - o-fMsin 4>) cos cb dcoaty. Jo,=o J=-irl2 If we substitute Equation 29.72 for cr, the variables are found to be separable and in accordance with the ordinary rules for integration of trigonometric and Legendre functions: we have then r _ 4ttR" +1 {In + 1) Integration of the other two equations of Equations 21.037 in the same way shows that the same rela- tion holds between the other coefficients, and we can write 29.73 d ah 4tt/\ , " +2 (c n (2n + l) \s„ If V is the external potential, we can write this equation in the notation of § 29-2 and with the physical definition of the potential as 29.74 w 4ttR" +2 (2n+l) {<'}■ which can be summed over in and // to give the required relation between the potential and the surface density. It is assumed that units have been chosen to make the gravitational constant G unity in Equation 21.035. However, Equation 29.74 would hold true if we consider cr to be the density multi- plied by the gravitational constant. 55. If we are given the surface density in spheri- cal harmonics, the potential is obtained as a series of spherical harmonics, convergent right down to the surface of the sphere, which in this case is the sphere of convergence defined in §21-11. However, if we preassign values to C„ m , Sum to represent the attraction potential of an actual body, such as the Earth, the corresponding series of solid harmonics is certainly convergent only outside the sphere of convergence for that body; it is only in that domain that the equivalent spherical coating, given by Equa- tion 29.74, can be said to give rise to the actual potential of the Earth. Equation 29.74 is accordingly limited in the same way as the expression of the actual potential in spherical harmonics, no more and no less. Subject to this limitation on Equation 29.74, we can always use Ecjuation 29.74 to find a spherical coating which will give rise to the actual potential. 56. The total mass M of the coating is 29.75 o-dS = 47rfl 2 c o = M=Coo, in agreement with Equation 29.73. Moreover, if the origin is at the center of mass of the actual body so that the first harmonics Cm. Cn, Su are zero, so then are Cm. Cn, andsn zero, and the center of mass of the coating is at the same origin. Equation 29.73 automatically ensures that the actual body and the coating have the same center of mass, whether this common center of mass is at the origin or not. 57. The potential at P (fig. 39), arising from the element of mass (rdS — (rR 2 dCl at Q, is -aR'dnil: the total potential at P is accordingly = + crR 2 I 2 i,dn where /, is a unit vector in the direction QP given by Equations 29.54 and 29.55. This equation can be contracted with the meridian, parallel, and normal vectors (which remain fixed during the integration) in the spherical polar system at P to give three components of force and thus the magnitude and direction of the total force. If V is the potential anomaly, the magnitude of the total force to a high degree of accuracy is the gravity disturbance. 59. Another formula, frequently found in the literature, connects the gravity anomaly g.\ with the density o" of a spherical coating, giving rise to the potential anomaly T. Both T and g\ have their values on the sphere. From Equations 29.74 and 29.32 we have (2-7TO- — g,\) 29.78 1 2 H=0 m=0 _3JT 2R (2» + l) 2R" +1 ra (»-l) R n+2 ra in which we have also used Equations 29.33 am 29.45. A similar formula for the gravity disturbanct gi> on the sphere is obtained from Equation 29.26 a: T_ 2R 29.79 (2tt missing in many of the main triangulation network of the world through fear of the effects of atmos pheric refraction and through the unrealized hop* that heights would eventually be provided b spirit leveling. It may be easier to remedy thi deficiency in regional, rather than in global, gravi metric surveys. 6. The usual method of utilizing the potential ha already been given in § 29-16 (fig. 38), but is re stated now for convenience in a slightly differen notation. In figure 41, Q is a current point on th (normal to S-surface) p-r (normal to coordinate surfaces) S-surface base coordinate surface Figure 41. Integration of Gravity Anomalies — The Green- Molodenskii Approach 329 S-surface, which is either the topographic surface smoothed as necessary or the Model Earth. In this chapter, we shall be dealing solely with (to, , //) coordinate systems, usually the spheroidal geodetic system: and v r is the straight normal through Q to the /i-coordinate surfaces. The point B on this normal is chosen to make the actual potential at Q{Wq) equal to the standard potential at B(Un). Because W is supposed to be known at all points of the S-surface, the locus of the point B is a known surface and is called the telluroid if S is the topo- graphic surface, or the terroid if S is the Model Earth surface. The height anomaly BQ = £, and the gravity anomaly ga — jB are as defined in §29-16. The unit normal to the S-surface is shown in figure 41 as v'\ 7. As usual, we shall provide full derivations of the basic equations, which are not easy to find ex- pressed in simple terms in the literature, and an outline only of the methods for approximating and solving the basic equations; descriptions of these methods are all too prolific in the literature. First, however, we must investigate some geometrical properties of the S-surface in (to, , h) coordinates to avoid breaking the argument later. THE S-SURFACE IN (to, , h) COORDINATES 8. The longitude and latitude (to, ) are constant along the straight /(-surface normal CQ (fig. 41) and therefore have definite values at Q so that (to, c/>) can be taken as S-surface coordinates. Coordinates of the space in which S is embedded will be taken as the (o>, , ). The x'a of the surface (§ 6-5) are then given by 30.02 x%=f a \ x' a =K (r=l,2) in which f a is the derivative of the scalar / with respect to x a , that is, with respect to to or cb, and Sq is the Kronecker delta. It should be empha- sized that / is interpreted as a height only on the S-surface; otherwise, /is simply a function of (to, (f>). The Metric Tensor 9. Substitution of the space metric tensor (Equa- tion 17.04) in Equation 6.06 gives the metric tensor of the S-surface as in which the overbar refers to the coordinate A-surface passing through Q (fig. 41), not to the base coordinate surface (A = 0) as in Equation 17.09, which, however, enables us to evaluate a a B in terms of the fundamental forms of the base surface. The determinant of the metric tensor by direct calcu- lation is a=(d 11 +/?)(a 22 +/!)-(a 12 +/ 1 / 2 ) 2 = a + aa n f\ + aa u p x + 2aa 12 fxf 2 =a(i+W) 30.04 =a(l + V/). The invariant V/(§3-9) is a surface invariant ob- tained from the metric of the h -surface passing through Q. but because / is a function of (to, (f>) only,/j = 0, and the invariant has the same value as the space invariant V/. The Unit Normal 10. We can obtain the covariant components of the unit normal to the S-surface by giving dissimilar values to the indices s and t in Equation 6.11. For example, for 5 = 2, r = 3, we have ^e 123 = e 2 ^ = e 21 /i which, using Equations 2.15, 17.05, and 2.30, gives Vl = -(a/a)" 2 /,; obtaining the other components in the same way, we have 30.05 vr=(ala) l l*{-fi, -/ 2 , 1}. Raising the index with the space metric tensor in Equation 17.05 gives the contravariant components as v' = {fl"i'i. a 22 V2, c.i} 30.06 =(dla.yl*{-a n f u -a 22 / 2 ,l}- But v r is a unit vector, and we must have 1 = v,V = {ala){& l f\ + a 22 fi + 1 } - (d/a) (1 + V/j, agreeing with Equation 30.04. But from Equation 17.28, the unit normal v r to the /i-surface is (0, 0, 1); if /3 is the angle between the two normals, as in figure 41, we have cos /3= v,v' '= {a/a) 11 ' 1 30.03 a a B — a a B +fafe so that 30.07 aid 30.08 1 + V/= 1 + V/= sec 2 (3 V/= V/ = tan 2 (3. 330 Mathematical Geodes Moreover, by making the space indices both 3 in Equation 6.10 for the S-surface, we have l = a^fcfp + (ala) or 30.09 V. s /= sin 2 f3 = V/cos 2 (3 = V/'cos 2 j8 in which the invariant V. s is obtained from the S-surface metric. All the Equations 30.01 through 30.09 hold true in a general (o>, $, h) system whose properties are given in Chapter 17. 11. The angle (3 is the zenith distance in the (&>, 4>, h) coordinate system of the S-surface normal and is also the maximum slope of the tangent plane to the S-surface in relation to the coordinate /(-sur- face, which usually will be the geodetic horizontal. If the azimuth of the direction of maximum slope is a, we can write the unit normal v, of the S-surface in terms of the usual parallel, meridian, and normal vectors {k r , fir, v>) as v r — k r sin a sin f3 + pL r cos a sin f3 + v r cos /3. 30.10 In a general (w, (/>, h) system, we can use Equations 17.26, 17.27, 17.28, and 17.13 to find the components of v r in terms of h and the curvatures of the base coordinate surface; we can compare the results with Equation 30.05 and so can express fi,f-> in terms of a, (3. For a spheroidal base surface with principal radii of curvature p, v, we can use Equations 18.11 and can express the unit normal to the S-surface as v r = {(v + h) cos 4> sin a sin (3, 30.11 (p + h) cos a sin/3, cos 0} so that we have f\ — — (v + h) cos <$> sin a tan (3 30.12 fz = -(p + h) cos a tan /3. In these equations, h is the geodetic height of the point Q on the S-surface (fig. 41). Instead of working in terms of the azimuth and zenith distance of the maximum slope, it is usual in the literature to make sin a tan /3 — — tan f3> and cos a tan j3 = — tan f3\ where /3i, y8-2 are, respectively, the inclinations to the geodetic horizontal in north-south, east-west direction of the tangent plane to the S-surface. However, by working in terms of the azimuth a and zenith distance (3 of the normal to the S-surface, with the sign conventions used throughout this book, we avoid any ambiguity as to whether an inclination or slope of the tangent plane means an elevation or depression. The Associated Tensor 12. The associated metric tensor of the S-surfaci is given by Equations 2.44 and 2.30 as = (ala)e a ye^(ay S +fyfs) 30.13 = cos 2 /3{a^ + i^e^fyfs}. An alternative expression is obtained from Equatioi 6.10 for the S-surface immersed in (w, , h) space By giving the indices r, 5 in Equation 6.10 th< surface values y, 8, we have 30.14 cF^a^ + i^ 8 , which can be shown to be equivalent to Equatioi 30.13. Normal Gradients 13. The component of the gradient of a scalar 1 along the normal to the S-surface is given by dF ds 30.15 = F r v r = cos ,3(- a n fiFi - d 22 f2F 2 + F 3 ) cos/3Jj|-V(F,/-) on substituting Equations 30.06 and 30.07. Th invariant V(F, /) is obtained from the metric o the h -surface passing through Q, and it may be mori convenient to calculate the S-surface invariant From the definition of the invariant in Equatioi 3.14, it is evident that the space invariant V(F,/) i the same as the A-surface invariant because / i a function of a), only. But if we set up anothe (u>, (f), h) coordinate system with S as a base surfac and use the metric in Equation 17.05, the S-surfac invariant \/s(F,f) is given by 30.16 V(F,/) = V. s -(F,/)+ (dFlds)(df/ds), whatever the surface coordinates on S may be Substituting/ for F in Equation 30.15 and remem bering that dfldh — because /is a function of w, (j only, we have 30.17 df/ds = -V/ cos j8 = — sin /3 tan (3, using Equation 30.08. From the last two equations we then have V(F,/) =V(F,/) = V 5 (F,/)-sin tan (3 (dF/ds) substituting in Equation 30.15, we have finally 30.18 f= 6 ec^{f-V s (F,/)}. Equations 30.15 and 30.18 have been obtained in i Integration of Gravity Anomalies — The Green- Molodenskii Approach 331 slightly different form by Moritz, 3 whose D(F,f) is V.s-(F, f) cos 2 (3 and is accordingly an S-surface invariant, although not the standard invariant used here. Moritz' D(F,f) is, however, the same as our /i-surface invariant V(F, /) or V(F, /). We can replace /by h in the S-surface invariant of Equation 30.18, but we are not entitled to do so in Equation 30.15 for the reasons given in §30-8. The Invariant V(T, f) 14. Next, we evaluate the invariants in Equations 30.15 and 30.18 when the arbitrary scalar F is taken as the potential anomaly T. We have already seen that the A-surface invariant in Equation 30.15 is the same as the space invariant, and therefore can be evaluated in the (o>, , h) space system as 30.19 V(T,f)=V(T,f)=gM(W p -U p )f q . If the components of standard gravity in the direc- tions (X r , fl r , V) of the coordinate system are (yi, 72, 7.3), the second term is -(7A 9 +72A 9 +73»*9/g in which the last term is zero because v Q =(0, 0, 1) and/3 = 0. In a symmetrical standard field 71 = 0, and the remaining term in the case of a spheroidal field is 30.20 -yi^ q f q = -y2JJ: 2 fi=yi cos a tan /8, using Equations 18.10 and 30.12. The component 72 is the g m of Equation 23.37. This term is ignored altogether in the literature on the assumption that there is no meridian component of standard gravity close to the spheroidal equipotential, but 72 can easily be computed and allowed for in extreme cases. 15. If, as in § 19-23, we define the vector deflec tion A 9 as A«=(jW) — pi in which (v Q ) , v q are, respectively, the unit vectors in the direction of the astronomical zenith and of the normal to the geodetic coordinate system, then again v 9 f q =0, and the first term on the right side of Equation 30.19 is to a first order g(v«)f q =g&%= (g cos )k% + (g84>)fiif q = — (g cos ) cos a tan /3, 3 Moritz (1964), The Boundary Value Problem of Physical Geodesy. Report No. 46 of the Institute of Geodesy, Photogram- metry and Cartography, The Ohio State University Research Foundation, Columbus, Ohio, 1-66. Republished with the same title in 1965 in the Annates Academice Scientiarum Fennicce, series A. III. Geologica-Geographica, no. 83, 1-48. using Equations 19.42, 18.10, and 30.12. In this equation, 8(0 (8dj) is the astronomical minus the geodetic longitude (latitude). In terms of the usual components of deflection £=8$, 17= cos (p 8w, we can finally write Equation 30.19 as V(7\/) = V(7\/)=-grjsina tan /8 30.21 - (g£-y 2 ) cos a. tan /3 in which 72 is the meridian component of standard gravity obtainable from Equation 23.37 (not to be confused with the curvature parameter 72 of either a general (co, c/>, N) system or the gravitational field as defined in § 12-17). 16. The use of Equation 30.15 and of the h -surface invariant V(7\ f) accordingly requires a knowledge of the deflection components f , rj. Suitable choice of a reference spheroid would make £, 17, and /3 small in flat country. In mountainous country, measurement of vertical angles in a triangulation network, combined with an open astronomical control, would provide sufficiently accurate deflec- tions by methods discussed in Chapter 26. Never- theless, it is a defect of the method to require a knowledge of deflections, in addition to potential and gravity, at all stations. For this reason, Moloden- skii uses Equation 30.18 with the S-surface invariant V.v(F, /) , which we shall evaluate next. However, Molodenskii 4 and Moritz 5 (following Molodenskii) use a special form of invariant instead of the stand- ard form of two-dimensional tensor invariants without reaping any apparent advantage. For ex- ample, Molodenskii's S-surface invariant A»/ for the function / of Equation 30.01 can be shown, not without some difficulty, to be A 2 /= sec 2 /8 A/ where A/ is the ordinary S-surface Laplacian a a,i f a p. We shall use the ordinary tensor invariants through- out this chapter, and we shall find that the gain in simplicity is considerable. The Invariant V S (T, f) 17. If we substitute Equation 30.18 for F=T in the basic integral equation for the potential 4 Molodenskii. Eremeev. and Yurkina, op. cit. supra note 1, 81-85. 5 Moritz, op. cit. supra note 3, 20. See also Moritz (1966). Linear Solutions of the Geodetic Boundary-Value Problem. Report No. 79 of the Department of Geodetic Science. The Ohio State University Research Foundation, Columbus, Ohio. 21-22. Republished with the same title in 1968 in Deutsche Geodatische Kommission bei der Bayerischen Akademie der Wissenschaften, series A, no. 58, 16-18. 332 Mathematical Geodesy anomaly, which we shall derive as Equation 30.50, we find that the term containing the S-surface invariant Vs(T,f) is 30.22 Vs(T,f) sec B/l. This term accordingly contains differentials of T which must be removed if the integral equation is to be linear in T. We propose to do this by trans- forming the term with the two-dimensional form of Green's Equation 9.18; that is, for any two scalars (/>, (//, we have {V(, i//) + c£Ai//}r/S = ifj«j a ds 30.23 connecting the surface integral on the left side with the contour integral on the right side, in which j a is a unit surface vector, normal and outward- drawn to the contour (away from the area covered by the surface integral). Using the same argument as in § 9—7, we can say also that over any closed surface, we have 30.24 {V(, ijj) + , i//) = a"^,^ of the first-order invariant, we can rewrite Equation 30.22 as V 5 (7\/) = v, ,f)-TV s I ,-,/ / cos B \lcosB' J J v s \l cos B" If we integrate this last equation over the closed S-surface and use Equation 30.24, we have Vs(T,f) I cos B dS T I cos B TVs A/as i ,f)dS, 30.25 \l cos B which removes differentials of T from the integral equation. 18. Next, we must express the S-surface Laplacian A/ in terms of known or measurable quantities. If we continue to overbar all quantities related to the coordinate h -surface passing through the point Q of figure 41, we have 30.26 V r X' =/«, using Equation 17.28 for the components of v r (the unit normal to the /i-surface) and Equations 30.02. Taking the tensor derivative of this last equation over the S-surface, we have VrsXlpcl+VrXlp^fap. According to § 17-18, there are no 3-components of Vrs in a (o», cf), h) coordinate system, and v a = — b a i3. From Equation 6.16, we have also x r a p = b a pv r so that we have 30.27 — Oap+baP COS B=f a /3 connecting the second fundamental forms of the /i-surface and the S-surface. Contracting this last equation with Equation 30.14 and using Equation 8.13, we have 30.28 Af=2H cos (3-2H+b afi p a i>f i , the last two terms of which can be evaluated from Equation 30.06 or 30.10 in any particular (o>, (f), h) coordinate system. In the geodetic system with principal curvatures p, v of the base spheroid in and perpendicular to the meridian. Equations 30.10 and 18.10 give components of the unit normal to the S-surface as , sec d> sin a sin B cos a sin B 1 v r — i ; — — n , — ; — — r^ — , cos B\ (v + h) ' (p + h) which, together with Equations 18.01, 18.02, and 18.05, give the last two terms of Equation 30.28 as 1 -+- 1 sin 2 a sin 2 B cos 2 a sin 2 B (v + h) (p + h) (p + h) (p + h) so that Equation 30.28 finally becomes A/=2// cos0 (1 — sin 2 a sin 2 B) (v + h) 30.29 (1 — cos 2 a sin 2 B) (p + h) In addition to the azimuth a and zenith distance B of the normal to the S-surface, that is, the direction and magnitude of the maximum slope of the tangent plane to the S-surface, this equation also contains the mean curvature H of the S-surface. There is no way of avoiding some expression for the curva- ture of the S-surface in a formula for A/, and this limitation must be considered the price for avoiding inclusion of the deflection components £, 17. If the S-surface is the actual surface of the Earth, (2H) may be obtained by estimating the sum of normal curva- tures in two perpendicular directions from contoured maps. Deformation of the S-Surface 19. Next, we consider a family of surfaces ob- tained by progressive deformation of the S-surface; each member of the family is obtained by reducing the /i-coordinates of points on the S-surface in the constant ratio k while retaining the (oj, (f>) coordi- Integration of Gravity Anomalies — The Green- Molodenskii Approach 333 nates so that corresponding points between two members of the family lie on the same normal to the /i-coordinate surfaces. For k=l, the member of the family is evidently the original S-surface; for k = 0, the member is the base coordinate surface. We can take any relation obtained in this section for the S-surface and simply substitute kh for h to obtain the corresponding relation for the deformed surface. If we enclose quantities relating to the deformed surface in parentheses, we have, for example, from Equations 30.01 and 30.02 30.30 30.31 (/) = ¥ (fa)=kf a . 20. In particular, we shall require the azimuth (a) and zenith distance (/3) of the line of greatest slope on the deformed surface relative to the normal of the h -coordinate surface. From Equations 30.12, we have in geodetic coordinates (v + kh) sin (a) tan (j3)=— (/i) sec = k(v + h) sin a tan (3 (p + kh) cos (a) tan ( J 8)=-(/ 2 ) 30.32 =k{p + h) cos a tan /3. In spherical polar coordinates (p=v = R), these equations reduce to (a) — a 30.33 ,„, k(R + h) tan 08)= p , ,,/ tan/8; R + kh if we neglect h/R, the last equation becomes 30.34 tan ()3) — k tan (3. 21. We shall also require an expression for the distance (/) between two points (P) and (Q) on the deformed surface. Quantities relating to Q are over- barred. From either Equation 25.18 or 25.19, after some manipulation with the formulas given in § 19-7, we have in geodetic coordinates U) 2 ^(i>+kh) 2 -2(v+kh)(v + kh) cos cr+(v+kh) 2 — 2e 2 k(v sin 4> ~ v sin — h sin cj>) + {e 4 — 2e 2 )(i> sin 4> — v sin ) 2 30.35 where a is the angle between the spherical repre- sentation of the points (P) and (Q). For the azimuth (a) and zenith distance (j8) of the direction (P){Q) at (()), we have from Equation 25.19 (/) sin (a) sin (/3) = (v + kh) sin a sin a* (I) cos (a) sin (j8)= (v-\- kh) sin sin <\> ~ v sin ) (/) cos 08) = — (y + kh) cos ar+(v + kh) 30.36 — e 2 sin 4>{v sin c/> — v sin ) where cr and a* are functions of the geodetic lati- tude and longitude of corresponding points of the deformed surfaces given by Equations 19.01, 19.07, and 19.08. In spherical polar coordinates {y=v—R, e = 0), Equations 30.35 and 30.36 reduce to for 2/? sin ?i// (/r = 0) to agree with the notation of Chapter 29. We have also (/) sin (a) sin (J3)=(R + kh) sin iff sin a* = (R + kh) cos sin (a> — w) (/) cos (a) sin ()8)= (/? + kh) sin i/* cos a* = (R + kh) (— sin 4> cos (/> + cos 4> sin (^ X cos (a) — a>)) (I) cos (j8) = (R+ kh )-{R + kh) cos 30.38 =2sin 2 iiK# + M) + &(£-/i). APPLICATION OF GREEN'S THEOREM 22. In this section, we shall apply Green's theorem in the form of Equation 9.19 to a volume bounded in part by an arbitrary surface S (fig. 42), which is somewhere near the actual surface of the Earth E, and shall obtain an expression for the potential at a point P on S. Later, we shall make S coincide with the actual surface of the Earth, but for the present we consider the more general case where there are masses external toS. One of the scalars in Equation 9.19 will be the reciprocal of the distance (1//) from the fixed point P to a current point Q within the volume, and the other scalar will be f(the attraction potential at Q) so that Equation 9.19 becomes ds \l J I els -'*)-> 30.39 334 Mathematical Geodesy Figure 42. in which dv is an element of volume and ds is the arc element in the direction of the normal to S, drawn outward from the volume considered. 23. In the first place, we shall consider the internal volume bounded by S, and shall consider only the attraction potential V\ arising from the matter out- side S (shaded in fig. 42) so that mathematically we may suppose all the matter inside S has been re- moved. But in applying Equation 30.39 to this case, we notice that the surface integrand at least becomes infinite when / = 0, that is, when Q coin- cides with P, as it must do for some, part of the integration; therefore, we cannot apply Equation 30.39 as it stands to the whole region bounded byS. To overcome this difficulty, we remove a small hemi- sphere 2 <>f radius e, centered on P, from the volume enclosed by S and integrate over the remain- ing volume, the surface S minus 2, and the curved surface of 2. For this area and volume, Equation 30.39 becomes u r, ds\l) I ds) 30.40 d /l ds\l 1 '/ I ds LV^tdv dS Throughout the volume, both 1// and l\ are har- monic and the volume integral is zero. Next, we consider the surface integral over 2. If (V,) is an average value of V\ over 2 and if dil is an element of solid angle at P, the first term is (V> 1 0. In regard to the first term of the volume integral, we can consider that (—1//) is the potential at Q of a particle of mass 1/G sit- uated at P (G is the gravitational constant). When Q is within 2, A(l//) is therefore zero except actually at the particle of mass 1/C, that is, when Q coincides with P. For physical reasons, we must suppose that the actual attraction potential V\ of the external masses is finite within 2. Accordingly, the first term of the volume integral becomes zero as e— * 0, provided we exclude the actual point Integration of Gravity Anomalies — The Green- Molodenskii Approach 335 P (e = 0) as we have already done in deriving Equation 30.41. The exclusion of the point P, and of any matter actually at P, from the volume and the surface, although mathematically necessary, makes no significant physical difference to the mass distribution and to the external field. Subject to these considerations, we are justified in assuming that the volume integral over the hemisphere be- comes zero as e— * 0. 25. Next, we shall consider the potential V 2 arising from matter within the same surface S (shaded in fig. 43), and shall evaluate the potential Figure 43. at the same point P on S by applying Green's theorem to the volume enclosed between S and a sphere S of infinite radius (not indicated in fig. 43). To isolate the singularity in 1// at P, we take a small hemisphere £ of radius e out of this volume, as shown in figure 43. Equation 30.39 for the re- maining area and volume then becomes U' ds 1 dV 2 I ds dS ds \lj 1 dV 2 I ds TsXi) 1 dV 2 I ds dS dS 30.42 -JM AF 2 dv As in the case of V\ arising from the external mat- ter, we find that the volume integral is zero, even when e — > 0; and so also is the integral zero over the infinite sphere S because V 2 behaves like 1// at infinity. The second term of the surface integral over 2 is similarly zero, and the first term is where (V 2 ) is an average value of V-> over the hemispherical surface. In this case, the positive direction of ds is inward away from the volume over which the volume integral is taken, as shown in figure 43, so that dl/ds = —l on the boundary of the hemisphere, and the surface integral as e — » is 2ttV-u> where V-ip is the value of V 2 at P. Equation 30.42 thus reduces to T\ 1 dV 2 ] 30.43 -2itV 2P {{ V 2 ds \l I ds dS, which has exactly the same form as Equation 30.41 for the potential Vn> arising from the external masses. A glance at figures 42 and 43 shows, how- ever, that the positive directions of ds are opposite in the two cases. If we wish to combine Equations 30.41 and 30.43, we must change the sign of ds in one of these equations. We shall do so in Equa- tion 30.43, thereby ensuring that the positive direc- tion of ds is away from the main mass of the Earth in both cases. Equation 30.43 then becomes 30.44 ^WKGHf}*- Because there are no volume integrals in Equations 30.41 and 30.44, the current point Q is now restricted to the surface S. The potentials {V\ or V 2 ) and the component of the forces of attraction in the direction of the normal to S, drawn toward the main mass of the Earth (dVJds or dV 2 /ds in accordance with Equation 20.05), must be evaluated or observed at points Q on S. The distance / is PQ between two points on S; one point is fixed at P and the other is the current point Q in the surface integration. 26. Next, we consider the potential (1 of the centrifugal force, given by Equation 20.08 as 30.45 £l=W{x 2 + f) in which to is the angular velocity of rotation of the Earth and (x 2 + y 2 ) is the square of the per- pendicular distance of the point considered from the axis of rotation. If W , V are. respectively, the geopotential and the attraction potential at the same point, then Equation 20.08 with suitable change of notation also gives 30.46 W=V-a=V 1 + V 2 -£l 336 Mathematical Geodesy in which we have applied the principle of super- position (§20-7) by making the total potential V= Vi + V 2 . It is clear from Equation 30.45 that ft has no singularities on or inside the surface S of figure 42 so that we can use Equation 30.40 simply by replacing V\ with Cl. The volume integral in Equation 30.40 is now 2cb 2 dv I I taken over the whole volume enclosed by S; allow- ing for the singularity of 1// at / = as in §30-23, we have -2-n-iV 30.47 ^(iHfW/T- 2 - By adding Equations 30.41 and 30.44, subtracting Equation 30.47, and using Equation 30.46, we have /{ -Ui 13 I ds dS- I dv = 2TT{V 2 p-Vu>)+27rn P = 2tt(W p + £I p -2V 1p ) + 2irfL,> 30.48 =2Tr(Wp-2V 1P )+2TTcb 2 (x P + f P ). This equation, which is fundamental to the subject, has been given explicitly by de Graaff -Hunter 6 and in various approximate or special forms by several other writers. 27. It can be argued that V\ and V 2 in Equation 30.46 are harmonic and therefore V= V\ + V 2 is harmonic — for example, in the sense that if V\ and V 2 can be validly expressed in solid spherical harmonics, so can V— even when P lies inside the attracting matter, where V should satisfy not the Laplace equation but Poisson's Equation 20.14. This question, which does not appear to be satis- factorily answered anywhere in the literature, requires an answer even though, for reasons given in §21—74, we are not concerned in geodesy with potential and force inside matter. We have already noted in §30-23 and §30-24 that the point P, and with P any particle of matter at P, must be excluded from both the surface and the volume considered in Green's Equation 30.39. The point P must there- fore be considered as lying within a small cavity where the potential is harmonic. This argument is even clearer if we consider the potential at P in the form of a volume integral V=- Gpdv I in which / is the distance between P and a current point Q, G is the gravitational constant, p is the density, and the integral is taken over the whole volume occupied by attracting matter. The inte grand becomes infinite when Q coincides witl P (1 = 0), and the integral does not therefore con verge unless we exclude the particle actually at P. If we do exclude this particle (and we can do sc without significant effect on either the mass dis tribution, the external field, or the internal field) V becomes a sum of harmonic functions and there fore itself a harmonic function. Nevertheless, we are not entitled to differentiate the relation V= V\ + V 2 — once to find the force, or twice tc find the density — unless the resulting differentials are continuous; differentiation implies a displace ment from the cavity at P into the surrounding matter, and we must expect some discontinuity to result from this process. In the case of a continu- ous distribution of matter — except in the cavity at P — we can argue physically from the principle of superposition (§20-7) of potential and of force, implying that a relation similar to V— V\ + V 2 exists for the force components, that the potential and its first derivatives are continuous at P, but the second derivatives are not continuous and Poisson's equation provides a measure of the discontinuity (§20-18). This question of discontinuous derivatives will become clearer when we consider single- and double-layer distributions in §30-31 through §30-41. We shall find that in the case of a single layer the potential is continuous across the surface, whereas the force is discontinuous and cannot be obtained by differentiating the potential; in the case of a double layer, both the potential and the force are discontinuous. 28. It is usual to simplify Equation 30.48 by intro- ducing the potential anomaly T as defined in § 29-10. To do this, we write Equation 30.48 for the actual geopotential W and for the standard geopotential U and subtract, remembering that the centrifugal terms are the same for both actual and standard fields and will cancel. We assume further that the mass giving rise to the standard field is entirely contained within the surface S so that V lP for the standard field is zero. The result is 30.49 2ir(T P -2Vip) ds 1ST] I ds] dS. 6 de Graaff-Hunter, op. cit. supra note 2. It is usually assumed that the mass giving rise to the standard field of Chapter 23 is contained within the equipotential base spheroid of this field, so that the field external to the spheroid can be ex- pressed in the form of a convergent series of sphe- Integration of Gravity Anomalies — The Green- Molodenskii Approach 337 roidal harmonics as discussed in §23-9 and §22-23. In that case, we must ensure that the spheroid lies entirely within the surface S of Equation 30.49. 29. If the surface S coincides with or lies entirely outside the topographic surface of the Earth, we have Vu> — 0, and Equation 30.49 becomes 30.50 «'-JteGH£}-- This equation applies also to the co-geoid, or regu- larized geoid, outside which all masses have been removed. On the other hand, if we do not remove the external masses, we must estimate their potential Vu> at P and substitute in Equation 30.49, which then applies to the actual or nonregularized geoid. In the same way, we can estimate the potential of masses external to the Model Earth (or the telluroid or the terroid), but with less drastic assumptions; Equation 30.49 can then be applied to the surface of the Model Earth (or the telluroid or the terroid). Unless S coincides with the actual topographic surface, and for reasons given in § 30-2 this situa- tion is not altogether possible, some estimate of the potential of masses external to S is necessary. 30. In applications involving satellites, we may require the attraction potential at a point P not on S but external to S, which, for this purpose, we shall assume contains all the matter. We apply Green's Equation 30.39 to the volume bounded by S and the infinite sphere. The integrand becomes infinite when the current point Q coincides with P (/ = 0), and to deal with this situation, we exclude a small sphere 2 of radius e centered on P from the area and volume considered. As in §30-25, we find that the volume integral is zero (even when e — » 0), the integral over the infinite sphere is zero, and the second term of the surface integral over £ is zero. The remaining integral over 2, as in §30-23, is V ds j\d:-. V e 2 da = tkrV,>. The positive direction of ds in Equation 30.39 is away from the volume considered, and we must accordingly change the sign of ds to follow the con- vention of §30-25, that is, positive ds away from the main mass of the Earth. We have finally 30.51 Mr= fL * m laF1 V ds \l I c>s dS in which we may substitute for V the harmonic potential anomaly T, so that Equation 30.50 holds for an exterior point P with the substitution of 4>tt for 2tt. If P is an exterior point, the singularity at P has to be removed by excluding a small sphere of radius e, whereas if P is confined to the surface S, a small hemisphere serves to remove the singularity. The difference in area between a sphere and a hemisphere accounts for the factor of 2 between Equations 30.51 and 30.50. Equations 30.41, 30.44, and 30.51 are equivalent to three of the six relations, usually known as Green's third identities, obtained from Equation 30.39 for the volumes external and internal to the closed sur- face S when P is outside, on, or inside S. The other three identities can easily be obtained similarly if required. If V in Equation 30.39 is not harmonic, the volume integral - 1 ¥ * I must be retained in deriving these identities as we have done in deriving Equation 30.47: V must still be finite and continuous, and have finite and con- tinuous first and second derivatives throughout the volume considered. Green's first identity is Equation 9.18, and Greens second identity is Equation 9.19 with Equation 30.39 as a special case. POTENTIAL AND ATTRACTION OF A SINGLE LAYER 31. We shall now develop further the ideas in §29-53 through §29-59 by considering the field arising from a layer or coating of surface density cr spread over an arbitrary S-surface. At External Points 32. Omitting the gravitational constant or con- sidering cr to be the surface density multiplied by the gravitational constant, the potential at an ex- ternal point P arising from an element of mass at Q on the surface is —crdS/l where / as usual is the distance PQ. The total potential at P is accordingly 30.52 I r- mIS I 0. The whole arrangement is accordingly analogous to a surface distribution of dipoles of moment density p., oriented in the direction of the surface normals. At External Points 37. The potential at an external point P arising from the elements ±adS is crdS adS p 1 1 dS. ds \r i, In the limit ds^> 0, this last expression becomes so that the total potential at P is 30.55 V P "M>" i which contains no singularities as long as P lies j outside the surface (/^0). Partial differentiation with respect to 5 implies differentiation along the ( surface normal at Q with P fixed, so that we have »••" sGHr 1 p VrV r ) in which l r is now a unit vector in the direction PQ. 38. The attraction at P is adS crdS _ p I 1 1\ ,. which becomes in the limit [ ^is{i) dS in the direction PQ = l r . The gradient of the total potential at P is the negative of the total attraction at P. 30.57 (V r )p=- ^|(^ /,/S 2/i (l t v%dS. At Points on the Surface 39. If P is on the surface, there are strong singu- larities when P and Q coincide (/=0) and the inte- grands become infinite. As in the case of a single layer, we consider small circular discs of radius e taken out of the two layers, as shown in figure 45, where we have located P on the outside of the top layer. The potential of the top disc is — 2ircre and ds Figure 45. the potential of the bottom disc is 277cr (e — ds) to a first order in ds/e, so that the total potential at P oi the double disc is — Irrcrds which becomes — 2irp in the limit ds — > 0. Instead of Equation 30.55 for the potential of the whole double layer, we have 30.58 V,- = -2np, '£©* If we had located P on the inside of the bottom layer, the potential of the double disc would be + 2irp.p; accordingly, there is a discontinuity of 4?Trpi> in the potential on crossing a double layer. 40. If we use the same method to evaluate the attraction of the double layer at a point P on the surface, we find that the attraction is indeterminate when both e and ds tend to zero. Moreover, the singularity in the potential of Equation 30.58 sug- gests that there must also be a singularity in the attraction containing differentials of the moment distribution p, which we cannot introduce if we suppose that the distribution is uniform over the small discs. We can overcome the difficulty by a device similar to that used in §29-41, which has also been used by Koch 8 in the present application. 41. We can evaluate the integral dS ds by the divergence theorem, Equation 9.17, pro- vided we remove the singularity at P. where the integrand becomes infinite, by the device used in § 30-23. Or, we can use Equation 30.41 for V\ equal to unity and obtain the result at once as 30.59 £ Hi) * 30.63 = f (M ~^' ) m tV <)(Lrm>)-m r v r }dS, remembering that the gradient / at P with fixed is in the direction QP and is therefore minus / As in § 29-41, we can show that there is no singi larity in this integral at P so that Equation 30.6! with the alternative expression in Equation 30.6c correctly gives the component in the direction m 1 i the potential gradient (or the negative of the fore vector) at P when P is on the surface. Because n, is an arbitrary vector which can be considere constant during the surface integration, we ca rewrite Equations 30.61 and 30.63 in the vector forr Vr)p — — 27T{fJLr)i n — y-p {S(l,v')lr-Vr}dS 30.64 in which /, is the unit vector in the direction PQ Because the moment density /jl can vary only alon the surface, its gradient /x, has no normal corr ponent. If we contract Equation 30.64 with an arb trary unit vector m'\ we must accordingly ignore th normal component of m 1 in evaluating the (yu,, )/<-terrr For example, if q' is a unit surface vector in th plane of v 1 ' and m'\ we may write /??'' = v r cos (3 + q' sin /3, and the value of the (/i r )p-term will then be — 27r(/x r q sin /3). THE EQUIVALENT SURFACE LAYERS 42. One object of the diversion on single an double layers in the last two sections, apart from th fact that these notions appear often in the literature is to show that any harmonic potential can be cor sidered as arising from a combination of single an double layers spread over a surface S which cor tained all the original mass distribution. If P is a external point and Q a current point on the surfac as usual and if we suppose that we are given th values of Vq and (dVjds)Q over the surface, we ca: take the moment density of the double layer to b fXQ — — VqI( of Q. As shown in figure 47, h is the geodetic height of Q. The element of solid angle dil from Equation 29.01 refers to the spherical representation of the base coordinate spheroid and can be expressed as 49. Substituting Equations 30.68, 30.71, and 30.' in Equation 30.50 gives 30.74 2ttT p + ) JT Q dn= J LdQ. where J = \ cos /3 + cos fa — a) tan (3 sin /3 +i d -^y- P +h){v+h)ip L = — {gA + grj sin a tan /3 30.75 +g(£ + x) cos a tan /3} (p + h) (v + h), Equation 30.74 is an integral equation for the u known potential anomaly with J as kernel. 1 quantities in the surface integrands J, L refer the current point Q, and the integrands conta the geodetic heights h. But if we know the geodel heights of all points as well as the potential, \ should also know the standard potential and th the potential anomaly; there would be no problei Accordingly, the integral equation can only be solvi by successive approximation. We could start wi assumed heights and solve the equation for a fii approximation to the potential anomaly T, whi< enables us to calculate the corresponding geodet height at P. This operation would have to be 1 peated for the height of a network of points P to 1 used in a second approximation, so that ultimate we should end with a network of consistent heigh which do satisfy the integral equation. Unfortu ately, this direct numerical method is excessive complicated and involves so much computation th it could hardly be used in preference to correspon ing data provided by satellite methods. 50. Many attempts have accordingly been mac to derive integral formulas for the second and high approximations. The most elegant method is parametric method proposed by Molodenskii 9 wl uses it to solve the simpler integral equation d rived for a single layer in the next section, althouj the method could be used to solve the present pro lem. The integral equation is rewritten for a d formed surface, as discussed in §30-19 throuj §30-21, which simply means substitution of tl formulas given in §30-19 through §30-21 ar expansion in powers of A\ In addition, the potenti anomaly is expanded in powers of k as 30.76 T=T i) + kT i + kiT-2 + k :i T-, + . . . . Because k is arbitrary in the sense that any d formed surface intermediate between the S-surfa< 30.73 (li\ = cos 4> h) which depend on the adopted standard field. These logical conse- quences of the spherical approximation are almost always ignored. Another common approximation is to neglect h/R in order to simplify further the expressions for /, a, /3, etc. The best reference for these approximate solutions is Moritz, 10 but other approximate solutions are given by de Graaff- Hunter n and by Levallois, 12 the latter using a spherical base surface in conjunction with gravity disturbances on the topographic surface. Iterative methods of approximate solution have been dis- cussed fully by Koch 13 for the case of a single equiv- alent layer, and could be extended to cover the case of both single and double layers. 52. If we evaluate anomalies over the base sphe- roid instead of over the S-surface, we have /8 = 0, h — 0, and Equations 30.75 become 3 In yl pv J= cos/3 + / dh I / 2 L = — g A pv/l. The resulting integral equation, with the usual assumptions relating to (9 In y/dh) unless gravity disturbances are used, can be used to solve the Zagrebin-Bjerhammar problem (§29—43) or to provide results equivalent to those derived in § 29—44 through §29-47. If, in addition, we assume that the base surface is a sphere so that p—v — R, l = 2R sin iifj, cos j8= sin hjj, d lnyldh^-2/R, the integral equation can be solved by spherical harmonics to give Stokes' Equation 29.32. Gradient Equations 53. Integral equations for the gradient of the potential anomaly can be obtained by the method outlined in § 30-45. For example, by adding Equa- tions 30.54A and 30.64 for the potential anomaly T at a point P on the S-surface and by using the layer 10 Moritz, loc. cit. supra note 5. 11 de Graaff-Hunter, loc. cit. supra note 2. 12 Levallois (1958), "Sur une Equation Integrale Tres Generale de la Gravimetric" Bulletin Geodesique, new series, no. 50, 36-49. 13 Koch (1967). Successive Approximation of Solutions of Molodenskys Basic Integral Equation. Report No. 85 of the Department of Geodetic Science, The Ohio State University Research Foundation, Columbus, Ohio, 1-34. 344 Mathematical Geodei densities given in §30-42, that is, 4-n- \ds)" 47r' we have 30.79 +i(f r )p-^ f ^y^ {3il'v t )lr-vr}dS where the overbar in the third term on the right implies the S-surface gradient of T for reasons given at the end of § 30-41. This equation can be con- tracted with any vector fixed at P during the inte- gration. The simplest results are obtained by contraction with the S-surface normal and with the surface vectors at P because we are then able to assimilate the nonintegral terms on the right. For example, if we contract with (v r )p (the S-surface normal at P), we have (2V)/ 30.80 ds) P -JLf 2tt ) 'ldT , v _l l as + ^-jr 1 {W , V>)lr(v r ) P - -V r (v r )p} dS where as usual quantities which are not suffixed P refer to Q, and /, is a unit vector in the direction PQ. Contraction with an S-surface vector (m'')p (line element dm) produces exactly the same result with the substitution of (m r ) P for (v r ) P and dT/dm for dT/ds. Equivalent unexpanded formulas not in vector form have been obtained by Koch 14 who has priority for these formulas. The vector Equa- tion 30.79 is, however, more general and can be contracted with the coordinate vectors at P to obtain the deflections and gravity disturbance at P, using Equation 29.23. Invariants in contractions of Equa- tion 30.79 are best calculated from Cartesian com- ponents which, of course, are the same at P and Q for parallel vectors. Theoretically, the deflections and gravity disturbance at P can be calculated from the three simpler components of 7Y, given by con- traction with the S-surface normal and surface vectors; but, if S is the topographic surface, the results would be completely invalidated by uncer- tainty in the slope of the surface. 54. If the S-surface is - a sphere, the equations are, of course, much simpler and are often given in the 14 Koch, op. cit. supra note 8, 18-21. literature, but they tell us nothing which has n< already been obtained even more simply in Chapt( 29. For example, Equation 30.80 can be reduce to Equation 29.60, using only results which ha\ been obtained in Chapter 29. THE EQUIVALENT SINGLE LAYER 55. The extra complication involved in the repr sentation of the basic Green's Equation 30.50 I both single and double layers, especially in tl gradient equations and when the point P is on tl surface, has led Molodenskii to propose using on a single layer, spread over the topographic S-su face, as an equivalent mass distribution giving ris to the actual potential anomaly. The density o" i this single layer is, of course, no longer (dV/ds)l(4<7i but has to be determined to agree with the actu; potential anomaly. We have seen in § 29—55 th< this arrangement is possible for a general Newtonia potential if the S-surface is a sphere containing a the attracting matter, but some justification i needed in the case of a more general surface. Th potential of a single layer at a point P, which i either on or outside the surface, is from Equatio 30.52 and § 30-34 30.81 V, adS f era If we hold the current point Q fixed so that cr an dS are fixed, differentiate twice covariantly for displacement of an external point P, and contrac with the associated metric tensor, all in Cartesia coordinates, we have 30.82 (An -J A(l//)o-dS = 0, provided cr is bounded. Moreover, the potenti; at P when P is at a great distance L from S tend toward 30.83 r, -II adS=- M L where M is the total mass of the coating. The coa ing accordingly does give rise to a Newtonian poter tial throughout the space outside the S-surface, an it is not unreasonable to suppose that we have suff cient freedom in the choice of cr to represent an Newtonian potential in this way. Similar justifies tion when P is on the surface presents more difl culty. Differentiating Equation 30.54A again i Cartesian coordinates, we have (V rs )p = 2ir(ds)p(vr)p + 2Tra-p(vrs)p + J (VDrsadS 30.84 Integration of Gravity Anomalies— The Green- Molodenskii Approach 345 in which the overbar implies the surface gradient of cr because cr can vary only over the surface. We are entitled to assume that the surface integral derives from a Newtonian potential in free space at P, which is analytic at P, because the integral is not taken over the small disc at P (§ 30-35), so that there are no singularities in any of the deriva- tives of the potential represented by the integral. If we contract Equation 30.84 with the metric tensor and use Equation 7.19, we have 30.85 {AV) r = — 47r///.o> in which Hp is the mean curvature of the surface at P. If the coating is to give rise to a Newtonian potential at a point P on the surface, we must assume, as indeed we have done in § 30—34 and § 30-35, that the small disc at P can be considered flat; in other words, that a small part of the surface at P, where we wish to find the potential, can be replaced by a plane. In practice, this conception presents no difficulty in the case of a sufficiently smooth surface; we do not avoid, and cannot expect to avoid, the necessity for some smoothing of the topographic surface by adopting the single layer device. 56. Brovar '"' has noticed that Equation 30.82 is satisfied if we use a more general harmonic function E instead of 1// and a more general bounded func- tion c/> of the position of tire current point Q on the surface instead of the surface density cr. In that case, 30.86 V,.= \EdS represents a harmonic potential. But if V p is to represent a general Newtonian potential throughout the free space external to the surface. Equation 30.83 must also be satisfied, and it has to be shown that a particular choice of E does so. For example, the spheroidal coordinate a does so (§ 22-35) and could be used in the representaion of a general Newtonian potential; so could Pizzetti's extension of the Stokes' function (Equation 29.14 for A=/?/r), provided the zero- and first-degree harmonics are omitted. The use of Pizzetti's function does, in fact, result in some simplification in the formation and solution of the basic integral Equation 30.90 for a single layer. But the Cartesian coordinates of P and many functions of the Cartesian coordinates, which become infinite at great distances, cannot be used. 15 Brovar (1963), "Solutions of the Molodenskiy Boundary Problem," American Geophysical Union translation of Geodeziya i Aerofotos"yemka, no. 4, 237-240. The Basic Integral Equations 57. Writing Equation 30.54A for the potential anomaly, contracting with the coordinate vectors at P (a point on the surface), and using Equations 29.23, 29.25, 29.30, 30.66, and 30.67, we have '<) In y\ 30.87 30.88 30.89 . cr cos Bp JC Ittctp cos f3i' — | T5 (IS (gr})p = 2irap sin ctp sin /3/> cr sin otp sin fit P dS {g(g + k) }i- — 2tt(Ti> cos «/• sin (3r a cos ap sin j8/« l- dS where [gp)p, {g.\)i> are as usual the gravity dis- turbance and gravity anomaly at P; «/•, /3/> are the azimuth and zenith distance of the normal to the S-surface at P, that is, the azimuth an of the greatest slope, a depression of ftp; and «/•, (3p are the azimuth and zenith distance at P of the line PQ obtained in geodetic coordinates from Equation 25.18. Equation 30.87 can be combined with Equation 30.81 and written as 27TCT/' cos fin 30.90 cos/3/ d In y \ 1 Bh Jpl adS = (g A which is the basic integral equation to determine cr from gravity anomalies. The corresponding equation for the gravity disturbance is obtained by omitting the term containing (d In yjdh)p. If P is outside the S-surface, we use Equation 30.53A instead of Equa- tion 30.54A, which amounts simply to dropping the terms containing cri> from Equations 30.87. 30.88. and 30.89. 58. The integral Equation 30.90 can be solved by any of the methods outlined in § 30-50 and § 30- 51. It is usual to solve the equation in spherical polar coordinates with the usual approximation d In yldh = — 2l(R + h) (Equation 30.78). In that case, the first- or zero-order approximation, obtained by ignoring all heights or by solving the integral equa- tion resulting from terms not containing A in Molodenskii's parametric solution, must be given by any of the results obtained for a spherical layer in §29-53 through §29-59. Equation 30.90 in spher- 346 Mathematical Geodes ical polar coordinates with all heights ignored (£,, = (), pp=hr+b}f, l=2Rsinhl/, dS = R 2 dD) becomes {g\),> = 2iT(Ti' — l\ {\ cosec \\\f)crdQ. 30.91 =2tto-p-| [ o- JT/ J „(cos^)(/n, using Equation 29.11 for k=l. Using Equation 29.10, this last equation can be solved in spherical harmonics as {«*?}i»= 2ir{crff}i»r 30.92 47r(» — 1 2ji + 1 3 4-77 ~* 2« -h 1 {o-;r}/.. {oW}, which is equivalent t< ^ 2// + 1 4m#}p 30.93 4tt J £5 n - 1 using Equations 29.10, 29.05, and 29.06 and noting that terms in the expansion of g.\ in spherical harmonics make no contribution to the integral except the term of the nth degree. Terms of zero and first degree must be dropped from the sum- mation, and are not included or determined in either gA or cr for reasons given in §29-32. How- ever, when these harmonics are suppressed in the potential anomaly, as always assumed, they will not appear in g,.\ (Equation 29.32) or cr. The function Mi/>)=>. f„(eosi//) n=2 n — \ is easily found by writing <^±Df- 4 „ + 2 + 3(2n + li I I Using Equation 29.11 (for k=l), Equation 29.11 differentiated (for A = l), and Equation 29.15, we have S ( — 9 cos i|/ In (sin ii// + sin 2 ii//). The final equation for determining the first appro: imation to the density a from gravity anomalie is 30.94 167T 2 crr= S(\ft)g A dn- The density would have to be calculated from thi equation if we wish to use Equations 30.88 an 30.89 for the deflections or to calculate the exterm field. However, if we merely require the potenti anomaly T at a surface point P, we can use Equ£ tions 30.92 and 29.32 and write 4t7{ct;;'} / 2/» + l (n-l){T '»} P 1 R" + i which is the same as Equation 29.74 for a sphericc layer. We then have R „ 477 '2i In + 1 {cr',!'}/' = -^ [{cr'//}/\(C0S, h ) coordinate sys- §17-4 tern. "Height" along straight normal-to-base coordinate surface. I stu •••(») nth-order inertia tensor § 21—13 i Inclination of orbit § 28-30 K Gaussian or specific curvature §5-14 A Normal curvature of a surface § 7-3 k A general parameter §29-5; §30-19 k\, k-z Curvature parameters. Normal curvature of /V-surface § 12^23; § 12-24 in parallel (A, •) and meridian (jAr) directions, respectively. / Distance between two points § 29-22 L r , M,-, N r Orthogonal triad of vectors § 26-36 /,, Mr, n r Orthogonal triad of vectors § 26-7 /,-, L, Covariant vectors (three dimensions) § 1-2; § 1—13 l a , L a Covariant vectors (two dimensions) § 1-3 l r ,L' Contravariant vectors (three dimensions) § 1-3; § 1-12 l a , L a Contravariant vectors (two dimensions) § 1-3 l r m r , l r m r Scalar products (three dimensions) § 1-4; § 1-5 l a m a , l a m a Scalar products (two dimensions) § 1-4 M (or W) Geopotential §20-10 M Total mass §21-13 M Mean anomaly §28-28 m Scale factor § 10-1; § 13-4 m Mass of a particle § 20-2 Index of Symbols 349 /V Third coordinate in a general (w, , N) coordinate sys- § 12-1 tem. N may be given various meanings in different chapters, for example, potential or geopotential in Chapter 20. N Angular momentum § 28-25 n Magnitude of gradient of /V in a (w, c/>, N) system. Various § 12-1 meanings may be given to n in different chapters, for example, gravity in Chapter 20. n Mean motion § 28-27 p Perpendicular from Cartesian origin to /V-surface tangent § 12-95 plane. p Atmospheric pressure § 24—34 Q Latitude and longitude matrix § 19—9 q u q-i Rectangular coordinates in an ellipse § 22-9; § 28-29 R Disturbing force § 28-41 R[jj k ., Rmijk Riemann-Christoffel tensors § 5-3: § 5-5 Rij Ricci tensor § 5-11 R, S Matrices of components of K r , jx' , v r and A. r , £i r , v r § 19—16 r, R Radial distance from Cartesian origin. Third coordinate §1-11; §29-22 in spherical polar coordinates. S Surface or area § 9-1 S Optical path length § 24-3 Sp«,S u Lame tensor §5-12; §5-13 S nm , Cnm Coefficients of Legendre functions of degree n and order § 21-25 m in spherical harmonic expression of the potential. \S(k, iff), S(k, \jj), Stokes or Pizzetti functions §29-6; §29-8; SW,S(*)Jw §29-9; §30-58 5 Arc of curve or contour § 9-3 T Atmospheric temperature °K § 24-46 T Potential anomaly § 29-10 t Geodesic torjsion § 7—5 t Atmospheric temperature °C § 24-45 t Time §28-3 h Curvature parameter. Geodesic torsion of A^-surface in §12-23 the direction of the parallel (\ r ). U Geopotential of standard gravitational field § 29-10 u Reduced latitude in spheroidal coordinates § 22-4 {u™}, {u n } Spherical harmonics §29-2 «„, v a Principal directions of a surface § 7-14 I Volume §9-2; §9-12 V Attraction potential §21-11; §30-22 350 Mathematical Geode v Volume §30-22 v Scalar velocity § 28-5 V Velocity vector § 28-5 W Geopotential §20-14; §23-9; §28-19 w Argument of perigee § 28-30 X, F, Z Cartesian coordinates § 27-3 x, y, z Cartesian coordinates § 27-1 x r Generalized coordinates (three dimensions). Position §1-9 vector in Cartesian coordinates. x r a dx r ldx a §6-3 a Azimuth § 12-11 « Eccentricity angle in spheroidal coordinates § 22-3 a Temperature coefficient of refractivity of air § 24-45 ft Zenith distance § 12-11 P Auxiliary angle in spheroidal coordinates § 22-6 Mj, [ij, k] Christoffel symbols § 3-1 7 Standard gravity § 29-13 7i, 72 Curvature parameters. Rate of change of (In n) in parallel § 12-17 (V) and meridian (/u. r ) directions, respectively, in a (to, 0, N) system. A Laplacian § 3-9 A Surface Laplacian §8-17 A Deflection or increment § 25-24; § 26-1 1 A'' Deflection vector § 19-23 V First differential invariant §3-10 V, V„. Surface V §8-18: §30-10 8 Deflection or increment, or intrinsic derivative § 26-10; § 4-1 8f Kronecker delta (three dimensions) § 1-21 &,"',", 8",'" Generalized Kronecker deltas (three dimensions) §2-27; §2-28 8yf , Sjj Kronecker deltas (two dimensions) § 2-40 e' s ', e,. s , e-systems or e-tensors (three dimensions) § 2-24 e rst A s B t , e rst A s B< Vector products §2-30 e rs 'A r B s C t Scalar triple product § 2-31 e rst F ts Curl of vector F, §3-11 e™' 3 , e U /3 e-systems or e-tensors (two dimensions) § 2-35 £ Height anomaly § 29-16 rj Parallel component of deflection § 29-13 k Standard curvature correction § 29-12 Index of Symbols 351 Ki, K2 Principal curvatures of a surface § 7-14 k Wavelength §24-43 \ r , p r , v r Parallel, meridian, and normal vectors in a (w, $, N) §12-9 system. p Dipole magnetic moment or moment density §21-101; §30-36 p Refractive index § 24-3 p Product of mass and gravitational constant (GM) § 28-25 p, p Principal radii of curvature of a spheroid in parallel and § 1 8-24 meridian directions. v r Unit normal vector §6-13 v' a Surface tensor derivative of unit normal § 6-22 £ Meridian component of deflection §29-13 p, v Principal radii of curvature of a spheroid in meridian § 18-24 and parallel directions. p Density § 20-16 p r , p r Position vector § 11-3; § 12-95 Latitude of the gradient of N and second coordinate in § 12-1 a (a). 4>, N) system. X First or principal curvature (curve in three dimensions)... § 4~3 i// Isometric latitude § 2 1-79 Spherical arc § 29-4 " Longitude matrix § 19-9 ft Solid angle § 20-16; § 29-2 ft Right ascension of ascending node § 28-30 w Longitude of the gradient of TV and first coordinate in a §12-1 (oi, (/>, N) system. f Angular velocity (usually of the Earth's rotation) § 20-10 Summary of Formulas Metric: 1.06 Chapter 1 ds 2 = grsdx'dx* (r, s— 1, 2, 3) (in three dimensions) ds 1 = a ali dx a dx ii (a,j8=l,2) (in two dimensions) Unit Contravariant Vector: 1.08 l> = dx>lds Unit Covariant Vector: 1-12 l x=gr jr = g jr Vector of Magnitude k: 1.09; 1.13 L>=kl>; L,=kl r Scalar Product: 1.17 L'M, = k /J, cosO Gradient of a Scalar V: 1.20; 1.21 Nr=dNldx r =nv r Transformation of Vectors: 1.18 i dx r j L' = — L s 3x s 1.19 — 3x s dx r Kronecker Delta: 8?=1 if s = t 1.24 8*=0 if s^t Chapter 2 Transformation of Tensors: dxf dx q 2.01 Ars=~ S r-A m dx r dx* ' ' 2.02 2.03 7 C)X' dx s . dx? dx" 7 dx r dxi . s dxP dx s " Addition of Tensors: 2.04 C' S , = A' S , + B' X , Multiplication of Tensors: 2.05 C' S , = A S ,B>- 2.06 C, = /*,,#' The Metric Tensor as Product of Unit Orthogonal Vectors: 2.07 k'K + /x'/x, + v r v„ = 8j = g rt g s , 2 . 08 k,K + Hr^s + V r V s = gr, 2.09; 2.10 k r k t + fi r fi t +v r v' = g rt =G rt lg Raising and Lowering Indices: g rs A l . t =A* l grsA s ., = A r , Determinants: 2.12 Ae rs , = eij k A).A^ 2.16 V.A =e i J k e r: «Ai r ApA kl 2.17 2L4 "•= e^ k e rst A jls A k , The e-Systems: 2.14 2.15 2.18 2.19 dx*> = Vg dx' e rs i = Vgerst cist= <>rst e rs '/Wg rsl c ^rst " "rat Simn = Simnp 353 306-962 0-69— 24 354 2.22 2.20 2.21 8f = £8™* o s , A mil ii = ^ xfp ^3 (sp °s< ^..p ^..p ^..p Vector Products: 2.24 e r * 2.26 e"%./z x i>, = 1 (X r , /a, ■, v r unit orthogonal right-handed vectors.) Further Properties of e-Systems: € rSt = k r {/JL s l>> - P s fJL<) + /jl'( p s K< — AV) 2.27 +v r (\ s n t -(A s k t ) 2.28 gne^'e^ = g'Jg k - g' k gj> Two-Dimensional Formulas: 2.29 j _ c>x y dx b 2.30 e^^e^/Va; e u0 =\4e o/j 2.32 e a ^ = \ a ^-fx a k^ (A", yfi unit orthogonal vectors.) 2.32 €a/3 = A a £l/3 ~ Ha^fi 2.37 fJ-a = €aaK" 2.34 CL a fS= \ a \$-\- lAalAQ 2.35 a a P = \. a kl 3 + fi a V< 13 2.36 a"< 3 a0y=8 y =A"Ay + /A < >y 2.38 8 y £=e«%y 6 2.39 8§=8$ 2.41 8y^y4 agpo- = A ygp, r — A gy po . 2.42 2L4 = e«V«4 a/ ^y 8 2.43 A°»=(^e» h Ay h 2.44 2.45 a a = e"V s ay S a a/3 — € ( »ye/isa Chapter 3 y8 Christoffel Symbols: 3.01 W,k]=^(^+^f- d ^i Mathematical Geodei 3.02 Covariant Derivatives of a Vector A'. A, 3.07 dA ( - Aj=^+rj^ J dxJ Jk 3.08 ^ST 1 ^ Covariant Derivative of a Tensor K r st : 3-09 A^ = ^+r;,Ai,-ri s A];-Fi,A^ (Rules follow Equation 3.09.) Covariant Derivative of a Scalar Gradient r : 3.10; 3.11 v ' dx r dx s rs ^ *'• Laplacian of a Vector F, ■: l a 3.12; 3.17 \F = g rs F r « = F s . s = =^~ s (VgF s \ g oX Other Differential Invariants: 3.13 V(F)=g rs F,F s 3.14 V(F, G)=g r *FrG s Curl of a Vector F, : 3.15 e rs 'F,,. Differentials of Determinant of Metric Tensor: B (In \ r g) 3.16 a.t» 1 fru Covariant Derivatives of Unit Perpendicular Vectors 3.19; 3.20 /,,, /' = 0; lr, s j r =-jr,s l r Chapter 4 Frenet Equations of a Curve in Three Dimensions m rs l s = — \l r + Tn r 4.06 n r J s = — Tm, Curvature of Orthogonal Surface Curves l a , j a : 4.11 latl = (TJ a ll3 + (T*j c Jl3 jap = ~ crlJn — a lajli Summary of Formulas 355 4.12 Chapter 5 Riemann-Christoffel Tensors: 5.03 R 5.02 5.04 5.05 5.06 ijk dxJ i : ri + r»ir/ _ r»ir/ ik Q x k ' U^ ' i* J >»J >J mk h,jk — h, kj — R'.iji^i Ri ■ijk WikJ R'. ijk + R[ jki + Rl kij = R mijk ■gl.nR 1 . ijk 5.07 Rmijk = — [ik, m] -— [y, m] + r(,.[mA, /]-n,[ m >, /] „ =1 ( Vgmk d 2 gij d 2 gmj &gik mijk 2 {dx^xJ dx'"dx k dx'dx k dx"'c)xJ 5.08 +gP*{[mk, p][ij, q]-[mj, p][ik, q]} 5.09 Rni uk = Rjk m i Ricci Tensor: 5.11 Ri j = g»»>R„ l i j k = R% k . ri r) 5.12 dxJ" dx l h u m} ' jl >»i Lame Tensor: 5.13 SPR h „ Ry — ^Rhapy — KiA u epy 5.24 e^k a ,ffY=Kfi a Riemannian Curvature: 5.25 ; 5.26 C = R mijk k m iJ. i V l x k = S™v p v q General: 6.02 6.06 Chapter 6 dx r = ,. dx a %a a a p = grsXa x [! Surface Vectors: 6.07 l r =x r a l a 6.08 fcc|=//B 6.09 x r a =l'l a +j<\j a The Unit Normal: 6.10 g rs = a a H T a x%+v r v s 6.11 i/r€ r "=e , *x«x' fl 6.12 fp=ie°% s ,x£4 Covariant Derivatives: 3 (In Va) 6.13 6.14 - V 13 dx a "* d-x r + H; *<,*$- 1^*$ " d% a d^ The Gauss Equations: 6.16 x T a y=b a yv r The Weingarten Equations: 6.17 v^-a^bpyxl The Fundamental Forms: 6.18 ««H = grs-X'aXfr C a p = gr8V£V$= d^baybfi?, 5.22; 5.23 The Mainardi-Codazzi Equations: 6.21 b a py=b a yi} (flat space) 6.22 b a ffY = b a yfi — R U rstV u X r a xlXy Gaussian Curvature: Kesa€0y = Rsa/fY = (baybps — bsvbap) (flat space) 6.26 356 Mathematical Geodes 6.27 aK — R\2vi = b (flat space) Kesaepy = Rsapy = b a yb($b — bdyb a p+ R U rstx^x r a x^x y 6.28 Chapter 7 Curvature (Meusnier's Equations): 7.03 l rs v r ls = -vrj r l*=b ali l a iv = x cosd = k 7.04 lrsj r l s = 7.22 K= Ki k 2 7.23 a a p= u a up + v a vp 7.24 bali= K\UaU$+ KtVaVjj 7.25 C a 0= KiM a M|8+ K|Wa^/3 7.26; 7.27 v r $u> s =— Ki?/ r ; v r sV s — ~k%v t Chapter 8 Contravariant Fundamental Forms: 8.01 Kb c " 3 = e^e'H yf> 8.02 =k*l a ^-t(l a f+j"l^)+kj a p 8.03 =K Z U a lll 3 +KiV a V 13 8.04 K 2 c°< , = e a V 8 cy8 K 2 c a < 3 =(k* 2 +t 2 )l a l( i -2Ht(l a ji 3 +j a li i ) + (Jfe»+«»)jV 8.05 = k l 2 u q ^ + k 2 ?;|8 8.16 + (ki — K2) (crwy+ crVy) {u a v$ + v a u.ii) C a py= (K 2 x )yU a U(}+ (l«|* 8.19 +(l/K'--l/K^)(o-«y+cr*?;y)(uV+2; a ^) a a eb al3 y=(2H)y a^c a/3 y=(4// 2 -2K) b^b al3 y=anK)y b<#CaPY= (4//)y 8.20 c^baffv=- (2H/K) y c^c a ff Y =2(lnK) y Mainardi-Codazzi Equations: (ki — k 2 )ct= (Ki )y?; y 8.22 (K,-K 2 )o-*=(K,)yU> o-(^-^*) = (A-)y; Y -(0y/ Y -2io-* 8.23 dS = -\ 0,/ Js Jc 9.06 9.07 9.08 9.09 9.10 9.12 9.13 L i'ds A(f)dS = (closed surface) Volume and Surface Integrals: 9.15 9.16 9.17 9.18 9.19 Tjjk.mdV— TijkVmdS J F% dV=j F'"v m dS= J F,„p>»dS j t (&lds)dS J {S7i4>,^)+4>^}dV= J (j)(dil/lds)dS {d>Aih-^d>}dV=! U^-^)dS Js \ ds dsj Chapter 10 Metrical Relations: 10.01 ds 2 = m 2 ds 2 10.02 grs=m*grs 10.03 \grs\ = m«\g rs \ 10.04 grs= m -2 g rs m 2 [ij, k] = [ij, k] + g ik ( In m )j + g jk ( In m ) , 10.05 - gij (\nm) k T;. - rj. + 8\( In m )j + Sj( In m ) , : - gij g lk ( In m ) , 10.06 m ~ 2 R qrs i — Rqrst = mgqs ( l//») rt ~ TTlgqt ( 1/ffl) rg — mg™ ( 1 / m ) 9 , + mg r , ( 1 / m ) gs 10.07 + m*(g„g*-grtgq,)V(llm ) 10.09 Rrs- Rrs =-m(l/m) ) ,+ (l/m)(Affl)g ( , 10.11 S,,. - S rs = - m(llm)rs ~ ( A In m )g rs Transformation of Tensors: 10.12 l< = m-H> 10.13 /~=m/ r 10.14 m " l J r , , = /r, , - ( In m ) , /, + g rs (In m),l< 10.15 ml r tS = l r tS + h T s {\am)tl t -g rt {h\m) t l s rs = 0rs — (/),( In m ) g — c/). s (In m ) , + gy-s V ( In m , <£ ) 10.16 10.17 m 2 A^ = A) ;h 2 A(/> = A0+ V(ln m, 0) (three dimensions) 10.18 Correspondence of Lines: 10.19 /„,/«=/„/«- (In m), + {(hi m),l'}l r Xn r = mxn, = {x- (\nm) t p'}p r - { ( In m),q'}q, 10.20 m\ cos 0= x — (In m ),p' 10.21 mxsmd=— {Inm^q 1 10.22 mf = T+(dBlds) mxidd/ds ) = — sin d(dx/ds ) + r(ln m ),- rc '' 10.23 + mQlm) rs b r l* m 2 XT = X T cos $ — sm d(dx/ds)+ m(llm),sb r l s 10.24 358 Mathematical Geodes m' z {dxlds)= cos d(dxlds) — T(\n m) r b' 10.25 +malm)rsn r l* Surface Normals: 10.27 N r = nv r 10.28 v TS = v sr + *A-(ln n ),• — v,(\n n ) s 1 0.29 v n v* = (In n ),■ - i>,{\n n ) s v* Transformation of Surfaces: a, a p = m 2 a a p \ r 11.02 < = P'a = * r a=K 11.03 a a p=c a p 11.04 b a p = — Cap = — d a p 11.05 c a p=c a p = a a p 11.06 \a a p\ — \c a p\ =K 2 \a a p\ =K\b a p\ 11.07 a a ^=c a ^=-b aP =c a ^ 11.08 b^x^b^Xy- (Scale factor m = ds/ds.) 11.11 m 2 = k 2 + t 2 11.12 l a = ml a 11.13 ml(i=capl a = m 2 lp + 2Htjp 11.14 €ap = e a p/K- e^=Ke a ^ 11.15 jfi={mlK)h Principal Directions and Curvatures: m = — K\ (in ^-direction) 11.18 m = — Ki (in ^-direction) 11.19 u a = — K\U a ; up = — uplK X 11.20 v a = — Kiv a ; vp = — vplK> Direction l a (l a u a = cos t//) : 11.21 m 2 = k 2 cos 2 p 10.47 11.22 COS l// : (k 2 cos 2 t// + K.f sin 2 cos a» — fir sin (/> sin w + C r cos (/> v r = /4 r cos cits a; + j/ r cos cf> cos to y r = B r =\r cos (0 — /JL, sin $ sin a; + pv cos $ sin to z, = C, = /Xr cos + i', sin t\> 12.009 x = — (* — xo) sin a> + (y — Jo) cos to y = — (x — Xo) sin tj> cos a» — (y~ yo) sin sin to + (z — zo) cos z= (x — xo) cos cos to 12.010 + (v— yo) cos (/> sin to+ (z — Zo) sin t}> (x — xo) = — x sin cd — y sin tj> cos to + z cos cos a; (y — yo) = x cos a> — y sin sin to +z cos c/> sin a» (z — Zo) =y cos (/) +z sin c/> 12.011 /A,\ /l \ /-sin to cos to 0\//4A I ju-c ) — I sin (j) cos $ II — cos a» —sin to Oil fi» \i/,-/ \0 -cos sin / \ l/\C,./ 12.012 /A,\ /-sin &) -cos to OW] \ /\, fi, = cos a> —sin to I sin — cos K r co,-v,4s 12.016 »Vs — COS $ Arto., + /A, .s 12.017 N„=n,v r + nv n 12.020 v rs v s = {(In fi) g \ s }X r +{(ln nls^'V' 12.021 =y,\r + y 2r ir 12.022 k Q = riaiop sin 4> 12.023 fJ-ap — — X a tofj sin <£ 12.024 v aii = -b a ti Components of Base Vectors: 12.025 12.026 $8 = — r l r*V r (COS $)to.s- = — X rs V r \r=(to s \ s , (/> S A S , iV s A s ) 12.029 =(-Ai sec , -f,, 0) /U.' = ( tos/i." . Sr A S , Vs/Lt.s ) 12.030 =(-ti sec) 12.041 Kk,=(—k> cos 0. +?,. A. sec (j>d(lln)ldco) 12.042 Kixr = (+t l cos . -/.,. Kd{lln)ld. +.,) 12.045 K/x„ = ( + /■ cos 0, -Ai) 360 12.046 (COS 4))(i) r = — kik r — UfXr + JxVr 12.047 4h=*-tikr-ktlJLr + yiVr 12.048 (In n),=yi\ r + y2Hr+{(\n n) s i>"}i>, Curvatures of /V-Surfaces: ft = k\ sin- a + 2^i sin a cos a+A"2 cos 2 a 12.049 t= (A'2 — Ai) sin a cos a-/, (cos- a -sin- a) 12.050 12.051 tan 2/f = + 2f,/(A 2 - A, ) K\ = A'i sin 2 4 + 2t\ sin ,4 cos A + A 2 cos 2 4 k> — k\ cos 2 4 — 2t\ sin 4 cos A + k 2 sin 2 4 12.052 12.053 H=H«i + K 2 )=i(ki + h) ( Ki — k-2 ) = ( A2 — Ai ) sec 24 = 2f i cosec 2A 12.054 12.055 K = K i K 2 = k l k 2 -fi 12.056 A = k. cos 2 (4 — a) + K2 sin 2 (4 — a) 12.057 t = i(#c,-/c 2 ) sin 204 -a) Ai = Ki sin 2 4 + k-j cos 2 /4 A2 = Ki cos 2 A + k> sin 2 4 12.058 ti= (ki — k 2 ) sin/4 cos A (cos (j>)do)/dl = — Ai sin a — fi cos a = — A sin a + f cos a d(f>ldl = — 1\ sin a — A-2 cos a = — A cos a — £ sin a (cos (f))do)/dj= Ai cos a — /, sin a = A* cos a— t sin a 12.060 B(f>ldj=t\ cos ot — kz sin a = — k* sin a — f cos a (cos )(o r = (~k sin a+ ? cos a)/ r 12.061 + {k* cos a— f sin a)j,+J\V r (/>,=— (A cos a+f sin a)l r 12.062 - (A* sin a + t cos — ay) Mathematical Geodes 12.064 crly + cr*jy=(oy sin — a> 12.065 (do>ldl) - (da/dl) cti = — Ai tan ^; cr 2 = — /, tan c/> 12.066; 12.067 o"= o"i sin a + cr 2 cos a— {da/dl) 12.068 Metric Tensor: gu = (iu — (kj + t'i) cos 2 $ fK' 2 gv> = ci\2 = — 2Ht\ cos 4>IK- g 22 = a 22 = (k\ + f\)IK- gn gza k-> 3(l/n) d cos (/> 3(l/n) = -[y.(A- 2 + / 2 )-2//y,r 1 ]/(«A' 2 sec 0) _ £i sec (f> d(l/n) Ai d(l/n) K da> K d(f> =-[yz(ki + f{)-2Hy 1 ti]lnK 2 £33 = sec 2 $ a(l/n)\ 2 , (B{Vn n- 12.069 12.070 12.071 12.072 12.073 dco ) \ = [y\(kl + f\)+yl(k\ + ti) -mt iyi y 2 + K 2 V(n 2 K 2 ) = sec 2 j8/n 2 £ = cos 2 $/(n 2 A. 2 ); « = cos 2 (j)/K- fi u = (k'i + fi + y'i) sec 2 g v2 = (yiyz + 2Hti) sec £ 22 =(Ai + / 2 + yi) g Vi — nyi sec A /2:, = «y 2 o"=(A-f + tf) sec 2 a 12 = 2//fi sec (/> a 22 =(As+f 2 ) |^'" s | = n 2 A.' 2 sec 2 : |a a ^| = X 2 sec 2 Second Fundamental Form: 12.074 12.075 (A-2 cos 2 0/A:. —t\ cos (/>/&. Ai/A) Summary of Formulas 361 12.076 b = cos 2 IK 12.077 b a > i = (/„ sec- , f, sec . /.,) 12.079 6, a =-(cos)\„; b> a =-fi a 12.080 b la =- (sec , 0, 1) 12.084 c = cos- 12.085 c^=(sec- 0,0.1) Coordinate Directions: Longitude: 12.086 f=(l/V>„,0, 0) cos «, = ?,/( A| + t'\ ) ''- = /,//»■_> 12.087 sin oc!=- fol {ki+.t*y l* = - folrn-, m-> = (&f + £f) 1/2 = (k! sin- // + kt cos- //)"- cos cci = (ki — k>) sin // cos /4/ma 12.088 sin ai=— (k> sin- A + Ki cos- .4)/w-, sin (-4 — ai ) = Ki cos .4/m^ 12.089 cos {A — a\)= — k 2 sin A/m 2 Coordinate Directions: Latitude: 12.090 j'= (0, 1/V^, 0) cos a-> = - /, ,/ (/.'t + ^ ) "- = - Ai/m, 12.091 sin a a = fi/(Af + «f) ,/2 = fi//n, mi =(AT + f'T)"-= (kt sin- ,4 + K3 cos- ,4 ) "* cos a> = — (k\ sin- .4 + k< cos- .4 )/mi 12.092 sin a>= [k\ — K>) sin A cos ,4/mi sin {A — a>) = — K\ sin .4/m, 12.093 cos M-a>)=-K2 cos /f/m, The Isozenithal: 12.095 A' = (0, 0, 1/V& sin a sin /3 = {sec d ( 1//? )/d )r> ( In n)/du> 12.097 cos a tan /3 = -d(ln /i)/d<£ 12.098 \,,A * = /Ar,A s = i/ M A-« = Laplacians of Coordinates: 12.099 /V,.., = n.,i/, + ni;, s 12.100 AN = dn/ds-2Hn 12.101 (cos 4>) V(a>, c/>) —2 cos V (w. In /i) 12.104 + (1//;)(A/V),V A, In ra) — sin cos <£ V(a>) 12.105 +(l/n)(A/V) r /a r A// = /({cos- V(co) + V() } + ( A/V),V 12.106 12.109 cos- 4> V(w) = /,f + /f + y'T 12.110 V(0)=A1 + fT + y3 12.111 cos V(ft), 0)=2///, + -y,y, cos V(a), In n) =-/.,yi — to* + 2//yi + (y,A.V)//; 12.112 =/.jy,-/,y,.+ (y.A.V)//; V ( 0, In /1 ) = - f , y, - /,,.y,- + 2/7 y, + ( y,A;V )/n 12.113 =kiy i -tiy l + (y 2 A;V)/n cos- V(w) + V()=4//--2/^ + (yf + yji) 12.114 =K? + K|+(yi + yi) (l//i)A/» = 4//--2A + (yf + yl) 12.115 + (l/n)(AAO r i/ r Surface Invariants: (cos )Aw = 2(sin 0)V(w, 0)- (2//)„\ a 12.118 L 362 12.119 V(a), (jj.„k y — k a /A y )+ bupv^ (space) =— 2\t+----m a** 3 c)x fi ^ nifl surface) =-H! \y+_c^ /u, y 12.130 F^ (space) -r*3 (surface) = —v a fjV y 12.131 =bapv y Mainardi-Codazzi Equations: dbu/d(f) — db\->ldoj + bu tan (j) + b>> sin $ cos (b~0 12.134 db V z/d(f> — db-z-zldo) — 6 t2 tan = 12.135 12.143 12.144 12.142 dfrg£ = a 2 (l//?) | p y 3(l//l) Cg/3 aTV a^a.r^ a/3 a* Y " n sL+'-M^-? T 2 , = sin cos ; T} 2 = — tan Mainardi-Codazzi Equations in Tangential Coor- dinates: 12.145 P = PrV r 12.146 p a = p r v'a l>ati=— bafi + p,\b yh b at ihv$ — c a pv r ) 12.147 12.148 12.151 bafi — — Pali + b yS b af i py ~ pC a d 2 P , -F dN n — b a = p a l3+ pC a Higher Derivatives of Base Vectors: d{l/n) cos (f> d(f> n 12.153 A-3a — I sin (j) 12.154 /* 3 a = -tan0^^ 8 a --5 2 12.155 v 3a =(lln)a Kpz = Aa:i/3 = — (sin $)p. a b ly {dbpYldN) Pafu = Pa-.ii3 = {sm cj>)k a b ly (dbpYldN) 12.159 v a03 = v Mli = dban/dN K:a = HaQ-ln) tan A 6 {a 2 (ln n)/dx s dN} p a3 3 = - K(lln) tan A 5 {a 2 (ln n)/dx & dN} Va33 =(lln){d-(\n n)ldx a dN} 12.160 12.161 Vapy = — 6«/3Y — b^ ( In n )„ The Marussi Tensor: N rs \'\ s = -nki Nrsp' p* = — nk-i N rs \ r pf = — nh N rs k r v s = nj\ Nrsp/v* — ny-2 12.162 NnvV = n(\n n ) s v s The Position Vector: 12.169 p r = (sec cos co + y cos —~x sin cos to— y sin sin oj+z cos 4> 12.171 (sec )c)p/c)ct) = — x sin w + y cos w 12.172 Summary of Formulas Chapter 13 Curvatures and Azimuths: 13.02 k = -m cos [a -a) 13.03 t = -m sin (a -a) m cos a = — k cos a — t sin a = — k> cos a — t\ sin a 13.04 =90/95 m sin a = — k sin a + ? cos a — — k\ sin a — 1\ cos a 13.05 =(cos )dQ)lds Geodesic Curvatures: ma — a = (a — a) pi 1 * 7r \ a (///,) 363 13.10 Covariant Derivatives: (mlK)l a n= lau+jja — a)u k*\d(tlk) 13.12 13.13 Double Spherical Representation: m A» cos a + /i sin a 13.19 13.20 13.21 m* A* cos a* + t* sin a* A i sin a + t] cos a tan a tan a * . a + 6 tana* c + (/ tan a* a + c tan a 13.22 13.25 6 + d tan a a=(fefT-fiA-?) : &=(A- 2 A*-*,rf) c=(tif?-A-iW) : rf=(fiA'?-A-itf) ad-bc = KK* (m/m*)K* sin a* = — (a cos a + c sin a) (m/m*)K* cos a* =(6 cos a + ^/sin a) (m*/m)K sin a = (« cos a* + /> sin a*) (m*lm)K cos a = — (c cos a* + a? sin a*) Chapter 14 (In some cases, the formulas have been extended or rearranged to give the isozenithal derivatives ex- plicitly and can be obtained at sight from the textual references on the left.) Fundamental Forms: da afi ldN= b yS any(db aS ldN) + b^aMdbfaldN) 14.03 Bb a pldN=bayrz t 12.127 12.144 12.143 & I bv & b a 0f> I n h Cgfi n d-(Mn) = HIM C °V d&dxP <* dxv n = — {lln) a p — Capln (Covariant derivatives refer to surface metric; over- bars refer to metric of spherical representation. Only nonzero values of spherical Christoffel symbols are 12.142 r?, = sin cos ; rj 2 = - tan 0.) 14.08 14.06 da^ldN-- [aovb^ + a^b^JidbyildN) fib" ^rr^-b^b^iabyslaN) 14.07; 12.127 =-W§, 14.08 dN 14.04; 3.16 d In Vq 5 In b dN dN d In A dN 1 a3 12.084 d In c _ „ d/v Curvature Invariants: (Overbars refer to surface metric. d(2H)ldN = -a"Hdb a0 ldN) = A(l/«) + (4// 2 -2A")(lM 14.32; 14.28 -6«*(2ff)«(l/n)0 a ( In A ) IdN = - b a V ( dbap/dN ) 14.32; 14.29 ■■b^(lln)afi + 2H(lln] -b a(i (\nK) a (lln)i3 364 Mathematical Geodes d(2HIK)ldN = c a e(db a0 /dN) 14.32; 14.30; 14.37 ■C*(l/n),tf-2(l/i0 ■b*(WIK)«{lln)f> Christoffel Symbols: _a dN •> \i X /3Y ^ 03'? =(r?s)/s 14.14; 14.15 b^\ dN J, Curvature Parameters: bapyk"^ = ( Ai ) Y - 2t 1 bapyk a iiP = (ti)y + (A:i — fa) (Oy sin 14.35 b a pyix a ijJ i = (fe 2 )y + 2ti dfa = _db S fi = db^ dN dN * dN dh db a p o , afr 12 — — = "f A a U/* = COS © — — dN dN ^ r a/V 14.34; 12.077 dh = _dba£.. a ..t3. dN dN ^ ^ db 22 dN dN\Kj db-ii Jn dN\KJ — sec 3(fa\ dN\K) sec 2 4> afeu aw dyi/dN=- - n^ COS (£ dy-z/dN =-n ,^ s 12.075 14.40 14.41 Principal Curvatures: bat}yii (X u l3 = (Ki)v b a pyV a vP = ( K 2 ) y 14.43 baiiyu a v ti = (ki — K 2 )(ojy sin (f)—Ay) dKi/dN = - {db a pldN)u a u^ dK->ldN = -(db a0 /dN)v a vi 3 14.42 do - k 2 ) (d.4ldN)=(db a(i ldN)u a vi i Miscellaneous Point Functions: 14.09 de af ildN = -c a vd(\n K)ldN 14.10 d€«< i ldN = + e u ' i d(\n K)ldN 14.11 d(Ke a0 )ldN=d(e^lK)ldN=O 13.14; 14.05 d(KdS)ldN=0 14.50 ax^/a/v=r^ y ' 14.51 dv r a /dN=0 14.52 dv'a ldN=- (I% i )0Vy (The space coordinates are Cartesian in the last three equations.) 14.16 dbauy_, , /db» OapOfJv dN dN Surface Vectors Defined in Space: 1,,^-jAda/dN) 14.17 jn= IridaldN) Projection of Surface Vectors: Length: 6 (In 8s) = 3 (In m) = d{\n (P + * 2 ) 1 ' 2 } dN ~ dN ~ dN 14.53; 14.54 14.56 d{\n(m/K)}ldN=Vy.,j°jy Azimuth: (daldN) = -n,J a jy = (k 2 lm?){d(t/k)ldN} 14.61; 14.67 Components: 14.59 d/ u ldN = -V y .Jvy° 14.64 dlJdN= T^-UdaldN) Curvatures: 14.65 dkldN=k{d(\n m)ldN}-t(daldN) 14.66 dt/dN = t{d (In m)ldN} + k(daldN) 14.69 go; a(lnm)__ga L a/v CT a/v ax^aA^ Covariant Derivatives: 14.70 r^A 7 "^ - ~dN~ Ja d^dN r&My 14.71 dFaj3_(dF dN \dNj ali '& )/»*"> (F is a scalar defined in space.) Chapter 15 Normal Coordinates: Metric: 15.02 ds- = aapdxrdxl* + ( 1/ii) 2 dN- Summary of Formulas 365 15.04 15.05 Unit Normals 15.07 g=(\ln*)u ,r S= („«^„2) Vr = (0,0, 1//I) 15.08 v'=(0,0,/j) 15.10 ^3 = 0; vg = - « ay />^y = - 6 a ^Y Surface Vectors: / 3 = 0; / 3 = Christoffel Symbols: (space) = F^ (surface) r 3 a 3 = ( l/« 2 )« a/3 (ln n fe ; rg :j = - ( 1 \n ) a^fyjy \ T^ = nb afi ; r 3 3 a =-(ln«) a ; r 3 3 3 =-3(ln n )/dN 15.11 i 15.12 r& _a_ a/v Derivatives of Unit Normal: 15.18 y a/3 = — 6 a/ s Va3 =— (l/«)a ^3u = 15.19 ^33 = 15.20 15.21 v%p UCali Mainardi-Codazzi Equations: 15.24 b lt 0y=b a y0 15.25 — — =n -I — c aj8 as \n/ ali Normal Differentiation: Fundamental Forms: 15.13 da a 0lds = — 2b a a& ai8 /1 ■ — - = 11 I — I -Cafi ds \ n kn 15.25 15.26 dCa ds da a0 ds 15.14; 15.15 15.27 na yS b a y[—\ +na y8 bpy , n ) nh 2a <*y a &by S = 4//a a/j - 2Kb <* a6 a0 /i\ os \n/ys i) 15.28 15.16 — — = - nb ay c^ (— ) - nb ay c" f ' ds \n/yf, \n/yf, d(\na) ds W 15.25 d(\n b)/ds = b at3 (db„i i lds) = nb ali ain) ufi -2H 15.30 d(ln c)/ds = c^dcafj/ds) = 2nb a0 ( \/n ) a/3 Surface Invariants: a(2#) 95 = nA(l/«)+(4// 2 -2/0 15.29 -^-^ = ,^^1 '// 15.31 a 5 d(2H/K) ds U/ aji T ,a/3 a/3 Christoffel Symbols: dV a —J*l = - a aS bpy S + a a % s (Innh ds 15.32 + o a8 6y S (ln n)p — a a8 6/jy(ln h. Curvature Parameters: (See § 15-35. + 2y^i tan <£ dA 2 /ds=ra(l/n) a/ 3juV J +(*f+*i) — 2yiti tan $ 5«i/9s = n ( l/n) a/ 8XV + 2Ht, 14.36 -yi(k 1 -k 2 )ta.n(b Principal Curvatures: (See § 15-35. 14.45 14.46 dKilds = n(lln) afi u a uf 3 +K 2 i aK 2 /a5=/!(l//Oa0r a (' /3 +K5 14.44 (K,-K 2 )(y, tan — d.4fds)= nil/n ) a /juV Miscellaneous Point Functions: as 2//e a 15.17 15.39 15.33 ae ng as = +2//€^ d(8S)/ds = -2H8S (area) a ( 6«/jv )lds — n(l/n) a )iy — C a /3 ( In n ) y — c^y(ln n)„ — cy ( ,(ln n)^ + a 8f ( In n) e ( 6„y6s^ + 6#y6s<, ) 366 Surface Vectors Defined in Space: dl a lds — — kla+ja(ji tan — t — da/ds) 15.36 dl a /ds = kl a +j a {y A tan + t — da/ds) Meridian and Parallel: d\ a /ds = (yi tan — ?i)/u, a — kik a dk a lds=(y l lan (t> + t l )/jL a + k l k a dfjia/ds = — (j\ tan + ?i ) K u — k 2 ^ a 15.35 dix a /ds = -{yi tan — dA / ds) Bu a /ds = + K\u a + v a {y\ tan — dA Ids) dv a /ds = — K 2 v a — u a (yi tan -dA/ds) Normal Projection of Surface Vectors: Length: 15.38 dQn8l)lds = — k Azimuth: 15.43 da/ds — y 3 tan + t Components: 15.40 15.41 dP ds = kl a dlalds = — kl a — 2tj a Curvatures: 15.44 dk/ds = n(l/n) a ^ + (k 2 -t 2 ) 15.46 dt/ds = n{\/n) ali i a l^+2kt da/ds = k)u)tj 15.49; 15.50 =Kbv & (\nn)yeim=e ya (ln n) y b a0 15.52 j a Qn = b ati (\n n ) yl y - b^(\n n ) u / 8 ^ =n (-¥) +Fy{b a0y -b a y(lnn) li 15.53 — bpy(\n n) a + b a p(\n n) y } Chapter 16 Darboux Equation: (l/n) a pu a v li (surface) =(l/n) rs u r v s (space) =0 16.03; 16.04 16.05 Mathematical Geodes' dA/ds = y\ tan Uft , d 2 (l/n) r 9(l/n) fl(l/n) 16.06 = 1 \ 2 + T 2 , (Surface coordinate lines are fines of curvature.) Particular Solutions: 16.07 a + bx + cy+dz + er 2 Chapter 17 The (co, 0, h) Coordinate System: Metric: 1 7.04 ds 1 = a a!3 dx a dx^ + dh 2 17.05 g rs = (a ali . 1) Fundamental Forms: a u = {k'i + t'j) cos 2 4>/K 2 b\ , = k 2 cos 2 /K a vl = — 2Ht x cos 0/A? b v > = — t ] cos 0//, --h 2 , 0) 17.25 ^=(0,0,1) 17.26 KK r = (-k, cos 0, +f,, 0) 17.27 A>, = (+/, cos 0, -A,, 0) 17.28 *V = (0, 0, 1) 17.29 (COS 0)o>,- = — fciXr— tl/Ar 17.30 0, = -f,A,-A:,/x, X«/3 = /u.oOJ/3 sin (JLaj3 = — k a (i) f 3 Sin 17.31 V a p = — 6 Q /3 = (COS 0)A a OJ0 + HX a 0,8 17.32 A:u= — cos ; M32 = — 1 Christoffel Symbols: T^ (space) = Tg^ (surface) Summary of Formulas 367 17.36 Vln = b ali r%--a^b^Y=v$ (All other 3-index symbols are zero.) Laplacians: 17.37 M = -2H 17.38 (cos )Aw = 2(sin (/>)V(oj, (f>) — (2H) a k a (space or surface) 17.39 Ac/> = -(sin c/> cos (f>)S7(a>)—{2H) a fJL a (space or surface) 17.40 with V(Q), cf)) = a [ - = 2Ht, sec 17.41 VM=(;" = (^ + (t) sec- AF = AF — 2// — r+ ttt = AF + A. rrh^rr r»/i d/r d/i \A. ri/i / 17.42; 17.43 The A-Differentiation: (References may be to formulas in (w, $, /V) or in normal coordinates from which the formulas given now are derived. See § 17-32. The corresponding formulas given now in (w, (f>, h) coordinates are not, of course, the same in all cases.) The Fundamental Forms: 15.13 da af }ldh = — 2b af } 15.25 dbafjldh= — C a fj 15.26 dc ali ldh = da^ldh = 2a Q ^ 8 6y 8 = \Ha<® - 2Kb^ 15.14; 15.15 15.27 db°*ldh = a a P 15.28 dcrt/dh = o 17.09 a«ti = dan — 2hbaii + h 2 c al s 17.10 b a $=bal3 — hCafl 17.11 Ca/8 = C a /3 17.17 u«»/k 2 = d^lK 2 - IhbrtjR + h^crt 17.18 //«V/v = />//\-/;c^ 17.19 c »tfi == £aji §17-14 dlnVa d In b ,„ (9 h dh §17-14 *jnc dh Curvature Invariants: 15.30 d(2H)tih = 4H 2 -2K 15.29; 17.22 d(lnK)/dh=2H 15.31 d(2HIK)dh=-2 K/K = l-2Hh + Kk i ={l-hK X ){l- hk, ) 17.14; 17.15 17.16 2HIK = 2H/K-2h §17-14 K- = c = Kb = K-d Christoffel Symbols: 15.32; 17.52 17.36 -V%=-"" h b i*y UYfr dh Y rt" d ., I '* = dh * 3 C al i ■■ — a a yc $y Curvature Parameters: dkjah= (/,-, + /-,) aki/Bh= (ki + t-) 14.34; 14.36 dt 1 /dh = 2Ht 1 kilK=kilK-h hlK=hlK 17.13 k 2 /K = k 2 IK-h Principal Curvatures: 14.44; §17-21 dA/dh = ±(L\± dh \kJ dh 14.45; 14.46 17.35 (1/ki) = (1/7ci)-A (1/k 2 ) = (1/k 2 )-A Miscell aneous Point Functions: de^/dh = - 2He afi 15.17 d€#ldh= + 2H** 15.39 d{8S)ldh=-2H8S (area 13.14 d{K8S)/dh = Q 14.11 d(Ke a0 ) Idh = d ( e^/K ) /dh = 14.50; 17.36 dx , a jdh=vl, 14.51 dv'Jdh = 14.52 dvydh = ay^b^yv'^ (Space coordinates in the last three equations are Cartesian.) 17.53 db„pyldh = Surface Vectors Defined in Space: dljdh = - /,/„ - /',, ( t + da/dh ) 15.36 dl a /dh = /,/" +>"(/- da/dh ) (The vector is defined in space; it is not projected.) 368 Mathematical Geodes Meridian and Parallel: dkjdh = -t t fjL it -kiK,= - b],^ dX a /dh = ti/JL a + k l k a = a a %,iyk y d/Xa/dh — — 1\ k a — k-ifx a = — bafjfd 3 15.35 dfJL a ldh= t ] k a +k- 2 /x a =a a ' i buy^ Principal Directions: dujdh = — K\ U a dvjdh = — K->V a 15.37 du a /dh = K 1 u a dv a ldh = KoV a Normal Projection of Surface Vectors: Length: (m = scale factor of spherical representation.) 14.53; 15.38 d (In m) / dh=-d (In 81) I dh = k 14.56; 17.36 *M*/*) = _ A * dn Azimuth: 15.43 t)a/c)h = t Components: 15.40 dl*ldh=kl a 15.41 dUdh = -kI a -2tja Curvatures: 15.44 8klc)h = k--f 1 15.46 c)t/c)h = 2kt 15.54 dcrlBh=ka—t^ Covariant Derivatives: 14.70; 15.51 dl a fildh = -k*lafi-j a tfi dF a B (dF aa "^h + a^F b b^ A(f)+V(2H,f) i)h \8h I a/3 dFad -/dF\ , ^ BF\ , = 17.54 17.55 ^M3= A f ^f ) + V(2//, F) + 4#AF - 2Kb^F ali r)h \c)n ) 17.56 Normal Projection — Integral Equations: (Overbars denote values on base surface.) Aa= (a — a) (dslds) sin Aa = — ht 17.47 (ds/ds) cos Aa = (l-hk) (dslds) 2 =l-2hk+h 2 (k 2 + P) (dslds) 2 k=k-h(k- + ?) 1 7.44 (dslds)Hk- + f-) = k 2 + t- 17.48 (dslds)H=t The Position Vector: 17.64 p r = p' + hv' 17.65 p' = (sec BplBoj)k' + (BplB)jji r + pi> r p' = (sec dpldo))X r + (dpld(/))fJL r + (p + h)v r 17.66 17.67 p = p + h Chapt er 18 Radii of C urvature: 18.01 Rx=- - 1/A. = — 1/k,= = /? + A 18.02 Ri = - -Vh=- 1/k 2 = = «v + h Fundamental Forms: 18.03 g™ = (a a M); g,s = (a a0 A) 18.04 aafi= {{Rx + h) 2 cos 2 0,0, (# 2 + /i) 2 } 18.05 6«*={-(i?, + A) cos- 0,0, -(R, + h) 18.06 c a/3 ={cos- 0,0, 1} 18.07 a^={sec^0/(ft, + /i) 2 , 0, l/(fl 2 + /0 2 } 18.08 &^={-sec 2 0/(/? 1 + /*),O,-l/(/?2 + /*)} 18.09 c^={sec' J 0, 0, 1} Base Vectors: k' = «'={sec 0/(/?, + /i), 0, 0} p' = i/' = {0, 1/(5, + h), 0} 18.10 J/'- {0,0,1} \ r = u, = {(/?,+/*) COS 0, 0. 0} jU. r = » r = {0,(Ri + h),0} 18.11 ^r = {0,0,1} 18.12 < cos )u r =Kl(R) + h) 18.13 r = p. r l(R-> + h) fsec sin a sin /3 cos a sin B n ~\ /'" = { -= — , = — , cos « \ /, ■ = { (/?i + h )cos sin a sin /3, 18.14 (Ri + h) cos a sin /3, cos B} Summary of Formulas 369 Derivatives of Base Vectors: 18.15 A,, = (R- + h) sin : Xgi— — cob0 18.16 /Xn = — (fli + /») sin 4> cos ; M:i-' = — 1 18.17 v n =(Ri + h) cos 2 <£ ; v T i = Ri + h Surface Curvatures: -k=l/R 18.18 = sin- a/(/?i + /0+cos 2 a/(/? 2 + /i) 18.19 {Ri — R\ ) sin a cos a ' (Ri + h)(R-i + h) 18.20 cr, = tan sin al(R\ + h) — daldl Coda; iu Equations: 18.22 ^= (R t -R») tanc/> 18.23 ^=0 18.24 dba0 18.34 Christoffel Symbols: r 2 , = (^, + A) sine/) cos <£/(/?•> + /i) r,',=-(«, + A) tan /(/?, + A) 9 In (/L + M rw a 18.35 r?, =-(«, + /i)cos-^ r, i 3 = i/(«,+/i) n 3 =i/(R 2 +/i) Higher Derivatives of Base Vectors: 18.36 X«is = -(& + &) sin /(«■ + M 18.37 yu M :( = sin cos <£ 18.38 i'tt/83 = -c« / 8 18.39 X„S3 = /A«83=l'aa3 = 18.40 I'npy = — b t ,0Y Vatiy = 8181 ( tf , ) y cos- + 8 2 8| ( fi > ) y 18.41 + (Ri-R>) sin cos ^(5^+5^)5; = fi, sin- [ (Ri-R->) sec p = i?i cos'- (/> + sin (/> I /?2 cos =-[ #2 sin <£<#> X = R\ COS (/> COS (D 18.28 y—R\ cos sinto 2= I /?2 COS (/> (/ = — R\ cos cot — I /?] cos cosec- (f> d<\> 18.30 = — sin (}> I /?i cos $ cosec- (f) d J (Rt-R-,) sec cos + cos (/> /?•> cos d$> — — R i cot — cos R] cos cosec- r/ = — cos(/)l {R]—R o ) sec (/> 18.32 18.33 p r ={dpld)^(p + h)v r Laplacians: 18.42 A/? =-2// 18.45 Ao> = A(/) = -tan (t>l(R^+h) 2 -(2H)„p" tan c/> 1 Mj 18.46 (Ri + h)(R, + h) (R> + h) :i Surface Ceodesics: r)a A', sin a + £i cos a COt TT = l , : ()(p kz cos a+ f , sin a A" sin a — t cos a 18.49 (any surface) k cos a + ? sin a Ri cos sin a = (^, + /j ) cos sin a = constant 18.50; 18.51 (surfaces of revolution) The Spheroidal Base: (Eccentricity e; semiaxes a, b.) 18.53 e =bla = +(l-e 2 ) u - 306-962 0-69— 2 5 370 Mathematical Geodesy 18.55 RA=N=v)=a(\-e 2 sin 2 )-V 2 RA=M = p) = ae~-(l-e 2 sin- <£)~ 3 / 2 18.54 = e 2 /j?/a 2 18.56 z=e 2 Ri sin 18.57 P = a 2 /R l 18.58 dp/ d4> = — e 2 R i sin

    x — x + h cos (f> cos a>= (/?i + h) cos cos cj y=y + h cos c6 sin w = (/?i + h) cos c/> sin w z=z+h sin c/> = (e 2 Ri + h) sin 18.59 p'=-(e 2 R x sin cos (j))^ + (a 2 /R x + h)i> r 18.60 Chapter 19 Auxiliary Spherical Formulas: cos cr= sin sin <^> + cos (f> cos cos Sou 19.01 cos t = cos 4> cos <£ + sin cf> sin (/> cos 8a> 19.02 = sin a* sin a* + cos a* cos a* cos cr cos 8a> = cos a* cos a* 19.03 +sin a* sin a* cos cr sin <£> sin 8a> = — sin a* cos a* 19.04 + cos a sin a cos cr sin sin 8w = cos a* sin a* 19.05 —sin a* cos a* cos o- sin o" cos a* = cos (f> sin $ 19.06 —sin c6 cos <£ cos 8co sin cr cos a* = — sin cos $ 19.07 + cos 4> sin ^ cos Sod 19.08 sin o" sin a* = cos sin 8w 19.09 sin o- sin a* = cos sin 8w cos 4> cos a* = — sin sin cr 19.10 +cos cos cr cos a* cos 4> cos a*= sin (/> sin cr 19.11 + cos (f) cos cr cos a' cos (j) cos 8w = cos (/) cos cr 19.12 — sin c6 sin cr cos a* cos cos 8oj = cos $ cos cr 19.13 + sin c6 sin cr cos a* cot a* sin 8a; = cos (/> tan 19.14 — sin cos 8a> cot a* sin 8co = — cos <£ tan $ 19.15 + sin <£> cos 8w sin cr da* = sin a* cos cr dcf> + cos cos a* rf(8w) 19.16 —cos (/> sec <£ sin a* c/ sin a da* = sin a* sec c/> cos d 19.17 +cos c& cos a*d(8ci)) — sin a* cos crc/c6 dcr — ~ cos a* c/(/> 19.18 + cos (/> sin a* d(8oj) + cos a* d(/> Rotation Matrices: /l 19.20 = sin cos (^ \0 — cos <£ sin (/> / — sin o> cos a) N 19.21 H= -cosoj -sinw \ 1 / —sin o> cos w Q = 4>fl = I — sin (f> cos o> — sin sin o> cos (/> \cos cos a> cos (/> sin co sin $/ 19.22; 19.26 (cos 8 —cos sin 8aA - sin <£ sin 8a> cos t — sin cr cos a* I cos sin 8cj sin cr cos a* cos cr I 19.25 Base Vectors: 19.24 {K, jir,i>r}=QQ T {K,P-r,Vr} Azimuths and Zenith Distances: {sin a sin /3, cos a sin /3, cos j8} 19.27 = QQ 7 '{sin a sin /3, cos a sin y8, cos /3} Orientation Conditions: (0 sin 8(d — cos $ 8wN -sin8w -8(/> cos $ 8w 8 8w + cot /3 (sin a 8(f) — cos a cos (f) Sou) 5/3 = — cos — cos a 8 19.29 The (co, 0, iV) Components of Base Vectors: k 1 K 2 \ 3 \ / — ki sec* -tj 0' — t\ sec 4> — k 2 /A 1 jU, 2 jU. 3 yi sec * y 2 n /\j A. 2 ^-3\ S=l /U-l fJ-2 fX 3 Vl V-2 VS/ /- k t cos 4>IK ti/K sec <£ d(l/n)lda)\ \ t, cos*/K -Arx/K d(\ln)ld4> d/«) 19.32 19.33 19.34 RS r =SR T =I R ^S 7 ; S-^Rr Tensor Transformation Matrices: ^*/dcu ax/3^ ^/a/V\ /AiA-,A :i \ dylda) dy/d a y /a/V =15, B, B, = Q r S ,, te \dzlda> dz/dQ az/a/V/ \Ci C 2 C 3 / /aw/ax a*/a* aiv/a*\ /a i a 2 A 3 \ doldy d/dy dN/dy\= I B l B 2 B 3 = Q^R ^cu/az aovaz a/V/az/ Xc'C 2 ^/ 19.36 ^&>/ao> a ao>/a r -^ r 19.41 A r / r =cosj3-cos/3 A r = (cos* sin8a>)A r 19.43 +(sincrcosa*)/Li r -2sin 2 (cr/2)^ r Change in Coordinates: 19.44 {w r , 4> r , N r } = R T QQ T S{w r , <}>r, N r } {(8 r ,0r,/Vr} 19.45 =(RiQQ»'-R T ){X r ,fi Pl i/ r } /a(Sco) d(8o>) a(5co)\ M = 19.46 19.47 ax a/u, dv 3(8*) a (8*) d(8ct>) ax d/JL dv a(8/V) d(8N) d(8N) R r QQ r -R a\ a/u. a j/ R + M r = QQ r R d(8w) a (8*) a(8/V) ] a/ ' az ' a/ J = (K T QQ T — R T ){s'ma sin/3, cos a: sin /3, cos /3} 19.48 Chapter 20 Attraction Potential (Free Space): 20.01 /V=^-Cm/r 20.03 A/V=0 Force: 20.05 F s = — N s = — n v. Geopotential: 20.08 M = N-%to 2 (x 2 + y 2 ) 20.09 A/W = -2w 2 (free space) 20.15 AM = 47rCp-2w- (at density p) Equations of Motion: 20.11 ^r=^r=-V r (fixed axes) ot ot ^=-W r -2erA) sin a cos a + t]{cus- a — sin 2 a) 20.36 — yi cos a cot /3 + y 2 sin a cot (3} 20.37 = -(2mn/ 2 sin /3) (i/ rs ; r / s ) 20.38 = (2mnl 2 sin /3){? sin j3-(ln n) r /' r cos (S\ = (2mnl 2 sin (3){t sin /3 + yi cos a cos (3 20.39 —72 sin a cos /3} Chapter 21 Generalized Harmonic Functions: 21.003 A'-' ■ • •<">//,„. . . ( „) 21.004 CL r M s N* . . . //,,■, . . . on B rst . . unp'p'p' ■ ■ ■ p (n) . {p l B k „. . . (») = 21.009 Potential at Distant Points: Maxwell's Form: y x ( — \n / 1 __= y ' Ljstu . . . («) ' G *•* n\ H=0 r/stu ...(«) 21.017 21.012 /*»..- (n>=2 mjrac'x" . . . x ( 21.019 pp*p'p" . . . p [ll) dv Successive Derivatives of (1/r): (-)"(l/r) W r»/...(«)r" +1 _ 1-3-5 ... (2/1-1) + {gpqVrVsVt ■ ■ ■ V{n)} (2n-l) (2n-l)(2n-3) 21.025 (-)"(l/r) wr .s7...(»)r»-' 1 -3-5 . . . (2n-l) = VpVqVrVgVt . . . I'CO n(n-l) 2(2n-l ■ g,,qV r V x V< . . . V(n) 21.026 ra(n-l)(n-2)(n-3) 2 -4(2/1-1) (2/i-3) g "" grs "' • • ■ " (,,) ... (symmetrical form) (-)"n\ n . = — 7IT~ "«(sin0) r/ 333... CO r" + ' 21.027 r/[\\... in) r — jj^ — r„(cos

    ) 21.028 — — — — "»(cos

    A 2 21.031 2(2n-l) 2-4(2/i-l)(2n-3) . . /„(*, y, z) Potential in Spherical Harmonics: ~^ = J 2 p »"( sin ^^ c cos mw » = o m=o 21.035 + S n m sin maj}/r" + 1 c,,o=2 ?nf "^ > ''( sin 0) 21.037 C»»A __ (n-m)\ Dii . . - /-cos row (rc + m) sin mco Normalized Coefficients: C„o = C»o/(2n + l)" 2 Jin (n + m)\ 2(2n+ l)(/i-m)! 21.038 Inertia Tensors (First and Second Orders): 21.062A; 21.062B 7 s = M^ = Mpfi 21.064 I=g r J rs =2,mr 2 21.065 IoP= : I r8 {grs-VrV s ) = I r »(X. r \ s +fXrfls) Summary of Formulas 373 Iop = I~I u cos 2 (f> cos 2 a) — / 22 cos 2 sin 2 (o — I 33 sin 2 — 21 12 cos 2 $ sin cos cos sin oj 21.066 / 12 = Zjmxy /' ,! = 2jihxz 21.067 /23=2myi" 21.073 J/*'(l/r) rf = (2/-3/or)/(2r ! ) Potential at Near Points: V P =V +(V s )op s +KVst)op s p' + . ■ ■ 21.085 +^(V 8l .., ,))op s p'. ■ ■ p M +. . . -£=2 2 r"/»j? (sin £){[<:,,-] cos ina» ** n = o m = o 21.086 + [5h«] sin mxa) [Co] =2 7777TTT P " (sin #) YSnm]/ r" + ll (« + m): \ sin mw 21.087 Potential at Internal Points: V _ " — — — — = ^ 2 ^'" * sin ^ ^ nm cos /na,_, "^" m sm ma *} " « = ll m = o 21.096 C,„„ = (C, l „,),/r l " t " + r''[C„,„] f ; 21.102 Alternative Expressions: P !!'(sin <£ ) /cos maA ^»+n \ sin mw/ 21.100 ( - )" d" "1 jV— z] "'I' 2 / cos mwY _r \r + z\ \sin mcu/ P',i' (sin ) + •„ (-)" 3" | — \ e -ni( + iu>) fill+l) (n — m) ! dz" 21.103; 21.104 Isometric Latitude: e l/ '= cosh i// + sinh i// = sec + tan = tan (477 + 2) _ 1 + sin /l + sin (/)\ 1/2 cos 21.101 1/1= I sec ^ 21.107 ^0=7 {/(* + io)+g(./f-i<»)} *+*V \ + (l\ r. ( x ~ [ y 21.107A Vo : The (£, 17, 2) System: £ = x+iy=r cos <^e'"' 7) = x—iy=r cos e~ iw r 2 = tr) + z 2 21.108 ^=(r+2)/(^-n) ,/2 ^+m- ( r +z)lr) = i;l(r — z) 21.109 e*- i »=(r + z)l{ = -nl(r-z) 3f~* r drj r 21.110 f)(ft+tQ>) _ 1 dr_z dz r d£ 2r ,-(!/!+ ICU) 3i9 2r d(ft + ia*) = l . 62 r 21.111 d(iji-ia>) = 1 „,,_,„,, d£ 2r fK_____l_J_ dr? ~~2r d ( i/> — ia> ) __ 1 dz r -(ill-iui) 21.112 21.113 2-Ui-ii- df r)x dy 2— =— +i — $17 r)x dy ds 2 = d£dr } +dz 2 gvi = h #33=1; |g| = — i; g 12 = 2; g*>=l 21.114 a 2 _aj_ d^d-q + d2 2 A = 4^r^r + -^ 21.115 2 — (-e- m W- Uo) )=——(-e' d{ \r J dz\r / dz \r J 374 21 116 2 — ( - e -m() \ —— J - e -(m + lMi//+ia.) 21.117 2—(- p-m(il)-iw)\—^_ (± p -(m + l)«<-!a.) | dl)\r J dz\r e J 21.118 2— (- e -mW+iu)\ = — A [1 e -(m-lX*+M di? \r / dz \r = (re-m + 2)(re-/re+l) ^f*'" ^ e' -0 "- 11 " 21.119 2 21.12 JL p»' I) j ^n+2) Gravity: g cos cos d> _ ^, 'Si, 1 P,'," +1 (sin cfo) G _ ^ -^ r ( " + 2) n = m = X (C(h + i), m cos mw 21.136 +S(» + i),fflSinmo)) C(n+ |),o = I'i(n + l)C n i C(„ + i), l=~C„o + 2«fn-l)C|,-i S(„ + i). i=ire(re — 1)S H2 (>(«+ 1 1, m 2t«, (»i-i) + i (re — m + 1) (re — m)C„, (,„ + ,, •S(n + i), m = — 2 O/,, (w-i) -, + £ (re — to + 1) (re — m)S„, (m+1) J (m = 2, 3, . . . (re-1)) Wn+ l ), n 2"»,(»-ll ■J(n+ l), n 2 -Jh, (h-1) Lr(n + i), (n+l) = it», /, ■J(n + 1), (« + l) == 2 ■%, n 21.137 g cos (ft sin o) = ^ !£,' PgVj (sin ) n=0 /n=0 X(C(«+d, »i cos men 21.138 + S(,,+i), m sin mo)) Mathemat ica/ Geodesy G{ll + {), o=ire(re+l)S„, C(n+\ », .=ire(re- 1 )S n , S(«+D, 1 == L ;,() - -hn(n-\)C, - f C(H+1), — J-S IN 2-J ll, {111- -ii 1 + i(n- - m + 1 ) ( n - m)S,,, (JW+I) S(H+1), ,„=-hc„. (iii-n 1 -h(n - to + 1 ) ( re - m)C„ , (w+l ) J ( re* = 2, 3, . . ■ (n -D) Mn+1) , n == 'ZOii, in -ii S(„+i) ll 2^ II (»-D G(»+l ), (M + l ) = 2-J//, ii S(n + 1), (n + 1) — — 2C n , /, 21.139 g_sinj> = - t+i P;," + ,(sin(/)) G 2< 2 r (n+2) 21.140 X (C(„ + d, „, cos /na> + S(„ + i), », sin mat) C(/i+l). m = — («— TO + l)C n m 21.141 5(H + n, „, = — (n — m + l)S„ m Spherical Harmonic Coefficients in Second Differentials of the Potential: dx 1 \ G, C{„+2), = — 2(11 + 1 ) (n + 2)C„o fi(re-l)re(re+l)(re+2)C, ( 2 C(« + 2), . =-fre(n + 1 )G„, + £(re - 2) (re- 1 )n(n + l)C»s S(„+2), 1= — £re(re + l)S„i +i(re-2)(re-l)re(re+l)S H 3 t>(//+2), 2 — fC«o — 2fi(n — 1 )C,i2 +i(n— 3)(re-2)(re-l)reC,M ■S(»+2), 2 — — in(re— 1)S„2 + i(re-3)(re-2)(re-l)reS, l4 C(h+2), m = 4C a, (in -2)— 2(n — m+l) (re— to+2)C„„, +1 (re— to— l)(n — to) (re— m+l)(re— to+2) X C«, (H| + 2) S(/,+2), in= iS„, (m-2)— 2"(re— TO+l)(re— m+2)S n m +{(re— to— l)(n— to)(h— m+l)(re— to+2) 21.145 xS«,( H1+2 ) (m>2) Summary of Formulas 375 £L(JL dy 2 \ G C (H+ 2), o = -i(n + 1 ) (n + 2)C„o - {-(n - 1 )n(n + 1 ) (n + 2)C„ 2 C(„ +2 ), i = — \n(n-\- l)C„i — i(n — 2)(rc — \)n{n+ l)C li:i i S(h+2), i =-fn(R+ l)S„i — i(n — 2)(n — l)n(n+ DS„ ;i ^(,,+2), 2 = — Wm) — 2n(n — 1 )C„i —i(n — S)(n — 2)(n — l )nC llA S(„+2), ■> = — ^n(n — l)Sn2 — i(n— 3)(n — 2)(n — l)nS„A C( ll+ 2),„,= — ^C,,,( m -2)—Un— m+\)(n—m+2)C ll m— 4(n—m—l)(n—m)(n— m-\-\)(n—m+2)C „,(!!,+■> > S( ll +z),m=—4S ll ,im-i)—2{n—m+l)(n—m+2)S ll m—4(n—m—l)(n—m)(n — m^rl){n — m+2)S ll A»i+2) 21.146 (m>2) d- ~dz- C( H +2), o—{n+l)(n+ 2)C,o C(„+2), i = «(« + l)C,n S(, l+ 2). i = n(n+ 1 )S„i C( n +i),%— n(n — \)C„i S(n+2),2 = n{n— l)S„2 C(„+2),m= (n — m+ l)(n — m + 2)C, im S( H +2), , n = (n — m+ l)(n — m + 2]S, im 21.147 ( m>2 ) -^— f— f1= C,„ + 2,,o = i(n-l)n(n + l)(n + 2)S„, dxdy \ G/ C(„+2), i = — in(n + llS/n + ifn — 2)(n — \)n(n + 1 >S„ :1 S( H+ 2), i= — in(n+ DC,n — ^(n — 2)(n — \)n(n + l)C, r . i C(„ + 2), 2 = 4,,(,„->) — \-(n — m — \){n — m)(n — m + l)(n — m + 2)C„, ( W +2) 21.148 (m>2) 376 Mathematical Geodesy jj- z ( -|d : C 0(+2)> = -in(/i + 1 ) (n + 2)S„, C( H+ 2), i = — 2 + Un — m) (n — m + 1 ) (n — m + 2)C„, ( ,„ +l ) 21.149 2) a 2 C(„+2), o = — ?n(n+ l)(n + 2)C„i C(n+2), i = (n+ l)C„o — Un — l)n(n+ l)C ir2 S(«+2), i = — 2(n — l)n(n+ 1)S„2 C( ll+ 2), 2 = |nC„i — i(rc — 2)(n — l)nC„ :! S( W+ 2), 2 = inS ll i — 2(n — 2)(n — l)nS„:i CiH+2),m = i(n — Tn+2)CnAm-i)—i(n — m)(n — m+l)(n — m+2)C H ,( m +i) S(„+2), m= 2(n — m + 2)Sn, (»i-D — 22) p = a cos 2 a see' 5 y3 Chapter 22 =Q gec a /(i+ tan -' u cos -; ^)3/2 The Meridian Ellipse (fig. 26): =a cos 2 a/(l —sin 2 a sin 2 (/>) :,/2 22.03 sin /3 = sin a sin (/> 22.12 = a sec a(l — sin 2 a cos 2 u)' !/2 tan /3 = tan a sin u 22.13 dfild<$> = sin a cos u .>.) a< , 22.14 dB/du = tan a cos 2 fl cos u ZlAtV tan u = cos a tan cp ^ ^ sin u = cos a sec sin 2215 ^(ln p)/aty = 3 sin a tan a sin u cos u 22.05 = cosasin/(l-sin 2 asin 2 )' /2 22.16 d{v cos cos u = sec /3 cos 22.17 d(f sin <£)/e^> = p sec 2 a cos 22.06 = cos 0/(1- sin 2 asin 2 ) ,/2 22.18 dvld(f>= (v-p) tan 22.07 (1 — sin 2 asin 2 (/>)(l — sin 2 acos 2 u) = cos 2 a K=l/(pv) 22.08 (l-sin 2 acos 2 u) ,/2 = cosasec/3 22.19 2// = -(l/p+ 1M i' = a cos « sec = a sec £ g , = OS sin/ = 6 sin E = a(\ - e 2 )" 2 sin E = a sec a/(l + tan 2 a cos 2 0)" 2 g2 = OS cos/= (a cos E-ae) = a(cos E-e) = o/(l — sin 2 a sin 2 ) l/2 22.20 = a sec a(l — sin 2 a cos 2 u) l/2 22.10 = a 2 /(a 2 cos 2 + 6 2 sin 2 <£) 1/2 (1 + ecos/) 22.21 r=OS = a(l-ecos£)- Summary of Formulas Spheroidal Coordinates: x = (ae ) cosec a cos u cos w y= (ae) cosec a cos u sin w 22.22 z = (ae) cot a sin u 22.23 r = (ae)(cos a+icota) ,/2 (cos u — icot a)" 2 22.24 geocentric latitude = cos a tan u ds' = (a- cos'- u)du)-+ (v- cos- a)du- 22.25 + (y 2 cot- a) da- g u = l/(a 2 cos 2 u) : g" = ll(v' 2 cos- a) : 377 22.26 22.27 22.28 22.29 22.30 22.31 22.32 22.33 22.34 22.35 22.36 22.37 g M = \j(v- cot 2 a) a r = nv r 3a n 9s tan a i^ 9 (In a) da da = — cot a ; 9(ln «) = 1 ds v sec a cos /3 sin (/) — = tan a sin q> cos m 9a d In (i> cos ) 9 In (« cos u) — = = — cot a da da 9 In v 9a 9 In p 9a cot a + tan a sin 2 (/> cot a — 2 tan a + 3 tan a sin 2 $ 0«0 = (— V cos 2 $, 0, — a 2 lv) c 1(/3 =(cos 2 (/), 0, « 2 K) r ,*; t = — cot a r| 3 = — a~l(v' 2 sin a cos a) = — \/(np) rjjj = — sec a sin $ cos r,', = tan a cos 2 (f> V:L = (a- tan a)/^ 2 a 2 9 In rc :»•$ = : — = — cot a- r'a = 2# i' 2 sin a cos a rc i ( 9 In n . . . I ;{- 2 = = sin a tan a sin

    ()U rj., = — tan « If, = sec a sin c/> cos (/> _,„ 9 In n 9 In v . . . 1 & — — : = sin a tan a sin cos

    + D,,m sin mw) 22.50 Internal Potential: F x n -tt= ^ ^ /J " M/ cot »)^{"( sin «*)( [/4„ m ] cos mw n=0 »i=0 + [/?„,„ J sin mw] 22.51 @»(i cot a) = — ;'a Q\ (i cot a) = a cot a — 1 (?•_•( i cot a) = ;h'(a + 3a cot 2 a — 3 cot a) 22.52 (n + l)Q ll+ i-{2n+l)i cot a (?„ + «(>„-, = 22.53 „( sin u) [A„ m ] \_^2l'(2n+l [B,,m] ae {n + m)\ m 22.38 22.65 X ;!'(sin u) ( COS \ sin cos moj ma> 378 Spherical and Spheroidal Coefficients: (An \B„ m \ l-3-5. . . (2n+l) , m+n+u J (n + m)\ {n—m)(n—m—l) 1 1 (ae)" +1 \S„„ _ /Qn-2), m\ "- 1 \S 0l -2),m) + 2-(2n-l) (ae) {n— m)(n — m — l){n — m—2)(n — m—3) 1 2-4(2n-l)(2/i-3) (ae)"- 3 VS ( (C(n-A), m\ , | \S(,,-4), m/ | 22.59 \$nm) 1-3-5... (2/1+1 (n + iw)! //<, (n-m)!\B„ + 2n+l (n + m-2) + 2 (n-m-2) (2n+l)(2n-l) (n + m-4) + 2-4 (n-m-4) (2n + l)(2n-l)(2n-3) (n + /n-6) 2-4-6 (n — m — 6) -4(n-2), 22.60 n-2), m\ n-2), m/ /A( n -4), m\ \fi(,,-4), m/ ^(h-K), »A B(„-e), ml + . . Inertial Properties: 22.61 M = C o = -i(ae)A I*IM=(CnlM, SnIM, CxolM) 00 22.63 i»'(«)»^.l«(«)-^-*(«)*^) C 2 o = / :,:i -i(/ n +/ L " 2 ) = iJ(ae )•'(§ Aw + An) C« = /»=t(ae)»^« S 2 , = / 23 = f (ae) :, fl,., C,, = 4-(/"-/-- 2 )=-!t(ae) :! ^2 S 2 2=i/ ,2 =-|i(ae) 3 fl 22 22.64 Differential Form of the Potential: 22.66 -g=2r---""(a)nr...oo Chapter 23 Symmetrical Models 23.01 JT = <§ C wO P w (sin0) |-dJ 2 U 2 + y 2 ) Mathematical Geodesy gcos = 2) -7^7^ + 1 (sin 0)-di 2 rP](sin 0) 23.02 23.03 23.04 g sin = 2, «7^ F«+i(sin 0) n = r g sin (0 — ^)=— £ 7777 ^«(sin 0) n= 1 + d> 2 r sin cos g cos (0-0)= ]£ — — — P„(sin 0) h=o r 23.05 d) 2 r cos 2 Standard Potential: -W = GMal(ae)+GA 20 Q 2 {i cot a)/ > 2 (sin u) 23.13 + { V« 2 - Va 2 P 2 ( sin u ) } — Wo = GMatol {do sin ao) + ^ u> 2 a% 23.11 23.12 23.14 _i 23.18 23.19 G4 20 = §ia> 2 al W 20 3 cot ao — ao(l + 3 cot 2 ao) (ae) =ao sin a a = ao sin cto cosec a (2n + l)iV*oo+(2n-2)i/*2o 2 H" (2n-l)(2n + l) (ae) 1} — / > 2n- 2 (sin 0) + A o=iMI(ae) W{x 2 + y 2 ) Standard Gravity: 23.24 g» g£_ge = b a 5or 23.28 __ age cos 2 + 6g> sin 2 (a 2 cos 2 + 6 2 sin 2 0) l/2 (1+^ tan 2 a„ — ^ tan 4 a ( .+ . . .) 23.33 2&l + ZlL = 4 TrGp -2a> 2 a 23.34 £ = g e (l+fl 2 sin 2 + fi4sin 2 20 + fl 2 =-/+fg— rW+W 23.35 B A = kf 2 -iaf f=(a — b)/a; q = u) 2 a/g e ■ ■) Summary of Formulas 3 sin u cos u v cos a [GA> () Q>(i cot a) — jara-] 23.37 23.39 g =(gl + g $)m 7 v> m w. (2n + l)t/foo+ (2n-2)/,4 2 o * cos = J) M") (2n-l)(2n + l) 23.40 x (ae) 2 "- 1 — P>„_,(sin 0)-w-r cos .7 ^ ri ,„ (2n + 1Moq+ (2n-2)t/l s gsin ^2 6H (2^TT) (ae) 2 "- 1 23.41 X U / 2 „ /'zn-iCsin 0) g sin (0 - 0) = £ C(-)" + (2n + l)tAo» + (2/i-2)t/i.„ (2n-l)(2n+l) (ae)-" -1 23.42 gcos<0-0)=£ G(-)" + aj 2 rsin cos (2n + 1 Moo + (2n - 2)L4 2 ,i 23.43 (ae) 2 "-' x (2/i+D Pa«_2 (sin (/>) — d)-r cos- . ^ ^/i+nt'/^m + ^/i^UXnlae) 2 "- 1 23.44 x Curvatures: (2n + l) 1-3-5 . . . (2^-3) 2-4-6 . . . (2n-2) (For particular values of the coefficients A,B,C, and Z) given below.) ^ + c S7^^ p » + - (sin ^ + CPj l + , (sin ) + DPl + ., (sin 0)} For g# 2 : /4 = a) 2 sin 2 fl = (/i+l)(/i + 2)(cos 2 0- 2 sin 2 0) C = — (n+\) sin 20 23.49 D=lsin 2 For gki : A=a) 2 B=-i(n+l)(n + 2) C = 23.50 £> = -£ 379 For g-y 2 : ^ =io) 2 sin 20 B=-i(n+l)(n + 2) sin 20 C=-(n + l) cos 20 23.51 D = isin20 For — : /4= — oi 2 cos 2 5 = - (n + 1 ) (n + 2) (sin 2 -± cos 2 0) C = -(n + l) sin 20 23.52 D = -icos 2 23.53 T797 — — tan — — h- dW * o0 g 23.54 ^=™+l a** 7 a0 2 g 23.55 9 a , 1 d(lng) d ds S dW ' p d0 00 23.56 to__Ii(l+l) + «aL + 3S ds p 0(f) \p v I g V V2 = a(lng) pd0 Geocentric to Geodetic Coordinates: rcos 0= (f+ /i) cos 23.58 r sin 0= (f cos 2 a + h) sin Chapter 24 Laws of Refraction: 24.01 (Jl = c/v 24.03 24.04 24.05 24.06 -" J pa's S r = fllr VS=p 2 (ln ix),m' = x (In p) r n' =0 Equations of Refracted Ray: 8(p/ r 24.07 OS p, pr sin /3 = constant (spherical symmetry) 24.11 380 Arc-to-Chord Corrections: (The (a), , In p.) system.) t— (k-2 — k\ ) sin a cos a — £i(cos- a — sin- a) + (-y, cos a — y 2 sin a ) cot yQ 24.23 = t — Vrsn r v s cot (3 24.24 24.25 (ln/x) r = -^ r 24.26 x= ( hi pj ,/»' = 1)a dh 760 dh 24.59 del + 55X10- Hi 7f dh 24.60 tr~m X = _(/*- -l)sin/3 dh g c 24.61 (e = 0) (optical waves) Astronomical Refraction: / XdS= "/l{U/sin i 8o) 2 1-1/2 ll + 6 cos a> cos a sin (3 = — a sin <£ cos co — b sin (/> sin oj + c cos cos co + 6 cos <£ sin w 25.06 + csin (sec )/| =— (k>/K) sin a sin /3 + {t]/K) cos a sin /3 /•_> = (ti/K) sin a sin /3 — (k\/K) cos a sin /3 _ aq//i) /.! — — sec )/' = — /fi sin a sin /3 — 1\ cos a sin fi + yi cos /3 l 2 = — t\ sin a sin (3 — k> cos a sin /3 + -y^ cos (3 25.08 / :i = n cos fi 25.10 {sin a sin fi, cos a: sin fi, cos (3} = Q{a, b, c} a = — sin a) sin a sin (3 — sin (/> cos a> cos a sin (3 + cos (/> cos a> cos )3 b — cos Q) sin a: sin )8 — sin sin a> cos a sin /3 + cos (/> sin oj cos /3 25.12 c = cos (b cos a sin fi + sin cos w y=yo(o>, sin w 25.14 2 = zo((o, cos oj y={v + h) cos <£ sin a> z = (e-v+h) sin $=(j/+/i)sin <$> — e l v sin $ 25.15 /, ■ = { ( j/ + h ) cos (/> sin a sin /3, (p + h) cos a sin /3, cos /3} __ __ , f sin a sin (3 secc/> cos a sin fi 25.17 /'= — -^- — -. ; _,,>" cos/3 (. (y + A) (p + h) {s sin a sin /8, 5 cos a sin fi, s cos fi} = (f + A)Q{cos 4> cos oD, cos $ sin aj, sin $} -eVsin {0, 0, 1} — (f + /i)Q{cos 4> cos &>< cos s i n w ' s ' n Q{0, 0, 1} = (v + h){s'm cr sin a*, sin a cos a*, cos cr} -(v + h){0, 0, 1} — e 2 (£ sin 4> — v sin <£){0, cos (/>, sin (/>} 25.18 {s sin a sin y8, 5 cos a sin fi, s cos j8} = (|/+ /i){sin cr sin a*, sin cr cos a*, —cos cr} + {v + h){0, 0, 1} — eHi> sin — v sin (/>){0, cos , sin <£} 25.19 {*, y, z} = {x, y, z} + Q T {s sin a sin fi, 5 cos a sin fi, 5 cos (3} 25.21 25.22 tandi = y/i (i> + A)cos = (3c 2 + y 2 ) 1/2 25.23 (e L ^ + ^)sin^) = 2 Taylor Expansion Along the Line: 25.31 (F-F)=| 5 (F'+F')+-^5 2 (F"-F") Expansion of the Gravitational Potential: (N - N)/n = s cos (3 + h 2 { ~ k sin 2 /3 - \ sin (3 + 2(ln n) s q s sin (3 cos fi 25.33 +(ln n) s i^cos 2 fi} N — N = h(n cos fi + n cos (3) 25.35 (first order) 382 Mathematical Geodesy Expansion of Geodetic Heights: h — h = \s( cos /3 + cos /3 ) + T L zs 2 (k sin 2 /3 + x sin 25.39 -k sin 2 /3-x sin 0) with ^_ , sin 2 a , cos 2 a 25.38 -k= - , — +• (y + A) (p + M Astro-Geodetic Leveling: i 5 (A + A) = (l/n){/V-yV}-{A-/i} 25.43 (first order) A = (cos c6 8(d) sin a sin )3+ (8) cos a sin /3 25.44 Deflections by Torsion Balance Measurements: Ip = + k\ sin /3 cosec /3 {sin a cos a(l + cos cr) — sin a cot fi sin cr cos a*} — &2 sin )8 cosec )3 { sin a cos a ( 1 + cos cr) — cos a cot P sin a - sin a*} + 1\ sin )3 cosec /3 {(cos 2 a — sin 2 a) ( 1 + cos a - ) — cot )3 sin cr cos (a + a*)} — yi cos/3 cosec /3 {cos a(l + cos cr) — cot /3 sin cr cos a*} + 72 cos /3 cosec /8 {sin a(l + cos cr) 25.48 — cot /3 sin cr sin a*} 1 1' — sin (5 cosec (5 ( 1 + cos cr) X {{k\ — k>) sin a cos a + t\ (cos 2 a— sin 2 a) 25.50 — yi cos a cot fi + y> sin a cot /3} //• — (1 + cos a) { (ki — k-z) sin a cos a 25.51 -+- 1\ (cos 2 a — sin 2 a)} 25.49 s(I P + h)=4 sin A ( \a — k<; ) — — cos a cosec /8 cos <£> ( a>..i — a>r, ) + sin ci cosec /3 (q>A ~(;) + cos a cosec /3 cos $ (a>i — cor,) 25.52 — sin a cosec /3 (<£i — r.) Chapter 26 The Triangle in Space: 26.02 5 12 /[ 2 +s 23 / 2 r 3-5,3/r 3 = si 2 (cos «i3, — sin a 13 , 0) X {sin c*i2 sin )8i2, cos a.\2 sin /812, cos (Siz\ — — S23 (cos «i3, — sin CX13, 0) X 0Q r {sin a 2 3 sin /3 23 , cos a 2 3 sin /3 23 , cos /3 23 } 26.04 512 (cos 0:23, — sin C*23, 0) X QQ r {sin ai2 sin /3i2, cos an sin /3i 2 , cos /3i 2 } = Si3 (cos «23, —sin a 2 3, 0) x QQ r {sin a 13 sin /3i 3 , cos ai 3 sin /3i 3 , cos y8i 3 } 26.05 Variation of Position: 26.08 ds = Irdx r - l r dx r sd{l r ) = ( m s dx s — m s dx s ) m r + ( n s dx* — n s c/x s ) n r 26.10 Variation of Position in Geodetic Coordinates: /' = A.' sin a sin (B + p.'" cos a sin /3 + i> r cos /3 26.11 m'" = A.' sin a cos /3 + p r cos a cos fi — v' sin /3 n r = — A.' cos a + /x' sin a 26.13 sd($ = m s dx $ — m s dx s — s cos <£ sin a da) — s cos a c/<£ 26.15 5 sin /3 da = — hsdx" + n s efa: s + s(sin sin /3 — cos cos a cos /3)c/oj 26.16 +5 sin a cos y3 c/(/> mi m> (v + h) cos ' (p + A)' = {sin a cos j8, cos a cos )8, — sin/3} (v + A) cos (/>' (p + h) 26.20 = {—cos a, sin a, 0} fe m 1 TO2 m 3 + /i) cos 0' (p+fc)' =QQ r {sin a cos /3,cos a cos /8,— sin /?} h] n.2 n 3 (v + h) cos ' (p + h) = OQ 7 {— cos a, sin a, 0} 26.21 Summary of Formulas 383 26.22 m. ■+77Tt^+^ =1 (v + h) 2 cos- (f> (p + h)' fh\ = m\+ s cos 4> sin a h\ = ri\ + s (sin $ sin /3 — cos cos a cos /3) 26.23 Observation Equations in Geodetic Coordinates: Horizontal and Vertical Angles: ( Observed Minus Computed) Zenith Distance = — A/3 + fhidibls + m 2 d(f>ls + rh 3 dh/s — midco/s — m 2 d(f>ls — m 3 dh/s — (d(o + 8(o) cos (/> sin a— {d + 8) cos a 26.24 (Observed Minus Computed) Azimuth = — Aa — h\dU) (cosec /3)/s — n 2 d<}> (cosec /3)/s ~n 3 dh (cosec ft) Is + riidw (cosec ft)/s + n 2 d (cosec ft) Is + n 3 dh (cosec ft) Is + (d(o + 8a>)(sin — cos cos a cot ft) + (d + 8) sin a cot /3 26.25 Reverse Equations: (Observed Minus Computed) Zenith Distance = — A/3 — fh\dl s — m 3 dh/s + m t da)ls + m 2 d$\s + m 3 dhfs + (da) + 8a>) cos 4> sin a 26.28 +(rf^ + S^)cosa (Observed Minus Computed) Azimuth = — Aa + hidcb (cosec /3)/s + n 2 d (cosec ft) Is -\- n 3 dh (cosec ft) Is — nido) (cosec ft)/s — n 2 d (cosec ft)/s — n 3 dh (cosec ft) Is + (da) + 8~u>) (sin (/> — cos cos a cot /8) + {d + S(t>) sin a cot )8 26.29 Lengths: (Observed Minus Computed) Distance = ( v + h ) cos sin a sin ft(da) — d(o) + (p + h) cos a sin ft d + cos /3 — cos ft dh Observation Equations in Cartesian Coordinates — Auxiliary Vectors: {/i,/»,/ 3 } = {/"., *»,/*} 26.33 = Q'jsin a sin/3, cos a sin/3, cos/3} {mi, m> , m :i } = { m i , m 2 , m : { } 26.34 =Q 7 {sin a cos /3, cos a cos /3, —sin ft] {ri\, n 2 , n 3 }={hi, h->, h 3 } 26.35 =Q 7 {-cos a, sin a, 0} Observation Equations — Hour Angle and Dec- lination: // = (cos D cos //)/T + (cos D sin //)Z3 r + (sin D)C r 26.36 W= (sin D cos //)/* r + (sin D sin H)B r -(cos D)C r 26.39 26.40 /V r = -(sin//M r +(cos//)£'- 26.42 sdD = -M s dx s + M s dx s 26.43 (s cos D)dH = N s dx s -N s dx s Observation Equations — Hour Angle and Dec- lination— Cartesian Coordinates: x — x — s cos D cos H y — y=s cos Z) sin H 26.44 z — 2 = 5 sin D M ' = M~ r = (sin Z> cos //, sin Z) sin //.— cos D) /V r = A 7 ' = (- sin //, cos //, 0) 26.45 Observation Equations — Hour Angle and Dec- lination—Other Coordinates: (Mi , Mj, M.O = (sin D cos //, sin D sin //, — cos D) 26.48 x Q'S (Mi,M 2 ,M :i ) = (sin DcosH, sin D sin//, — cos D) 26.50 x Q'S (/V, , /V 2 , N-,) = (- sin //, cos //, 0)Q 'S 26.51 (A 7 , . N t , A 7 :! ) = (- sin // , cos H , 0)Q ' S Satellite Triangulation — Directions: Basic Photogrammetric Equations: 26 * (vr^ M fc: 384 Mathematical Geodesy (cos k sin k 0\ — sin k cos k J 1/ /l \ /- sin H c cos H r 0\ X sin D f cos Z) c I _ cos // f — sin H c \0 - cos D c sin D c / \ 1/ 26.56 «V- (x-Xo) 2 +(y-y ) 2 +f VAJ (*-Z ) 2 +(F-F ) 2 +(Z-Zo) 2 x — jcd m n (X —Xp) + m v2 (Y—Y {) ) + m v AZ — Zn) / ~m.n(X- X ) + m :i2 ( Y- Y ) + m 33 (Z - Z„) y — yo _ m-iilY — Xi) + mggQK ~ F () ) + m L »,i(Z — Z») / ~ to:, , (X - Jo) + m :! ,( F - F„) + m :t ,(Z - Z„) 26.58 26.57 inn m i ■> m i :i M= ( m 2 i m 2 2 m-23 .m 3 ) m.i2 m :i ;( Photogrammetric Equations — Star Images: /cos Z) cos // ./' sin Z) rf 2 =(x-JCo) 2 +(y-y n ) 2 +^ x — Xd _ ffii i cos D cos H+m\> cos Z) sin ZZ+mi :i sin Z) / ira.-n cos D cos H-\-m-M cos Z) sin H+m M sin Z) y — y () _ n?2i cos D cos H-\-m?> cos D sin H+m-^ sin Z) / ni: ( i cos Z) cos H+m.i> cos Z) sin H+m X i sin D 26.60 (cosZ)cos/Z\ /sin a sill /3\ cos ZJ sin A/ 1= N cos a sin /3 sinZ) / \ cos/3 / /—sin co —cos co 0\ /l \ N = l cos co —sin co sin c/> —cose/) \ 1/ \0 cos (j> sin / 26.53; 26.54 Alternative Photogrammetric Equations: X — Xo _ n ii sin a sin B_+ n v > cos a sin fl+ n i:i cos /3 / rt.u sin a sin /3 + n 3 2 cos a sin (3 + n M cos /3 y — yp _ H2i sin a sin (3-\-n>-> cos a sin /3 + ii2:i cos (3 / n : u sin a sin (3 + n :! 2 cos a sin /3 + n-.u cos /3 26.62 "ii "i2 ^13' 11 : l n-n n-n n-n n-.u n-M n.ni tan H _ mvzix - go) + W2-»(y ~ y»> + /n :> 2(/) m, ,(x — ^(,) + m2,(y — y ( i) + ni:n(/) tan D = sin ZZ X m i:iU ~ *"> + ma » ( r = & ± "^ mi 2 U — *o)+ffi22(y — y»)+ m-xAf) = cos ZZx 26.64 m VA (x — xu) + m> n (y — yo) + m :t:< (/) mn(jc — jc ( ,) + /7i2i(y — y ( )) + m :t i(/) Chapter 27 Change of Spheroid: c/co = (p + /i )d cos c/> cZa + (e/e 2 )(p + t»e 2 ) sin c/> cos c/> cZe 27.04 dh=-(alv)da + ev sin 2 cZe Change of Origin: (^ + /; ) cos c/) cZco = (sin w)dX — (cos co)cZF (p + h)d(f> — (sin c/> cos co)cZA^)+(sin c/> sin co)cZF — (cos c/>)c/Zn cZ/i = — (cos cos cotcZYo 27.06 — (cos sin co)c/F — (sin )dZ Change of Cartesian Axes: ( v + h ) cos c/> cZco = — (1)3 (v + h) cos c/> + (coi cos C0 + CO2 sin co) X (Pv + h) sin c/> (p + /i)c/c/)= (CO2 cos to — coi sin co) x (/i + fl» dh = (C02 cos co — coi sin co) 27.14 x {e 2 v sin <$> cos 0) Change of Scale and Orientation: /sin a sin /3 sin a cos (3 —cos a A— cos a sin /3 cos a cos /8 sin a \ cos (3 — sin /3 27.18 {(f + A) cos <£ c/co, (p + h)d, dh} 27.21 =sA{cZs/5, cZ/3 . -sin/8 cZa } Summary of Formulas Extension to Astronomical Coordinates: Change of Origin : 27.22 {do>, (1$, dN} = R T Q{dx, dy, dz} where l—k\ sec — 1\ sec 4> y\ sec <$\ W \ ~h ~k 2 y, ,i Change of Cartesian Axes: 27.23 {d, dN}=sR T A{ds/s, rf/8 , -sin (3 da } 27.24 Chapter 28 Equations of Motion — Inertial Axes: d 2 x d{mv x ) 385 28.001 28.002 28.003 28.004 28.005 28.006 28.007 m dt 2 dx r dt = F, - , _ d*' dx s _ dx' _ ds dx' _ . dx s dx s dt dt dt ds 8p^ 8t vi 8'-p, 8v, 8{mvl,) r - „ = m —r- = — r , 8t< 8t ot °"P' _8v r _ 8{vlr) _ r, ~8t T ~^t~ 8t ~~ ' 8 2 p r 8v, 8(vl r ) 8t 2 8t ot F,=-V, v _ F _ d_ ( dx?\ _ i dgkq dx k dx' 1 '" ' ' dt\ g ' s dt) 2 dx r dt dt aw at dV_d ,= , M2 Ji 1 r>{(^, + /i)^cos-(/)} .., . a{(ft. + / t ) 2 } ,., aF J r -, . d{(Ri + h) 2 cos- <£} ., a{(ft> + W , dh 28.009 dV d . t ..... — - — = — (r- cos -

    CD dt W d , , , , . . . ... - = — (rd>)+ r~ sin cos cit- ric/) dt dV .. ., , ... ;., = r — r cos- (p o» — r0- Equations of Motion — Moving Axes: A r = A r cos d>t-\- B, sin ai/ B,— — A, sin d)t + B, cos oif 28.010 C=C, cW, - D dfl, _ , dC, n — j——<0tf r ; — j— = — coA l : — i - = U tff dt dt 28.011 28.012 F,=-^-+2ioe, vq C»v»-fa 2 (x 2 + y 2 )r 28.013 -r,=^7+2o>e,p,C^9 of x - 26jy = F. r + oj-x = - a JF/ri.v y + 2wi = F y + w'-'y = - d W\'dy z =F, =-dW/dz 28.014 Inertial Axes — First Integrals: 28.015 d(v-) n 8v, : = lv' ~r~ dt 8t 28.016 28.020 28.021 28.027 8(Vr) = ~Vr 8t f dV ?v 2 + V= — dt + constant J a/ vr sin /3— constant Moving Axes — First Integrals: 28.028 hv 1 + W= constant The Lagrangian: 28.029 28.030 * = A I V-2 (± 2 + y 2 + z 2 )-f / (.v. y, 2. t) dt \dq r )~ 8q r 28.008 The Canonical Equations: 28.032 H* = Hx 2 + y 2 + z 2 ) + V(x. y, z, t) 3H^ = dx^ dH* = dir d.ir dt ' dx r ~ dt 28.033 306-962 0-69— 26 386 Mathematical Geodes The Kepler Ellipse: 28.035 h 2 -(Mlr=H* 28.036 vr sin ft = N drjds = cos ft 28.037 rdf/ds = sin ft r = i; cos /3 28.038 rf^v sin ft r 2 f=N 28.039 (r) 2 +(rf) 2 = v 2 = 2(»lr + H*) 28.040 /V=vVa(l-e 2 ) 28.041 H* = -\x\2a 28.042 \r a) ar 28.043 2n n — — 28.044 n = fi^a- 3 ' 2 28.045 A^=V/ia(l-e 2 ) = na 2 (l-e 2 ) 1 / 2 dr_ . „ dE _ ae(l —e 2 ) sin f df _ r^e sin/ jV — -aesint dt - {1 + ecosf y dt~ a(\-e 2 ) r 2 28.046 28.047 dE na dt r (1 — e cos E) 28.048 (E-e sin E) = n(t-t») = M e sin E 28.049 cot)8 = 28.050 cot ft (l_e2)i/2 e sin/ _ re sin/ (1 + ccos/) a(l-e 2 ) a(l-e 2 ) 28.051 r=a(l-ecosE)=,. (l + e cos/) q\ = r cos /= a(cos E — e) 28.052 qt = r sin/=a(l-e 2 )'/ 2 sin £ /u, 1 ' 2 ^ sin/ _/x 1/2 a 1/2 e sin £ 28.053 ^cos^ = al/2(1 _ e2)1/2 - - M,n ^ = 7 = P = a^l-e 2 )" 2 28.054 28.055 cos £ = cos /= cos /+ e 1+e cos/ cos E — e 1 — e cos £ _ jit 2 (l + 2e cos/+e 2 ) _ /"-(1 + e cos £) W 2 r 28.056 v cos (/+/3)=9i- v sin (f+ft)=q 2 = 28.057 rca sin/ na 2 sin £ ~(l-e 2 ) 1/2 ~ r na(e+ cos/) _ na 2 (l— e 2 ) 1 ' 2 cosi: (1-e 2 )" 2 r~~ Auxiliary Vectors: / fc = r* cos ft + t k sin /3 28.058 r* sin (3 + t k cos j8 / fc = \ fc sin a sin ft + fJ. k cos a sin ft+v k cos /3 m fc = \ fc sin a cos ft + i^ k cos a cos ft — v k sin )8 \ fc cos a + //. fe sin a t* = ffc — ^fc s j n a _|_ ^t cos a 28.059 (cos (u;+/) cos ft — sin (w+f) sin ft cos i\ cos (w+f) sin ft + sin (w+f) cos ft cos i J sin (w+f) sin i / 28.060 (— sin (w+f) cos ft — cos (w+f) sin ft cos i\ — sin (w+f) sin ft + cos (w+f) cos ft cos i cos (if+/) sin i / 28.061 28.062 SI ,..<■■ I- nft sin i I cos ft sin j cos i j 28.063 28.064 (cos (w+f+ ft) cos ft — sin (w+f+ ft) sin ft cos i\ cos (w+f + ft) sin ft + sin (w+f + ft) cos ft cos i J sin (w+/+/3) sin i / (— sin (w+/+/3) cos ft — cos (w+f+ (3) sin ft cos i s — sin (w +f+ ft) sin ft+ cos (w+f+ ft) cos ft cos i cos {w+f + ft) sin i / Summary of Formulas 387 /r k \ /cos (w+f) sin (w+f) 0\ /l \ / cos ft sin ft 0\ /i*\ f fe = -sin (w+/) cos (u>+/") cos i sin i - sin ft cos ft 0|[fi fc ) W V 1/ \0 - sin i cos t/ \ 1/ \CV 28.065 = K{i*, B k , C*} cos (w+f) cos fi — sin (w+/) sin fi cos i cos (w+/) sin fi + sin (w+f) cos fi cos i sin t sin (w+f)\ K = [ —sin (if +/) cos fi — cos (u/+/) sin fi cos i —sin (w+f) sin fi + cos (w+/) cos fi cos j sin ( cos (w+f) sin fi sin i —cos fi sin i cos i 28.066 28.067 {A k , B k , C k ) = K T {r k , t k , n k } Variation of the Elements: 28.068 {gi,92,0}=K/= {*, y, z] 28.082 F r = --„p r + R r r 3 ' 28.069 {*.**>-!M«..*.0} 28>og3 ^ =Fr/r = _^os^ + /?/r / ■ \ efr r 2 / cos/ sin/ 0\ ,, ,. , s , r > K= -sin, cos/ ,L 2 «-« 8 * ^^T^^W, \ o o 1/ rfr 28.070 = FK /=0 28.085 -j- = p r p r \r = v cos /3 28.071 {/*, rf, „*} = K,W A *. C*} Semimajor Axis; / COS ^ Sin ^ °\ 28.086 (£-^ = vR r lr = RrF 28.072 K w+/+i8 = -sin/8 cos/? OK \2aV dt \ 1/ |=2a!{ esin/(/?rrr) + «iiz-!l (/?r , r) } p k =vl k =(v cos /3, i; sin /3, 0)K{/f*, fi fc , C fr } 28.087 28.073 p*(i*, #*, C k ) = (p«A k , P k B k , p k C k ) = (i, y, z) 28.074 = ( V cos B, v sin B, 0)K Angular Momentum: 28.088 (vr sin B)n r = Nn r = e r Pip p p q {*, y,i} = K r {^ cos /3, i; sin j3, 0} 28.089 -^ n r ' + N jp = e r ™p p R q = KI =0 F T {v cos )8, u sin B, 0} <#V 28.075 =KJ =0 {t;cos(/+/3), i,sin sin a 28.078 cos (w+/) = cos $ cos (a) — ft) Eccentricity: +/) = sin ( cot (w+f) = sin sin a cot (a) — fi) J e a 1/2 (l — e 2 ) 1/2 = sin t cos (di-n) = sin i sec cos (w+f) dt p. 1 ' 2 \ sm n g) 28.080 28.093 + (cos /+ cos £)(*«/?,)} 388 de_ (l-e 2 ) [ 2 cos E(l"R q ) sin E(m"R q ) } dt v [ 1-e cosE (l-e 2 ) 1/2 J 28.094 Zenith Distance: 28.095 vr -£ =fi sin 13 (~^] + r(m^Rg) True Anomaly: df_ N , (N cos f)(R q r«) fxe N(2-cos 2 f- cos E cos/Kggg) dt r 2 + 28.096 fxe sin/ TV | (N cos f)(R q r 7V_ M 1/2 Q 1/2 (1 -e 2 yl 2 = n(l + e cos/) 2 ^B.098 r 2 a 2 (l-ecos£)2 (1-e 2 ) 3 ' 2 Eccentric Anomaly: dE na a 1 l 2 (co sf-e)(R r r r ) 28.099 28.100 (a 1 ' 2 sin E) (2 - e 2 + e cos f)(R r t r efi^d-e 2 ) 1 ' 2 na (2smf)(R r l r ) ~~ r ~~ ve(\-e 2 y' 2 (1-e 2 )" 2 cos E(R r m r ) ve Mean Anomaly: dM {(\-e 2 ) cosf-2er/a}(R r r r ) —jT=n-\ at nae 28.101 {(1 - e 2 ) 1 ' 2 sin E(2 + e cos/)} (R r t r ) = n + {a(l-e 2 ) cos/-2er}(/? r r r ) eM i/2 a i/2 28 . 102 - '"-'■'rd'w u efJL V2 a H2 1-e 2 dM dt ~~ 28.103 Mathematical Geodesy 2 sin £(l+e cos f+e 2 )(R r l r ) ev (1 -e 2 V' 2 r cos f(R r m r ) vae Inclination: 28.104 -j = jj(R q n«) cos (w+f) Right Ascension of the Ascending Node: 28.105 -j-=^{R q ni) sin (w+f)coseci Argument of Perigee: dw df v sin /3 . dil —j- + -j: — cos i —r- dt dt r dt 28.106 N r n = 3 — 7v yR+/) cot i dw_ (N cos f)(R q ri) dt + 1+- 28.107 ^e (N sinf)(R q t j cos (w+f+fS) cos fl — sin (w+/+/3) sin ft cos t (v cos i)m r — v sin t cos (w+f+(l)n r N ,,_ 2 , {— cos i(l+e cos f)r r + (e cos i sin/)/ r 28.126 + [sin i sin (w+/) +e sin t sin w]n r } 389 Argument of Perigee BF 28.127 B£ Bl r -f~- = v — — = v m r Bw Bw Bw = rF r t r 28.128 /2„l/2 p^a 1 {- (\-e 2 ) l ' 2 r r +(e sin E)t r } Relations Between Partial Derivatives: - BF r(l-e 2 )" 2 aF (ae sin /) —— = —ri J Ba a BM — (1 + e cos f) { e (l- e 2)i/2 s in£} — = (e + cos£) — ac aw; aF aw BF + (l-e 2 )" 2 (e-cos£)^ / i-n^ F aF ^ - dF sin i cot (m; + /) -tt = — —^ + cos t — — Bi Bil Bw 28.129 Derivatives of Cartesian Coordinates: Bp r p r rr r Ba a a Bp r Be (a cos f)r r + ( a + j) f sin/ Bp r -—=r sin (i^+/)n r a/ ap r _ ( ae sin f)r r a(l + e cos f)t r _p^ B~M~ (1-e 2 ) 1 ' 2 (1-e 2 ) 1 ' 2 "n a«; ap r r/ r 28.130 ttt— (r cos i)t r — r sin i cos (w+f)n Bil d -f = -^rH e sin ^)r r + (i -e 2 )" 2 r} aa 2a v 'r Bp r V f- = 71 FT {- r r sin /+ r cos E} Be r(l—e) B£^ pn r -Ft = —rr {cos ( w +J ) + e cos w) dp r _ pr r BM~ nr 2 B6 r u ll2 a 112 JL =vm r = ' ± — — {- (l-e 2 )"V+(e sin E)t r ) Bw r N Bp r _ an~ a(l-e 2 ) 28.131 '— cos i ( 1 + e cos /) r r + (e cos i sin/)f r + [sin i sin (zc+/) + e sin i sin w]n r } h 390 Mathematical Geodes (2ae sin/) ^= (1 -e 2 ) x ' 2 (l -e 2 cos 2 E) %, da dM q(l-e 2 ) dp r 2r dw {ed-e 2 ) 1 ' 2 sinE} ^-=cosE dp r _ de dp r dw + (l-e 2 )" 2 (e-cos£) dp r dM [sin i sin (w+f) + e sin i sin w] -+r 9 1 28.132 { cos (w+f) + e cos w} X -"=- — cos i - c - The Lagrange Planetary Equations: da = 2 dR dt na dM de = (l-e 2 )dR (l-e 2 ) l l 2 dR dt na 2 e dM na 2 e dw di _ cot i dR cosec i dR dt~ N ~dw N. dO, dM = 2 dR (l-e 2 ) dR dt na da na 2 e de dw_ N dR cot i dR dt /xae de N di dCt _ cosec i dR dt~ N di 28.134 dN = dR dt dw 8n r t r dR 28.135 8t N sin (w+f) di df = N_ N_dR dt r 2 p.ae de dE = na 1 f dR ae(\ + e cos f) dR dt r na 2 e\ de (I — e 2 ) da e 2 sinfdR] + (1-e 2 ) dw Curvature and Torsion of the Orbit: dv " 28.136 28.137 28.138 dt dt trl v 2 x~ F r m' 28.139 28.140 28.141 28.143 m r = m r cos y + n r sin y v 2 x~ (F r m r ) cosy + (F r n r ) siny n r — n r cosy— m r siny F r n r tany : F r m> 28.144 F r m r = v 2 x cosy= (^t sin /3) / r 2 + R r m 28.145 F r n r =v 2 x siny = R r n r v 2 xrhr = F r -(F s l s )l r = (F s m s )m r + (F s n $ )n r fx sin /3 28.146 = + R s m s )m r + (R s n s )n r 28.147 F rs n r l s = TF r m r = v 2 XT 28.148 T = (dylds)-n rs m r l s n rs m r l s = — x Sin 7 cot fi 28.149 T=(dy/ds)+x sin y cot /3 The Delaunay Variables: Canonical Equations: 28.163 dL = dt dl _ dt 28.164 H* = -R = ~^~-R dH* dl ; dH* dl ; 2a dG dt dg_ dt dH* ' dg ; dH* dG ; dH dt dh dt dH* ' dh dH* dH First Integrals of the Equations of Motion — Furthe: General Considerations: 28.166 28.167 d(N cos i) dt \v 2 + V — (dN cos i = constant dV \ dt dt = a>N cos i + constant — rR Q {t q cos i — n q sin i cos (w+f)} 28.168 = (r cos 4>)R q \i N cos i = € rst C r p s pi d(N cos i) dt 28.169 28.170 28.171 = e rs N cos i = constant 28.174 h 2 + V-6Je rs 'C r p s pt = constant Integration of the Gauss Equations: The Hamilton-Jacobi Equation: 28.192 "*(<'.f,<) + f=0 dS(q r , Otr, t) Pr = 28.180 Cn0 = — (a e ) n Jn 28.181 3ira 2 e cos ij 2 Aid = a 2 (l-e 2 ) 2 A ia = Ak? = A,i = 28.193 Q r = p 28.194 V + WS-. "" AlW= a 2 (l-e 2 ) 2 ( * l) [ idM \ 37ragy 2 ,_ , . 2 .. 28.182 Integration of the Lagrange Equations: dq r _ dS(q r , Oir, t) as dt 28.195 VS = g rs S r S s 28.196 S=W*-a,t 28.197 //* = F + |Vr* = a, 28.198 />,=«. 28.199 Q r = 8 r l t + p r P, = a, aw* da, 28.200 Q r =8';t+p r = Rnm— „ + 6 j ^ Fnmp(i) 2j G npq {e) a p = 9=-x 28.184 xs„™(w, m, n, e) Jnmpq n-m even n-m odd cos [{n — 2p)w The Vinti Potential: 28.202 (ae) 2 = -C,„ = + a 2 J, - f 1 1 - (— V ^ ( sin *> > + (-V ^ (^n *) ^nmj n- (n-2p + 9 )M + m(n-e)] _/M' ; y,p 4(sin dda) {«;,"} =P; ( "(sin (f})(C„iii cos maj + Snm sin m ;; , (sin 4>){C nm cos moj + Snm sin mw) 29.03 29.04 {u„} = C lia P n (sin(t>) I {u%}{u%}d£l = if (m, p) are different 29.05 or if (n, g) are different { «£»} {a;,"} dil= 7^— r — r; (C n mC nm + S„ m S nr , (2n+l) {n — m)\ 29.06 (m #0) (m = 0) { «„} { u„} dO = -r— — - C„ C„o J (Zn+ 1) J [Pjffsin 0) cos ma>] 2 dn= I [P^(sin ) sin mojfdD, 29.07 2tt (n + m)! (2n+l) (n-m)! 29.08 29.09 (m #0) f [P„(sin )]= dO = 477 (2n+l) r 477 - {«;;'}P„,?}p 1 (l-2k cosip + k 2 ) (1-A 2 ) lni(l-A cos )K r + (Scf))^ = r)K r + £ Jl' 29.19 29.20 i> T = v r cos k — pt r sin k Gradient of the Potential Anomaly: 29.21 Tr=W r -Ur = gVr-yV r T,^(giq)Kr + (gi + y sin k )fZ r + (g — y cos K)v r 29.22 29.23 7" r *>(gr))lr+g(g+K)&r+(g-y)Vr Gravity Disturbance: 29.24 g„ = g-y 29.25 g,, = T r v r = dTldh 29.26 g/;^-2 («+l){r»'}/r» +2 Gravity Anomaly: 29.27 g A =g P -y H 29.28 Tp^-y B i, 29.29' y^-yC <9r /d In y 29.30 29.31 ^4 = dh V aA ar 2r gA^ — + — dr r / mi- (" + D{ry} 2{r;;'} = (/i-D{ry} lo.-IJl/ _»+•> _nj-9 _»i-f> 29.32 29.33 = _ v (n-\){T>?} & A Zj r n+2 Summary of Formulas The Spherical Standard Field: — I/— 2, 3Tm h3 ~ w 2 r 2 -^ w z r 2 / , 2(sin ) n = ' 29.34 -f/o = GC o//?+id) 2 /? 2 29.35 0=GC 2 olR 3 -to*R* Poisson's Integral: «" 29.36 " /3 = 2 (2»+D-^ T / ? n(cosi/>) n=0 r 29.37 #«=2 2 {//;r}//?" +l n = m = (i 29 . 38 (J^l^^yy^iM J /! £o £o r " + ' R " +l 29.39 H P =^j^^P-H Q dn 29.40 47T 4vr r ,/!> 29.43 (r r ) P 29.44 g> = / 3 _ R(r 2 -R 2 ) \tt R(r 2 -R 2 ) 477 29.45 ^=2 2 {^!'}/ r " +l n = m = 29.46 r£=-]T 2 (" + l){f / "'}/''" + ' 29.47 g P R 2 (r 2 -R 2 ) \-nr dn gu-p o 0/ ,q i \ R 2 (r 2 -R 2 ) f dU 29.48 (*,),= — I (^)Q-p- 393 Stokes' Integral: 29.49 r s = y y 1 7 ^-r n=2 m=0 29.50 29.51 7V 29.52 N 29.53 -£;| S(R/r,il,)(g A ) Q dn -! S(Rlr, ij))(g A ) Q da \it( Deflection of the Vertical: 29.57 R ( dS ■ i wo £(£+*)=-£;] H cos «(^) Q dn Gravity and Deflection From Poisson's Integral: 3R(r 2 -R') fsin/Scosa «£= gi7 : 4tt 3R(r--R-) ( sin /3 sin a 47T /' zwn R f f 2r 3 ( r 2 -R 2 ) cos ft ] (g " )/, = 4^ 7» F < TQdn 29.58 dT\ R = _K f[2r 3 ( r-' -/?-) cos j8 dr p r 2 '' 4ttJ I/ 3 /' 29.59 *(T (i -Tr)di\ dT\ , Tp R 2 [( Tv-Tp) / :1 2960 <777j„+K=^ 712 Extension to a Spheroidal Base Surface: - T/G= 2 2 ^»" u cot a )P »" (sin " ' n = m = 6 29.61 X {A nm cos ma) + B„„, sin mw) _dT _ tana^T 1 rte t" da G tan a * " _, , = 2 2 (l ' cosec a )Q'" d cot a) v n=0 m=0 29.62 X ^;T (sin u ){/*„,„ cosmco + fi„ m sin ma>} 394 Mathematical Geodes ^=-2 2 (" +1 H^'} n = m = _ » " i tana(n — m+l)Q™ +1 (i cot a) , ^ ^ m (i cot a) " 29.63 cos t//= sin u sin u + cos u cos u cos (w — to) 29.64 (^)S(.//)){T™}dn ^=-2 2 ("-iHn m } n = m = + 2 2 sin 2 acos 2 u{r™} n = m = + 2 2 (2^ 2 Wy){r-} n = o m = _^ " i tana(n—m + i)Q^ + Ai cot a) , , £.£. QZUcota) ~ Uni 29.67 Bjerhammars Method: /? 2 (r 2 -/? 2 ) fU,) Q 29.68 (&,),>= I 4tjt J I 3 29.69 (*.)q=2 { """ } rfO [ Rn + 2 (2n+ \) J n tn Dn + 2 29 ™ -2-pMrOff}' U.4)« = 2 C » /? " 29.71 (^)/'=2 c »J 7 A" K 2 (r 2 -R 2 ) 477T rfn The Equivalent Spherical Layer: X /I cr= V V / J ;"(sin c/>) {c„„, cos ma> + s„„, sin mw} H=0 »1=0 29.72 29.73 C„,i,\ 4ttH" + - (c„ S,,J (2 \irR"+- (c, m \ 2» + l) \S„„,/ 29.74 29.75 29.76 29.77 J Ov/S = 47r/? 2 C„o = /tf=Coo , h) Coordinates: 30.01 h Q =f(w,4>) 30.02 x»=/«; *£=«£ (r=l,2) The Metric Tensor: 30.03 a a = a a +/<*/# 30.04 a = a(l + y7) = a(l + V/) The Unit Normal: 30.05 v r ={alayi*{-f u -/*, 1} 30.06 =(a/a) , / 2 {-a n /i, -a 22 /.,!} 30.07 a/a= 1 + V/=l+V/= sec 2 /8 30.08 V/=V7=tan 2 )3 30.09 V.s/= sin 2 (3 = S/fcos- j8= V/cos 2 j8 iv = X, sin a sin j8 + /I, cos a sin /3+ v r cos /3 30.10 IV = { ( ^ + /' ) cos (/) sin a: sin j8. 30.11 (p + /») cos a sin /3, cos /3} /, = — ( f + /) ) cos sin a tan /3 30.12 /»=— (p + /») cos a tan)8 The Associated Tensor: = (d/a)e«v^ 8 (ay 8 +/y/ s ) 30. 1 3 = cos 2 y8{a a ^ + i^i^fyfs} Summary of Formulas 30.14 d y6 = a y *+vVv 6 Normal Gradients: dF ds 30.15 = F r v r = cos j8(- a 11 /,/ 1 , - a 22 f 2 F 2 + F 3 ) = cos/3{f-V(F,/)} 30.16 V(F,/)=V.s(F,/)+ (BFIds)idflds) 30. 1 7 df/ds = - Vfcos /3 = - sin tan (3 as 30.18 -; sec/3{^-V. s (F,/) The Invariant V(T, /): 30.19 V(T,f)=V(T,f)=g™(W p -U p )f q V(7 , ,/) = V(r,/)=-g7j sin a tan /3 30.21 -(g£-y,)cosa tan/3 The Invariant Vs7\ /): v. s -(r,/) dS A/as 7 rfS / cos /3 J / cos /3 30-25 -J r7 '(r^F- / ) 30.26 i> r x r a = xl=f a 30.27 - b a + bap cos /3 =/ aJ 3 30.28 A/=2// cos (3- 2H + b a i3V a vi 3 . „ „,„ „ (1 — sin 2 a sin 2 6) A/=2// cos /3 + - : — -rr — J (y + /i) (1 — cos 2 a sin 2 /3) 30 - 29 + " (p + *) Deformation of the S-Surface: 30.30 (/)=¥ 30.31 (/«)=*/« (i> + £/i) sin (a) tan (j3)=— (/i) sec = /c(i^ + /i) sin a tan /3 {p + kh) cos (a) tan (/3) =- (/ 2 ) 30.32 =£(p + /i) cos a tan /3 («; = « 30.33 tan (/8)= p , rr tan ft 30.34 R + kh tan (/3) — A; tan /3 395 (l) 2 =(v+kh?-2(v~+kh)(v+kh) cos cr+^ + A/t) 2 — 2e 2 A:(t< sin — v sin <£)(A sin c/> — /i sin ) + (e 4 — 2e 2 )(P sin — t> sin $) 2 30.35 (/) sin (<5) sin (J3) = (v + kh) sin cr sin a* (/) cos (d) sin (f3)=(v + kh) sin cr cos a* — e 2 cos c6(P sin (j> — v sin 4>) (I) cos (j8) = — (v+A/i) cos a- + (p + kh) 30.36 — e 2 sin { v sin — v sin c6) (Z) 2 = 4 sm 2 \x\t(R + kh){R + kh) + k 2 (h-hf = H 1 + k(h + h), k 2 hh + ■ -\ + k 2 {h-hf R R 2 30.37 (/) sin (a) sin (J3)=(R + kh) sin i// sin a* = (R + kh) cos sin (d) — a») (/) cos (a) sin {(3)=(R + kh) sin i// cos a* = (R + kh) (— sin c6 cos c6 + cos c6 sin c6 X cos (a> — a>)) (/) cos 08) = (R+ kh)-(R + kh) cos h 30.39 30.40 =[fr,Aft)-yAFiJrfi> 30.41 - 2 „^/{r,A(I)-I^},,s LKGHfl- a /i\ l dF 2 ■/,{■ H ir ^l7)-7l7 1 2 (x 1 + y 2 ) w=v-n=v 1 +v i -ci JteGHf}-/ 1 ?* = 2-n-(F 2 p-F 1P )+27rll/» = 2tt( W P + Ctr - 2Vw ) + 277-0,, = 27r(r / .-2F 1/ .)+2 7 ra) 2 (^ + yf.) 2 „ (r ,- 2 r„> -J { r £ (])-}£}* *»-JteGH3}« SCSI „,./{,• (J)-^* Potential and Attraction of a Single Layer: At External Points: 30.52 ^=-/^f=-/f/^ 30.53 (Vr)r = + j ^f '' = + jfp '' (ln 30.53A (V r ) P = -j^~l r ^-j-~l r dn At Points on the Surface: — | M ±(l)dS 30.56 d /l\ /L {V r ) P = - j (X J s (jj\ l,dS = j ^ (l,v')l r dS 30.57 At Points on the Surface: 30.58 V P = -27T t ji l ,-j ixy s (jj 30.60 r,-- J !>-,,,) ±@) dS t/S r')/» r)m ds \/ (/S — >££&* 30.61 30.62 -2^ am ^-^i^iw 11 -j / '/. = | (fx — [Mr) ' ,., dS i^-rt-ki®'* 30.63 (M ~ /X/ ' ) {3(/,^')(/,m') -m,.|/'-}rfS / :; M~M-/' (L,) / . = -277( M ,)r + J ^f 11 {m,v')lr-v r }dS 30.64 The Equivalent Surface Layers: jHq = -Fq/(47T) r cos /3 30.67 Su m m a ry of Form u his d (1\ l,v r ds \l) ' I 2 30.68 — -= T r v r —gT) sin a sin B as ■|2 {cos B cos B + cos (a — a) sin B sin /8} 30.69 + #(£ + /<) cos a sin 0+ (#-y) cos j8 30.70 'iT 1 — = gr) sin a sin /3 + ,§:(£ + k) cos a sin /3 a In "y 30.71 + gu+— r J dA n cos B cos /3 dS = (i>+ h) (p + h) cos $ dcudcf) 30.72 =(? + A)(p + A)rfn 30.74 f /7v/n = [ wn 2777V + jr^/n 7 = j cos B + cos (a — a) tan /3 sin B L = — {g A + grj sin a tan /3 30.75 +g({ + K ) cos a tan B}(p + h)(j} + h)/l Gradient Equations: i ct-t,. 30.79 + t(T r )p- 477 [3(/'v,)/,-i/,.}dS **>- a J_ 277 J& ds Uv'lf 30.80 + ^T fi {3(/'v f Kr(^ r )p - vAv r )r] ds 397 The Equivalent Single Layer: 30.87 = 277CT/' COS /3/ cr cos Br 30.88 (gri)i> = 2iT(Tp sin a P sin /3/< J / 2 I- dS dS {#(£ + k) }/- = 277ct/> cos a/, sin /3/> 30.89 f cr cos a/' sin /3/» / 2 277(T/' COS 30.90 f [ cos g/> / j In y \ 1] (g-i)/- = 277cr/' — f (icosec|t|/)ov/n 30.91 =2t7ot-| I o-jrP„(cos«/>)dO {^;;'}/'=2t7{o--}p-i 477 2n + l {o-!!'}/ 30.92 477(» — 1 2/i + l {o-!!'}/ ^ 2« + 1 4770-,- = ^^— j-{^!!'}/< 477 J ^2 « ~ ] 30.94 16ir 2 o-/»= J S(ty)g A dn 30.93 ,(cos l//) General Index Adiabatic formula, 222 Adiabatic lapse rate, 221, 222 Allan, R. R., 299 American Ephemeris and Nautical Almanac, 257, 258 American Society of Photogrammetry, 253 Anderle, R. J., 299 Angle of refraction, 215, 226, 243 Angular momentum, 273, 276, 282, 283, 387 Angus-Leppan, P. V., 220, 222 Anomaly: eccentric, 189, 277, 284, 388 gravity, see also Gravity anomalies height, 329 mean, 277, 284, 287, 389 potential, 311, 312, 316, 317, 319, 321, 343, 344, 392, 393, 397 true, 189, 276, 281, 283, 284, 388 Area, 49-53, 357 Arsenault, J. L., 304 Associated metric tensor, see Metric tensor Astro-geodetic leveling, 233, 234, 382 Astronomical Ephemeris, 257, 258 Astronomical refraction, 223, 224, 254, 380 Atmospheric pressure, 216 Atmospheric refraction: adiabatic formula, 222 adiabatic lapse rate, 221, 222 angle of refraction, 215, 226, 243 arc-to-chord corrections, 213, 214, 380 atmospheric pressure, 216 Barrell and Sears formula, 218, 219, 225, 226 Bender-Owens proposal, 226 Cauchy dispersion formula, 218 coefficient of refraction, 215, 216, 380 curvature, 220, 221,380 Atmospheric refraction: — Continued Dalton's law of partial pressure, 217 dispersion formula, 218, 220 distance measurements, 225-226 distance measurements, electronic, 215, 216, 225, 227 Dufour's formula, 212 eikonal, 210 eikonal equation, 210, 307, 379 equations for moist air, 217 equations of state, 216-218, 380 Essen and Froome formula, 219, 220, 225 Fermat's principle, 209, 210 flat curves, 211,212 general remarks, 209 geodetic model atmosphere, 214, 215, 220, 380 geodetic model corrections, 214, 215, 380 geometrical wave front, 210 gravity, mean, 217 humidity, 221 hypsometric formula, 217, 218, 380 index of refraction, 218-220, 379, 380 International Association of Geodesy, 218, 219 isopycnics, 214 lapse rate, see also Lapse rate laws of, 209, 210, 379 mean gravity, 217 measurement of refraction, 225, 226 microwaves, 219, 220 model atmosphere, geodetic, 214, 215, 220, 380 moist air, 217 optical, see also Optical parallax correction, 224 refracted ray equation, 210, 211, 379 refraction correction, 251 refractive index, see also Index of refraction refractivity, 217, 218, 219 residual, 254, 255 satellite triangulation, 223, 224, 254, 380 399 400 Mathematical Geodes Atmospheric refraction: — Continued Smithsonian Meteorological Tables, 217, 220, 221. 222 spherically symmetrical medium, 211 temperature, 219, 221 torsion, 213, 379 vapor pressure, 217 velocity corrections, 215, 216, 380 velocity of light, 209 wave front, geometrical, 210 wavelength, short, equations, 210 wave number, 218 Attraction: external points, 337, 338, 339, 396 force of, 143, 144, 371 points on the surface, 338, 339, 340, 396 potential, 143, 144, 155, 172, 174, 315, 371, 373 S-surface, double layer, 338-340, 396 S-surface, single layer, 337, 338, 396 Azimuth: definition, 71 isozenithal projection, 99, 364 Laplace, 134 normal projection, 121, 368 jV-surface, 76 /V-systems, 133, 134, 370 observation equations, geodetic coordinates, 243, 244, 383 principal directions, 76, 360 surface vectors, 99, 110, 111, 364, 366, 368 symmetrical (o», , h) coordinates, 126 transformation, 133, 134 vector in space, 109 ((o,, h) coordinates, 121 (a>,<£, N) coordinates, 70, 71, 76, 89, 90, 363 B Baker-Nunn camera, 302 Barrell, H., 218, 219, 225 Barrell and Sears formula, 218, 219, 225, 226 Base vector: (A r , B'\ C r ) system, 70, 359 Cartesian coordinates, 70 contravariant components, 73, 74, 359 covariant components, 74, 75, 359.360 curvature parameters, 74 derivatives, 72, 73, 126, 359 derivatives, higher, 85, 86, 127, 128, 362, 369 derivatives, symmetrical (a>,,/i) coordinates, 126, 127, 128, 359, 369 matrices, 135 /V-surface, 70-75, 133, 135, 359, 370, 371 N-systems, 70-75, 85, 86, 133, 135, 359, 370, 371 relations between, 71, 72 symmetrical (co,$, h) coordinates, 126, 368 transformation between, 71, 72, 133, 135, 359 zenith distance, 71 Base vector: — Continued (\ r ,/x r ,^0 system, 71, 359 (co,,h) coordinates, 119, 366 (w,, N) coordinates, 69-75, 135, 371 Bateman, H., 175, 176, 315 BC-4 camera, 302 Bender, P. L., 225 Bender-Owens proposal, 226 Bianchi, E., 114 Binormal: definition, 21 n'\ 39 q\ 57 transformation, 57 Bjerhammar, A., 320, 323, 324, 394 Bjerhammar: the Zagrebin-Bjerhammar problem. 343 Blades' equation, 194, 196 Bomford, G., 220, 221 Bonnet, P. O., 61 Bonnet: Gauss-Bonnet theorem, 61, 358 Bonnet's theorem, 37 Born, M., 210, 218 Brand, L., 49 Brouwer, D., 302 Brovar, V. V., 345 Browne, W. E., 246 Bruns, E. H., 80. Bruns' equation, 80, 148, 313, 318, 361, 371, 392 Canonical equations: contact transformations, 299, 300, 391 Delaunay variables, 293, 390 formation of, 275, 276, 385 Hamiltonian //*, 275, 276, 293, 299, 385 Hamiltonian K*, 299 Hamilton-Jacobi equation, 300, 301, 307, 391 Heine's theorem, 193, 301 integration, 299-302, 391 Vinti potential, 301, 302, 391 von Zeipel transformation, 302 Cauchy dispersion formula, 218 Centrifugal force, 169 Centripetal force, 272 Chandler wobble, 168 Chovitz, B. H., 230 Christoffel symbols: first kind, 17, 354 ^-differentiation, 121 isozenithal differentiation, 94, 95, 363, 364 normal coordinates, 105, 107, 108, 365 second kind, 17, 354 spherical representation, 65, 66, 359 symmetrical (w, cj), h) coordinates, 127, 369 General Index 10 1 Christoffel symbols: — Continued (to, cf), h) coordinates, 120, 367 (to, , N) coordinates, 81, 82, 362 Clairaut's: equation, 128, 203, 230, 378 first-order result, 204 formula, 202, 203 Codazzi equations, see Mainardi-Codazzi equations Coefficient of refraction, 215, 216, 380 Co-geoid, 320 Conformal map projection, 60 Conformal transformations, 55-61, 357, 358 Conjugate metric tensor, see Metric tensor Contour integrals, 49-51, 357 Contravariant: components, see also Contravariant components curl of a vector, 19, 354 fundamental forms, 43, 44, 356 metric tensor, 78, 360 space metric tensor, 33, 78, 355, 360 surface metric tensor, 33, 78, 355, 360 tensor, second-order, 9 vector, three-dimensional, 3, 5 vector, two-dimensional, 3 vector, unit, 5, 353 Contravariant components: base vector, 73, 74, 359 isozenithal projection, 99, 364 metric tensor, 77, 78, 360 surface vector, 99, 364 Cook, A. H., 174 Coriolis force, 146, 272 Courant, R., 177 Covariant: components, see also Covariant components derivatives, see also Covariant derivatives differential invariants, 19 differentiation, 17-20, 34 Laplacian of a scalar F, 19, 191 Laplacian of a vector, 19, 354 Riemann-Christoffel tensor, 26, 355 tensor, second-order, 9 vector, see also Covariant vector e-system, 13 Covariant components: base vectors, 74, 75, 359, 360 isozenithal projection, 100, 364 metric tensor, 77, 78, 360 surface vector, 100, 364 Covariant derivatives: fundamental forms, 44, 356 /i-differentiation, 121 isozenithal differentiation, 100, 101, 364 isozenithal projection, 100, 101, 364 metric tensor, 94 rules, 19, 20, 34 Covariant derivatives: — Continued scalar gradient , 18, 354 spherical representation, 90, 91 surface, 33, 34, 355 surface vectors, isozenithal differentiation, 100, 101, 364 surface vectors, normal projection. 111, 112,366,368 tensors, 18, 354 unit perpendicular vector, 20, 354 vector, 18, 20, 354 (to, $, h) coordinates, 121, 122 (w, , N) coordinates, 86, 90, 91, 363 Covariant vector: generalized, 6 three-dimensional, 3, 6 two-dimensional, 3 unit, 6, 353 Curl, 19, 354 Curvature: atmospheric refraction, 220, 221, 380 binormal n r , 39 correspondence of lines, 57, 58, 357 equipotential spheroid, 207 first, 21, 29, 41, 42, 57, 58, 65, 76, 97. 119, 120, 356, 357, 358, 360, 364, 367 Gaussian, 27, 28, 36, 37. 41, 46, 60, 76, 91, 355, 356, 360 geodesic, 22, 23, 39, 46, 60, 76, 77, 90, 100, 112, 126, 358, 360, 363, 364, 366, 369 gravity anomalies, 311, 312 gravity field, 180-183, 205-207, 311, 312, 392 /i-surface, 118 intrinsic, 27, 28, 36, 37, 41, 46, 60, 76, 91 , 355, 356, 360 invariants, 41 , 96, 97, 107, 1 19, 189, 356, 363, 366, 367, 376 isozenithal projection, 100, 364 Lame tensor, 27, 56, 355, 357 lines, correspondence of, 57, 58, 357 lines of, 42 locally Cartesian systems, 26 mean, 41 meridian, 183 Meusnier's equations, 40, 356 normal, 40, 41, 42, 60, 75, 76, 89, 126, 358, 360, 363, 369 /V-surface, 75, 76, 360 orbit, 290, 291,390 orthogonal surface curves, 23, 354, 355 parameters, 74, 97, 118, 199, 207, 366, 367 parameters, differentiation of, 97, 364 principal, 21, 29, 41, 42. 57, 58, 65, 76, 97, 119, 120, 356, 357, 358, 360, 364, 367 principal, differentiation of, 97 principal radii, 125, 129, 368. 370 Ricci tensor. 27, 28, 56, 355, 357 Riemannian, 28, 29, 60, 355, 358 satellite orbit, 290, 291, 390 second, 22 specific, 27, 28, 36, 37, 41, 46, 60. 76, 91 , 355, 356, 360 306-962 0-69— 27 402 Mathematical Geodesy Curvature: — Continued spherical harmonics, 205-207 standard correction, 312 standard gravity field, 205-207, 311, 312, 392 surface curves, orthogonal, 23, 354, 355 surface, symmetrical (a>, , h) coordinates, 126, 369 surface vector, 100, 111, 364, 366, 368 symmetrical (co, , h) coordinates, 126, 369 tensor, see also Lame tensor, Ricci tensor, and Riemann-Christoffel tensor torsion, 40, 41, 356 two-dimensional, 27, 28 umbilic, 44 vector, 21, 22 velocity components of curvature correction, 216 (to, 0, h) coordinates, 125, 129, 311, 312, 392 (w, (/>, N) coordinates, 76, 89, 90, 363 Curve: binormal, 21 extrinsic properties, 39-47 flat, geometry, 211, 212 Frenet equations, 22, 354 intrinsic properties, 21-23 normal to, 22 orthogonal surface, 23, 354, 355 osculating plane, 21 principal normal, 21, 39 second curvature, 22 three-dimensional, 21, 22, 354 torsion, 22, 40, 41, 356 twisted, 21 two-dimensional, 22, 23 vector curvature, 21, 22 Curved space, 27, 45-47, 57, 356 D Dalton's law of partial pressures, 217 Darboux equations: solutions, 114, 115, 366 triply orthogonal systems, 113-115, 366 (to, (/>, h) coordinates, 118 Deflection vector, 136, 371 de Graaff-Hunter, J., 212, 221, 327, 336, 343 Delaunay variables: canonical equations, 293, 390 definition of, 291-293 H, 293, 294, 390 time derivatives, 292 de Masson d'Autume, G., 222 Derivatives with respect to the elements: argument of perigee, 288, 389 Cartesian coordinates, 288-290, 389 eccentricity, 286, 287, 388 general, 285-290, 388, 389 inclination, 287, 288, 389 mean anomaly, 287, 389 Derivatives with respect to the elements: — Continued partial derivatives, 288, 389 perigee, argument of, 288, 389 right ascension, 288, 389 semimajor axis, 286, 388 Differential invariants, 19, 354 Dispersion formula, 218, 220 Distance measurements, 225—226 Distance measurements, electronic, 215, 216, 225, 227, 256 Disturbing: force, 282, 305 potential, 281 Divergence, 19 Divergence theorem: tensor form, 52, 357 two-dimensional, 51, 123, 357 Doppler tracking system, 302, 305, 306 Douglas, B. C, 299 Dufour, H. M., 212, 224 Dufour's formula, 212 Dupin, F. P. C, 113 E Eccentric anomaly, 189, 277, 284, 388 Eccentricity, 283, 286, 287, 387, 388 Edlen, B., 218 ' Eikonal, 210 Eikonal equation, 210, 307, 379 Eisenhart, L. P., 103 Electronic distance measurements, 215, 216, 225, 227. 256 Eotvos': deflection, torsion balance, 234 double torsion balance, 151 Hungarian plains experiment, 151 torsion balance, 150 Ephemeris, 257, 258 Eremeev, V. F., 327, 331, 342 Erickson, K. E., 225 Essen, L., 219, 220 Essen and Froome formula, 219, 220, 225, 380 Eulerian free nutation, 168 Euler-Lagrange equations, 307 Euler's angles, 262 Extrinsic properties: curves, 39-47 surface, 31-37, 43-47, 60, 61, 358 e-system: Kronecker delta, 13, 14, 15, 353, 354 scalar, 14, 354 tensors, 13, 14, 353, 354 three-dimensional, 13, 14, 353, 354 General Index 403 e-system : — Continued two-dimensional, 15, 354 vectors, 14, 354 Fermat's principle, 209, 210 Fermi, E., 26 Ferrers' definition, 193 Figure of the Earth, 266, 267 Finzi equations, 55, 59, 61, 357, 358 Flare triangulation, 246, 247 Flat space, 25 Force: attraction, 143, 144, 371 centrifugal, 169 centripetal, 272 Coriolis, 146, 272 disturbing, 282, 305 gravitational, flux of, 149, 150, 372 magnetic, 185 tube of, 149 Forsyth, A. R., 37, 56, 114, 115, 274 Frenet equations: curve, 22, 354 surface, 22 three-dimensional, 22, 354 two-dimensional, 22 Froome, K. D., 219, 220 Froome: Essen and Froome formula, 219, 220, 225, 380 Fundamental forms: contravariant, 43, 44, 356 covariant derivatives, 44, 356 differentiation, 93, 94, 363 differentiation, isozenithal, 93, 94, 95, 363 differentiation, normal, 105, 106, 107, 365 first, surface, 35, 355 normal coordinates, 105, 106, 107, 118, 119, 365, 366 normal differentiation, 105, 106, 107, 365 principal curvatures, 42, 356 second, 43, 64, 358 second, isozenithal differentiation, 95 second, /V-surface, 78, 360, 361 second, of spheroids, 190 second, surface, 35, 60, 63, 64, 355, 358 symmetrical (co, (/>, h) coordinates, 125, 126, 368 third, 43, 64, 94, 358 third, /V-surface, 78, 79, 361 third, surface, 35, 43, 60, 64, 94, 355, 358 (a), <£, h) coordinates, 118, 119, 367 Gangetic Plain, India, 324 Garfinkel, B., 223 Garfinkel's theory, 223, 255 Gauss': divergence theorem, 52, 357 equations, see also Gauss characteristic equations, Gauss planetary equations, and Gauss surface equations Gauss-Bonnet theorem, 61, 358 Poisson's equation, 146, 147 spherical representation, 63—66 Gauss characteristic equations, 36, 356 Gaussian curvature, 27, 28, 36, 37, 41, 46, 60, 76, 91, 355, 356, 360 Gauss planetary equations: corrections to, 304 first-order, 282, 285, 290, 387 integration, 294-298, 302, 391 Kepler elements, 295, 296 second-order perturbations, 296—298 Gauss surface equations, 31, 35, 355 Gedeon, G. S., 299 Geocentric latitude, 157. 189, 194, 377 Geocentric longitude, 157, 194 Geodesic: curvature, 22, 23, 39, 46, 60, 76, 77, 90, 100, 112, 126, 358, 360, 363, 364, 366, 369 parallels, 59 principle, 307 space, 21, 28 three-dimensional, 21 torsion, see also Geodesic torsion triangle, 61, 358 Geodesies: family of, 59 space, 21, 28 surface, 22, 29, 46, 128, 369 three-dimensional, 21, 22 two-dimensional, 22 Geodesic torsion: balance measurements. 234-237 conformal space, 60, 358 TV-surface, 74, 75, 76, 360 surface curves, 40, 41, 356 symmetrical (w, , h) coordinates, 126, 369 (w, , N) coordinates, 74, 75, 76, 89, 363 Geodetic: heights, 233, 382 latitude, 233 longitude, 233 model atmosphere, 214, 215, 220, 380 model corrections, 214, 215, 380 Geodimeter, 216, 218, 225 Geoid: co-geoid, 320 definition, 200 regularized, 320 404 Mathematical Geodesy Geoid: — Continued spheroidal model, 200 Geometrical wave front, 210 Geopotential: Laplacian, 145, 371 Newtonian gravitational field, 144, 145, 146, 147, 371 rotation of the Earth, 169 second differential, 183 spherical coordinates, 201 spheroidal coordinates, 200, 201, 378 spheroidal harmonics, 200 standard gravitational field, 311 total time differential, 274 Goldstein, H., 165, 299 Gradient: equations, 343, 344 normal, S-surface, 330, 331, 395 potential anomaly, 312, 392 scalar, 7, 18, 70, 353, 354 surface, 31, 355 Gravimeter, 151 Gravity: anomalies, see also Gravity anomalies Newtonian gravitational field, see also Newtonian gravitational field Newtonian potential, see also Newtonian potential potential, see also Potential standard gravity field, see also Standard gravity field Gravity anomalies: attraction, see also Attraction Bjerhammar's method, 323, 324, 394 Bjerhammar, the Zagrebin-Bjerhammar problem, 343 Bruns' equation, 313, 318, 392 co-geoid, 320 curvature, 311, 312 deflection, geocentric, 319, 320, 341 deflection of the vertical, 318, 319, 393 density of a surface layer, 325 equivalent layer. Green, 341 equivalent single layer, 344-346 equivalent spherical layer, 324-326, 394 equivalent surface layer, 340, 341, 396 fundamental equation of physical geodesy, 313, 392 Gangetic Plain, India, 324 geoid, regularized, 320 gradient equations, 343, 344 gravity disturbances, 316, 319, 320, 321, 341, 393 Green-Molodenskii, 327-346, 394-397 Green's, see also Green's height anomaly, 329 integral equations, 341-344 integration, 309-346, 391-397 Model Earth, 327, 328, 329, 337 Molodenskii's integrals, 327 Pizzetti's extension of Stokes' function, 311,318,345, 392 Pizzetti's extension of Stokes' integral, 317, 321, 393, 394 Gravity anomalies: — Continued Poisson's integral, 309, 315, 316, 319, 320, 324, 325, 393, 394 Poisson-Stokes approach, 309-326, 391-394 potential anomaly, 311, 312, 316, 317, 319, 321, 343, 344, 392, 393, 397 potential, double layer, 338-340, 396 potential, single layer, 337, 338, 396 series expansions, 310, 311, 392 spherical polar coordinates, 318 spheroidal base surface, 320-322, 393 S-surface, see also S-surface standard field, 199, 313, 314, 392 Stokes', see also Stokes' surface integrals, spherical harmonics, 309, 310, 391 telluroid, 314, 328, 329, 337 terroid, 328, 329, 337 The Royal Institute of Technology, Stockholm, 323, 324 upward continuation integral, 316, 317, 323, 393, 394 Vening-Meinesz',see also Vening-Meinesz' Zagrebin-Bjerhammar problem, 343 Green-Molodenskii, gravity anomalies, 327-346, 394-397 Green's: equation, 332, 333-337, 341, 344 equivalent layer, 341 first identity, 337, 357 second identity, 337, 357 theorem, 49, 327, 328, 333-337, 395, 396 third identities, 337 H Haalck horizontal pendulum, 151 Hamiltonian: //*, 275, 276, 293, 299, 385 K*, 299 Hamilton-Jaeobi equation, 300, 301, 307, 391 Hamilton's: characteristic function, 301 principal function, 301 Hann, J. F., 221 /i-differentiation, 121-123, 367 Height anomaly, 329 Heiland, C. A., 185 Heine's theorem, 193, 301 Heiskanen, W. A., 314, 317, 319, 320, 324, 338 Helmert, F. R., 296 Hilbert, D., 177 Hiran, 258 Hirvonen, R. A., 314 Hobson, E. W., 155, 158, 174, 176, 177, 192, 193, 194, 196 Hobson's formula, 158, 159, 171, 175, 372 Hopcke, W., 216 General Index 405 Hungarian plains experiment, 151 Hypsometric formula, 217, 218, 380 I Index of refraction: Barrell and Sears formula, 218, 219, 225, 226 eikonal, 210 eikonal equation, 210, 307, 379 Essen and Froome formula, 219, 220, 225, 380 geometrical wave front, 210 laws of refraction, 209, 210 measurement, 220 microwaves, 219, 220 optical, see also Optical refractive index, 209, 379 velocity of light, 209, 379 Inertia: MacCullagh's formula, 167, 373 moment of, 165, 166, 167, 195, 373, 380 principal axes, 166 principal moments, 166 products of, 165, 373 tensor, see also Inertia tensor Inertia tensor: first-order, 164, 165, 195, 372, 378 nth-order, 156, 372 second-order, 165-168, 195, 372, 378 zero-order, 156, 195, 378 Integrals: contour, 49-51, 357 Molodenskii's, 327 Pizzetti's extension of Stokes' integral, 317, 321, 393, 394 Poisson's, see also Poisson's integral Stokes 1 , 317, 318, 320, 321, 322, 346, 393 surface, 49-53, 357 surface, spherical harmonics, 309, 310, 391 upward continuation, 316, 317, 323, 393, 394 Vening-Meinesz', 319, 320 volume, 51-53, 357 International Association of Geodesy, 204, 218, 219 International Astronomical Union, 204 International Geophysical Year 1957-58, 257 International Latitude Service, 168 International Polar Motion Service, 168 Intrinsic: curvature, 27, 28, 36, 37, 41, 46, 60, 76, 91, 355, 356, 360 derivatives of a tensor, 21 properties of a curve, 21-23 properties of a surface, 28 Invariants: curvature, 41, 96, 97, 107, 119, 189, 356, 363, 366, 367, 376 differential, 19, 354 Marussi, 205 Molodenskii's, S-surface, 331 Invariants: — Continued scalar, 7, 21 space, 45, 81,356, 361 S-surface, 331,332, 395 surface, 45, 81, 107, 356, 361, 362, 365 tensor, 10, 13 vector, 4 V(r,/),331,395 V. s (7\/),331,332, 395 Isometric latitude, 174, 175, 373 Isopycnics, 214 Isozenithal projection: azimuth, 99, 364 contravariant components, 99, 364 covariant components, 100, 364 covariant derivatives, 100, 101, 364 curvature, 100, 364 double spherical representation, 91 geodesic curvatures, 100, 364 length, 98, 99, 364 A-surface, 93 surface vector, 98-101, 364 Isozenithals: definition, 66 differentiation, 93-101, 363, 364 normal, 118 projection, see also Isozenithal projection vector, differentiation, 95, 363 (to, 4>, h) coordinates, 118 (to, , A) coordinates, 80, 85, 361 Izsak, I. G., 302, 304 Jacchia, L. G.. 304 Jeffery, G. B., 196 Jeffreys, B. S., 295, 323 Jeffreys, H., 168, 295. 323 K Kaula, W. M., 160, 262, 290, 298, 299, 302, 304, 305 Kellogg, O. D., 173 Kepler's: elements, 281,295, 296 ellipse, see also Kepler ellipse equation, 277, 386 orbit, 281 second law, 277, 386 third law, 277, 386 Kepler ellipse: angular momentum, 276 apogee, 279 argument of perigee, 279 ascending node, 278, 279 auxiliary vectors, 279-281. 386, 387 406 Mathematical Geodesy Kepler ellipse: — Continued descending node, 278 description of, 276-281 eccentric anomaly E, 189, 277 inclination, 279 line of nodes, 279 mean anomaly M , 277 mean motion, 277 orbital geometry, 189 osculating ellipse, 281 perigee, 278, 279 potential, 189 rectangular coordinates of the satellite, 277 right ascension, 279 satellite, 277, 278 true anomaly, 189, 276 unperturbed orbit, 281 variation of elements, 282 vector, auxiliary, 279-281, 386, 387 Kinetheodolite, 302 King-Hele, D. C, 304 Koch, K. R., 339, 343, 344 Koskela, P. E., 304 Kozai, Y., 298 Kreyszig, E., 32 Kronecker delta: generalized, three-dimensional, 13, 14, 353, 354 generalized, two-dimensional, 15, 16, 354 mixed tensor, 11 three dimensions, 8, 353 e-system, 13, 14, 15, 353, 354 Lagrange equations: corrections to, 304 Euler-Lagrange, 307 integration, 298, 299, 302, 391 planetary, 290, 390 resonance, 299 Lagrangian, 275, 385 Lambert, W. D., 317 Lame tensor, 27, 56, 355, 357 Laplace equation, classical geodesy, 134 Laplace equation, potential theory: Cartesian coordinates, 161 geopotential, Newtonian gravitational field, 145 potential, 183 solutions, 175 spherical harmonics, 183 spherical polar coordinates, 174, 175 surface tensor equation, 115 Laplacian: geopotential, 145, 371 of a scalar, 19, 191 of a vector, 19, 354 Laplacian: — Continued surface, 45, 356 symmetrical (w, (f>, h) coordinates, 128, 369 (o>, <£, h) coordinates, 120, 367 (a), , N) coordinates, 80, 81, 361 Lapse rate: adiabatic, 221, 222 constant, 222 humidity, 221 recent work, 222, 223 Smithsonian Meteorological Tables, 222 temperature, 218, 221 Lasers, 218, 302 Latitude: geocentric, 157, 189, 194. 377 geodetic, 233 isometric, 174, 175, 373 Mercator, 174 /V-systems, 134, 371 reduced, 187, 188 sign conventions, 70 spherical isometric, 174, 175, 373 spheroidal, 187, 188 transformation, 134 {fit, , IV) coordinates, 70, 79, 361 Legendre functions: associated, 192 coefficients, potential, spherical harmonics, 159, 170, 372, 373 in infinite series, 310, 392 second kind, 193, 377 spherical harmonics, 159, 170, 192, 193, 372, 373 spheroidal coordinates, 193 Legendre harmonics, 177, 179 Levallois, J. J., 222, 343 Leveling, astro-geodetic, 233, 234, 382 Levi-Civita, T., 12, 26, 28, 29, 42, 55 Line-crossing techniques, 258-260 Line of observation, see Observation line Lines of curvature, 42 Longitude: geocentric, 157, 194 geodetic, 233 jV-systems, 134, 371 sign conventions, 70 spheroidal, 188 transformation, 134 (co, (t>, IV) coordinates, 70. 79, 361 Lunar observations: declination, 258 equations, 258 geocentric coordinates, 257 International Geophysical Year 1957-58, 257 Markowitz' moon camera, 257 Markowitz' system, 257 origin-hour angle, 258 right ascension, 258 General Index 407 Lunar observations: — Continued spherical polar coordinates, 257 The American Ephemeris and Nautical Almanac, 257, 258 The Astronomical Ephemeris, 257, 258 U.S. Naval Observatory, 257 Lunisolar perturbations, 304, 305 M MacCullagh's formula, 167, 373 MacDonald, G. J. F., 168 Magnetic anology, 184, 185, 186 Magnetic potential, 184, 185 Magnetometer, 185 Mainardi-Codazzi equations: flat space, 35, 36, 44, 355, 356 A-surface, 123, 126, 129, 369 isozenithal derivatives of second fundamental form, 93 normal coordinates, 106, 107, 123, 126, 365 TV-surface, 82-85, 362 potential, 189 space, 126 spheroid, 189, 376 surface, 35, 36, 355 symmetrical (to, , h) coordinates, 126, 127, 369 tangential coordinates, 84, 85, 362 (a>, , h) coordinates, 106, 107, 123, 126 (a), , N) coordinates, 82-84, 362 Manual of Photogrammetry, American Society of Photo- grammetry, 253 Markowitz': moon camera, 257 system, 257 Marussi, A., 60, 86, 151,207 Marussi invariants, 205 Marussi tensor, 86, 150, 183, 205, 362 Matrix: rotation, 72, 133, 135, 359, 370 tensor transformation, 135, 136, 371 Maxwell, J. C, 154 Maxwell's: form of the potential, distance points, 155, 156, 372 theory of poles, 176-179, 185, 186 McConnell, A. J., 165 Mean: anomaly, 277, 284, 287, 388, 389 curvature, 41 gravity, 217 motion, 277 Mercator latitude, 174 Mercury, 147 Meridian: curvature, 183 ellipse, 187-189, 376 normal coordinate system, 109, 366, 368 trace, 77, 360 (&), 4>, N) coordinates, 71 Merson, R. H., 297 Metric: normal coordinates, 103, 104, 364 space, 5, 353 Metric tensor: associated, 12, 330, 355, 394 conjugate, 12, 353 contravariant, 33, 78, 353, 360 contravariant components, 77, 78, 360 covariant components, 77, 78, 360 definition. 5, 11 determinants, 13, 55, 353, 357 differentials of determinant, 19. 34, 354, 355 indices, 12, 353 normal coordinates, 103, 104, 105, 118, 119, 329, 365 product of unit orthogonal vectors, 11, 12, 353 space, surface relation, 45, 356 sphere, 63, 358 S- surface, 329, 330, 394 symmetrical (a>, , h) coordinates, 125, 126, 368 three-dimensional, 5 two-dimensional, 7, 15, 16. 354 (o), , h) coordinates, 103, 104, 105, 118, 119, 329 (o>, 4>, N) coordinates, 77, 78, 360 Meusnier, J., 40 Meusnier's equations, 40, 356 Microwaves, 219. 220 Mixed tensor, 9, 353; see also Kronecker delta Model: geodetic atmosphere, 214, 215, 220, 380 geodetic corrections, 214, 215, 380 standard gravity field, 199, 200, 378 Model Earth, 327, 328, 329, 337 Moist air equations, 217 Molodenskii's: basic integral equations, 342, 345 equivalent surface layer, 344, 345 gravity anomalies, 327, 331. 342, 344, 345 Green-Molodenskii, gravity anomalies, 327-346, 394- 397 integrals, 327 integration of gravity anomalies. 327 invariants, S-surface, 331 potential anomaly. 344 Moment of inertia, 165, 166, 167, 195, 373, 380 Monge's: S-surface equation, 329, 394 surface equations, 31, 32 Morando, B., 299 Moritz, H., 173, 209, 314, 317, 319, 320, 324, 331, 338. 343 408 Mathematical Geodesy Morrison, F. F., 174 Mueller, I. I., 151,234 Munk, W. H., 168 Musen, P., 304 N Nabauer, M., 226 Networks: adjustment, external, 261-267 adjustment, internal, 239-260 adjustment procedure, external, 265, 266 astronomical coordinates, 265, 385 Cartesian axes, change of, 262, 263, 265, 384, 385 Cartesian rotations, 267 change of Cartesian axes, 262, 263, 265, 384, 385 change of orientation, 264, 265, 384, 385 change of origin, 261, 262, 265, 384, 385 change of scale, 264, 265, 384, 385 change of spheroid, 261, 384 Euler's angles, 262 external adjustment, 261—267 Figure of the Earth, 266, 267 flare triangulation, 246, 247 general remarks, 239 geocentric coordinates, 262 geodetic coordinates, 239-242, 261, 382, 383, 384 hiran, 258 internal adjustment, 239-260 line-crossing techniques, 258-260 lunar observations, see also Lunar observations observation equations, see also Observation equations orientation, change of, 264, 265, 384, 385 origin, change of, 261, 262, 265, 384, 385 satellite triangulation, see also Satellite triangulation scale, change of, 264, 265, 384, 385 spheroid, change of, 261, 384 stellar triangulation, see also Stellar triangulation Straits of Florida, 247 triangle in space, 239, 240, 382 variation of position, Cartesian coordinates, 240, 241 \ 382 variation of position, geodetic coordinates, 241, 242, 382 (w, (j>, h) coordinates, 239-242, 261, 382, 383 Newcomb, S., 223 Newton, I., 274 Newtonian: equations of motion, 146, 269, 270, 272, 273, 274, 275, 285, 290, 307, 371, 385, 390 equipotentials in free air, 327 gravitational balance, 151 gravitational field, see also Newtonian gravitational field law of attraction, 173 potential, see also Newtonian potential system, 275 Newtonian gravitational field: attraction force, 143, 144, 371 Bruns' equation, 148, 371 Cartesian vectors A r , B r , C r , 144, 145 central field, 143-144 Coriolis force, 146 differentials, 148-150 equipotential surfaces, central field, 143 flux of force, 149, 150, 372 force of attraction, 143, 144, 371 general remarks, 147 geometry, 143, 148, 149 geopotential, 144, 145, 146, 147, 371 gravitational equation, 149 gravitational force, flux, 149, 150, 372 gravity differentials, 148, 149, 150, 371, 372 Hungarian plains experiment, 151 Laplacian, geopotential, 145, 371 laws of gravity, 145, 148, 150 Marussi tensor, 86, 150 mechanical principles, 143-146 parameters, measurement of, 150, 151 Poisson's equation, 146, 147 potential, see also Newtonian potential principle of superposition, 144 rotating Earth, 145 rotation, effect of, 144-146 satellite, equations of motion, 146, 371 superposition of fields, 144 symbols used, 144 test particle,' 143 torque, 151, 372 torsion, 151 tube of force, 149 ((o,(j),N) system, 143-151 Newtonian potential: attraction, 143, 144, 155, 174, 371. 373 basic equation, 156, 372 constants, relations between, 160-162 continuous distribution of matter, 156-157 convergent series, 154 derivatives of (1/r), successive, 157-159, 372 distant points, 155-168, 372 distribution of mass, 196, 197 distribution of matter, 156, 157 energy, 143 general, free space, 345 generalized harmonic functions, 154, 372 harmonic functions, generalized, 154, 372 Hobson's formula, 158, 159, 171, 175, 372 homogenous polynomials, tensor form, 154 inertia, moments of, 165, 166, 167, 195, 373, 380 inertia, principal axes, 166 inertia, products, 165, 373 inertia tensor, see also Inertia tensor invariance, distant points, 162-164 invariance, near points, 171 Laplace equation, see also Laplace equation, po- tential theory General Index 409 Newtonian potential: — Continued Legendre coefficients, 159, 372 Legendre functions, spherical harmonics, 159, 170, 372, 373 MacCullagh's formula, 167, 373 mass distribution, 196, 197 Maxwell, distant points, 155, 156, 372 moment of inertia, 165, 166, 167, 195, 373, 380 near points, 169-171, 373 normalized functions, 160 primitive, 175 products of inertia, 165, 373 spherical harmonics, 159, 160, 170, 171, 372, 373 spheroidal coordinates, 191 successive derivatives of (1/r), 157-159, 372 Normal: curvature, 40, 41, 42, 60, 75, 76, 89, 126, 358, 360, 363, 369 differentiation, 96 gradients, S-surface, 330, 331, 395 isozenithals, 118 principal, 21 projection, azimuth, 121, 368 projection, surface vector, 110-112, 366, 368 to a curve, 22 unit, see also Unit normal /V-surface: azimuth, 76 base surface, 104, 117 base vectors, 70-75, 133, 135, 359, 370, 371 base vectors, derivatives, 72, 73, 85, 86, 359 Christoffel symbols, 81, 82, 362 coordinate directions, 79, 80 curvature, normal, 75, 76, 360 fundamental form, second, 78, 360. 361 fundamental form, third, 78, 79, 361 Gaussian curvature, 76, 91, 360 geodesic curvature, 76, 77. 360 geodesic torsion, 74, 75, 76, 360 isozenithal projection, 93 Laplacians of the coordinates, 80, 81, 361 Mainardi-Codazzi equations, 82-85, 362 Marussi tensor, 86, 362 meridian trace, 77, 360 metric tensor, 77, 78 normal curvature, 75, 76, 360 parallel trace, 77, 360 position vector, 86—88 principal curvature, 76, 360 spherical representation, 79 /V-system: azimuth, 133, 134, 370 base vector, 70-75, 85, 86, 133, 135, 359, 370, 371 coordinates, changes in, 136, 137, 371 definition, 69 deflection vector, 136, 371 directions, transformation of, 132, 133, 370 latitude, 134, 371 /V-system: — Continued longitude, 134, 371 matrices R andS, 135, 371 matrices, tensor transformation, 135, 136, 371 orientation conditions, 134, 135, 370, 371 parallel transport of vectors, 136, 371 transformation of directions, 132, 133, 370 transformations, 131-137, 370, 371 vector, deflection, 136, 371 vector, parallel transport, 136, 371 zenith distances, 133, 134, 370 O Observation equations: angle of refraction, 243 angular equations, 244, 245 azimuth, geodetic coordinates, 243, 244, 383 Cartesian coordinates, 246, 249, 383 Cartesian rotations, 267 differential, 302-306 direction. 248, 249, 302-305 disturbing force, 305 Doppler tracking system, 302, 305, 306 drag, 304 geocentric coordinates, 250, 383 geodetic coordinates, 242-246, 250, 383 horizontal angles, 242, 243, 383 initial values, 246 length, geodetic coordinates, 245, 383 lunisolar perturbations, 304, 305 perturbation, 305 radiation pressure, 304 range, 302-305 range rate, 305, 306 reverse equation, 243, 244. 383 rotation of the Earth, 168 satellite observations, 302 satellite triangulation, 256 solar radiation pressure, 304 station correction, 243 stellar triangulation, 247-250, 383 vertical angles, 242, 243, 383 zenith distance, geodetic coordinates, 243. 383 (w. <£, h) coordinates, 242-246, 250, 383 Observation line: astro-geodetic leveling, 233, 234, 382 Cartesian coordinates, 228, 229, 381 Clairaut's equation, 230 deflection of the vertical, 234—237 direct problem, 230, 231 general equations, 227-229, 380 general remarks, 227 geodetic coordinates, 229-231, 233. 381 geodetic heights, 233, 382 gravitational potential, 231, 232, 381 latitude, geodetic. 233 longitude, geodetic, 233 plane of normal section, 227 410 Mathematical Geodesy Observation line: — Continued reverse problem, 230 Taylor expansion, 231, 381 torsion balance measurements of deflection, 234— 237, 382 (to, , h) coordinates, 229-231, 233, 381 (to, 0, N) coordinates, 228, 229 Optical: path length, 210 path length equation, 210, 379 wavelength, 215, 218, 219, 380 waves, 220, 221 Orbit, see Satellite orbits Orbital geometry, 189 Osculating plane, 21 Owens, J. C, 225 Owens: Bender-Owens proposal, 226 Palmiter, M. T., 299 Parallax correction, 255 Parallel: direction (to, $, N) coordinates, 71 normal coordinate system, 109, 366, 368 trace, geodesic curvature of, 77, 360 transport of vectors, 136, 371 vectors, 131 Pellinen, L. P., 298 Permutation symbols: three-dimensional, 13 two-dimensional, 15, 16 Perturbing potential, 281 Photogrammetric equations, 251-255, 383, 384 Pizzetti's extension of Stokes' function, 311, 318, 345, 392 Pizzetti's extension of Stokes' integral, 317, 321, 393, 394 Pizzetti's formula, 203 Planetary equations: Gauss, 282, 285, 290, 294-298, 302, 304, 387, 391 Lagrange, 290, 390 Plummer,H. C.,301 Poisson,S. D.,309 Poisson's equation, 146, 147 Poisson's integral: alternate, 325, 394 Bjerhammar's method, 324 deflection, 319, 320, 393 gravity, 319, 320, 393 integration of gravity anomalies, 309, 315-317, 393 potential anomaly, 319 Poisson-Stokes, gravity anomalies, 309-326, 391-394 Position vector: Cartesian coordinates, 5 TV-surface, 86-88 spherical representation, 63, 358 symmetrical (to, , h) coordinates, 127, 369 (to, cf), h) coordinates, 124, 368 (to, $, N) coordinates, 86-88, 362 Potential: analytic continuation, 172, 173 anomaly, 311, 312, 316, 317, 319, 321, 343, 344 392, 393, 397 attraction, 143, 144, 155, 172, 174, 371, 373 attraction potential ///>, 315 Blades' equation, 194, 196 Cartesian differentials, 180-183 centrifugal force, 169 Chandler wobble, 168 convergence, 194 curvatures of the field, 180-183 differential form, 196, 378 differentials, 180-183, 374-376 disturbing, 281 double layer S-surface, 338-340, 396 energy, 143 Eulerian free nutation, 168 external, 174-176, 192, 373, 377 external points, 337, 338, 339, 396 external (£, tj, z) coordinates, 175, 176, 373 geopotential, see also Geopotential gravitational, 176, 185, 231, 232, 381 gravity representation, 179, 180, 374 harmonic functions, generalized, 153, 154, 372 Hobson's formula, 175 inertia tensor, see also Inertia tensor internal, 192, 193, 377 internal points, 173, 174, 196, 373 International Latitude Service, 168 International Polar Motion Service, 168 Kepler ellipse, 189 Laplace equation, 183 Legendre functions, spheroidal coordinates, 193 Legendre harmonics, gravity representation, 179 Legendre harmonics, spherical harmonics, 177 magnetic, 184, 185 magnetic analogy, 184, 185, 186 Mainardi-Codazzi equations, 189 Marussi tensor, 183 mass distribution, 193, 194, 196, 377 Maxwell's form of the potential, 155, 156, 372 Maxwell's theory of poles, 176-179, 185, 186 meridian ellipse, 187-189, 376 near points, 196 Newtonian, see also Newtonian potential perturbing, 281 rotation of the Earth, 168, 169 satellite triangulation, 168 second differentials, spherical harmonics, 181-183 single layer S-surface, 337, 338, 396 General Index II Potential: — Continued spherical coefficients, 194-196, 201, 378 spherical harmonics, 153-185, 201, 202, 372-376, 378 spheroidal coefficients, 194-196, 201, 378 spheroidal coordinates, 189-193, 201, 377 spheroidal harmonics, 187-197, 200-202, 376-378 spheroid of convergence, 194 S-surface, double layer, 338-340, 396 S-surface, single layer, 337, 338, 396 surface points, 338, 339, 340, 396 symbol convention, 199 total, 169 Vinti potential, 301, 302, 391 (£, 7), z) coordinates, 175, 176 Priester, W.,304 Principal: curvature, 21, 29, 41, 42, 57, 58, 65, 76, 97, 119, 120, 356, 357, 358, 360, 364, 367 directions, 29, 42, 64, 65, 109, 120, 126, 358, 366, 368 moments, inertia, 166 normal, 21 R Radiation pressure, solar, 304 Rainsford, H. F., 218 Range measurements, 302 Range, observation equations, 302-305 Range-rate measurements, 302 Range rate, observation equations, 305, 306 Refraction, atmospheric, see Atmospheric refraction Refractive index, see Index of refraction Reit, B. G., 323 Ricci tensor, 27, 28, 56, 355, 357 Riemann-Christoffel tensor: covariant form, 26, 355 definition, 25, 26, 355 special forms, 26, 27 two-dimensional, 27, 28 Riemannian: curvature, 28, 29, 60, 355, 358 geometry, three-dimensional, 307 space, 25 Roemer, M., 304 Rotation matrix, 72, 133, 135, 359, 370 Rotation of the Earth, 168 Routh, E. J., 168 Royal Institute of Technology, Stockholm, 323, 324 Saastamoinen, J. J., 216, 217 Satellite, equations of motion: first integrals, 293, 294, 390 inertial axes, 269-271, 385 moving axes, 271, 272, 385 Newtonian, 146, 269, 270, 272, 273, 274, 275, 285, 290,307,371,385,390 Satellite geodesy: action, least, 307, 308 angular momentum, 276 angular momentum vector, 273 Baker-Nunn camera, 302 BC-4 camera, 302 canonical equations, see also Canonical equations centripetal force, 272 Coriolis force, 272 Delaunay variables, see also Delaunay variables derivatives with respect to the elements, see also Derivatives with respect to the elements disturbing force, 282, 305 disturbing potential, 281 Doppler tracking system, 302, 305, 306 dynamic, 269-308 eikonal equation, 307 energy integral, 294, 390 energy, law of conservation, 294 equations of motion, see also Satellite, equations of motion Euler-Lagrange equations, 307 Gauss planetary equations, 282, 285, 290, 294-298, 302, 304, 387, 391 geodesic principle, 307 geopotential, 274 Hamiltonian #*, 275, 276, 293, 299, 385 Hamiltonian K*, 299 Hamilton-Jacobi equation, 300, 301, 307, 391 inertial axes, first integrals, 272-274, 385 Kepler elements. 281, 295, 296 Kepler ellipse, see also Kepler ellipse Kepler orbit, 281 Lagrange equations, 290, 298, 299, 302, 304, 307, 390, 391 Lagrangian, 275, 385 least action concept, 307, 308 moving axes, first integrals, 274, 275. 385 Newtonian equations of motion, 146, 269, 270, 272, 273, 274, 275, 285, 290, 307, 371. 385, 390 Newtonian system, 278 observation equations, see also Observation equa- tions orbit, see also Satellite orbits perturbed orbits, 281, 282 perturbing potential, 281 planetary equations, Gauss, 282, 285, 290, 294-298, 302, 304, 387, 391 planetary equations, Lagrange, 290, 390 principle of least action, 307, 308 range measurements, 302 range-rate measurements, 302 true anomaly, 276, 281, 283, 284 412 Mathematical Geodesy Satellite geodesy: — Continued variational method, 306, 391 variation of elements, see also Variation of the elements velocity vector, 270, 385 zenith distance, 283 Satellite, Kepler ellipse, 277, 278 Satellite orbits: curvature, 290, 291,390 Kepler, 281 perturbed, 281,282 torsion, 290, 291,390 Satellite triangulation: American Society of Photogrammetry, 253 astronomical refraction, 223, 224, 254, 380 atmospheric refraction, residual, 254, 255 camera calibration, 251, 253, 254 coordinate system, choice, 251, 252 declination, 255 definition, 250 direction, 250, 251 direction to satellite, 255 distances, 256, 257 electronic distance measurements, 256 Garfinkel's theory, 255 Manual of Photogrammetry, American Society of Photogrammetry, 253 net adjustment, 255, 256, 257 observation equations, see also Observation equa- tions parallax correction, 255 photogrammetric equations, 251-255, 383, 384 potential, 168 radiation pressure, 304 refraction correction, 251 rotation of the Earth, 168 SECOR, 256 star calibration, 252 swing, 252 U.S. Coast and Geodetic Survey, 255 Scalar: general, 285 gradient of AT, 7, 70,353 gradient of , 18, 354 gradient, surface, 31, 355 invariant, 7, 21 Laplacian, 19, 191 product, 4. 7, 14, 353, 354 velocity, 270 e-system, 14, 354 Schild, A., 26, 51 Schmid, H. H., 224, 255 Schols, C. M., 60 Sears, J. E., 218, 219, 225 Sears: the Barrell and Sears formula, 218, 219, 225, 226 SECOR, 256 Sign conventions: latitude, 70 Sign conventions: — Continued longitude, 70 (to, , N) coordinates, 69, 70 Smithsonian Meteorological Tables, 217, 220, 221, 222 Solar radiation pressure, 304 Somigliana's equation, 203, 204 Somigliana's formula, 202, 378 Souslow, C, 60 Space: curved, 27 flat, 25 Riemannian, 25 Space metric, (to, <$>, h) coordinates, 118. 366 Specific curvature, 27, 28, 36, 37, 41, 46, 60, 76, 91, 355, 356, 360 Spherical excess, 61, 358 Spherical, isometric latitude, 174, 175, 373 Spirit levels, 245, 246 Springer, C. E., 49, 51 S-surfaee: associated metric tensor, 330, 394 attraction, double layer, 338-340, 396 attraction, single layer, 337, 338, 396 basic integral equations, 341-344, 345, 346, 396, 397 deformation, 332, 333, 395 diagram, 328, 341 double layer, 338-340, 396 equivalent single layer, 344-346, 397 equivalent surface layers, 340, 341, 396 geodetic coordinates, 341-344, 396, 397 gradients, normal, 330, 331, 395 Green's, see also Green's integral equations, basic, 341-344, 345, 346, 396, 397 invariant V(T,f), 331, 395 invariant V.v(7\/),331, 332, 395 metric tensor, 329, 330, 394 Model Earth, 337 Molodenskii's invariants, 331 Monge's equation, 329, 394 normal gradients, 330, 331, 395 potential, double layer, 338-340, 396 potential, single layer, 337, 338, 396 single layer, 337, 338, 396 Stokes', see also Stokes' unit normal, 328, 329, 330, 341, 394 (a>, . h) coordinates, 329-333, 394 Standard gravity field: anomalies, see also Gravity anomalies Bruns' equation, 313, 392 Clairaut's, see also Clairaut's convention, symbol, 199 curvature, standard correction, 312 curvatures, 205-207, 311, 312, 392 deflection, 199,311,312,392 disturbances, 199, 312, 313, 317. 318, 319. 323, 392, 393 General Index 413 Standard gravity field: — Continued equipotential spheroid, 202-204, 207 flattening of the spheroid, 203 free air correction, 206 general remarks, 203, 204 geocentric coordinates, 207, 379 geocentric deflections, 319 geodetic coordinates, 207, 379 geodetic system, 311 geoid, 200 geopotential, 311 gradient of potential anomaly, 312, 392 gravitational flattening, 203 height correction, 206 International Association of Geodesy, 204 International Astronomical Union, 204 Marussi invariants, 205 Marussi tensor, 205 mean gravity, 217 model, spheroidal, 200 models, field, 199 models, symmetrical, 199-200, 378 Pizzetti's formula, 203 potential anomaly, 311, 312, 316, 317, 319, 321, 343, 344,392,393,397 potential, see also Potential Somigliana's equation, 203, 204 Somigliana's formula, 202, 378 space, 204, 379 spherical harmonics, 204, 205, 379 spherical standard field, 314, 393 standard curvature correction, 312 symbol correction, 199 (to, , h) coordinates, 200, 311, 312 Star calibration, 252 Stellar triangulation: declination, 247 observation equations, Cartesian coordinates, 247- 249,383 observation equations, geodetic coordinates, 250, 383 origindiour angle, 247, 249, 383 time correction, 249 Sterne, T. E., 304 Stokes, G. G., 309, 313, 320, 328 Stokes': Bruns' equation, 318 equation, 313, 343, 392 function, 31 1 , 317, 318, 345, 392 integral, 317, 318, 320, 321, 322, 346, 393 Pizzetti's extension of Stokes' function, 311, 318, 345, 392 Pizzetti's extension of Stokes' integral, 317, 321, 393, 394 theorem, 50, 122,357 Strain, 29 Straits of Florida, 247 Surface: base coordinate, 328 Surface: — Continued conformal space, 60,61, 358 covariant derivatives, 33, 34, 355 curvature, see also Curvature curves, extrinsic properties, 39-47 curves, torsion, 40, 41, 356 equations, forms of, 31,32 equations, Gauss' form, 31 , 35, 355 equations, Monge's form, 31, 32 equations, third functional form, 32 extrinsic properties, 31-37, 43-47, 60, 61, 358 family of, 32,66 Frenet equations, 22 fundamental form, first, 35, 355 fundamental form, second, 35, 60, 63, 64, 355, 358 fundamental form, third, 35, 43, 60, 64, 94, 355, 358 Gauss equations, 31, 35, 355 Gaussian characteristic equation, 36, 356 Gaussian curvature, 27, 28, 36, 37, 355, 356 geodesic, 22, 29, 46, 128, 369 geodesic torsion, see also Geodesic torsion gradient, 31, 355 integrals, 49-53, 357 integrals, spherical harmonics, 309, 310, 391 intrinsic properties, 28 invariants, 45, 81 , 107, 356, 361 , 362, 365 Laplacian, 45, 356 layer, density, 325 Mainardi-Codazzi equations, 35, 36, 355 metric tensor, see also Metric tensor minimal, 65 Monge's surface equation, 31, 32 normal curvature, see also Curvature normals, 58, 59, 358 /V-surfaee, see also /V-surface orthogonal, unit vector, 28, 355 projection, 98-101 Riemannian curvature, 28, 29, 60, 355, 358 space relation, tensor, 45, 356 spherical, equations of, 31 S-surface, see also S-surface tensor derivative, unit normal, 34 tensors, 34, 45, 115, 191,356 transformation of normals, 58, 59, 358 transformations, 59, 60, 358 vector, see also Surface vector Weingarten equations, 35, 43, 66, 104, 355, 356 Surface vector: azimuth, 99, 110, 111, 364, 366, 368 components, normal coordinate system, 104 components, normal projection, 110, 366, 368 contravariant components, 99, 364 covariant components, 100, 364 covariant derivatives, isozenithal differentiation. 100, 101,364 covariant derivatives, normal projection, 111, 112, 366, 368 curvatures, 100, 1 1 1 , 364, 366, 368 curvatures, normal projection, 111, 366 414 Mathematical Geodesy Surface vector: — Continued differentiation, space, 108, 109 general, 32, 33, 355 geodesic curvature, normal projection, 112, 366 isozenithal projection, 98-101, 364 length, 98, 99, 110, 364, 366, 368 normal coordinates, 104 normal projection, 110-112 orthogonal, 15, 354 space, differentiation, 108, 109, 366, 367 spherical representation, 64, 358 unit normal, 7, 33, 355 unit normal to coordinate surface, 328, 341 Sutton, O. G., 222 Swing, 252 Synge,J. L.,26,51 Taucer, G., 60 Telluroid, 314, 328, 329, 337 Tellurometer, 219 Tengstrom, E., 226 Tensor: absolute, 13 addition, 9, 353 antisymmetric, 10 associated, see a/50 Metric tensor character, 11, 12 conjugate, see also Metric tensor contracted, 10, 353 contravariant, second-order, 9 covariant derivative, 18, 354 covariant, second-order, 9 covariant, e-system, 13 curvature, see also Curvature determinant, 13, 16,353,354 divergence theorem, 52, 357 dummy index, 10 equations, 10, 25, 26, 356 general rules, 9, 10 indices, 12,353 inertia tensor, see also Inertia tensor intrinsic derivative, 21 invariant, 10, 13 Kronecker delta, see also Kronecker delta Lame, 27, 56, 355, 357 Laplace surface tensor equation, 115 Marussi, 86, 150, 183, 205, 362 metric, see also Metric tensor mixed, second-order, 9, 353 multiplication, 9, 10, 353 order, 9 relative, 13 Ricci,27,28,56,355,357 Riemann-Christoffel, 25, 26, 27, 28, 355 skew-symmetric, 10 space, surface relation, 45, 356 Tensor: — Continued surface, derivative, 34 surface, equation, 115 surface, space relation, 45, 356 surface, spheroidal coordinates, 191 symmetric, 10 transformation, 9, 56, 57, 353, 357 transformation matrices, 135, 136,371 e-system, 13, 14, 353, 354 Terroid, 328, 329, 337 Test particle, 143 The American Ephemeris and Nautical Almanac, 257, 258 The Astronomical Ephemeris, 257, 258 Theodolite, 227 The Royal Institute of Technology, Stockholm, 323, 324 Thompson, E. H., 244 Thompson, M.C., Jr., 225 Torque, 151,372 Torsion: alternate expression, 213, 380 atmospheric refraction, 213, 379 balance, 150, 151 balance measurements, 185,234—237 correspondence of lines, 57, 58, 357 curvature, 40, 41 , 356 curve, 22, 40, 41, 356 deflection, 234-237, 382 EotvoV double torsion balance, 151 Eitvos' torsion balance, 150 geodesic torsion, see also Geodesic torsion gravimeters, 151 Hungarian plains experiment, 151 Newtonian gravitational field, 151 satellite orbit, 290, 291, 390 surface curves, 40, 41 , 356 Transformation: azimuth, 133, 134 base vectors, 71, 72, 133, 135, 359 canonical equations, 299, 300, 391 conformal space, 55-61, 357, 358 directions, 132,133,370 latitude, 134 longitude, 134 /V-systems, 131-137, 370, 371 space, 55-61, 357, 358 surface normals, 58, 59. 358 surfaces, 59, 60, 358 tensors, 9, 56, 57, 135, 136, 353, 357. 371 vectors, 7, 8, 353 von Zeipel, 302 zenith distance, 133, 134 Triangulation: flare, 246, 247 satellite, see also Satellite triangulation stellar, see also Stellar triangulation True anomaly, 189, 276, 281, 283, 284, 388 Tube of force. 149 General Index 415 U Umbilic, 44 U.S. Coast and Geodetic Survey, 255 U.S. Naval Observatory, 257 Unit normal: coordinate surface, 328, 341 normal coordinates, 104, 105, 106, 365 space derivatives, 106, 365 S-surface, 328, 329, 330, 341, 394 surface tensor derivative, 34 vector, 33, 355 v r , 7 (o>, (/>, h) coordinates, 329 Unit vector: contravariant, 5, 353 covariant, 6, 353 orthogonal surface, 28, 355 perpendicular, 20, 354 symmetrical (w, , h) coordinates, 126 (a>, 4>, N) coordinates, 75, 80 Upward continuation integral, 316, 317, 323, 393, 394 Vaisala, Y., 247 Variation of the elements: angular momentum, 282, 283, 387 argument of perigee, 285, 388 disturbing force, 282 eccentric anomaly, 284, 388 eccentricity, 283, 387 inclination, 284, 388 mean anomaly, 284, 388 perigee, argument of, 285, 388 right ascension, 284, 388 semimajor axis, 282, 387 true anomaly, 283, 284, 388 zenith distance, 283, 388 Vector: angular momentum, 273 azimuth in space, 109 base, see also Base vector Cartesian, 3 contravariant, 3, 5 contravariant, unit, 5, 353 convention of indices, 4 covariant, 3, 6, 353 covariant derivatives, 18, 20, 354 covariant, unit, 6, 353 curl, 19, 354 curvature, 21, 22 curvilinear coordinates, 4^7 deflection, 136,371 differentiation, space, 108, 109 dimension, 9 divergence, 19, 354 geodesic curvature, 22, 23, 39, 46, 60, 76, 77, 90, 100, 112, 126, 358, 360, 363, 364, 366, 369 Vector: — Continued indices, convention, 4 invariant, 4 isozenithal differentiation, 95, 363 Kepler ellipse, 279-281, 386, 387 Kronecker delta, 8, 353 Laplacian, 19, 354 line element, 5, 353 magnitude, 6, 353 nonunit, 6 orthogonal, unit, surface, 28, 355 parallel, 131 parallel transport, 136, 371 position, 5, 63,86-88, 124, 127, 358, 362, 368, 369 products, 14, 354 scalar, see also Scalar space, differentiation, 108, 109 summation convention, 4 surface, see also Surface vector tangent, 39 transformation, 7, 8, 353 unit, see also Unit vector velocity, 270, 385 e-system, 14, 354 Velocity: light, 209, 379 scalar, 270 vector, 270, 385 Vening-Meinesz': equations, 318 function, 318 integrals, 319, 320 Vinti, J. P., 301, 302 Vinti potential, 301,302, 391 Volland, H., 304 Volume, 49-53 Volume integrals, 51-53, 357 von Zeipel transformation, 302 W Wagner, C. A., 299 Walters, L. G., 304 Watson, G. N., 194, 196, 301 Wave front, geometrical, 210 Wave number, 218 Wayman, P. A., 270 Weingarten equations, 35, 43, 66, 104, 355, 356 Whittaker, E. T., 194, 196, 301 Wolf, E., 210, 218 Wood, L. E., 225 Yionoulis, S. M., 299 Yurkina, M. I., 327, 331, 342 416 Mathematical Geodesy Zagrebin, D. W., 320 Zagrebin-Bjerhammar problem, 343 Zenith, astronomical, 145 Zenith distance: dynamic satellite geodesy, 283 Zenith distance: — Continued equations, geodetic coordinates, 243, 383 N-systems, 133, 134, 370 symmetrical (to, $, h) coordinates, 126 transformation, 133, 134 (to, , N) coordinates, 71 Zhongolovitch, I. D., 298 U.S. GOVERNMENT PRINTING OFFICE 1969 OL — 306-962 i^P^H^^i PENN STATE UNIVERSITY LIBRARIES lllllllllllllllllllllllllll ADDDDTEDIDSTM mm iUIB HHF