AY ~ ~ ~ ~ ELEMENTS OF ANALYTICAL GEOMETRY. BY ALBERT E. CHURCH, A. M., PROFESSOR OF MATHEMATICS IN THE U. S. MILITARY ACADEMY; AUThIOR OF ELEMENTS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS, NEW-YORK: GEO. P. PUTNAM, 155 BROAD-WAY. 1851. En;ered, according to an Act of Congress, in the year 1851, by ALBERT E. CHURCI, In the Clerk's Office of the District Court for the Southern District of New-York. I. P. W T:;utr, S ereotyper and Printer. 74 Fultonl street, Now-York. PRE FACE. No branch of pure Mathematics presents more to interest and improve the mind of the mathematical student, than Analytical Geometry. Uniting the clearness of the geometrical reasoning, with the brevity and generality of the algebraic, it not only satisfies the requirements of the closest reasoner, but gives continued and increasing pleasure, by the elegance with which its varied results are deduced and interpreted. In preparing this treatise the Author has endeavoured to preserve the true spirit of Analysis, as developed by the celebrated French mathematician, Biot, in his admirable work on the same subject, while he has made such changes, both in the arrangement of the matter and the methods of demonstration, as he believed would render the whole more attractive, and easily acquired by any student possessing a knowledge of the elementary principles of Algebra and Geometry. In discussing the Conic sections he has preferred to consider the Parabola first, not only for the reason that the properties of this curve are more simple and more easily deduced than those of the others, but because, by this course, iv PREFACE. he was enabled to treat of the Ellipse and Hyperbola together, thus avoiding much of the repetition of words, which necessarily arises from their separate discussion. Although the treatise has been prepared with special reference to the wants of the Author's own classes at the Military Academy, he trusts that it will be found acceptable and useful to all, who are disposed to advance in the higher branches of Analysis. Those who desire to make the subject as practical, as may be, will find in the last article of the work a large number of examples. U. S. Military Academy, West Point, N. Y., July 1 1851. CONTENTS. PART I. DETERMINATE GEOMETRY. Page Definition and division of the subject............................... 1 Mode of representing geometrical magnitudes................... 2 Linear expressions and equations................................... 3 Construction of algebraic expressions............................... 4 cc of the roots of equations of the second degree............. 8 Determinate problems.............................................11 General rule for the solution of determinate problems.................23 PART II. INDETERMINATE GEOMETRY. Mode of representing points in a plane.............................. 24 Definition of rectilineal co-ordinates and co-ordinate axes............. 25 Equations of a point...............................................25 Expressions for distance between two points, in a plane...............26 Definition of polar co-ordinates..................................... 27 General definition of co-ordinates..................................28 Of the right line in a given plane..................................28 Manner of constructing a line from its equation......................31 Definition of the equation of a line..............32...................3 Every equation of the first degree between two variables represents a right line................................... 33 Vi CONTENTS. Page Manner of determining the intersection of lines.......................36 Angle included by two right lines...................................37 Conditions that two right lines be parallel or perpendicular............ 38 Equation of a right line passing through one point..................39 It" " " two points................40 Every equation between two variables, the equation of a line..........42 Classification of lines...................................... 42 General equation of the circle......................................43 Equation and discussion of the circle referred to its centre.............. 44 When a line passes through the origin of co-ordinates................46 Definition of co-ordinate planes, &c........................47 Equations of a point, in space.................................48 Expression for the distance between two points, in space...............49 Polar co-ordinates, in space................0............50 Equations of the right line, in space................................ 51 Intersection of lines, in space................................. 54 Angle included between two lines, in space..........................56 Condition that two lines, in space, be parallel or perpendicular........ 59 Equations of a right line passing through a point, in space............ 60 LL" " "L two points.............61 Equations of curves, in space..6.................................... 62 Equation of a plane...............................................64 Equations of the traces of a plane................................. Every equation of the first degree, between three variables, is the equation of a plane.....................................................67 Intersection of lines and planes..................................... 68 Conditions that a right line shall be perpendicular to a plane..........70 Angle between a right line and plane...............................71 Intersection of two planes..................7................ 72.Conditions that two planes shall be parallel.........................73 Angle between two planes..........................................74 Condition that two planes shall be perpendicular..................... 75 Equation of a plane passing through one point.......................77 e"'" " two points............78 Transformation of co-ordinates..................................... 78 Formulas for passing from one system to another.....................81 " " to a system of polar co-ordinates................84 Polar equation of the circle....................................85 Two classes of propositions in transformation of co-ordinates..........87 Formulas for transformation in space................................ 89 CONTENTS. Vii Page Formulas for transformation to polar co-ordinates, in space............90 Of the cylinder..............................91 General equation of cylinder...................................92 Of the cone................................................93 General equation of the cone..................94 Equation of right cone with a circular base........................... 97 Intersection of right cone with a circular base, and plane......98 Classification of conic sections....................10'2 Equation of the parabola...............................103 Definition of the axis of the parabola and discussion of equation... 104 Manner of constructing the parabola...................1...........105 Definition of the parabola; of its focus and other modes of construction.107 Squares of the ordinates proportional to the abscissas................108 Equation of the tangent line to the parabola.... 108 Expression for the subtangent...1......................111 Different methods of constructing tangent lines to the parabola........111 Polar line to the parabola...........................114 Properties of tangents at the extremities of a chord passing through the focus.........................................................117 Equation of a normal to the parabola..............................118 Parabola referred to oblique axes...............................119 Definition of a diameter, and mode of constructing it.................121 Parameter of any diameter................1.2....................122 Area of the parabola............................................124 Polar equation of the parabola..................................... 125 Equation of the ellipse and hyperbola.............................129 Discussion of the equation of the ellipse referred to its centre and axes.132 Equation of the hyperbola referred to its centre and axes............133 Equilateral hyperbola.13 Parameter of the ellipse and hyperbola................. 38 Foci and eccentricity of the ellipse and hyperbola...................139 Definition and construction of an ellipse........................... 141 t" it of an hyperbola.................. 142 Property of the foci of ellipse and hyperbola.......................145 Equations of ellipse and hyperbola referred to the principal vertex.... 146 Relation of the squares of the ordinates...................147 Mode of constructing ellipse with a ruler...........................149 Equation of tangent to the ellipse..................................150 Property of subtangent, and construction of' tangent.................. 152 Equation of tangent to hyperbola..................................153 Hii CONTENTS. Page Properties of two lines drawn from the foci to the point of contact of a tangent....................................................... 154 Construction of tangent lines......................................155 Corresponding properties and constructions for the hyperbola......... 156 Equation of condition for supplementary chords..................... 157 Properties of supplementary chords, and construction of tangents...... 162 Polar line of the ellipse........................................... 163 " of the circle and hyperbola.............................. 165 Equation of a normal to the ellipse...................1..............16 Area of the ellipse................................................ 167 Conjugate diameters of the ellipse and hyperbola...................I(68 Equation of condition for conjugate diameters in the ellipse..........171 " " " " in the hyperbola.......173 Parameter of any diameter.......................................... 175 Construction of the ellipse and hyperbola, two conjugate diameters being given...................................................176 Construction of diameters and tangents.......................... 177 Parallelogram on conjugate diameters, &c...182 Equal conjugate diameters of the ellipse............................183 The asymptotes of the hyperbola...................................184 Equation of the hyperbola referred to its centre and asymptotes...... 187 Power of the hyperbola....... 88 Equation of tangent referred to the asymptotes......................190 Polar equations of the ellipse and hyperbola..... 192 Discussion of the general equation of the second degree..............199 Equation of a diameter....................................2....... 01 Equation of second degree when a =- 0, c - 0.......20f) Classification of the conic sections represented by the equation........207 General discussion of the parabola.................................207 Limits of the parabola............................................208 Particular cases..................................................209 Practical examples................... 210 Construction of the parabola from its equation......................213 General discussion of the ellipse.....................213 Limits of the ellipse......................................15 Particular cases.................................................. 15 Practical examples...............................................2t16 Construction of the ellipse from its equation........................2!7 General discussion of the hyperbola.........219................!9 Particular case..................................................220 CONTENTS. iX Page Practical examples...........................................221 Construction of the hyperbola from its equation......................221 Definition and discussion of centres of curves......................222 Application to lines of the second order.......................223 Definition and discussion of diameters............................225 Of loci.......................................................... 230 Of surfaces of revolution....................238 General equation of surfaces of revolution...................239 "' " r when axis of Z is the axis of revolution...........241 Examples.......................................................241 Classification of surfaces of the second order....................... 244 Intersection of every such surface is a line of the second order.......245 Every system of parallel chords of such surface may be bisected by a plane............... 246 General equation of surfaces of the second order....................247 Discussion of such equation......................................248 Equations of the three classes.....................................250 Centres of surfaces of the second order.............................250 Definition of diametral and principal planes........................251 Of the ellipsoid.............................252 Its particular cases............................................255 Of the hyperboloid of two nappes...................256 Its particular cases.............................................259 Of the hyperboloid of one nappe..................260 Its particular cases.........................................262 Of the elliptical paraboloid.....................................2G63 Its particular case..............................................264 Of the hyperbolic paraboloid...................................... 265 Intersection of surfaces of the second order by planes.................266 When the intersections are right lines..............................268 Descriptive classification of surfaces of the second order............272 Circular sections of surfaces of the second order......................273 Subcontrary sections in a cone.......................276 Intersection of surfaces of the second order..........................278 Equation of tangent planes to surfaces of the second order......280 Line of contact of cone and surface of the second order..............283 Tangent plane passing through a right line.........283 Equations of normal line to surfaces of the second order............284 Practical examples...................................285 ANALYTICAL GEOMETRY. PART I. DETERMINATE GEOMETRY. 1. GEOMETRY, in its most general sense, has for its object, not only the measurement, but the development of the properties and relations, of lines, surfaces, and solids. This object may be attained, either by operating directly upon the magnitudes themselves; or, by representing them and their parts, by algebraic symbols, and operating upon these representatives by the known methods of Algebra, thus deducing results essentially the same as those which would be obtained by the direct method. As the reasoning employed is much generalized, and operations are much abridged by the application of Algebra, the latter method evidently possesses many advantages over the former. This latter method, which is Analytical Geometry, may be defined to be: That branch of Mathematics, in which, the magynitudes considered are represented by letters, and the properties and relations of these magnitudes made known by the application of the various rules of Algebra. Analytical Geometry may be Determinate, or Indeterminate. 2 2 DETERMINATE GEOMETRY. ZDeterminate, when it has for its object the solution of determinate problems, that is, of problems, in which, the given conditions limit the number, and afford the means of deducing the values, of the required parts. Indeterminate, when it has for its object the discussion of the general properties of geometrical magnitudes. 2. Geometrical magnitudes may be represented algebraically, in two ways. First. The magnitudes may be directly represented by letters; A - rj as the line AB, given absolutely, may be represented by the symbol a. Likewise, the.0 a _ square AC, may be represented by the symA bol A; or better by the symbol a2, a being or the representative of the side AB. Also, the rectangle ABC'D' may be represented by the symbol B; or by the product ab, a and b be-'A ing the representatives of the sides AB and 3 BC'; or more properly by c2, c representing B the side of a square equivalent to the rectanal or | gle. In the same way, a cube would be re_A a B presented by a3, a being the representative of one of the edges; and a rectangular parallelopipedon by abc or by d3. And in general, we thus represent a definite portion of a line, whether straight or curved, by a single letter or expression of the first degree; a surface by the product of two letters or an expression of the second degree; and a solid by an expression of the third degree. Second. Instead of representing the magnitude directly, the algebraic symbol may represent the number of times, that a given or assumed unit of measure is contained in the magnitude; as, for the line AB, a may represent the number of times that a DETERMINATE GEOMETRY. 3 known unit of length is contained in it; and a2 and ab or c2, the number of times that a square whose side is the unit of length, is contained in the given square or rectangle; and a3 and abc, the number of cubic units contained in the given cube or parallelopipedon. Since, in this case, the algebraic symbols represent abstract numbers, any algebraic expression, thus composed, is called an abstract expression or equation, to distinguish it from one in which the direct representatives of the magnitudes enter. Since a line is always represented by an algebraic expression of the first degree, such expression is called linear. Also, a linear equation is an equation of the first degree. 3. From what precedes, it. is evident, that any abstract expression may be changed into one in which the direct representatives of the magnitudes enter, by substituting, for the representative of each abstract number, the representative of the magnitude divided by the representative of the unit of measure. Thus in the expression, x = a + b, x, a and b representing numbers; if we substitute for them, their X A B equals -, -I-, X, A and B being the direct representatives of the magnitudes, and I that of the unit of measure, we have X A B or X = A + __= or *X=A+B. { l In the same way, the abstract expression x = ab + c, may be changed into the corresponding one, X A B C _ + C or Xl = AB + Cl. I I I I 4 DETERMINATE GEOMETRY. It should be remarked, that every expression of this kind must be homogeneous, else we should have magnitudes of different kinds added or subtracted or equal, which can not be. 4. After having deduced a result, by the application of algebra to a geometrical proposition, it will be necessary to explain this result geometrically, that is, to draw a geometrical figure, each of the parts of which shall have its representative in the algebraic expression, and also have the same geometrical relation to the others, as that indicated in the expression. This is called constructing the expression. E.xamples. 1. Let x = a + b. If a and b are the direct representatives of right lines, x will be the representative of their sum. To construct it, take the line represented by a, in the dividers, and fiom any point A, on the indefinite line X'X as a point of beginning, or origin, lay off AB equal to this distance, then from B lay off BC equal to the line represented by b, the line AC = AB + BC will evidently be represented by x. Or if a and b represent numbers, lay off from A, a times the unit of length, then from B, b times the same unit, and, as before, AC will be the line represented by x. 2. Let x = a - b. From A lay off AB = a, then from B lay off, towards A, the distance BC = b; AC = AB- BC will be the line represented by x. DETERMINATE GEOMETRY. 5 If a -= b, x will be equal to 0, the point C will evidently fall on A, and there will be no line. If b > a, x will be essentially negative, the point C will fall on the left of A, as at C', and AC', laid off from A to the left, will be represented by x. Thus, we see an illustration of the principle taught in Trigonometry, that if lines having the positive sign are estimated or laid off in one direction, those having the negative sign must be estimated in a contrary direction. 3. Let x ab C In this case x is a fourth proportional to c, a and b, and is thus constructed. Draw any two right lines making an angle; on one, from their point of intersection, as an. origin, lay off the distances AC = c and AB = a; A C B on the second, lay off AD = b; join the points C and D, and through B draw BX parallel to CD; AX will be the line represented by x. For, we have AC: AB:: AD: AX or c: a:: b: AX whence ab AX = = =. 4. Let =ac de This may be put under the form ab c x - x -. d e 6 DETERMINATE GEOMETRY. ab Place a g, and construct g as above, then we have d ge which may be constructed in the same way; and so with any expression, in which the number of factors in the numerator is one greater than in the denominator. 5. Let x = a or x2 = ab. In this case, x is a mean proportional between a and b. To construct it: On any right line, lay off AB = a; from B lay off BC = b; on the sum, AC, describe a semi-circle, and at the point B erect BX perpendicular to AC. The part ]BX, ineluded between the diameter and circumcA B ference, will be the -line represented by x. For from a known property of the circle, we have BX2 = AB x BC or BX = d/ab = x. 6. Let x = d x c. ab Place =- g and construct it as in example 3, then we have d = V-C, which may be constructed as above. C1. Let. = N/a2 + b2 or 22 = a2 + 62. DETERMINATE GEOMETRY. 7 In this case, x is the hypothenuse of a right angled triangle, the two sides of which are a and b. Therefore, draw two lines forming, with each other, a right angle: From the vertex, A, on one, lay off AB = a; on the other, lay off AC = b; join B and C, the line BC will be represented by - x. For we have J BC = -AB2 + -AC~ or BC = a+ = x. 8. Let x =- Va2 - b2 From A, in the last figure, lay off AC - b; then from C as a centre, and with CB = a as a radius, describe an arc cutting AB in B; the distance, AB, will be represented by x. For AB = VB2 X- C= - /a2 - b2 = X. 9. Let x _= v/a2 + b2 c2. Place a2 + b2 = g2, and construct g as in example 7; then we have X = /g2 _ 2 which may be constructed as above. 10. Let x - Va2 + ac. 11. Let x abc + g2d f2 12. Let x = +Va2 -bc. 8 DETERMINATE GEOMETRY. 5. Let us now construct the roots of the four forms of equations of the second degree. The first, x2 + 2ax = b2, gives the roots x= a + V/b2 + a2 x = - a /b2 + a2. From any point, as A, lay off AB = b; at B, erect the perpendicular BC = a, then as in example 7 / I) ~ AC = Vb2 + a2. B Now from C, as an origin, lay off CD = a, then AC - CD = /b2 + a2 - a = AD will be represented by the first value of x. From E, layoff EC = a, also CA = /b2 + a2; then - EC - CA = a - a /b2 + a2 = - EA will be represented by the second value of x. The given equation may be put under the form x(x + 2a) = b2, from which we see that b is a mean proportional between x and x + 2a, and this relation will be satisfied by either of the above lines AD or - EA. First, by substituting AD for x, we have AD(AD + 2a)= 62 or AD(AD + DE) = AB3, as it should be, since AB is a tangent, AD + DE = AE, a DETERMINATE GEOMETRY. 9 secant, and AD its external part. Second, by substituting - EA for x — EA(- EA + 2a) = b2 or EA x AD AB - The second, X2 _ 2ax = 62, gives the roots x = a + vb2 + a2, x= a- 62 a2. Construct as before, AC = V/b2 + a2; then from C lay off CE = a, and AC + CE = V/b2 + a2 + a = AE, will be represented by the first value of x. From D, lay off DC = a; then from C in a contrary direction lay off CA = 1/b2 + a2, and DC - CA = a - /b2 + as = - DA will be represented by the second value of x. The given equation may be put under the form x(x - 2a) = b6, which will evidently be verified by the substitution of either AE or - AD. It should be observed that the values, just constructed, are the same as those for the first form, with their signs changed. This should be so, since the first form will become the second by changing x into - x. The third x2 +- 2ax = - b2 10 DETERMINATE GEOMETRY. gives the roots x= -a+ /a2 2, x=-a- /a -2. From A as an origin, on the line AA' lay off the distance - AD = - a; at D erect the perpendicular DC _ b; c ~' from C as a centre, with CB = a, as a radius, describe the are BB'./ I \ cutting the line AA' in B and.A' BX - i,- -' B'; join these points with C and Nwe shall have DB = -/a2 -- b2 and AD + DB = - a + a2 _ b2 - AB - AD - DB = - a - /Va-2 - = - AB' will be the lines represented by the values of x. The fourth, -2 _ 2ax - b2, gives the roots x = a + V2 - b2, = a - a2 - bx. From A', as an origin, lay off A'D - a, and make the same construction as for the third form. We thus have A'D + DB = a + %/a2 - = A'B A'D - DB' = a - a/a2 - b2 = A'B' for the lines represented by the values of x. If a = b, both values of x reduce to a = A'D. In this case, the circle does not cut the line AA', but touches it at the point I), and the distances BD and B'D become 0. The same DETERMINATE GEOMETRY. 11 supposition, in the third form, reduces both values of x to -AD. If a < b, the values of x become imaginary in both forms; the circle neither cuts nor touches the line AA', and the imaginary roots admit of no construction. DETERMINATE PROBLEMS. 6. A thorough knowledge of the preceding principles, will render the solution of all determinate problems simple and easy. Problem, 1. In a given triangle, to inscribe a square. Let ABC be the triangle. Represent its base, AB, by b, and its altitude CG by h. Suppose the problem to c be solved, and that ODEF is the required square, its unknown side DE = EF being -i represented by x. Since the side DE is parallel to AB, we must have A O A F B AB: DE:: CG: C or b: x:: h: h - x; whence hx = bh - bx and x b b - h hence x is a fourth proportional to b + h, b and h, and may be constructed as in example 3, Art. (4). Or better thus: Produce the base AB until BL = h; at B and L erect the perpendiculars BN and LM; make LM = h and X join M and A; the part BN cut off on the first perpendicular will be A 0 R B represented by x. For, since BN is parallel to LM, we have AL: AB:: LM: BN or b + h: b:: h: BN whence 12 DETERMINATE GEOMETRY. BN bh X. b + h Therefore, through N draw ND parallel to AB; let fall the perpendiculars EF and DO, and the square ODEF will be the required square. The value of x, and the construction of BN, will evidently be the same for all triangles having the same base and equal altitudes. If all the angles of the triangle are acute, the square will lie wholly within the triangle as r in the above figure. If there is'u~~~ \ff*Zone right angle, two sides of the square will lie upon the sides of the triangle as AD'E'F'. If there is -A 0.' 0'" a one obtuse angle, part of the square will lie within and part without the triangle, as O'D"E"F". 7. Problemn 2. In a given triangle, to inscribe a rectangle, the ratio of whose adjacent sides is known. Let ABC be the triangle. Let AB = b and CG = h, and let the known ratio of the sides of the rectangle be denoted by r. Suppose the problem, F to be solved, and that ODEF is the required / ~_5, A rectangle. Denote the unknown side DE, by y, -I o c 2B and DO by x; then by the given condition, we have Y =............() X Since DE is parallel to AB, we have AB: DE:: CG: CH or b: y:: h: h -- whence hy = bh - bx. From this, by substituting the value y = rx, taken from DETERMINATE GEOMETRY. 13 equation (1), we deduce bh rhx = bh - bx or x= b + rh To construct this value of x; produce the base AB until BL = rh; through L draw A LM parallel to BC until it meets CM parallel to AB, in M; join M and A; at the point E, let fall EF perpendicular to A.B, it will be the re- A O B P h quired line. For, since the triangles AEB and AML are similar, their bases will be to each other as their altitudes, and we shall have AL: AB:: MP: EF or b- +rh: b:: h: EF whence bh EF= - x. b + rh Therefore, through E draw ED parallel to AB, and let fall the perpendiculars EF and DO; ODEF will be the required rectangle. If r = 1, the sides are equal, the rectangle becomes a square, and we have the same value for EF as in the preceding article. 8. -Probleml 3. To drawz a straight line tangent to two yiven circles. Since the two circles are given, both in extent and position, we know their radii and the distance between their centres. Let us denote the radius, CM, of the first circle by i, that of the second, C'M', by. r', and the M distance between their centres, CC', by a, and suppose that MMI' is the required C tangent and denote the distance CT by x. 14 DETERMINATE GEOMETRY. There are two cases: First; when the tangent does not pass between the circles. Since the radii drawn to the points of contact, M and M', must be perpendicular to the tangent, we have CM parallel to C'M', and hence the proportion CM C'M':: CT: C'T or r: r'':x: x — a, whence ar r'= rx - ra and x - r - r To construct this value of x: Through the centres C and C', draw any two parallel radii CN and C'N', on the same side of CC'; join their extremities by the C. line NN' and producee it until it meets CC' in T; CT will be thie line represented by Z. For, draw N'O parallel to CC', we then have NO: NC ON': CT or r- r': r:: a: CT whence ar C - rx Therefore, through the point T, draw TM tangent to one of the circles, it will be tangent to the other. If r > r', the value of x is positive, and the point, T, will be on the right of C. If r = r', the two circles are equal, the value of x reduces to ar = co DETERMINATE GEOMETRY. 15 the point T is at an infinite distance, and the tangent is parallel to CC'. If r < r', the value of x is negative, and the point T is on the left of C. If r = 0, x will be 0, the first circle becomes a point, and the tangent is drawn from this point to the second circle. If r' - 0, z will reduce to a, the second circle becomes a point, and the tangent is drawn from this point to the first circle. 0 If r = 0 and r' = O, the value of x reduces to -, an indeterminate quantity, each circle becomes a point, and the tangent coincides with CC'. Second; when the tangent passes between the circles. In this case as in the other, the lines CM and C'M' are parallel, hence, the triangles MCT and M'C'T are similar, and we have the proportion CM: C'M':: CT: C'T, or r:':: x: a - x, whence ar r'x = ar --' and x = ar r + r' To construct this: Through C and C' draw any two parallel radii, on different sides of CC'; join their extremities by the g2Z t line NN'; CT will be the line represented by x. For, through T B N', draw N'O parallel to CC', then we have the proportion NO: NC: ON'': CT, or r + r':: a CT, 16 DETERMINATE GEOMETRY. whence CT = ar _ r + r' The value of x is positive for all values of r1 and r'; reduces to a when r = r'; to O when r =-; to a when r' = 0, and to - - when r and r' are both equal to 0. 0 9. Problem 4. To construct a rectangle, knowing its area and the difference between its adjacent sides. Let a2 denote the given area, Art. (2), and d the difference between the sides. Let x denote the least side, then x + d will denote the greatest, and since the rectangle of these two sides must equal the given area, we have x (x + d) = a2 or x2 + dx = a2; whence d - a2 + d2 2 4' If we take the first value 2 4 a = - A ~V/a2 + -4, and add d to it, we have for the greatest side x + d =d + 2 2 To construct these values: M-:fake AB - a; at -B, erect tl:,e perpendicular BC = -, we shall 2 have, Example 7, Art. (4), A AC =\ aI -+ d. DETERMINATE GEOMETRY. 17 From AC, take CD - d and we have 2 d AD- a2 / d+- x = the least side. 2 4 AEL =d + 5/-a2- x + x + d - the greatest side. 2 4 and the rectangle AE x AD = a2 will be the required rectangle. If we take the second value d 1 d2 x - - /a + -4 2 4 and add d to it, we have for the greatest side x+d d V2 d2 Tc + d = d _ at + a By examining these values, we see, that the expression for the least side, taken with a negative sign, is the same as that for the greatest side, in the first case. Also, that the expression for the greatest side, taken with a negative sign, is the same as that for the least side, in the first case. Therefore we have, in this case, - AE, for the least side, - AD, for the greatest side, the product of which is evidently positive and equal to AB = a2. It should be observed, that it is only in an algebraic sense, that - AE is less than - AD, its numerical value being evidently the greatest. 3 18 DETERMINATE GEOMETRY. It is also evident, that the two rectangles thus determined are absolutely equal, or that in reality there is but one rectangle which will fulfil the required conditions. Why then, it may be asked, do these conditions lead to an equation of the second degree? To this it may be answered, that in Algebra, properly applied, not only are problems solved in their most general sense, every possible solution being given by the equation, which is the algebraic statement of the problem; but also, whenever the conditions of a problem, expressed in two independent ways, give rise to the same equation, this equation must give an answer corresponding to each nmode of expressing the conditions; that is, must be of the second degree, and it will thus be impossible to arrive at one solution disconnected from the other. Thus, in the above problem, should we represent the greatest side by - x, the least would be represented by - - d, and their product give x (- x - d) = a2 or x2 + dx = a2 the same equation found before; hence, this equation ought not only to give the least side, as at first proposed, but also another value of x, which taken with a negative sign, will represent the greatest side of the rectangle. 10. Problem 5. To divide a given straight line into extreme and mean ~ratio. Let AB -= a be the given line. It is to be divided into two parts, such, that the greater shall be a mean proportional between the whole line and less part. Denote the greater c part by x, then a - z will denote the less part, -A B and the condition will give _x = a (a x), or X2 + ax = a2, DETERMINATE GEOMETRY. 19 whence a / a2 a / a~ x = + Va2 +.- x = a2 + 2 4 2 4 which may be constructed precisely as in the preceding problem, the first being AD and the second - AD'. With A as a centre, and AD as a radius, describe the arc DF, the line will be divided in the required ratio at F, AF being the greater part. The second value of x = - AD' is nunmerically greater than AB. It can then form no part of it, and can not be an answer to the proposed question. But if we substitute it for z in the first equation, we have (- AD')2 = a [C - (- AD')] or AD'= a (a + AD') that is, AD' is a mean proportional between AB and AB + AD'. Since this second value of x is negative, we lay it off to the left of A, and thus construct the point F', the distance from which to A, is a mean proportional between its distance from B and the length of the given line. Moreover, we see that the second, as well as the first value of x, is a solution of the more general proposition, " Two points, A and B, being given, to find, on the indefinite line which joins them, a third point, the distance from which to the first shall be a mean proportional between its distance from the second and the distance between the two." To this proposition there are evidently two solutions, F on the right of A. being one of the points, and F' on its left, the other. Thus, the problem at first proposed being a particular case of a more general one, its solution, in accordance with the principle laid down in the preceding article, must necessarily ldraw with it that of the other case, thus giving rise to an equation of the second degree. 11. Problem 6. Through, a given point without a given angle, 20 DETERMINATE GEOMETRY. to draw a straight line, cutting the sides of the angle, so that the suem of the distances fronm the points of intersection to the vertex, shall be equal to a given line. Let YAX be the given angle, and M the given point. Produce AX to the left, and let the two distances MP' and MP be represented by a and b. Denote the given line by c. Suppose MN to be the re/ 0,_/ quired line, and denote the two unknown distances, AN by z and AO v A JY X by y. Then from the condition of the problem, we have AN + AO = c or x + y =............(l). But since MP is parallel to AO, we have PN: AN:: PM: AO, or a + x: x:: b: y; whence y (a + - )= b..........(2). Substituting the value of x, deduced from equation (1), we have y (a + c - y) = b (c - y) or y (a + c + b - y) = be. This being an equation of the second degree, its roots may be deduced and constructed as in Art. (5). But by examining it, in its present form, we see that V/bc is the ordinate of a circle whose diameter is a + c + b, and the corresponding segments of the diameter, y and a + c + b - y, which leads to a simple construction of the value of y. Thus: From P', lay off, on AY, P'B - c; also BC = a; on AB describe the semicircle ALB; at P' erect the ordinate P'L, it will be DETERMINATE GEOMETRY. 21 represented by V'bc, Example 6, Art. (4). Through L draw.. y LC' parallel to AY; then through A and C draw the perpendiculars AA' and CC'; A'C' will be equal to a + c + b; on this line describe the semi-: 0 o' circle C'OA'; the distance from -K the point 0, in which it cuts AY, to A will be represented P f by y. For 00' - P'L, and the segment A'O' - AO, fulfils the required condition. 12. Problem 7. Through a given point, without a given angle, to draw a straight line, so as to cut of a given area. Let the given point and angle be, as in the first figure of the preceding article. Let h2 represent the given area, and 3 the given angle. The expression for the measure of the required triangle will be -OT x AN. From the right angled triangle OAT, we have OT = OA sin YAX = y sin f; hence the area will be expressed by -xy sin /. Substituting the value of x, taken from equation (2), of the preceding article, and placing the result equal to h2, we have 1 ay 1 y sin 6 = ih2, which, by reduction, becomes 2h2y 2h2b ym a sin 3 a sin 3 22 DETERMINATE GEOMETRY. Solving this equation, we obtain h2 / 4 21h2b a sin a/ a2 sin2 3 a sinll To construct these values: Through A draw AA' perpendicular to AY; then since the angle A'PA = /3, we shall have AA' = AP sin / = a sin 3. -----— o Upon AY, in a negative direction, lay off AB -h l; join A' and B, <-~i A -7' X andll at B erect BC perpenlicular to A'B, then y0' /AB2 = AA' x AC or AC a sinl Since this expression is negative in the above values of y, we lay it off from A to D. The radical part of these values may be put under the form I asin a asin To construct it, we lay off P'S = b; on SD describe a semi-circle, the chord DE will be the value of the radical, for h2 AD = a sin DS = 2b + AD, and DE is a mean proportional between them. From D lay off DO = DE, and AO will be represented by the first value of y. From D lay off DO' = DE, and AO' will be represented by the second value of y. Through the points 0 and O', draw MO and MO', and either triangle cut off will fulfil the condition of the problem. DETERMINATE GEOMETRY. 23 13. By an examination of the manner in which the preceding problems have been solved, we may derive the following general rule for solving determinate problems. Conceive the problem to be solved geometrically, and draw a figure containing the given and required parts, and such other lines as may be necessary to show the relation between them. Represent the known lines by the first, and the unknown by the last letters of the alphabet. Consider the geometrical relations existing between these lines, and express them by equations, taking care to deduce as many equations as there are unknown quantities. Solve these equations and construct upon a single figure the values thus deduced. By an application of this rule the following problems are readily solved. 8. Through a given point without the circumference of a circle, to draw a straight line intersecting it, so that the chord included within, shall be equal to a given line. 9. To draw a line parallel to the base of a triangle, so as to divide it into two equal parts. 10. To inscribe, in a given triangle, a rectangle whose area is iknown. 11. Through two given points, to describe a circle tangent to a given right line. PART II. INDETERMINATE GEOMETRY. 14. THE second branch of Analytical Geometry, which has for its principal object the analytical investigation of the general properties of lines and surfaces, is much more extended in its application, and interesting in its results, than that which we have just examined. It is called Indeterminate Geometry, from the fact, that, in the equations used, the unknown quantities admit of an infinite number of values, or are indeterminate, and are therefore called variables; while from the nature of the problems discussed in the first branch, they admit of a finite number of values only, and must be determinate. OF POINTS IN A GIVEN PLANE. 15. Let AX and AY be two fixed right lines, indefinite in extent, and M any point of their plane within the angle YAX. Through this point draw MR and MP parallel respectively to AX;Y and AY; then if the distances MR and MP are given, it is evident that the position of the point M, will be known, and may be X.P A/d i X constructed, by laying off on the line AX, beginning at A, AP.' —--.'"' = RM, drawing PM parallel to A' AY; then on AY, laying off INDETERMINATE GEOMETRY. 25 AR = PM and drawing RM parallel to AX; the point of intersection of these parallels will be the required point. The distances MR and MP are called the rectilineal co-ordinates of the point. The first, or the distance of the point from AY, is the abscissa; and the second, or the distance of the point from AX, is the ordinate of the point, these distances being measured on lines parallel respectively, to AX and AY. The fixed lines, to which the point is thus referred, are called the axes of co-ordinates, or co-ordinate axes. Their point of intersection A, from which both abscissa and ordinate are estimated, is the origin of co-ordinates. 16. The abscissas of points, the position of which is indeterminate, are, in general, denoted by the letter x, and the ordinates by y, though other letters are sometimes used. The co-ordinates of points, the position of which is known, are usually denoted either by the first letters of the alphabet, or by the symbols x', y', xt", y", &c. If we denote the co-ordinates MR by a, and MP by b, the equations x -= a y = b...........(), are called the equations of the point M, and the values of a and b being known, the point is said to be given, and may be constructed, in the first angle, YAX, by laying off AP = a and AR = b, as in the preceding article. If, at the same time, we consider the point M', having AP' - AP, and P'M' = PM, it becomes necessary to adopt some notation, by which the two points may be distinguished from each other. This notation is at once suggested, by a reference to that which is used in a similar case, for the cosine of an are in Trigonometry, and the abscissa AP' is regarded as negative. Thus the equations of a point in the second angle, YAX', are x= -a y= b. 26 INDETERMINATE GEOMETRY. If the point is below the axis of abscissas, its ordinate, from analogy to the sine of an arce, is regarded as negative. Thus the equations of the point P"I, in the third angle Y'AX', are x=-a y = - b; and in like manner, the equations of the point M"', in the fourth anyle, Y'AX, are x=a y=- b. Thus it appears, that by assigning proper values and signs to a and b, equations (1) may be regarded as the representatives of any point in the plane of the co-ordinate axes. If the point is on the axis of X, (the axis of abscissas), its ordinate must be 0, and its equations x. = a y = O. If it is on the axis of Y, its abscissa must be 0, and its equations x = O y = b. By the essential signs of a and b, in these equations, we ascertain whether the points are on the right or left of the origin, above or below the axis of X. if the point is on both axes at the same time, that is, at the origin, its equations, or the equations of the origin,z become = O y = 0. 17. Let x', y', and x", y", be the co-ordinates of any two Y points, as M and M', in the plane - YAX. Join M and M', and draw l-R parallel to AX, then in the triangle MM'R we have, from Trigo_ X - - nometry, MM' MR2 + R2 2MR x M'R cos MRM', INDETERMINATE GEOMETRY. 27 the radius being supposed equal to unity. But MR = PP' = x" - x', M'R — = y" - yI; hence, denoting MM' by D, the angle YAX by A, and observing that cos MRM' = - cos 3, we have D -(x" )2 I (y1 _ y)2 + 2(x -') (y - y') cos...(). If /3 = 90~, cos f = 0, and this formula reduces to D = <(x", - x')2 + (y/- yl)2...........(2), that is, if the axes of co-ordinates are perpendicular to each other, the distance between two points, in their plane, is equal to the square root of the sum of the squares of the differences of the abscissas and ordinates of the points. If one of the points, as M, is at the origin, x' and y' will be 0, and the last formula reduce to D = i/x"2 + y"2 18. Let P be a fixed point, PS a fixed right lite, and M any point of a plane containing PS. If the length of the line PM, which we represent by r, and the angle v, made by this line f with the fixed line, are given, the position of the point will be fully cleterminecl, and may', R, the values of x will be imaginary, and there will be no point of intersection. Each position of the circle is y shown, in the accompanying figure. By making x = 0, we deduce y yi X 2X2 z/2, for the ordinates of the points, in!__ which the circle intersects the axis _ INDETERMINATE GEOMETRY. of Y, and these will be real, equal, or imaginary, according as xt is less, equal to, or greater than R. Solving equation (1) with reference to y, we have Y = y' VR - (x - ) By assigning values to x, in succession, we deduce the corresponding values of y, and thus determine as many points of the curve as we please, Art. (32). Every value of x, which makes (x -- x')2 < R2, will give two real values of y. For every such value, there will, consequently, be two corresponding points of the curve. If x = x' + R or x' - R, the values of y will be y' 0i O; the two points will unite, and the corresponding ordinate will be tangent to the curve, as SM or S'M'. If x > x' + R or < x' - R, the values ofy will be imaginary, and there will be no corresponding points of the curve. We thus see that the curve is limited, in the direction of the axis of X, by the two lines, SM and S'M'. In the same way, by solving the equation with reference to x, we may obtain the limits in the direction of the axis of Y. 35. If x' and y' are both equal to 0, the centre of the circle will be at the origin of co-ordinates, and equation (1), of the preceding article, will reduce to X2 + y2 = R2............(1). To discuss this equation, Art. (33); make y - 0, we thus obtain x = R, which shows, that the curve cuts the axis of X, in the two points, B and C, at distances, on the right and left of the origin, each INDETERMINATE GEOMETRY. 45 equal to R. Making x = 0, we obtain,,. ywhich shows that the curve cuts the C A which shows that the curve cuts the c5 -/ e As axis of Y, in the two points D and E. Solving the equation with reference i to y, we have y = i V't/R2 x-2 from which, we see that every value of x, positive or negative, and numerically less than R, gives two real values of y, equal with contrary signs; hence, for each of these values there are two corresponding points, one above, and the other below the axis of X, at equal distances from it, and the ordinates of these points, taken together, form a chord, which is bisected by the axis of X. This proves that the curve is symmetrical with respect to the axis of X. If x = -~ R, y becomes equal to ~: 0, which proves that the corresponding ordinates, produced, are tangent to the curve. If x is numerically greater than R, either positive or negative, the values of y are imaginary, and there are no corresponding points of the curve. The curve is therefore limited in the direction of the axis of X, by the two tangents at B and C. In a similar way, it may be proved, that the curve is symmetrical with respect to the axis of Y, and that its limits are two tangents at D and E. 36. For every point of the curve, as M, in the figure of the preceding article, we have y = 2 = (R + x)(R- x) = CP PB, a well known property of the circle. 46 INDETERtMINIATE GEOMETRY. 37. If y represents the ordinate of any point, as M', without the circle, in the figure of Art. (35), we have M'P > MP or y2 > R2 - X2. For any point, as M", within the circle, we have M"P < MP or y2 < R2 - x2 Hence we deduce the three analytical conditions y2 + x2 _ R2 = 0, for a point on the circle, y2 +'2 - R2 > 0, " " without the circle, y2 + x2 - R2 < O, " " within the circle. 38. If the origin of co-ordinates is at C, in the figure of Art. (35), the co-ordinates of the centre will be x' = R y' = 0, and the general equation (1), Art. (34), will reduce to x2 _ y2 - 2Rx = 0, or y2 = 2Rx - x2. This equation has no absolute term, or term independent of ax and y; the substitution of x = 0 and y = 0, will therefore satisfy it, which verifies the fact, that the origin of co-ordinates is on the curve; and, in general, if the equation of a line has no absolute term, the line passes through the origin of co-ordinates. OF POINTS IN SPACE. 39.' By space is to be understood, that infinite extent, in which all bodies are situated. As the absolute places of points and magnitudes, in this indefinite space, can not be determined, we have only to seek their situation, with reference to certain other objects, INDETERMINATE GEOMETRY. 47 which do not change their position with respect to eachl other. In a plane, we have seen that the situation of points and lines, is thus determined by a reference to two fixed objects, Art. (19). In space it is found necessary to refer them to three, the means of reference, as before, being called the co ordinates. 40. Let XAY, XAZ, and YAZ, be any three fixed planes, indefinite in extent, intersecting each other in the lines, AX, AY and AZ; and let M be any point within the solid angle formed by these planes. / / Through this point, draw the lines MP, MP' and MP", respectively i / parallel to the lines, AZ, AY and / I// AX, terminating in the planes. If the distances MP, MP' and MP", or their equals, AR, AR' and AR" are given, it is evident that the position of the point will be fully determined, and may be constructed, thus: On AX lay off AR" = MP"; on AY lay off AR' = MP'; through their extremities draw the lines R"P and R'P parallel respectively to AY and AX; through their point of intersection, P, draw PM parallel to AZ, and on it lay off the given distance, MP; the extremity will be the required point. The planes XAY, XAZ and YAZ, are called the co-ordinate planes. The first is designated as the plane XY; the second, as XZ; and the third, as YZ. The lines AX, AY and AZ, are the co-ordinate axes. The first is the axis of X, and the distances parallel to it are denoted by x. The second is the axis of Y, and the distances parallel to it are denoted by y. The third is the axis of Z, and the corresponding distances are denoted by z. The point A is the 48 INDETERMINATE GEOMETRY. origin of co-ordinates, and the distances MP, MP' and MP", are the rectilineal co-ordinates of the point M. 41. If the distances of a point, from the co-ordinate planes, YAZ, XAZ and XAY, are respectively denoted by a, b and c, we have for this point -= a y = b Z = C, which are the equations of the point; and when these equations are given, the point is said to be given, and-may be constructed as in the preceding article. The point M is in the first angle, that is, in the angle to the right of YZ, in front of XZ, and above XY. Those points which are on the left of the plane YZ, are distinguished from those on the right, by giving the minus sign to x; those behind the plane XZ, from those in front, by giving the minus sign to y; and those below the plane XY from those above, by giving the minus sign to z. Thus, for a point in the second angle, that is, in the angle to the left of YZ, in front of XZ, and above XY, the equations are =- - a y = b Z = C. For a point in the third angle, which is immediately behind the second, x = - a y -- b z = c. For a point in the fourth angle, immediately behind the first, X = a y = - b Z c. For a point in the fifth angle, under the first, - =a y = b z = - c. For a point in the sixth angle, under the second, INDETERMIN ATE GEOMETRY. 49 Z = - a y - b z --. For a point in the seventh anyle, under the third, x = - a y -b z = - c. For a point in the eighth angle, under the fourth, x = a = -- b z = - c. If a point is in the plane XY, the value of z for this point is 0, and the equations of a point, in this plane, are - a y =- b z = o; and there are similar equations for ~points in each of the other coordinate planes. If a point is on the axis of X, the values of y and z, for this point, are both 0, and the equations of a point on this axis are x = a y-O z= =O; and there are similar equations for points on each of the other coordinate axes. The equations of the origin of co-ordinates are x- O y=O z= O.,42. It is found most convenient, in practice, to take the coordinate planes at right angles to each other, and they are always considered to be in this position, unless it is otherwise indicated. Let x', y', z', and x", y", z", be the co-ordinates of any M' two points in space, as M and 2 I M'. Then x' = AT, y' =TP, z' = MP. "= AT', y" T'P', z"= MIP'. / Join P and P', and draw 4 50 INDETERMINATE GEOMETRY. MR parallel to PP'. Then from the triangle MRM', right angled at R, we have MM' = V MR2 + M-i'R2. But, Art. (17), MRa'= - 2-.= (x, x-)2 + (y", - y1)2, and M7R2 (z". I)2; hence, denoting the distance MM' by D, we obtain D = ~/(x.- xl)x + (y.- y')2 + (Z//.X)2; or, the distance between two points, in space, is equal to the square root of the sum of the squares of the differences of their co-ordinates. If one of the points, as M, be placed at the origin, x!, y' and z' become 0, and D = X/"2 + y"2 + z"2. 43. The position of points, in space, may also be determined by referring them to any other three Z fixed objects. For instance, let A be a fixed point, and AX a fixed line in the given plane YAX, and let I M be any point in space. If the dis/ X tance AM, and the angles, MAP and Vp PAX are given, the position of the point is known, and.may readily be constructed.. This method, in which points are referred to a fixed point, a fixed plane, and a fixed line of the plane, is called the system of INDETERMINATE GEOMETRY. 51 polar co-ordinates in space; in which the point A, is the pole, and the distance AM, the radius vector. The three variable co-ordinates are, the radius vector, the angle which it makes with the plane, and the angle which its projection on the plane makes with the fixed line. This, and the method of rectilineal co-ordinates, discussed in the preceding article, form the two principal systems of co-ordinates in space. OF THE RIGHT LINE IN SPACE. 44. Let x = az + -o be the equation of a right line, B'C, in the co-ordinate plane XZ, and 7Z y = bz + f the equation of B"C', in the plane / YZ. If through each of these /'-' ~, lines, a plane be passed perpen- / clicular to the planes XZ and YZ / respectively, these planes will in- S'J B tersect in a right line, BC, which will thus be completely determined. The two equations x = az +- a....... 1) y = b +............(2), taken together, may then be regarded as the equations of the right, line in space, and when they are given, the right line will be given, and may be constructed! by points. For, if a value be assigned to either variable, in these equations, the values of the other two can at once be deduced, and the three, taken together, will be the coordinates of a point of the line. For instance, assume a value ft[Or z = RP'; this, with the corresponding value of x deduced frlno 52 INDETERMINATE GEOMETRY. equation (1), will determine a point, P', on the line B'C, through which if a perpendicular, P'M, be, drawn to the plane XZ, it will intersect the given line in a point, M. This same value of z, with the corresponding value of y, deduced from equation (2), will determine a point, P", on B"C', through which, if a perpendicular be drawn to YZ, it will intersect the line, in space, at the same point, M, since no two points of this line can have the same value of z. The two planes passing through the line in space, perpendicular to the co-ordinate planes, are called the projecting planes of the fine; and the lines B'C and BI'C', in which they intersect the coordinate planes, are the projections of the given line. In equation (1), a represents the tangent of the angle which the projection of the given line, on the plane XZ, makes with the axis of Z, and a the distance cut from the axis of X, by the same projection, Art. (25). In equation (2), b represents the tangent of the angle which the projection on YZ, makes with the axis of Z, and f, the distance cut from the axis of Y. If we combine equations (1) and (2), and eliminate the variable z, we deduce Y (r: =.(3............ (3), a which, expressing the relation between y and x for points of the line, is evidently the equation of its projection on the plane YX. 45. The principle that the constants in the equation of a line, serve to determine it, Art. (23), may be well illustrated by supposing the four constants in equations (1) and (2) of the preceding article, to be given in succession. Thus, if a alone is given, the line is subjected to the single condition, that its projection on the plane XZ, shall make a given angle with the axis of Z, that is, it INDETERMINATE GEOMETRY. 53 may lie in either one of a system of parallel planes, perpendicular to XZ, and making, with the axis of Z, an angle, the tangent of which is a: If a is now given, the distance cut off on the axis of X is known, and the line may have any position in one of the before described planes: If b is also given, the other projection must make a given angle with the axis of Z, that is, the line in this fixed plane must make an angle with the axis of Z, the tangent of which is b, or it may occupy any one of an infinite number of parallel positions in this plane: If /3 is also given, the line is absolutely fixed. If a and /3 are 0, the line will pass through the origin of coordinates, and its equations become x = az, y = b.......... (). If in these, a = 0, and b = 0, the line will coincide with the axis of Z, and the equations become x = 0O y = 0, z indeterminate. If the value of z be taken from the first of equations (1), and substituted in the second, we obtain 1 b a a for the equations of the projections of the right line, passing through the origin, on the planes ZX and YX. If in these, 1 b - -_O and 0, a a the line will coincide with the axis of X, and the equations of this axis be z = O0 y = O, x indeterminate. 54 INDETERMINATE GEOMETRY. In a similar way, if the line coincide with the axis of Y, we have I a =0 and = 0, b b and the equations of this axis will be z = O, x = O, y indeterminate. 46. For the point in which a line pierces the plane XY, z must be 0. Substituting this value in equations (1) and (2) of Art. (44), we have x = a, y =; hence, a and 3, taken together, are the co-ordinates of the points in which the right line pierces the plane XY. In a similar way, the co-ordinates of the points in which the line pierces the other co-ordinate planes, may be determined. 47. Let x = az - a..........(1), y = bz /............(2), Z = a'/ + a'........(3), y = bz +..........(4), be the equations of two right lines. If these lines intersect, or have a point in common, the co-ordinates of this point must satisfy the equations at the same time; or for this point, x, y and z must be the same in all of the equations. Hence, if we combine these equations and find proper values for x, y and z, they will be the co-ordinates of the common point. These four equations, containing but three unknown quantities, can not be satisfied by the same set of values if they are independent of each other. If the lines intersect, there must then be such a relation existing between the INDETERMINATE GEOMETRY. 55 known quantities of the' equations, as to make one dependent upon the other three, and the equation which expresses this relation will be the equation of condition that the lines shall intersect. Equating the second members of (1) and (3), we deduce Z - M a a' and in a similar way, from (2) and (4), b - b Placing these values equal to each other, we have a'-a /'f. a - a b - b' or (a' a)( - 6') = (' - )( - a')............(5), for the equation of condition that the lines shall intersect. This equation contains eight arbitrary constants, any seven of which may be assumed at pleasure, and the remaining one thus determined, so as to cause the lines to intersect. Substituting the first of the above values of z in equation (1), and the second in equation (2), we find ac' - a'ac b6f'- b'i a a' b -b' These values of x and y, with either value of z, will give a point of intersection when equation (5) is satisfied. If a = -a' and b = b', equation (5) is satisfied, and the values of x, y and z become infinite. The point of intersection is then at an infinite distance, that is, the lines are parallel. a = a' b = b' 56 INDETERMINATE GEOMETRY. are then the analytical conditions that two right lines, in space, shall be parallel. But a = a' is the condition that the lines represented by equations (1) and (3) shall be parallel, Art. (28), and b = b', the condition that the lines represented by (2) and (4) shall be parallel. Hence, if two right lines, in space, are Fparallel, their projections on the same co-ordinate plane will be 1parallel, and conversely. If at the same time a = a' and / = f', the above values of z, x and y become indeterminate, as they should, since the two lines then coincide. 48. Since the angle included between two right lines, in space, is the same as that included between two lines passing through a z,P' common point and parallel respectively to the first; let the lines AP and AP' p be drawn through the origin of co-ordinates, parallel to any two given lines, /X making with each other an angle denoted by V. The equations of AP and;:Y' AP' will be x = az, y = bz; x = a'z, y = b'z; in which a, b, a' and b', are the same as in the equations of the given lines, Art. (44), and the included angle is equal to V. Denote the angles, made by the first line with the axes of X, Y and Z respectively, by X', Y' and Z', and let X", Y" and Z" represent the corresponding angles made by the second line. Take any point, as P, of the first line, and denote its co-ordinates by x', y' and z', and its distance from A, by r', and let x", y" and z", be the co-ordinates of any point, as P', of the second line, and I" its distance from A, and let D be the distance PP'. Then from Trigonometry, we have INDETERMINATE GEOMETRY. 57 12 + r2- _- D2 or D2 - r'2 - r"/2 + 2rl'r" cos V = 0............(1), in which, Art. (42), D2 = (x' - xi)2 + (yi - yI)2 + (Z _ -Zl))2. (2). But if from P, lines be drawn perpendicular to the axes of X, Y and Z, respectively, right angled triangles will be formed, from which we have' = r' Cos X', y = r' cos s', z'- r cos Z'......(3). In a similar way, we find x" = r" cos X"', y" = r" cos Y", z" = r" cos Z". Substituting these values in equation (2), developing and arranging, we have 2 = (COS2XI +COS2Y +cos2Zi)r'2+ (COS2Xii +COS2YII +cos2ZII)r/. 2 - 2 (cos X' cos X" + cos Y' cos Y" + cos Z' cos Z") r'r", and substituting this in equation (1), we have (cos2X' + cos2Y + cos2Z'-1 ),.,2 + (COS2XlI + COS2Yll + cOS2Z//- ).ii2/ + 2[cos V-(cos X' cos X" + cos Y' cos Y" cos Z' cos Z",)]r'r" = 0. Now since the points P and P' were taken at pleasure, and since the angles V, X', X", &c., are entirely independent of the distances r' and r", this equation will be true for any value of r' and r"; it is therefore an identical equation, in which the coefficients of r'2, r'2, &c., must be separately equal to 0; hence cos2X' + cos2 Y' -+ os2Z' = 1, cos2X" ~ cos2Y"+ cos2Z" = 1...(4), 58 INDETERMINATE GEOMETRY. COS V = COS X' Cos X" + cos Y' cos Y" +- cos Z' cos Z"...... From equations (4), we see that, the sum of the squares of the cosines of the angles, which any right line makes with the co-ordinate axes, is equal to unity, or radius square. From equation (5), we see that, the cosine qof the angle formed by two right lines in space, is equal to the sum of the rectangles of the cosines of the angles formed by these lines with the co-ordinate axes. 49. Since the point P is on the line AP, its co-ordinates x', y' and z', must satisfy the equations of AP and give az', y bz'............(1) Substituting these values of x' and y' in the equation, Art. (42), r12 = x'2 + yt2 + z'2 and deducing the value of z', we have z = _ V/a2 + b2 + 1 and this value of z', in equations (1), gives ar', br' x _ a+,+ y = _ 1/a2 9- b2 -+ I Va2 + b2 - 1 Substituting these values of x', y' and z', in equations (3), of the preceding article, we deduce a b Cos X' -, COS Y - V/a2 + b2 + 1 /a2 + b2 + 1 Cos Zi -- V/a2 + b2 + 1 In a similar way, we may deduce INDETERMINATE GEOMETRY. 59 cos XI" __ cos Y", V/a'2 + b'2 + 1.V/a/2 + b2 + 1 Cos Z" 1 /a'2 + b'2 + 1' Substituting these values in equation (5) of the preceding article, we have cos V = aa + bb+.....(3), V/a2 + b2 + 1 Va2 + b' + 1 giving the double sign as the angle may be acute or obtuse. If V 0, cos V = 1, hence aa' ~ bb' -+ 1 iV/a - +'b2 - I V a2 + b/2 + 1 Squaring both members, transposing and reducing, we obtain (a - a')2 + (b - b')2 + (ab' - a'6)2 = 0 and since the first member is the sum of three positive terms, it can not be 0, unless each term is separately equal to O; hence a = a/, b = b', ab' = a'b, conditions deduced in article (47), the third evidently resulting from the other two. If V = 900, cos V 0 O; hence aa' + bb' 4- 1 = 0, which is the equation of condition that two right lines, in space, shall be perpendicular to each other. This equation being entirely different from, and independent of equation (5), Art. (47), shows that two lines may be perpendicular in space, without intersecting. 60 INDETERMINATE GEOMETRY. The angle, which the line AP makes with the plane XY, is evidently the complement of that which it makes with the axis of Z, and so with the other co-ordinate planes; hence if we denote these angles by U, U' and U", we have sin U = cos Z', sin U' = cos Y', sin U" = cos X', or sin U - sin U' b V/a2 + b2 + 1 V/a2 + b2 + 1 sin U" = -V2a2 + b2 + 1 expressions from which the angles, made by a given right line with the co-ordinate planes, may be determined. 50. Let x = az + a, y = bz + /, be the general equations of a right line, in which a, b, a, and /, are undetermined, and let x', y', z' be the co-ordinates of a given point. If the line represented by the above equations passes through the given point, its co-ordinates must satisfy the equations and give the equations of condition of = az' + a, y' = bz' +. If we subtract the last equations, member by member, from the first, we shall introduce the conditions thus expressed into the first, eliminate a and j, and obtain - x' = a (z -')......(1), y - yI = b (z-z')......(2), which are therefore, the equations of a right line pcassing through a given point in space. In these equations a and b are still undetermined, as they ISND-.'1ERINIIINATE GEOMETRY. (i1 should be, since an infinite number of lines may pass through the given point. If the line is required to be parallel to a given line, the equations of which are x = a'z + a', y = b'z + if', a and b will become known, since we must have, Art. (47), a - al, a b b', and by the substitution of these values, the line will be fully determined. Find the equation of a right line, which shall pass through the point xi = 2, y_ - 3, z' 1, and be parallel to the line of which the equations are x = 2z + 3, y =- - z +1. 51. If the line, represented by equations (1) and (2) of the preceding article, be subjected to the additional condition that it shall pass through the point whose co-ordinates are x", y" and z", these co-ordinates must satisfy its equations and give the equations of condition x,' - X' = a(Z" - -'), yll - y = b(z" - z'), from which we deduce a - ] ba' Z _ Z/ = Z/ -- ZI Substituting these values in the equations (1) and (2), we have x = - x' /- / (z' (z z'), y y = - Z'); 62 INDETERMINATE GEOMETRY. which are the equatzbns of a right line passing through two given points in space. Find the equations of a right line which shall pass through the two points x' 2, y' = O, z' = O; x" O, y" = 3, z"- - 1. 52. Curves, in space, may be represented in the same manner as the right line has been represented in Art. (44). Thus, if through a curve, cylinders be passed whose elements are perpendicular to the co-ordinate planes, these cylinders will be the projecting cylinders of the curve, and their intersections with the coordinate planes, the projections of the curve, either two of which being given, by their equations, the curve may be constructed by points, as in Art. (22). 53. The points of intersection of two curves, in space, may also be determined as in Art. (47), by combining their equations. But as there will always be four equations, involving but three unknown quantities, proper values for the variables belonging to a common point, can not be found, unless an equation of condition, deduced as in that article, by eliminating x and y and equating the values of z, shall be satisfied. To illustrate the intersection of two curves, let us take the equations 2z2 — 3x -0............(1)) 1 1st curve. z- 3y = 0............(2) z2 + 3x2 - 12x + 9 = 0............(3) ( 2nd curve. z2 + 3y2- _ y =................... (4) If we combine equations (1) and (3), and deduce the values of x and z, we have INDETERMINATE GEOMETRY. 63 x-2, Z 3 A 3 3 2 2 These values of x and z are evidently the co-ordinates of the points M and M', in which the projections of the curves on the plane XZ intersect. Combining equations (2)and (4), we obtain y=, z-+ ~3; y = O, z =' + O3 and these are the co-ordinates of the points, A and N, common to the projections of the curves on the plane YZ. The second values of z, in the two cases, being unequal, can not, with the corresponding values of x and y, satisfy all four equations at the same time and therefore do not belong to a point common to the two curves. The first values of z, viz. z = - ~t /3, are the same in both cases and therefore taken with x = 2, and y 1, are the coordinates of two points in which the curves intersect, one of these points being above, and the other the same distance below the plane XY, at P. The same result may be otherwise obtained thus: Combine equations (1) and (3) and eliminate x, thus deducing an equation involving z. Combine equations (2) and (4) and eliminate y, thus deducing another equation in z; and since there can be no common point unless these equations give equal values for z, it follows (the second member of both being 0), that for each equal value of z the first members will have a common divisor of the form z - a; hence, if we seek the greatest common divisor of these first members and place it equal to 0, the roots of the resulting equation 64 INDETERMINATE GEOMETRY. will give all the values of z which will satisfy both equations. Those which give real values of x in (1) and (3), and real values of y in (2) and (4), will correspond to points of intersection. By applying this process to the above equations we find for the greatt common divisor Z2 - 3, which placed equal to 0, gives the same values before found. If only the form of the equations of two curves should be given, the constants which enter them being arbitrary,. x and y may be eliminated, as above, and then such values may often be assigned to these constants, as to give the first members of the resulting equations in z, a common divisor of the first or higher degree, thus causing the two curves to intersect in one or more points. OF THE PLANE. 54. The equation of a sufface is an equation which expresses the relation between the co-ordinates of every point of the surface. A plane suiface may be generated, by moving a straight line, so as to touch another straight line, and have all of its positions parallel to its first position. The moving line is called the generatrix; and the line on which it moves, or which directs its motion, the directrix. 55. Let y = a'x + b'...........(1), be the equation of any right line, DB, in the plane XY, and let x = az +- a, y = bz +-........... ( ), INDETERMINATE GEOMETRY. 65 be the equations of a right line in space, which is to be moved on the line DB, so as to generate a plane. Since the moving line must always be parallel to its first position, a and b will remain the same in all of its positions, while a and / will change, as the line is moved 1 X from one position to another. But a and ( are the co-ordinates of the H point, in which the line pierces the co-ordinate plane XY, Art. (46), and since this point must be on the line DB, the values of a and (, deduced from equations (2), must, in all positions of the generatrix, satisfy equation (1), when substituted for the variables. The values, thus deduced, are c = x - az, 3 = y- bz, and these, substituted for x and y in equation (1), give y - bz = a'(x - az) + b'............(3), which expresses a relation between the co-ordinates of the different points of the generatrix, in all of its positions; it is, therefore, the equation of a plane. If this equation be solved with reference to z, and the coefficients of x and y be placed equal to c and d, respectively, and the absolute term equal to g, we have z = cx + dy + g............(4), a form analogous to that of the right line, Art. (24). Since this equation contains three variables, either two may be assumed at pleasure and the corresponding value of the third deduced; the three, taken together, will be the co-ordinates of a point of the plane, which may be constructed as in Art. (40), and as any number of its points may be determined in the same way, the plane will evidently be given when the constants which enter its equation are known. 5 66 INDETERMINATE GEOMETRY. And, in general, any surface will be given, analytically, when the form of its equation and the constants which enter it are known. 56. The intersection of a plane with either co-ordinate plane is called a trace of the plane. For every point of the plane, which lies in the, co-ordinate plane XZ, y must be equal to 0. Substituting this value for y, in equation (4) of the preceding article, we obtain z = cx g............(l), in which x and'can only belong to points of the plane lying in the plane XZ. This is then the equation of the trace, BC, on the plane XZ. In the same way, for all points of the plane, in YZ, x must be equal to 0; whence z = dy + g............ (2), is the equation of the trace, DC, on the plane YZ. By making z = O, we obtain cx + dy +- g = 0, for the equation of the trace, BD, on the plane XY. For all points in the axis of Z, x and y must be equal to 0. Substituting these values for x and y in equation (4), we find z -g which is the distance AC, cut off by the plane on the axis of Z. In a similar way, we find the distances cut off on the axes of X and Y g -- = ABD. c d INDETERMINATE GEOMETRY. 67 If g = 0, these distances become 0, the plane will pass through the origin, and its equation become z= cx + dy, without an absolute term, as it should be, since the co-ordinates of the origin will then satisfy the equation. If c = 0, the distance AB becomes infinity, and the plane is parallel to the axis of X, or perpendicular to the co-ordinate plane YZ, and its equation becomes z = dy + g, the same as that of the trace on ZY. It should be remarked, however, that for the plane, x may have any value, or is indeterminate, since its coefficient c is 0; while for the trace, x must be~ equal to 0, as we have seen. If d = 0, the distance AD becomes infinity, and the equation of the plane perpendicular to XZ, z = cx + g, y indeterminate. In the same way, if equation (3), Art. (55), had been solved with reference to y or x, it might be shown that the equation of a plane perpendicular to XY, would be the same as that of its trace, z being indeterminate. 57. Every equation of the first degree between three variables, will be a particular case of the general equation Ax + By + Cz + D = 0, and this, when solved with reference to z, gives A B D z -- — X - — y C C C an equation of the same nature and form as 68 INDETERMINATE GEOMETRY. z = cx + dy +g..............(1), and will therefore represent a magnitude of the same kind; that is, every equation of the first degree between three variables is the equation of a plane, and when solved with reference to z, will appear under the form (1). 58. Let x = az + a, y = bz +...........(1), be the equations of a right line, and z = cx + dy + g............ (2) the equation of a plane. Those values of x, y and z which, when taken together, will satisfy these three equations at the same time, must be the co-ordinates of a point common to the line and plane. Therefore, by combining the equations and deducing the values of x, y and z, we shall obtain the co-ordinates of the point in which the line pierces the plane. Substituting the values of x and y, from equations (1), in equation (2), we find ac + 3d + g, 1 - ac - bd and by the substitution of this value of z in equations (1), we may deduce the corresponding values of x and y. If I - ac - bd = 0, the values of z, x and y will become infinite, the point in which the line pierces the plane will be at an infinite distance, and the line will be parallel to the plane. The last equation is then the analytical condition that a right line shall be parallel to a plane; or, that a right line, having one point in a plane, shall lie wholly in the plane. INDETERMINATE GEOMETRY. 69 Ill the same way, the points in which any line, in space, pierces a surface may be found; since the two equations of the line, with the equation of the surface, will always give three equations, by the combination of which, values of the three variables may be deduced which will satisfy the equations at the same time. The number of sets of real values thus found will indicate the number of common points. 59. Let z = cx + dy + g, be the equation of a plane, and suppose any straight line to be drawn perpendicular to the plane. If through the point where the plane cuts the axis of Z, a line be drawn parallel to the given line, its equations will be of the form = az + a, y = bz +, in which a and b are the same as in the equations of the given line, MJ Z Art. (47). Since this second line is also perpendicular to the plane, it must be perpendicular to the traces, BC and DC, which are two / X lines of the plane passing through its foot. The equations of the trace BC, Art. (56), may be put y under the form X -Z _, y O.z, c c since the projection of BC, on the plane YZ, coincides with the axis of Z. The general equation of condition that the right line shall be perpendicular to the trace is, Art. (49), 70 INDETERMINATE GEOMETRY. 1 + aa' + bb' = O............(1), in which, from the above equations of the trace, we must have aI b' = — 0. c Substituting these values in equation (1), we obtain 1 f a = 0...........(2), or a = - c for the condition that the line shall be perpendicular to the trace. In a similar way, for the trace DC, we have 1 b —I a/I = 0, and these, in equation (1), give b b I + - = 0...........(3), or b = - d. a = - c b - d are then the analytical conditions that a straight line shall be perpendicular to a plane. Condition (2) proves also that the projection CM is perpendicular to the trace BC, Art. (28); and condition (3) proves that the projection CM' is perpendicular to DC. Hence, if a right line is perpendicular to a plane, its projections are perpendicular to the traces of the plane, respectively. 60. Let x', y', z', be the co-ordinates of a given point, and z -= cx +- dy + g............(1), the equation of a given plane. The equations of a right line passing through the given point will be, Art. (50), - -- x' a( -')= y -' b(z z)............(2). INDETERMINATE GEOMETRY. 71 If this line is required to be perpendicular to the plane, we must have, by the preceding article, a - c, b = - d. Substituting these values in equations (1), we have - x'= -c(z - z'), y - y -d(z- z').....(3), for the equations of a right line passing through a given point and perpendicular to the plane. The point, in which this perpendicular pierces the plane, may be found as in Art. (58), by combining equations (3) with equation (1); and the distance between this and the given point, or the length of the perpendicular, by means of the formula of Art. (42). Find the equations of a straight line passing through a point whose co-ordinates are = - 2, y' z = 3, and perpendicular to the plane whose equation is 2x - 3y + 4 4z + 1 = 0. Find also the point in which the line pierces the plane, and the length of the perpendicular. 61. The angle, made by a straight line with a plane, is the same as the angle included between the line and its projection on the plane. Therefore, if through any point of the line a perpendicular be drawn to the plane, this perpendicular, a portion of the line and its projection on the plane, will form a right angled triangle, of which the angle at the.base will be the angle made by the line and plane, and the angle at the vertex, its complement. Denote the first angle by A, and the angle formed by the given line and the perpendicular by V. Then, the line being repre 72 INDETERMINATE GEOMETRY. sented by equations (1) and (2), Art. (44), and the plane by equation (4), Art. (55), the perpendicular will be represented by equations (1) of the preceding article, and by substituting - c for a', and - d for b' in the formula (3), of Art. (49,) we have Cos V = d= = sin A /1 r+ a2 + b2 /1 + c2 + d2 from which we determine the sine of A, and thence the angle itself. If 1 - ac- bd = 0, the angle becomes 0, and the line is parallel to the plane, a condition before determined, Art. (58). 62. Let z = cx + dy +g...........(1), z = c'x + d'y + g'............(2), be the equations of two planes. Those values of x, y and z which will satisfy both of these equations, at the same time, must belong to points common to the two planes. If then we combine these equations, x, y and z in the result can only belong to the line of intersection; and if one of the variables, as z, be eliminated, we have (c -') + (d - d')y + g -g' =...........(3), which must be the equation of the projection of this line of intersection on the plane XY. In the same way, if the equations be combined and x be eliminated, the result will be the equation of the projection of the line of intersection on the plane YZ. Two projections being thus determined, the line will be known. If such a relation exists between c, c', d and d', that no values INDETERMINATE GEOMETRY. 73 of x and y will satisfy equation (3), the planes can not intersect, but must be parallel. This can only be the case when c = c' and d = d', as we shall then have.g -' = 0, which can not be if the planes are different; hence, C =', d= d', are the analytical conditions that two planes shall be parallel. By referring to the equations of the traces of these planes, we see that c = c' is the condition that the traces on the plane ZX shall be parallel, Art. (28), and that d = d' is the condition that the traces on the plane ZY shall be parallel; hence, if two planes are parallel, their traces are parallel. If the plane represented by equation (1) is parallel to the coordinate plane XY, its traces on XZ and YZ must be parallel, respectively, to the axes of X and Y; hence, by a reference to the equations of these traces, Art. (56), we see that c = 0, d = 0, and that equation (1) reduces to z = g, x and y indeterminate, for the equation of a plane parallel to the co-ordinate plane XY. If g = 0, also, we have z = O, x and y indeterminate, for the equation of the co-ordinate plane XY. If the plane represented by (1) is parallel to the co-ordinate plane YZ, its traces on XZ and XY must be parallel to the axes of Z and Y, which requires I d - 0, -0. ~ ~ 74 INDETERMINATE GEOMETRY. These values, substituted in equation (1), placed under the form 1 d g X -- - - y - - C C C give x = - or x = h, y and z indeterminate, for the equation of a plane parallel to YZ, and at a distance from it equalto - = h. If g = 0, also, we have x - 0 y and z indeterminate, for the equation of the plane YZ; and similar equations may be found for a plane parallel to XZ, and for the plane XZ itself. The preceding method of finding the intersection of two planes is applicable to any surfaces whatever. Thus: Combine the equations. of the surfaces, and eliminate one of the variables, the result will be the equation of the projection.of the intersection on the plane of the other two variables. Combine the equations again and eliminate another variable, the result will be the equation of the projection on another plane, and the intersection will be thus determined. Find the intersection of the two planes whose equations are 2x - 3y 4- 2z = 0, x + 2y - 3z + 1 = 0. 63. If through any point, within the angle included by two planes, a line be drawn perpendicular to each plane, the angle included by one of these lines and the prolongation of the other, will be equal to the angle included by the planes. Let the equa INDETERMINATE GEOMETRY. 75 tions of the planes be the same as in the preceding article, then the equations of the perpendiculars will be, Art. (60), x - x' = - c (Q - z'), y -y'= - d ( -'), -' = - c' (z- z'), y y = -d' (z -'), If we denote the angle which these lines make, by A, and then substitute - c and - c' for a and a', and - d and - d' for b and b', in formula (3), Art. (49), we have cos A =........... (1), /i+ C2 + d2 Vi + C12 + dc 2 from which we deduce the value of cos A, and thence of A itself, which will express the number of degrees, &c., contained in the angle of the planes. If the two planes are parallel, we have A = 0, cos A = 1. By substituting this value of cos A, clearing of denominators, &c., as in Art. (49), we may deduce the same equations of condition as in the preceding article. If the two planes are perpendicular to each other, we must have A = 900, cosA = 0, which requires 1 + cc' + dad = O, the equation of condition that two planes shall be perpendicular to each other. If the first plane coincides with the plane XY, we have, from the preceding article, c,= O d = O, and cos A reduces to cos X" I,,' I ~7G INDETERMINATE GEOMETRY. for the cosine of the angle made by the second plane with the co-ordinate plane XY. If the same plane coincides with the plane YZ, we have = 0, = and these values substituted in equation (1), first placing it under the form, 1 c' + dd' C C cos A --:E 9 VT2 + 1 + C2'V + c/2 + d/2 reduce it to cos Y"- c V1 + c'2 + d'2 for the cosine of the angle made by the second plane with the plane YZ. If the plane coincides with XZ, we have - 0-~ 0, and equation (1) may be reduced tod and equation (1) may be reduced to cos Z" -- d' V/1 +- c'2 - d'2 In the same way, if the second plane be made to coincide, in succession, with each co-ordinate plane, we may deduce for the angles X', Y' and Z', made by the first plane with the co-ordinate planes 1 C cos X=._ -, cos Y' -= - _ i + c2 + d2 1+ c2 + d2 INDETERMINA'ITE GEO1METRY. 77 COS Z' /i1 + C2 + d2 If both members of these three equations be squared and the results added, member to member, we find cos2X' + cos2Y' -t-os2Z = 1. If the values of cos X' and co XI" be multiplied together, also cos Y' and cos Y", cos Z' and cos Z" and the three products added, we obtain cos X' cos X" + cos Y' cos Y" + cos Z' cos Z" = cos A, an expression for the cosine of the angle formed by two planes, in terms of the cosines of the angles made by the planes with the co-ordinate planes. 64. Let x', y', z', be the co-ordinates of a given point, and z = cx + dy + g............(1), the general equation of a plane, in which c, d and g are arbitrary constants. If the given point is in this plane its co-ordinates must satisfy the equation and give the equation of condition, z' = cx' + dy' + g. Subtracting this equation, member by member, from (1), we introduce the condition into that equation and obtain, z - z' = C(x - X') + d(y - y'), for the equation of a plane passing through a given point, in which c and d are still arbitrary. 65. If the plane, represented by equation (1) of the preceding '78 INDETERMINATE GEOMETRY. article, be required to contain the three given points x', y', z:, x", y", z", and x"', y z'", these co-ordinates, when substituted in succession for the variables, must satisfy the equation and give the three equations of condition, z' = ca' + dy' + g, z" = cx" + dy" + g, z'"- cx"' +- d"' + g. From these three equations, the values of the three constants c, d and g may be determined, and substituted in equation (1). The result will be the equation of a plane passing through three given points. Find the equation of a plane passing through the three points, x' = 1, y' = 0, z' = - 3; x" - 2, y" = 1, Z"/ - 1; x"' = 0. yl' _ 2,'z"' = O. TRANSFORMATION OF CO-ORDINATES. 66. In developing and discussing the properties of lines and surfaces, it is often of great advantage to change the reference of their points from one system of co-ordinates axes or planes to another. The system from which the change is made is called the primitive system; the one to which it is made is the new system; and changing the reference of points, from one system of co-ordinate axes or planes to another, is called the transformation of coordinates. If a line or surface be given by its equation, and it be required to change the reference of its points to a new system of co-ordinate axes or planes; it is only necessary to deduce values for the primitive co-ordinates in terms of the new, and to substitute these values INDETERMINATE GEOMETRY.'79 for the variables in the given equation. The result, expressing a relation between the new co-ordinates of the points, will of course be the equation of the line referred to the new system. From the nature of this operation, it is evident that no change whatever takes place, either in the nature or extent of the line or surface. 67. Let AX and AY be any set of co-ordinate axes, and AX' and AY' any other set having the same origin. Denote the angle included between AX and AY bye, Y and let a and a' denote the angles made by AX' and -AY', respectively, with AX. Let AP = x and MP = y be the co-ordinates of any point, as M, when referred to the first set, and let AP' = x' and A S P X MP' - y' be the co-ordinates of the same point referred to the second set. Through P' draw P'R parallel to AX and P'S parallel to AY. In the triangle ASP', the angle AP'S = P'AY = -/ - a, sin ASP' = sin YAX = sin /, and since the sides are as the sines of their opposite angles, we have the two proportions, AS: AP':: sin (/3 - a) sin ASP' or sin /, P'S: AP':: sin a: sin /; whence AS x'sin (/3 a) x' sin a)' sin sin /3 sin,8 In the triangle P'RM, 80 INDETERMINATE GEOMETRY. P'MR = YAY' = a - a', MP'R = Y'AX = a', MRP' = P'SA, and we have the proportions P'R: P'M::sin (/3 - a'): sin /3, MR: P'M:: sin a' sin /3; whence P'R y' sin (8i - a'') M' sin a' sin ll sin p We have also AP = AS + P'R, MP = P'S + MR. Substituting, in these equations, the values above deduced, we have x' sin (3 - a) + y' sin (/ - a') sinl x' sin a +- y' sill a' sin f in which the values of the primitive co-ordinates are expressed in terms of the new and constants; and these are the formulas, for passing from any system of rectilineal co-ordinates to another having the same origin. If the new origin is different from the primitive, at A', for instance, it is evident that we havs M simply to add to the above values, a' and b', the co-ordinates of the new / A'<-;/ origin referred to the primitive sysea / / tem. We thus obtain A P = a x'J, x, sin (3 - a) + y' sin (3 - a') X -at q- INDETERMINATE GEOMETRY. 81 x' sin a -+ y' sin a' y b' + ~, sin /3 general formulas for passing fiom one system of r ectilineal co-ordinates to any other, in the same plane. If the new axes of co-ordinates are required to be parallel to the primitive, we have = 0, a' = /, sina = 0, sin a' = sin 3, and the above formulas reduce to = a' + x', y = b' + y'............(2), formulas for passing from any set of co-ordinate axes to a parallel set, in the same plane. If the primitive axes are perpendicular to each other, we have 3 = 90~, sin / = 1, sin (i3 - a) = cos a, sin (/ - a') - cos a', and formulas (1), reduce to x = a' + x' cos a + y' cos a'....... (3), y = b' + x' sin a + y' sin a' formulas for passing from a system of rectangular co-ordinate axes to an oblique system, in the same plane. If the primitive axes are perpendicular to each other, and also the new, we have 3 = 900, sin/3 - 1, a = 90~ + a, sin o -a' cos cc, sin (/ -) = cos a, sin ( - a')=sin- - sin a, and formulas (1) reduce to x = a' + x'cosa - y' sin a............ (4), y = b' + x' sin a + y' cos a 6 82 INDETERMINATE GEO-METRY. formulas for passing from a system of rectangular co-ordinate axes to another system, also rectangular, in the same plane. If the new axes, only, are perpendicular to each other, we have = _ 900 + -a, sin a' = cos a, cos c' - - sin a, and formulas (1) reduce to X _ a,' + x' sin ( - a) - y cos (3 - a) sin /..... (5), x' sin a + y' cos oa sin/3 formulas for passing from a system of oblique co-ordinate axes to a rectangular system, in the same plane. If the new origin be the same as the primitive, a' and b' in each of the above formulas will be equal to 0. 68. We may illustrate the use of the formulas of the preceding article by the following Examples. 1. Let x2 + y2 R2............ (1 ) be the equation of a circle referred to its centre and rectangular co-ordinate axes, Art. (35), and let it be proposed to change the reference to a parallel set having the origin at the point C. D1 The co-ordinates of the new origin will be a' R, b! = O, a A and these values, in formulas (2), reduce them to INDETERMINATE GEOMETRY. 83 x -- — R + xI', y = y'. Substituting these last values for x and y in equation (1), and reducing, we obtain y(2 = 2Rx',._2x an equation before found in Art. (38). 2. Let y = ax + b........(2), be the equation of the right line A'B, referred to the rectangular axes AX and AY, and let it be proposed B to find the equation of the same line YJ referred to the axes A'X' and A'Y', also at right angles, the axis of X' making an angle of 450 with the axis of X and having the new origin at A', the point A X where the given line cuts the axis of Y. The general formulas to be used in this case are formulas (4), in which a' = 0 b =- b, sin a = cos a. These values reduce the formulas to x = (x' - y') cos a, y = b + (x' + y') cos a, and substituting these values for x and y, in equation (2), we have b + (x' + y') cos a = a(xl - y') cos a + b, or reducing, I a -1, a +set of rectan 69. Let AX and AY be a set of rectangular co-ordinate axes, 84 INDETERMINATE GEOMETRY. - M and M any point referred to them by the co-ordinates AR = x, and MR = y; and let P be the pole, and __ ~ PS the fixed line, to which the point is 1P T referred by the radius vector PM = r A-l o - 0 X and the angle MPS = v, Art. (18). Ljet AO - a', OP = b', SPT = a. In the right angled triangle, MTP, we have PT = r cos (v + a), MT = r sin (v +- a). Substituting the above values in the equations AR = AO + PT, MR = OP + MT, we have a' + r cos(v + ), y = b' + r sin (v + )......(l), which are general formulas, for passing from a system of rectacngular co-ordinates to a system of polar co-ordinates, in the same plane. The fixed line is generally taken parallel to the axis of X, in which case ca = 0, and formulas (1) reduce to x = a' +- rcos v, y = b' + r sin v............(2). If the pole is at the origin, we have a' = O, b' = 0. From the second of equations (1), we deduce y - b' sin (v + a) in which, if y > b', y - b' is positive, the point M is above the line PT, and sin (v + a) also positive; hence, the value of r will be essentially positive. INDETERMINATE GEOMETRY. 85 If y < b', y - b' is negative, the point M is below PT, sin (v + a) also negative, and the value of r positive. The value of the radius vector is therefore always positive. Hence, if in discussing the equation of a line referred to a fixed point and fixed right line, usually called the polar equation of tlhe line, a negative value of the radius vector is found, it must be rejected, as there can be no corresponding point. 70. To illustrate the principles of the preceding article, let it be proposed to determine and discuss the polar equation of the circle. Its equation referred to the rectangular axes AX and AY, is X2 + y2.RS.... (1)....(1..p. Suppose the fixed line PS, from which the angle v is estimated, is parallel to the axis of X, we must then use formulas (2). Squaring the values of x and y, we have =2 = at2 + 2a'r cos v + r2 cos2 v, y2 = b12 + 2b6r sin v + r2 sin2 sv. Substituting these values in equation (1), recollecting that sin2v + cos2v - 1, and reducing, we obtain r2 + 2(a' cos v + b' sin v)r + a'2 ~ b'2 R-R2 = 0............(2) for the general polar equation of the circle. By attributing particular values to a' and b', the pole may be placed at any point in the plane of the circle. If the pole be placed at C, we must have 86 INDETERMINATE GEOMETRY. a'= -R, b = O, and these, in equation (2), give q'2 - 2R cos vr = 0. This equation gives two values of r, r = 0, r = 2R cos v. These two values of r represent the distances from the pole to the points in which the radius vector, making any angle v, cuts the circle. Since the pole is on the curve, one of these values is necessarily 0, whatever be the angle v. The second may then represent any radius vector as CM. If in this second value v = 0, we have cos v = 1, and' D r = 2R = CB, 1 which gives the point B. As v increases, cos v will remain positive until v = 900, in whichcase cosv = 0, r becomes 0, and the radius vector takes the position CM' tangent to the circle at C. As v increases beyond 90~, its cosine becomes negative, the value of r is negative and gives no point of the curve, until v becomes equal to 2700, when cos v = 0 and r = 0, taking the position CM"'. As v increases beyond 2700, its cosine is positive, 2r is positive and gives points of the curve until v = 3600, when we again have r = CB. From this we see that as v increases from 0 to 900, we obtain all the points in the semi-circumference BDC, that no points of the curve are on the left of the line M'M"', and that as v increases from 2700 to 3600, we obtain all the points in the other semicircumference. The second value of r is readily verified, since in the right angled triangle CMB, we have INDETERMINATE GEOMETRY. 87 CM = CB cos BCM or r = 2R cos v. If the pole is placed at B, we have a' = R,' - O, and equation (2) gives the two values r O, r = - 2R cos v. The second value of r will be negative for all values of v less than 900 or greater than 2700, and positive for all values from 900 to 270~. If the pole isplaced at the centre, we have at = 0 b' = 0, and equation (2) reduces to r = R. v being indeterminate, since its coefficient is equal to 0. 71. By reflecting upon the discussion contained in the three preceding articles, we see that two classes of propositions may arise in the transformation of co-ordinates. First; when it is proposed to change the reference from a given set of co-ordinate axes to another set, the exact position of which is known. In this case the constants which enter the values of the primitive co-ordinates are given. Second; when it proposed to change from a given set to another, the position of which is to be determined, so that the resulting equation shall assume a certain form, or the new set fulfil certain conditions. In this case, the constants above referred to are arbitrary, and by assigning values to them, as many reasonable conditions may be introduced as there are such constants, and the position of the new co-ordinate axes thus determined. 88 INDETERMINATE GEOMETRY. 72. Let AX, AY and AZ, be three co-ordinate axes at right z z angles to each other, and AX', M-J4' —/ —./'-'-'-lr' IAY' and AZ', three oblique axes having the same origin. Denote the angles made by the new axis AX' with the three - -X primitive axes of X, Y and Z, respectively, by X, Y and Z, those made by the axis AY' y/i with the same, by X', Y' and Z', and those made by AZ', by X", Y" and Z". Let M be any point, in space, referred to the primitive planes by the co-ordinates x, y and z. Through this point draw the line MP parallel to AZ', until it pierces the new plane X'Y', in the point P; through this last point, draw PR parallel to AY', until it intersects the new axis of X', in R; then AR = x', PR = y', MP = z', are the co-ordinates of the point M referred to the oblique co-ordinate planes. Through the points M, P and R, pass planes parallel to the plane XY, intersecting the axis of Z in M', P' and R'. AM' is equal to z, and the lines AR, RP and PM, are the hypothenuses of right angled triangles, the bases of which are AR', RR" and PPl", and the angles at the bases, Z, Z' and Z". From these triangles we have AR' = AR cos Z, RR" - RP cos Z', PP" = MP cos Z". Substituting these values for their equals in the equation AM' = AR' d- R'P' + P'M', and for AM', AR, RP and MP, their values, we have, Z = x' cos Z + y' cos Z' + z' cos Z". INDETERMINATE GEOMETRY. 89 In a similar way, by drawing lines through the point M respectively parallel to the new axes of X' and Y', we may deduce x = a' cos X - y' cos X' +-' co s X", y = x' cos Y + y' cos Y' + z' cos Y". These three equations taken together express the values of the primitive co-ordinates in terms of the new, and are the formulas for changing the reference of points from a set of co-ordinate planes at right angles, to another set oblique to each other, having the same origin. If the origin be also changed to a point whose co-ordinates are a, b and c, these formulas become z - a + x' cos X + y' cos X' + z' cos X", y b +- x' cosY + y' cos Y' - +' cos Y",............(1). Z = c + x' cos Z + y' cos Z' + z' cos Z", In these formulas there are twelve constants; but since the angles X, Y, Z, &c., made by each of the new axes with the primitive, must fulfil the condition expressed in equation (4), Art. (48), thus forming three equations of condition, we can, by means of these constants, introduce only nine independent conditions. If the new axes are also perpendicular to each other, we shall have the cosines of the angles, included between each set of two, equal to 024wI Placing the expressions for these cosines, Art. (48), each equal to 0, we have three more equations of condition existing between the arbitrary constants. If the new axes are parallel to the primitive, we have X = O, Y' = 0, Z" - 0 and each of the other angles equal to 90~, hence the above formulas reduce to x = a + x', y b + y', z = c + z'......(2), 90 INDETERMINATE GEOMETRY. which are the formulas for passing from a set of planes at right angles, to a parallel set. 73. Let M be any point, in space, referred to the three Z rectangular co-ordinate planes, by the co-ordinates / I AR x, AR' = y, MP = z, H /R X and to the fixed plane XY, the line V._ _XZ AX and the point A, by the polar Y/ co-ordinates, Art. (43), AM - r, MAP = v, RAP = u. The right angled triangles ARP and MPA, give AR = AP cos u,'RP = AP sin u, MP = r sin v, AP = cos v. Substituting the value of AP, the first three equations give x --' Cos v Cos u, y = r cos v sin u, z = r sin v......), which areformulas for passing from a system of rectangular coordinates to a system of polar co-ordinates, in space. From the last of the above equations, we have z r =,____ sin v and since z and the sin v will always have the same sign, the radius vector will always be positive. The equations of the radius vector in any one of its positions, will be of the form, Art. (45), x = an, y = bz......(2), whence INDETERMINATE GEOMETRY. 91 a = -, ^ b=y. Z Z Substituting the values of x, y and z, taken from formulas (1), we have a = cot v cos u, b = cot v sin uZ and these, in equations (2), give X = cot v cos UZ, y = cot v sin uz, which will be given, when v and u are known. OF THE CYLINDER. 74. A cylindrical surface or cylinder, may be generated by moving a straight line, so as to touch a given curve and have all of its positions parallel to its first position. The moving line is called the generatrix; and the given curve the directrix of the cylinder. The different positions of the generatrix are called elements of the surface. The curve of intersection of the cylinder, by any plane, may be regarded as the base of the cylinder; and when the elements are perpendicular to the base, the surface is a right cylinder.'75. If the directrix of the cylinder is a plane curve, its plane may be taken for the co-ordinate plane XY, and its equation may be represented, generally, by f(x, y) = 0............ (1), which is read, a function of x and y equal to zero; the first member being a symbol to indicate an expression containing x, y and 92 INDETERMINATE GEOMETRY. constants; or that x and y are so connected that one can not vary without the other. Let x = az + a, y = bz + f, be the equations of a right line which is to be moved so as to generate the surface. Since the different positions of this generatrix are parallel, a and b remain constant, while, as the line is moved z from one position to another, a and j must change. But a and: are the co-ordinates of the point in which the generatrix pierces / the plane XY, Art. (46), and / C since this point must be on the directrix CD, the values of a and,3, when substituted for x and y, must satisfy equation (1). These values are = x - az, 3 = y - bz, and when substituted in equation (1), give f(x - aZ, y - bz) = 0, an equation expressing the relation between the co-ordinates of the different points of the generatrix in all of its positions. It is, therefore, the general equation of a cylinder, of which the directrix may be regarded as the base. In order then, to obtain the particular equation of a cylinder, whose directrix is given, we have simply to substitute, for x and y in the equation of the directrix, the expressions x - az, y - bz. 76. If the direct;rix is a circle, whose equation is x2 + y2 = R2 INDETERMINATE GEOMETRY. 93 the origin being at the centre, we have, by making the substitutions above referred to, (x - aZ)2 + (y - b)2 = R2............(1), the equation of an oblique cylinder with a circular base. If this cylinder be intersected by a plane parallel to XY, the equation of which, Art. (62), is z = g, x and y indeterminate, we have, by combining the equations, Art. (62), (X - ag)2 + (y - by)2 = R27 for the projection of the curve of intersection on XY. But this is evidently the equation of a circle, whose radius is R, Art. (34), and therefore equal to the base. But since this intersection is parallel to the plane XY, its projection is evidently equal to the line itself. WTe therefore conclude, that if a cylinder, with a circular base, be intersected by a plane parallel to the base, the intersection will be a circle equal to the base. If a and b are equal to 0, the generatrix becomes parallel to the axis of Z, or perpendicular to the base, the cylinder becomes right, and equation (1) reduces to x- 2 + y2= g2, the same as the equation of the base, z being indeterminate. OF THE CONE. 77. A conical surface, or cone, may be generated by moving a straight line, so as, continually, to pass through a fixed point and touch a given curve. 94 INDETERMINATE GEOMETRY. The fixed point is the vertex of the cone, and the parts of the surface separated by the vertex are called nappes. The intersection of the cone by any plane may be regarded as its base. The right line drawn from the vertex to the centre of the base is the axis of the cone, and if this axis is perpendicular to the plane of the base, the cone is a right cone. 78. If the directrix of the cone is a plane curve, its plane may be taken as the co-ordinate plane XY, and its equation be represented as in article (75), by f(x,y) = o........(1). If x', y' and z' are the co-ordinates of the fixed point, or vertex, the equations of the generatrix will be, Art. (50), X - x' = a(z - ), y b(z -')............(2), in which a and b change as the generatrix is moved from one position to another. These equations may be put under the form, x = az + (x' - az'), y =bz + (y' - bz'), in which the absolute terms, x' - az', y bz', are the co-ordinates of the point, in which the line pierces the plane XY, Art. (46), and since this point is on the directrix, whatever be the position of the generatrix, these values, when substituted for x and y in equation (1), must satisfy it, and give f(x' - az', y' - bz') = O. INDETERMINATE GEOMETRY. 95 Substituting in this equation, the values of a and b, in terms of x, y and z, deduced from equations (2), X - - x we have after reduction, =o............(3), Y z'~',, - ~,j f - zX z - X' an equation expressing the relation between x, y and z, for all positions of the generatrix. It is, therefore, the general equation of a cone, of which the directrix may be regarded as the base. In order then to obtain the particular equation of a cone, whose directrix is given, we have simply to substitute for x and y, in the equation of the directrix, the expressions, x'z -z'x y'z- z'y z -' z- 5'79. If the directrix is a circle, whose equation is z2 + y2 = R2, we have, by making the substitutions above referred to, ('/ - ZIX2 + y _' R or (.' z- Zx)2 + (yz - zly)2 R2 (Z - Z)2............(1), for the equation of an oblique cone with a circular base. If this cone be intersected by a plane parallel to XY, the equation of which, Art. (62), is 96 INDETERMINATE GEOMETRY. z = g, x and y indeterminate, we have, by combining the equations, (x'g - Z'x)2 + (y'g - zy)2 = R2 (g -')2, for the projection of the curve of intersection on XY. By dividing both members by z'2, this equation may be put under the form X $3 + C y)2 = R (g I 2Z)21 which is the equation of a circle, the co-ordinates of whose centre are zg and Yg, and the radius, the square root of the second Z,' Z, member, Art. (34). This projection being equal to the curve itself, we conclude, that if a cone, with a circular base, be intersected by a plane parallel to the base, the intersection will be a circle. The radius of this circle will decrease as g increases, until g = z', when the radius becomes 0 and the equation takes the form (x' - x)2 + (y' - y)2 = 0, which can only be satisfied, Art. (49), by making. = x Y= and the circle becomes a point. 80. If X =0, Y = O, -- = h, the vertex of the cone is on the axis of Z, at a distance, from the origin, represented by h; the cone becomes right, and equation (1), of the preceding article, becomes INDETERMINATE GEOMETRY. 97 (X2 + y2) h2 = RX (z la)2, or (x2 + y2) = (Z -,.)2.(1). If the angle, made by the elements of the cone with the plane of the base, be denoted by v, we have in the right angled triangle VAB', AV h tang AB'V = AV, or tang v= AB' R and equation (1) becomes (x2 + Y2) tang2 v = (z - h)2............(2), for the equation of a right cone with a circular base. 81. Through the axis of Y, in the figure of the preceding article, let a plane be passed intersecting the cone. This plane being perpendicular to the plane XZ, its equation will be the same as that of its trace on XZ, y being indeterminate, Art. (56). Let the angle, 7 98 INDETERMINATE GEOMETRY. which this plane makes with XY, be denoted by u, the equation of its trace, AR, will be, Art. (24), x = tang uz. The equations of the curve of intersection of the plane and cone may now be found, as in article (62). But as the different curves, obtained by changing the position of the cutting plane, form a class possessing very remarkable properties, the discussion of which is much simplified by referring the intersection to lines in its own plane, the latter method is chosen. Let us then take the right lines AX' and AY, as a new system of rectangular co-ordinate axes, and let us estimate the positive values of x' from A to X', and the positive values of y' from A to Y. Let M be any point of the curve of intersection. ~ Its co-ordinates, referred to the primitive planes, are x = AP, y = MR, z = RP, and referred to the new axes, AX' and AY, - x' = AR, y' = MR. From the right angled triangle APR, we have AP = AR cos u, RP = AR sin u, or x = - X' cos X, z = - X' sin u. We have also Y = y. If these values of x, y and z be substituted in equation (2) of the preceding article, the result expressing a relation between x' and y' for points common to the plane and cone only, will be the equation of the intersection. Making the substitution, we obtain (x'2 cos2 u + y'2) tang2 v - (- x' sin u - h)2, INDETERMINATE GEOMETRY. 99 or performing the operation indicated in the second member, and transposing, y'2 tang2 v = x12 sin2 U - X'2 COS2 U tang2 v + 2x'h sin u + h2, or recollecting that Sin2 u = cos2u tang2 u, and omitting the dashes of the variables, y2 tang2 v = X2 cos2 u (tang2 u - tang2 v) + 2xh sin u + h2...(), for the equation of the line of intersection of a plane and right cone with a circular base. In this equation, h may now be regarded as the distance from the vertex of the cone to the point in which the plane cuts the axis. 82. If in the above equation, v remain'ing the same, all values be assigned to u from 0 to 900, and all values to h, from 0 to infinity, it will represent, in succession, every line which it is possible to cut, from a given right cone with a circular base, by a plane. There are three distinct cases. First, when u = v, or tang u = tang v. In this case, the cutting plane makes the same angle with the base that the elements do, or is parallel, to one of the elements, and since tang2 ut = tang2 v, the coefficient of x2 becomes 0, the equation / — reduces to Y2 tang2 V= 2xh sin u + h2, and the curve represented by it is called a Parabola. 100 INDETERMINATE GEOMETRY. If in this equation, h = 0, the cutting plane passes through the vertex, and the equation reduces to y2 tang2 v = 0, which can only be satisfied by making y = 0, which, since x is indeterminate, is the equation of the axis of X, Art. (21). A right line is therefore regarded as a particular case of the parabola. Second, when u < v, or tang u < tang v. In this case, the cutting plane makes a less angle with the base than the elements do, or is parallel to none of the elements, see figure of Art. (80); and since, tang2 u < tang2 v, the coefficient of x2 is essentially negative and the curve represented by the equation is called an Ellipse. If in this case u = 0, the cutting plane is parallel to the base, cos u = I, sin u = 0, tang = 0, and the equation reduces to y2 tang2 u= - X2 tang2 v + h2, or dividing by tang2 v and transposing 2 + x2 __7 tang2 v which is the equation of a circle, Art. (35). If h = 0, u being still less than v, the plane passes through the vertex, and the equation reduces to INDETERMINATE GEOMETRY. 101 y2 tang2 v - x2 cos2 u (tang2'u - tang2 v), the first member of which is essentially positive and the second negative; it can therefore be satisfied for no values of x and y, except x = O, y = O, which are the equations of the origin of co-ordinates, Art. (16). A circle and point are therefore regarded as particular cases of the ellipse. Third, when Mb > v, or tang ut > tang v. In this case, the cutting plane makes a greater angle with the base than the elements do, or is parallel to two of the elements, viz. those cut from the cone by passing a plane through the vertex parallel to the cutting plane, and since i tang2'u > tang2 v, the coefficient of X2 is essentially positive, and the curve represented by the equation is called an Hyperbola. Ifin this case, h -0, the equation reduces to y2 tang2 v = x2 cos2'a (tang2 u - tang2 v), both members of which are essentially positive. Dividing by tang2 v, and placing cos2't (tang2 u - tang2 v) tang2 v we obtain y2 = r2X2, y -4- rx, 102 INDETERMINATE GEOMETRY. which, evidently, represents two right lines intersecting at the origin of co-ordinates, Art. (24), the equations of which are y = + rX, y = - rx. Two right lines, which intersect, are therefore regarded as a particular case of the hyperbola. 83. Resuming equation (1), Art. (81), dividing by tang2 v, and denoting the co-efficient of x2 by r2, as above, we have h sin u h2 y r2x2 + 2x.............(1). tang2v tang2 v Now let us transfer the reference of the points of the curve to a set of parallel co-ordinate axes, having their origin at D, the point in which the curve is cut by the axis of X, [see figure of Art. (80)]. Formulas (2), of Art. (67), become for this case, x = a + x', y = y, a representing the distance - AD, and b being equal to 0. Substituting these values in equation (1), we have h sin u h2 +- r2a2 + 2 sin a tang2 v tang2 v The origin of co-ordinates being on the curve, the absolute term 2raP + 2h sin u h2 r a q- a + tang2 v tang2 v * NOTE. It should be observed, that by placing the absolute term r2a2 + 2 + sin a+ =0, tang2 v tango v INDETERMINATE GEOMETRY. 103 must be equal to 0, Art. (38), and the equation, after omitting the dashes and placing h sin r2 +- r~a = P, tang2 v reduces to 2 = r2x2 + 2px..(2), a general equation, which may represent either of the above named curves; the parabola when r2 O0, the ellipse when r2 < 0, and the hyperbola when r2 > 0. OF THE PARABOLA. 84. If r2 = 0, equation (2) of the preceding article, becomes = 2............(1). This equation being of the second degree, the line represented by it is of the second order, Art. (33), and 2p being the only conwe have an equation of the second degree, and therefore two values of a, which will fulfil the required condition. Solving the equation, substituting the value of r2 and reducing, we find h (tang it f tang v) COs2 U (tang2 qb - tang2 v) In the parabola, u being equal to v, the first value reduces to 0 and the second, to infinity, but by striking out the common factor, tang u - tang v, the first value becomes finite and negative, as it should be to give the point D. In the ellipse, the first value is negative, the other positive, the negative value being used. In the hyperbola, both values are negative, the one which is numerically the least being used. 104 INDETERMINATE GEOMETRY. stant, the line is given when 2p is given, Art. (23). This constant is called the parameter of the parabola, and since from equation (1), we may deduce the proportion x: y:: y: 2p, we say, the parameter is a third proportional to the abscissa and ordinate of any point of the curve. 85. If equation (1), of the preceding article, be solved with reference to y, we have y = ~ /2px For every positive value of x, there will be two corresponding X real values of y; hence, the curve is continuous and extends from the origin, A, to infinity, in the direction of the positive abscissas; and since these values of y are.....x: - X equal with contrary signs, it follows that for each assumed abscissa, as AP, there will be two corresponding points of the curve, one above and the other below the axis of X, at equal distances, and the two values of y taken together will form a chord, as MM', which will be bisected by the axis of X; hence, the curve is symmetrical with regard to the axis of X. The line AX is called the axis of the parabola, and the point A, in which it intersects the curve, is called the vertex; and, in general, any straight line, which bisects a system of- chords perpendicular to it, is an axis of the curve in which the chords are drawn. If x = 0, we have y =-: 0, which proves that the curve is tangent to the axis of Y, at the origin, Art. (34). INDETERMINATE GEOMETRY. 105 If x is negative, the values of y are imaginary; hence, there is no point of the curve on the left of the axis of Y, If y = 0, we have X = 0, and the curve cuts the axis of X in one point only, at the origin. 86. The curve may be constructed by points from its equation as in Art. (22). This is clone geometrically, thus: Let AX and AY be two co-ordinate axes at right angles. Lay off from the origin in XD the direction of the negative ab- scissas AP' = 2p, and take any positive abscissa, as AP; on the P A \ j X line PP' as a diameter, describe a / circle, and from the points in which it intersects the axis of Y, draw lines parallel to the axis of X until they intersect the perpendicular erected to AX, at P. The points of intersection, M and M', will be points of the curve. For, from a known property of the circle, we have AD2 = AP' x AP = PM, or y: 2219x. 87. If a point whose co-ordinates are x and y, is on the curve, we must have the condition, Art. (23), y2 = 2px, or y2 - 2px = 0. If the point is without the curve, since its ordinate will be greater than the corresponding ordinate of the curve, we must have y2 > 2px, or y2 _ 2pz > 0. 106 INDETERMINATE GEOMETRY. If the point is within the curve, y2 < 2px, or y2 _ 2px < O. 88. If in equation (1), Art. (84), we make x = 23, we have Y2 2= p2 y = p, 2y = 2p. Hence, if a point, as F, be taken on the axis of the parabola, at >~- a distance from the vertex equal to one fourth of the parameter, the double ordinate, or thie chord, perpendicular to the axis at this point, will be equal to the parameter of the -:_B -:A: curve. If F be the point and M any point of...' the curve, the right angled triangle FPM will give FM ='FP2 + 1PM2, or, since FP AP - AF = x - P PM y, 2 we have FM (X - P) + y27, or squaring x _ and substituting for y2 its value 2px, 2 FM = / + px + + 4 2 INDETERMINATE GEOMETRY. 107 If from the vertex A, we lay off AB -, and draw BC 2 perpendicular to the axis, we shall have MC = BP = BA + AP =- x + P FM. 2 Hence, the distance from any point of the curve to the line BC, is equal to the distance from the same point to the point F. This remarkable property enables us to define a parabola to be a curve, such, that each of its points is at the same distance fr3om a given point and a given straight line. The given point, F, is called the focus, the given line BC, the directrix, and a straight line drawn through the focus perpendicular to the directrix, is the axis of the parabola. This property, also, gives another simple method of constructing the curve by points, when the directrix T and focus are given. Let BC be the directrix and F the focus. Through F draw FB perpendicular to BC, it will be the axis. At any point of the axis, as 2 X P, erect a perpendicular; with the focus F as a centre, and radius BP, describe arcs cutting the perpendicular in M and M'; these will be points of the curve, since FM = BP = MC. The curve may also be constructed by a continuous movement. Place one side DC, of a right angled triangular rule DCE, against the directrix; fasten one end of a string equal in length c to the other side EC, at the point E, and the other end at the focus; press a pencil Z'i against the string and rule, and as the rule is moved along the directrix, the point of the pencil will describe l;he parabola; for we always have 108 INDETERMINATE GEOMETRY. FM = MC. 89. Let x', y' and x", y" be the co-ordinates of any two points of the parabola. Since these are points of the curve, their co-ordinates will satisfy its equation and give the two conditions, Art. (23), y-2 = 2px', y"2 = 2px", from which, omitting the common multiplier 2p, we obtain the proportion r y /,2 X' X/ that is, the squares of the ordinates of any two points of the curve are proportional to the corresponding abscissas. 90. Let x", y" be the co-ordinates of any point, as M, on the curve, and through this point conceive any straight line to be drawn; its equation will be of the form, Art. (29), Y _ y " = d(x - X'')............(), in which d is undetermined. Since the given point is on the curve, we must have the condition, y12 = 2px" Subtracting this, member by member, from the equation \y2 = 2px, we have y2 y"/2 = 2p(x - X), or (y + y")(y - -y") = 2p(x - a"), which is the equation of the parabola, with the condition introduced that the given point shall be on the curve. Combining this INDETERMINATE GEOMETRY. 109 with equation (1), by substituting the value of y - y", taken from (1), we obtain (y + y") d(x- x") = 2p(x- "), or [(y + y") d - 2p](x - ") = 0............(2), in which x and y must represent! all the points common to the right line and curve, Art. (27). This equation being of the second degree, there are two such points, and only two; and the equation may be satisfied by placing! the factors separately equal to 0. Placing x - Xi/ = 0 we have x = x"/ and this value in (1) gives y - y". The values thus obtained are the co-ordinates of the given point, which is one of the points common to the two lines. By placing the other factor equal to O, we have (y + y")d - 2p = 0............(3), in which y must be the ordinate of the second point of intersection, M'. If now, the right line be revolved about the point M, so as to cause the point M' to approach M, y in equation (3), becomes. nearer and nearer equal to y", and finally, when the two points coincide, we shall have y = y", the line will be tangent to the curve, and equation (3) reduce to 2y"d = 2p, whence d = P. which is the value d must have when the assumed line becomes a tangent. Substituting this value of d in (1), we have y - y" P (x -- X) or 110 INDETERMINATE GEOMETRY. yy,, _ yY" 2 = pz - pzX which by the substitution of 2px" for y"/2, becomes yy = p(x + x")............(4), for the equation of a tangent line to the parabola at a given point. 91. If we multiply both members of the last equation by 2, and subtract the result, member by member, fiom the equation y 12= 2px", we have y 2yy" = - 2px, adding y2 to both members, y"2 - 2yy" + y22 = -2 2px, or (y - y)2 = y2 2px. The first member being a. perfect square, is positive for all values of y except y = y"; y2'- 2px, is therefore positive for all values of y and x, except y =- y", x - x", when it will be 0; hence, since x and y are the general co-ordinates of the tangent, all points of the tangent, except the point of contact, are without the curve, Art. (87). 92. If in equation (4), Art. (90), we make y = 0, we find o = p (~- xI+x), or x =-", for the distance AT, to the point in which the tangent cuts the axis; hence, we have PT = TA + AP = 2x". INDETERMINATE GEOMETRY, l The distance PT is called the subtangent, which, in general, is the distance from the foot of the ordinate of the point of contact, to the point in which the tangent cuts the axis, to which the ordinate is drawn; and in the parabola, is equal to double the abscissa of the point of contact. This property gives a simple method of drawing a tangent to a parabola at a given point. Let M be the point. From the vertex lay off, on the axis without the parabola, a distance AT, equal to the abscissa of the given point; draw a right line from the extremity of this distance to the point of contact, it will be the required tangent. 93. If the point M be joined with the focus F, we have, Art. (88), FM = x" +2. 2 But since AT = x", and AF P we also have FT = — x 2 7 2 hence, FM = FT, the triangle TFM is isosceles, and the angle FMT = FTM. Hence, if a right line be drawn from the focus of a parabola to the point of contact of a tangent, this line will make an angle with the tangent equal to that which the tangent makes with the axis. This property enables us to make the following constructions. First. To draw a tangent to the parabola at a given point. Draw a right line from the point, as M, to the focus; with this line as a radius and the focus as a centre, describe an arc cutting 112 INDETERMINATE GEOMETRY. the axis, without the curve, in a point, as T; draw a right line from this to the given point, it will be the required tangent, as the triangle MFT will be isosceles. Or otherwise, thus. Draw a right line through the given point perlpendicular to the directrix; join the point C, in which it intersects the directrix, with the focus, and through the given point draw a right line perpendicular to this last line, it will be the tangent. For, since MF - MC, the tri_C' -angle CMF is isosceles and therefore the angle FMT = CMT; but CMT = MTF; hence, FMT - MTF. Second. To draw a tangent from a point without the curve, as N. Join the point with the focus; with this distance as a radius, and the given point as a centre, describe an arc cutting the directrix in the points C and C'; through these points, draw lines parallel to the axis, cutting the curve in the points M and M'; join these points with the given point and we shall have the tangents NM and NM'. For, since MF = MC, and NF = NC, the line NM has two of its points equally distant from the points F and C, is therefore perpendicular to FC at its middle point and bisects the angle FMC. Let the co-ordinates of the given point N, be denoted by x' and y'. Since this point is on the tangent, we must have the equation of condition, Art. (23), y'y" = p (x' + x")............(l), and since the point of contact is on the parabola, we also have the equation of condition, y _= 2px". INDETERMINATE GEOMETRY. 113 In these equations x" and y" are unknown, and since one is of the first and the other of the second degree, their combination will give an equation of the second degree, and there will be two values of x" and two corresponding of y". Combining these equations by substituting the value 11 = y 12'2p in the first, we obtain y//2 - 2y'y" = - 21x'............ (2), from which we deduce the two values y = y' t 7('2- 2px). The values of y" will evidently be real, when y12 - 2px'> o0 that is, when the given point is without the curve, Art. (87), and there will be two tangents, as appears by the geometrical construction. The values of y" will be equal when the point is on the curve and there will be but one tangent. They will be imaginary when the point is within the curve and there will be no tangent. The corresponding values of x" being found, each set of co-ordinates may be substituted, in succession, in equation (4), Art. (90), and the equations of the two tangents thus determined. Third. To draw a tangent parallel to a given line as SR. Produce the line until it intersects the axis at S, with the focus as a centre, and the distance FS as a ra- / dins describe an arc cutting the /given line in R, join this point with the focus, the point M, in which the I last line intersects the curve will be A T F the point of contact, through which draw MT parallel to the given line, 8 114 INDETERMINATE GEOMETRY. it will be the required tangent. For, since MT is parallel toRS, and FS = FR, we have FM = FT. 94. Since the triangle FMT is isosceles, the line FD, drawn per-:r 1-: pendicular to the base MT, will pass through its middle point; and since AT = AP, Art. (92), the line AD,_~ —I -A — X also passes through the middle point of MT: Hence, iffrom the focus of a parabola, a right line be drawn 2pependicular to any tangent, it will intersect this tangent, on the tangent at the vertex; and conversely. Since the triangle FDT is right angled at D, we have FD2 = AF X FT, and since AF is constant and FT = FM; the square of the perpendicular FD, will vary as the first power of the distance from the focus to the point of contact. 95. If in equation (1), Art. (93),'y"y/ = p(x' + x").. (1), we regard x" and y" as variables, it will be the equation of a right line, Art. (25); and since both /Ef values of x" and y" deduced from equation (2), Art. (93), must satisfy this equation, the x line represented by it will pass through both points of contact, and will therefore be the indefinite chord which joins these INDETERMINATE GEOMETRY. 115 points. If any point as O, be taken upon this chord, its co-ordinates which we will denote by c and d, when substituted for x" and y, will satisfy the equation and give the condition y'd = p (' + c)............(2). Now it is evident, that every set of values for x and y' which will satisfy this equation, will give a point from which, if two tangents be drawn to the parabola and the points of contact be joined by a chord, this chord will pass through the point 0. Hence, if y' and x' be regarded as variables in this equation, it will represent a right line every point of which will fulfil the above condition. This line is called the polar line of the point 0, which is called the pole. If through the point O, a line be drawn parallel to the axis AX, the ordinate of the point in which it intersects the curve will be equal to d, and the equation of a tangent to the parabola, at this point, will be, Art. (90), yd = p (x + x"), and this tangent'is evidently parallel to the line represented by equation (2), that is to the polar line, Art. (28), If the line OA' be further produced till it intersects the polar line in N, the ordinate of this point will be d, which substituted for y' in equation (1), will give y"d = p (x + X"), for the equation of the chord corresponding to this point N, and this is parallel both to the polar line and tangent. These properties give the following constructions: 1. The pole being given, to construct the corresponding polar line. Through the pole draw a line parallel to the axis of the parabola; at the point in which this intersects the curve, draw a 16 INDETERMINATE GEOMETRY. tanogent; through the pole draw a chord parallel to this tangent, and at its extremity draw another tangent; through the point in which this intersects the line first drawn, draw a line parallel to the chord, it will be the polar line. 2. The polar line being given, to construct the pole. Draw a tangent parallel to the polar line; through the point of contact draw a line parallel to the axis; from the point in which this intersects the polar line, draw another tangent; through the point of contact thus determined, draw a chord parallel to the polar line, it will intersect the line parallel to the axis in the required pole. 96. If the focus be taken as the pole, the co-ordinates of which are d = O, c = 2 equation (2) of the preceding article reduces to O p, or x'= - 2 2' y' being indeterminate, which is the equation of the directrix, Art. (21). The directrix is then the polar line of the focus. Hence, if any chord be drawn through the focus of a parabola and two tangents be drawn at its extremities, these tangents will intersect on the directrix. 97. If in the general equation of a right line passing through a given point, Art. (29), we substitute for x' and y', the co-ordinates of the focus, we shall have y = a( -.......... INDETERMINATE GEOMETRY. lT for the equation of any chord passing through the focus. Combining this with the equation of the parabola, y2 - 2px, by substituting the value x Yp, we have ( Y2p ) or ay2 - 2y = p2. Denoting the two roots of this equation by y' and y", we have from a well known principle of Algebra, y/y"- - p2, and if d and d' denote the tangents of the angles made with the axis, by two tangents drawn at the extremities of this chord, we have, Art. (90), d _ P d' =P y' y" whence, dd' - P. yyl or substituting for y'y" the above value, dd' -- 1, or dd' - 1 = 0. Hence, Art. (28), if at the extremities of a chord passing through the focus of a parabola, two tangents be drawn, they will be perpendicular to each other, and intersect on the directrix, Art. (96). And conversely, if two tangents to the parabola areperpendicular to each other, the chord joining their points of contact will pass through the focus. For, let S'M and S'M" be the two tangents. If the chord MM" does not pass through the focus; through the focus and the point M, draw MM'; at M' draw the tangent M'S. From what precedes, it must be perpendicular to MS'; hence, SM' 18 INDETERMINATE GEOMETRY. and S'M" must be parallel. But since the tangent of the angle which a tangent to the parabola makes with the axis is ~, Art. y,1 (90), no two tangents can be parallel, for no two points have equal ordinates. It is then absurd to suppose that MM" does not pass through F. 98. Through the point of contact of a tangent, let any other straight line be drawn, its equation will be of the form, Art. (29), - Y" dl( - x")............(1). If this line is perpendicular to the tangent, we must have, Art. (28), dd'- 1 = O, or d' d' But, Art. (90), d = P whence, d P Substituting this in equation (1), we have Y - Y" = - ($ - x")............(2), for the equation of a straight line perpendicular to the tangent at the point of contact. This line is called a normal. If we make y = 0, in equation (2), we have X - x" = - -X! —p~ INDETERMINATE GEOMETRY. 119 in which, x is the distance AR from the origin to the point in which the normal intersects the axis, and x-x" =AR - AP = PR = p. The distance PR, from the foot of the ordinate of the point of contact, to the point in which the normal cuts the axis, is called the subnormal. Hence, the subnormal in the parabola is constant and equal to half the parameter of the curve. This property enables us to construct a tangent at a given point. Draw the ordinate of the point; from its foot lay off a distance equal to one half the parameter; join the extremity of this distance with the given point, through which draw a perpendicular to the last line, it will be the required tangent. OF THE PARABOLA REFERRED TO OBLI4QOUE CO-ORDINATE AXES. 99. It was observed in Art. (71), that two' classes of propositions might arise in the transformation of co-ordinates. As an example of the second class, let it now be proposed to ascertain if there are any other co-ordinate axes, to which if the parabola be referred, its equation will be of the same form as when referred, to its axis and the tangent at its vertex. For this purpose, let us take the general formulas (3), Art. (67), x = a + x' cos a + y' cos a', y = b + x' sinr a + y' sin a', and substitute the values of x and y in the equation y2 ='2p........... ( ). We thus obtain 120 INDETERMINATE GEOMETRY. b2 + 2bx' sin a-+ x12 sin2 a1 + 2by' sin o,' + 2x'y' sin a sin c.' + y'2 sin cU' = 2pca + 2_px' cos a + 2py' cos a', or transposing, arranging and omitting the dashes of the variables, y2 sin2 a, + x2 sin2 a + 2xy sin a sin a' ~ 2(b sin ca' - p cos a')y + b2 - 2pay = 2(p cos a -b sin a)x...(2), which is the equation of the parabola referred to any oblique axes. In order that this equation shall be of the same form as equation (1), the absolute term, in the first member, and the terms contain. ing X2, xy, and y, must disappear, which requires that 62 - 2 pa 0............(3); sin2 a 0............(4); sin a sill' 0............(5); b sin a' - p cos a' = 0............(6). These equations contain four arbitrary constants, it is therefore possible to assign such values to the constants as to satisfy the four equations, and thus reduce equation (2) to the proposed form. Equation (3) is the equation of condition that the new origin shall be on the curve, Art. (87). Equation (4) can only be satisfied by a = 0, or = 180~; hence the new axis of X must be parallel to the axis of the curve. Equation (5) is satisfied by sin a = 0, without introducing any new condition. Equation (6) can be put under the form sin a' tang -- - tang ac' COs aI b and therefore, Art. (90), expresses the condition that the new axis of Y shall be tangent to the curve. INDETERMINATE GEOMETRY. 121 Since we have thus far introduced but three independent conditions, and since a, b and a' are still undetermined, we may assign a value at pleasure to either of them, whence the other two will become known, and an infinite number of sets of co-ordinate axes be thus determined, which will fulfil the required condition, each of which will be subjected to the three conditions expressed by equations (3), (4) and (6). Substituting the above conditions in equation (2), and observing that, since sinla = 0, cosa = 1, we have y2 sin2 a' = 2px, or y2 2p sin2 a' or, denoting the coefficient of x by 2p' y2 = 2p'x............ (7) an equation of the same form as (1). 100. Solving the last equation, with reference to y, we find y = - V2p'x, and we see, as in Art. (85), that every positive value of x, gives two real values of y, equal with contrary signs, and that these two values taken together form a chord, as MM', parallel to A_ the axis of Y, which chord is bisected by the axis of X, at P. The line A'X is therefore called a diameter of the parabola; and, in general, any straight line which bisects a system of parallel chords is a diameter, of the curve in which the chords are drawn. The points in which a diameter intersects the curve are called the vertices of the diameter. Since condition (4) of the preceding article requires the new axis of X to be parallel to the axis of the curve, it follows that all the diameters of the parabola are parallel to each other. 122 INDETERMINATE GEOMETRY. Since condition (6) requires the new axis of Y to be tangent to the curve at the origin, it also follows that each diameter bisects a system of chords parallel to the tangent at its vertex. If the parabola is given, traced upon paper, a diameter may be found by drawing any two parallel chords as MM' and bisecting them by a straight line as PP; this line will be a diameter. Draw any two chords perpendicular to this diameter and bisect c them by a straight line, this will be /. ~~, the axis, Art. (85). At the vertex of the first diameter, A', draw a line / A 27 1 _ parallel to the chords which it biB A~ //!i SXX- - sects, it will be a tangent to the o. ~I g| curve. At the vertex, A, of the parabola, draw a line perpendicular to the axis, it will also be a tangent. At the point D, where these tangents intersect, draw a perpendicular to the first, it will intersect the axis in the focus F, Art. (94). The property, that each diameter bisects a system of chords parallel to a tangent at its vertex, suggests the following construction for drawing a tangent parallel to a given line, as BC. Draw two chords parallel to the given line; bisect them by a straight line PP, and at the point A', where this intersects the curve, draw a line parallel to the given line, it will be the required tangent. 101. The coefficient 2p' in equation (7), Art. (99), is called the parameter of the diameter A'X, and, as in Art. (84), is a third proportional to any ordinate and its corresponding abscissa. If we represent the distance FA' by r, and recollect that the angle FA'D = FTD is denoted by a', Art. (99), we shall have in v AD~j/ the right angled triangle FDA' 2 4 A; FD = r sin a', or INDETEPRMINATE GEOMETRY. 123 -FD2 - r2 sin2 a'. But we also have, Art. (94). Fib2 = FA x FA', or FD2 Pr. Equating these two- values of FD2, we have r2 sin2 c = -- r; whence sin2 a! - a. 2 2r Substituting this value of sin2 ai, in the expression, Art. (99), p 2p sin2 a' it reduces to 2p' = 4r that is, the parameter of any diameter of the parabola, is equal to four times the distance from the vertex of the diameter to the focus. 102. Let x" and y" be the co-ordinates of any point of the parabola referred to the diameter A'X and the tangent A'Y. The equation of a right line passing through this point will be X y -yl = d(x - x"), in which d will represent the ratio of the sines of the angles which the line makes with the co-ordinate axes, Art. (20). By a process identical with that pursued in Art. (90), we can find the value of d, when the line becomes a tangent, and thus deduce the equation of the tangent, 124 INDETERMINATE GEOMETRY. yy = p'(x + x"). By making y = 0, we find x = - x" = A'T; hence as in Art. (92), the subtangent PT, is equal to twice the abscissa of the point of contact. And a tangent may be drawn at a given point by drawing the ordinate MP, of the point, parallel to the axis A'Y, laying off the distance A'T equal to the abscissa A'P, and joining the extremity of this distance with the given point. 103. Let AM be an arc of a parabola, in which inscribe any polygon, as AM'......MP. At the points M, M', &c., draw the z......... tangents MT, M'T', &c. o Through the middle points of the chords MM', &c., draw the diameters RS, ~T' T'A' Pr' -' R"S', &c., and draw the ordinates MP, M'P', &c. It is evident that for each chord there will be a trapezoid, as MM'P'P, within the parabola, and a corresponding triangle, as OTT', without. Since the points of contact M and M', when referred to the diameter RS and tangent line at its vertex, have the same abscissa VR, the subtangent will be the same for each, Art. (102), and the two tangents MO and M'O will intersect the diameter VS, at the same point O; hence the altitude of the triangle OTT' will be equal to the line RR', drawn through the middle points of the two inclined sides of the trapezoid P'M'MP; and since, AP = AT, and AP' = AT', we have PP' = TT'; INDETERMINATE GEOMETRY. 1' 25 hence, the area of the trapezoid, which is measured by RR' x PP', is double the area of the triangle, which is measured by!RR' X TT'; 2 and so for each trapezoid and corresponding triangle, and the sum of all the interior trapezoids will be double the sum of the corresponding triangles. If now, the number of sides of the polygon be increased indefinitely, the sum of the trapezoids will be the same as the curvilinear area AM'MP, and the sum of the triangles the same as the exterior area TMMI'A; hence the first area is double the second. But the sum of these two areas is equal to the area of the triangle MTP, therefore AM'MP = 2MTP. 3 But since TP = 2AP, we have triangle MTP = rectangle ALMP. Therefore, the area of a portion of the parabola is equal to twothirds of the rectangle described on the ordinate and abscissa of t~he extreme point. OF THE POLAR EQUATION OF THE PARABOLA. 104. Let us resume the equation y2 = 2px, and substitute for y and x, their values taken from the formulas (2) of Art. (09); x = a' + r cosv, yv b + r sin v. 126 INDETERMINATE GEOMETRY. We thus obtain b/2 + 2 b6r sin v + r2 sin2 v = 2p(a' + r cos v), or transposing and arranging, r, sin2 v + 2 (b' sin v - pcos v)r + b'2 - 2pa' = 0......(1), which is the general polar equation of the parabola, Art. (69). By assigning particular values to a' and b', the pole may be placed at any point in the plane of the curve. First. If it be required that the pole shall be on the curve, we must have, Art. (87), b-2 - 2pa' = 07 and equation (1) reduces to [r sin2 v + 2(b' sin v - cos v)]r = 0, which may be satisfied by placing r = 0, or r sin2 v + 2(b' sin v - p cos v) = 0............(2). Since the pole is on the curve, as at P, it is evident, that one value of r should be 0, whatever be the value of v; and that the other value, deduced from equation (2), should, as v is changed, give the distance of each point of the curve from the pole P. If the point M is moved along the curve until it coincides with P, the second value of r will become 0, and equation (2) will reduce to b' sin v -p cos v = 0, or sin v p sin - tang v - p cos v b' INDETERMINATE GEOMETRY. 12'7 as it should, Art. (90), since the radius vector will now coincide, in direction, with the tangent PT. Second. If the pole be placed at the focus, we must have a' = b' = 0, 2 and these values, in equation (1), give q.2 sin2 v - 2p cos vr - p2 = 0, and after transposing p2 and dividing by sin2 v, 2p cos v - p2 sinll v sin2 v Solving this equation, we have p cos v 2 + p2 Cos2 v sin2 v Sil' V Si1 v or r pcos v /p2 (sin2 v cos2 v) p COS v p sin2 V sin4 v sin2V since sin2 v + cos2v = 1. As the cos v must be less than radius or unity, we have p cos v < p, and the second value of r is always negative, and must therefore, be rejected, Art. (69). The first value may be placed under the form = p (cos v + 1) Sill2 V and since 128 INDETERMINATE GEOMETRY. sin2 v 1 - cos2 v = ( + Cos)1 - cos v), it may be still further reduced to........... ), 1 - vcos which is positive for all values of v. If v = 0, cos v = 1, and the value of rbecomes 00 and the radius vector takes the direction AX, and gives that point of the curve which is at an infinite distance. If v = 90~, cosv = 0, and the value of r becomes r =p = FM. If v = 180, cos v = - 1, and r = FA. Thus by varying v from 0 to 360~, all the points of the parabola may be determined. If we wish to estimate the variable angle fiom the line FA, to the right, instead of in the usual way, from the line FX to the left, we have simply to change v into 1800 - v', in which case cos v = - cos v', and the value of r', equation (3), becomes 1 + cos v' in which v' 0, gives r- P and v' 180, gives Tr =-2 m ir 0"' O INDETERMINATE GEOMETRY, 129 OF THE ELLIPSE AND HYPERBOLA. 105. We have seen, Art. (83), that the equation y2 = r2x2 + 2px, or y2 = 2px + r2x2......(1), represents the ellipse when r2 is negative, and the hyperbola when it is positive. This equation being of the second degree, the ellipse and hyperbola are both lines of the second order, Art. (33). If in the equation, we make x = 0, we find y- = 0O; hence the axis of Y is tangent to each curve, at the origin of coordinates, Art. (34). If we make y = 0, we have 2px + r2X2 = 0, or x(2p + r2x) = 0, which may be satisfied by making or 2p ~+ r2 =;x whence x hence each of the curves intersects the axis of X in two points, one at the origin, and the other at a distance from it equal to 2p Now let us transfer the origin of co-ordinates, to a point on the axis of X, at a distance P -, equal to half the distance from r the origin to the second point in which the curve cuts the axis; the new axes being parallel to the primitive. In formulas (2), of Art. (67), we must then have 9 130 INDETERMINATE GEOMETRY. a = P b 0, r2 and the formulas become x=x_ I y y=. r2 Substituting these values of x and y in equation (1), we have y'2 2p(- ) + r2(/2 +- P r2, ~ r or reducing and omitting the dashes, 2- r2X2 -2 r2 If in this we make y = 0, we have X2 = P....(3), or = P; hence, each curve now intersects the axis of X in two points, one on the right and the other on the left of the origin, at equal distances from it. If x = 0, we have 2 / 2 y2 = - 4..(4), or y i - and these values of y will be real for the ellipse, and imaginary for the hyperbola; hence, the ellipse intersects the axis of Y in two points, at equal distances from the origin, one above and the other below the axis of X; and the hyperbola does not intersect the axis of Y. Giving to r2 its negative sign for the ellipse, expressions (3) and (4) will be essentially positive, and we may write INDETERMINATE GEOMETRY. 131 a, ~ b2; -4 = a2 2 = —r r from which, by deducing the values of p2 and equating them, we have a2r4 -r2b2 or ~2 a2 and substituting this in either of the above equations, we find b4 b2 p2 _ or p -- a2 a By the substitution of these values of r2 and p2 in equation (2), and reducing, we have the equation of the ellipse, a2y2 + b2X2 = a2b2............(e). r2 For the hyperbola is essentially negative, we must then place it equal to - b2, while the expression for a2 will remain:unchanged. If then, in the above equation, we simply change bP into - b2, we obtain the equation of the hyperbola, a2y2 - b2x2 = - a2b2............(h). Furthermore, it is evident from the preceding discussion, that any expression containing b, belonging to the ellipse, will become the corresponding one for the hyperbola, by changing 6b into - b2, or b into b V — 1. 106. Solving equation (e) with reference to y, we have b2 y2 (2 - 2), = a2.... ]. 32 INDETERMINATE GEOMETRY. in which every value of x numerically less than a, whether positive or negative, gives two real values of y equal with contrary signs: y Hence, C being the origin, CX and CY the axes of co-ordinates, and CB and CA each numerically equal — ~ A (. ]. ~j to a, the curve is continuous beA< i le ] ) B X tween the points A and B; and since each set of the equal values of y forms a chord as MM', which is bisected by the axis of X, the curve is symmetrical with respect to the line AB. x = + a or - a, gives y= 0; hence the ordinates at the points A and B, when produced, are tangent to the curve. And as every value of x numerically greater than a, positive or negative, gives imaginary values for y, there are no points of the curve without the tangents at A and B. y = 0 gives x = i~ a = CB or CA; and since the line AB bisects a system of chords perpendicular to it, it is an axis of the curve, Art. (85), and A and B are its vertices. x = 0 gives y = - b = CD or CD'. Any number of other points of the curve may be constructed by assigning values to x in equation (1), deducing and constructing the corresponding values of y, and the curve in form and position will be as in the last figure. If equation (e) be solved with reference to x, we have X ~ -at/bV2 _ y2, INDETERMINATE GEOMETRY, 133 from which it may be shown as above, that the curve is symmetrical with respect to the axis of Y, and does not extend beyond the tangents at D and D', and that the line DD' is an axis of the curve. The definite portion of the line AB, included within the ellipse, is called the transverse axis, and the portion DD', the conjugate axis; the transverse axis being the longest of the two. The point C, in which the axes intersect, is the centre of the ellipse. The vertices of the transverse axis are also called the vertices of the curve. Equation (e) is called the equation of the ellipse referred to its centre and axes; in which a represents the semi-transverse, and b the semi-conjugate axis. 107. If we solve equation (h), Art. (105), with reference to y, we have 62 y2 = (_- a2), y = b V 2- a2, a2 a in which every value of x numerically less than a, positive or negative, gives imaginary values of y: Hence, C being the origin, CX and CY the axes of co- I ordinates, and CA and CB each I A numerically equal to a, there, are no points of the curve be- / tween A and B. x = + a or - a, gives y = i O; hence, the ordinates at the points A and B, when produced, are tangent to the curve. Every value of x greater than a, positive or 134 INDETERMINATE GEOMETRY. negative, gives two real values of y equal with contrary signs; hence, the curve is continuous and extends to infinity in both directions beyond the points A and B, and is symmetrical with respect to the axis of X. y - O gives x =:1: a = CA or CB, and since the line BA produced, bisects a system of chords perpendicular to it, it is an axis, and the definite portion BA = 2a, included between the points A and B, is called the transverse axis of the curve, the points A and B being its vertices or the vertices of the curve. x = 0 gives y2 = - b2, y = t4- b / - 1; hence, the curve does not intersect the axis of Y. A sufficient number of other points being constructed from the equation, the curve may be drawn as in the figure, the two branches being equal, since values of x which are numerically equal with contrary signs, as CP and CP' give the same values for y. If equation (h) be solved with reference to x, we have X = ~ a y2 + b2, in which every value of y gives two real values of x, equal with contrary signs; hence, the line CY is an axis of the curve. This line, as seen above, does not cut the hyperbola, but if we lay off on it from C, distances above and below each equal to b, the portion DD' = 2b is called the conjugate axis, the point C being the centre of the hyperbola. Equation (h) is called the equation of the hyperbola referred to its centre and axes, in which a and b represent the semi-axes. INDETERMINATE GEOMETRY, 135 108. If in equations (e) and (h), and, in general, in the equation of any curve, we change x into y Bn and y into x, the effect is to change the line which at first is regarded as the / _ axis of X, into the axis of Y and the converse; or if the axes are at right A C I/ angles, to revolve the curve 90~ about the origin. Thus if the equation A a2y2 + b2x2 - a2b62 represents the ellipse as indicated by the full line, the equation a2x2 + b2y2 = a2b2, will represent it as indicated by the broken line. 109. If a point is on the ellipse, its co-ordinates must satisfy the equation of the ellipse, Art. (23), and we must have a2y2 + b2x2 - a2b2 = 0. If the point is without the ellipse, y will be greater than the corresponding ordinate of the ellipse, Art. (37), and we have a2y~ + b2X2 + a2b2 > 0. If it is within the ellipse a2y2 + b2X2 - a2b2 < 0. 110. The corresponding conditions for the hyperbola, by changing, in the above, b2 into - b2, Art. (105), will be a2y2 - bSX2 + a2b2 -- 0. a2y2 - b2.2 ~+ a2b2 > 0. a2y2 - b2x2 + a2b2 < 0. 136 INDETERMINATE GEOMETRY. 111. If a - b, the axes of the ellipse are equal, and equation (e) becomes y2 + x2 = a2 which is the equation of a circle, the radius of which is equal to either semi-axis, Art. (35). 112. Under the same supposition, equation (h) becomes y2 _ X2 = a2, and the curve is called an equilateral hyperbola. 113. If through the centre of an ellipse any right line be ~Y drawn, its equation referred to the axes CX and CY, will be y = d'x............(l,) 9 ]:f/'E P- e X in which d' represents the tangent /iMK( I of the angle which the line makes with CX, Art. (24). Combining this with equation (e), by substituting for y2 its value d'2x2, we obtain d'2a2x2 + b2x2 = a2b2; whence, for the abscissas of the points of intersection, Art. (27), we have a 2b2 d'2,a,2 q-b2 and by the substitution of this in equation (1), y = i d' /. _7o o I 7 INDETERMINATE GEOMETRY. -137 Since these values of x and y are real for every value of d', it follows that whatever be the position of the line CM, it will intersect the ellipse in two points; and since the co-ordinates of these points are equal with contrary signs, they will be on opposite sides of the origin, and at equal distances from it, as at M and M'. Hence, every straight line pcassing through the centre of an ellipse and terminated by the curve is bisected at the centre. 114. If in the above expressions we put - b2 for b2, the corresponding values for the hyperbola are _ a2b2 a a22 d'/2a2 - b y d'2a2 4 b2 which are real, whenever d'2a2 - b2 < 0 01o dO,< or a that is, whenever d', either positive or negative, is numerically less than b_, the line will cut the hyperbola in two points and be bisected at the centre. If d- b the values are both infinite, and the points of intersection are at an infinite distance from C. If 4 d' > -a a the values are imaginary, and the line will not intersect the curve. Hence, if at the point A, we erect the perpendiculars AE and AE', p each equal to b, and draw the lines CE and CE', these lines will just limit the curve, since 138 INDETERMINATE GEOMETRY. tang ACE = d' AE CA a b2 115. If we multiply both members of the expression p -, a Art. (105), by 2, we have 2p = 2ba which as in the parabola, Art. (84), is called. the parameter, and gives the proportion a: b:: 2b5: 2p, or 2a: 2b:: 2b: 2p. Hence, the parameter of the ellipse or hyperbola is a third proportional to the transverse and conjugate axes. 116. If in equation (e), we substitute for y the expression 5, a we find 2 = a2 - b2, or x = i V/a2 - b2; and conversely, if either of these values be substituted for, we shall find b2 y =:1: -; a firom which we see, that there are two points on the transverse axis of the ellipse, at which, if an ordinate be drawn, it will be equal to one half the parameter of the curve; hence, the double ordinate, or the chord perpendicular to the transverse axis, at each of these points, is equal, to the parameter of the curve. INDETERMINATE GEOMETRY. 139 These points are called the foci of the ellipse, and may be constructed thus: With either extremity of the conjugate axis as a centre, and the semi-transverse axis as a radius, describe an arc, the points in which this are cuts the -,,' transverse axis will be the foci. For in the right angled triangle DCF or DCF', we have, Art. (4), CF2 = CF'2 = a2 - b2, OF = CF' = i /a2 - b2. 117. For the hyperbola, the values of x, in the preceding article, become x=~V'ai M+ b24- V..(1), either of which substituted in equation (h), will give Y - i -, a and the points determined on the transverse axis, by laying off the above values of x are the foci of the hyperbola, and may be constructed thus: At either vertex of the hyperbola erect a perpendicular equal to b; join its extremity with the centie; with the last line a CE, as a radius, and with the point C as a centre, describe an are; the points in which this are cuts the transverse axis produced, will be the foci. For we have CF2 = CE2 a2+ b2 CF = CF =:= Va2 + b2. 118. The distance from the centre to either focus of the ellipse, 140 INDETERMINATE GEOMETRY. divided by the semi-transverse axis, is called the eccentricity of the curve, the expression for which is I/a2 - b2 a If a = b, this reduces to 0; hence, the excentricity of a circle is nothing, and the foci are at the centre. 119. The expression for the eccentricity of an hyperbola, is v a2 + b2 a which, when a = b, becomes for the equilateral hyperbola, Art. (112), a a/2 2. a 120. If we denote the distance CF by c, and the distance from D any point of the ellipse, as M, to the focus F, by r, the general expression for the 4 square of this distance will be, Art. (17), r2 = (x- x')2 + (y- y') - in which x' - C, ye l 0; whence -2= r2 = (x - c)2 + y2, and this, by the substitution of the values INDETERMINATE GEOMETRY. 141 Y2 b2 (a2 2), c2 a2 _ 2 becomes b2 r2- X= 2 2cx + a2 _ __x2 a2 or a4 - 2ca2x + c2x2 r2 a2 and extracting the square root, a2 -c CX r= = a...........(1), a a using the plus sign of the root only, as we require merely the expression for the length of FM. Since CF' = - c; if in the above expression (1), we put - c for c, we shall evidently obtain the distance F'M, which we denote bv r'; hence, r' a +...........(2). Adding equations (1) and (2), member by member, we have r + r' = 2a; hence, the sum of the distances from any point of the curve to the two foci is equal to the transverse axis. This remarkable property enables us to define an ellipse to be, a curve such, that the sum of the distances, from any point to two fixed points, is always equal to a given line. It also gives the following construction, of the curve by points, the foci and transverse axis being given. Divide the transverse axis into any two parts, the point of division being between the two foci, as at E; with one part EB A F, F 142 INDETERMINATE GEOMETRY. as a radius, and one focus F, as a centre, describe an arc; with the other part AE, as a radius, and the other focus as a centre, describe a second arce; the points of intersection of these arcs will be points of the ellipse. For we have FM + F'M = EB + AE = 2a. The curve may also be constructed by a continuous movement, thus: Take a thread, in length equal to the transverse axis, and fasten an end at each focus; press a pencil against the thread so as to draw it tight; the point of the pencil as it is moved around will describe the ellipse; for the sum of the distances, from this point to the foci, is always the same and equal to the transverse axis. 121. If F and F' are the foci of the hyperbola, and the dis-tances FM and F'M be denoted by r and r', we may deduce exL pressions for them from expressions (1) and (2) of the preceding article, by changing 12 into - b2, the only effect of which will be to make c = /a2 + b instead of /a2 - b0, and as for all points of the curve x must be greater than a, Art. (107), c- must also be greater than a, and the expression for the nua merical value of FM = r will be r a..........(3). a The form of the expression for r' will remain unchanged; hence, r'I = cx + a........... (4). INDETERMINATE GEOMETRY. 143 Subtracting the first of these from the second, we have A,- r = 2a; hence, the difference of the distances from any point of the curve to the twofoci, is equal to the transverse axis; and the hyperbola may be defined to be, a curve such, that the difference of the distances from any point to two fixed points is equal to a given line. The curve may be constructed by points thus: With one focus F' as a centre, and any radius F'E, greater than the distance to the farther vertex, describe an arc; with the other focus and a radius FM, equal to the first radius minus the transverse axis, describe another arc; the points of intersection will be points of the curve. For, we have F'M FM = 2a. It may also be constructed by a continuous movement. Take a rule of sufficient length as F'L, and fasten one end at the focus F'; at the other end of the rule fasten one end of a string shorter than the rule by the transverse axis; fasten the other end of the string at the other focus, F; press a pencil against the string and rule; as the rule revolves about the focus F', the point of the pencil will describe the branch AM. For, we have F'L - 2a = FM + ML, or F'L - ML - FM = 2a; hence F'M - FM = 2a. By placing the end of the rule at F, the other branch may be described. 144 INDETERMINATE GEOMETRY. 122. By a reference to equations (1) and (2), Art. (120), it is seen that the distance from any point of the curve to either focus is expressed rationally in terms of its abscissa. This remarkable property of the foci is possessed by no other points in the plane of the curve. For, if there is any other point, let its co-ordinates be x' and y'; x and y denoting the co-ordinates of any point of the curve. The square of the distance from x, y, to x', y', Art. (17), is = ( )2 + (y y)2 or squaring x - x' and y - y', and substituting for y its value ~ _ a2 _ 2 a we have D2 2- x2 _ 2xx' + x'2 + b2 =F 2y' /a2 / _ 2 + y'T a2 a It is evident that the value for D can not be rational, in terms of x, unless the term containing the radical disappears. But this can not be unless y' -- 0, that is, the required point must be on the axis of X. Substituting this value for y', D2, after changing the order of the terms, becomes a 2 _ b2 D2 = (62 + x'2) _ 2xx' + - 2 a2 Now no value of xa' can make this expression a perfect square. unless it makes the first and last terms perfect-squares, and twice the square root of their, product equal to the middle term, that is, we must have 62 + x'2 = m2, a2 - X 2- n2 - 2xx' = 2mn, a2 INDETERMINATE GEOMETRY. 145 m2 and n2 being two perfect squares. From the last expression we have X2X2 n2 in which, substituting the value of n2 taken from the second, we have a2 - b2 Substituting this in the first, we have 62 + x12 = a2 _ b2 which can be satisfied for no values of x', except x = I /a — b, the abscissas of the two foci. In a similar way it may be shown, that the foci of the hyperbola and parabola alone possess the above named property. 123. If in equation (h) we substitute b for y, we deduce x2 - 2a2 z = a V2; therefore, the abscissa of that point of the hyperbola, whose ordinate is equal to the semi-conjugate axis, is equal to the diagonal of a square, the side of which is the semi-transverse axis. Hence, the curve and transverse axis being given, the conjugate axis / A - may be constructed thus: At the vertex A, erect a perpendicular AR = a; join the extremity 10 146 INDETERMINATE GEOMETRY. with the centre; with C as a centre, and CR as a radius, describe an arc cutting the transverse axis in O, at which erect the ordinate OM; it will be equal to the semi-conjugate axis. For we have CO = CR = CA 2 a 1/2. 124. If the values 62 b2 P = r2 a a Art. (105), be substituted in equation (1) of the same article, giving to r2, first the negative and then the positive sign, we obtain the two equations b2 Y2 = - (2ax - x2)............(1), a2 Y2 _62 (2ax + x2)............ (2), which are the equations of the ellipse and hyperbola referred to the axis and principal vertex A. See figures of Arts. (106), (107). 125. Let x', y', and x", y", be the co-ordinates of any two points of the ellipse. These co-ordinates, when substituted for x and y in equation (e), must satisfy it, Art. (23), and give the two equations of condition a2y'S + b2x'2 = a6b2, a2Y"2 + b/x"2 = a22, or 12 b2 12 P2 y2 x' b (a2 - X,). a2 a Dividing the first by the second, member by member, we have INDETERMINATE GEOMETRY. 147 y,2 a2_ - 12 (a + x1)(a - x,) y1,2 a - x"2 (a + x")(a x")whence, we deduce the proportion yl: yl2 (a + x')(a -') (a + x ")(a - xI). But a + x' = AP, a - x' = PB, D a + x" = AP', a - x" = P'B. I A I P JB Therefore MP2: MP':: AP X PB: AP' x P'B; that is, the squares of the ordinates of any two points of the ellipse are to each other as the rectangles of the segments into which they divide the transverse axis. For the circle, a = b, and equation (1) reduces to, Art. (36), y, = a2 -'2 = (a + x')(a - x'). 126. By using equation (h) and pursuing the same method as in the preceding article, we shall find for the hyperbola y12 y"2:: (x' + a)(x' -a): (x" + -a)" - ), or MP2 A'P',: AP X BP: AP' x BP', that is, the squares of the ordinates of any two points of the hyperbola, are to each other as the rectangles of the distances from the foot of each ordinate to the - p.ip I" vertices of the curve. 148 INDETERMINATE GEOMETRY. 127. If with the centre of the ellipse as a centre, and CA = a as a radius, a circle be described, its equa-,/ I____ ~ tion, Art. (111), may be put under the _//ATvAV'.tq I,\ ~ form A 1,-C P i _ 2 _ - x2...... (1), ID' in which Y represents the ordinate of any point of the circle, as M'P. From equation (e) we have. _ (a2 - X2). (2), a2 in which, if x have the same value as in equation (1), y will represent the ordinate MP, of the ellipse. Dividing equation (2) by (1), member by member, we have y2 b2 y b i2' a2 Y a=_ whence Y y:: a: b, that is, if a circle be described on the transverse axis of an ellipse, any ordinate of the circle will be to the corresponding ordinate of the ellipse as the semi-transverse to the semi-conjugate axis. If with C as a centre, and CD = b as a radius, a circle be described, its equation may be put under the form X2 = b2 - y2........... (3), in which X represents the abscissa of any point of the circle as RN'. If we obtain the value of x2 from equation (e) and divide by equation (3), we may deduce the proportion X x:: b: a, that is, if a circle be described on the conjugate axis of an ellipse, INDETERMINATE GEOMETRY. 149 any abscissa of the circle will be to the corresponding abscissa of the ellipse as the semi-conjugate to the semi-transverse axis. From the first of the above proportions, it appears that the ordinate of any point of the circle described on the transverse axis, is greater than the corresponding ordinate of the ellipse; hence, all the points of this circle are without the ellipse, except the vertices A and B. From the second proportion, it also appears that every point of the circle described on the conjugate axis is within the ellipse, except the vertices D and D'. We also conclude, that of all straight lines, passing through the centre, and terminating in the ellipse, the transverse axis is the longest, and the conjugate the shortest. Upon the above properties, the following constructions of the ellipse depend. First. On each of the axes as a diameter, describe a circle; at any point of the transverse axis, as P, erect a perpendicular and produce it, till, it meets the outer circle in M' join this' /1 "", point with the centre by the line M'C; / friom the point R, where this line meets the inner circle, draw a line parallel to the transverse axis, the point in which it meets the perpendicular will be a point of the ellipse. For, we have M'P MP:: MI'C: RC:: a: b. Second. Take a rule MO, in length equal to the semi-transverse axis; from the extremity M, lay off MS equal to the semi-conjugate axis; move the rule so that the extremity O and,/ the point of division S shall remain, the /' first on the conjugate and the second on the transverse axis; the point of a pencil at M, will describe the ellipse. For, draw OP' parallel to CB, 150. INDETERMINATE GEOMETRY. until it meets the produced ordinate MP in P'; join M'C, then the two equal right angled triangles M'CP and MOP' give MP' = M'P, and the similar triangles MPS and MP'O give MP' MP:: MO MS:: a: b. 128. Let x", y", be the co-ordinates of any point of the ellipse, as M, and through this point conceive any straight line to be drawn; its equation will be of the form A C P X B T y - v" = d(x - x,)...(1), in which d is undetermined. Since the given point is on the curve, its co-ordinates must satisfy equation (e), and give the condition a2y"2 + b2%x12 = a2b2. Subtracting this, member by member, from equation (e), we have a2(y2 - y12) + b2(X2 - x12) = 0 or a2(y + y,")(y - y") + b2(x + X")(x - X") = 0. Combining this with equation (1), by substituting the value of y - y" taken from equation (1), we obtain [da2(y + y") + b2(X + x,,)] (x - x"s) = o, in which x and y are the co-ordinates of all the points common to the right line and curve. This equation being of the second degree, there are two such points, and only two. These points may INDETERMINATE GEOMETRY. 151 be determined by placing the factors, separately, equal to 0. Placing x - x" = 0, we have x = x", which in equation (1), gives Y = ", and these values evidently belong to the given point M. Placing the other factor equal to 0, we have da2(y + y") + b2( + x") = 0............(2), in which x and y must be the co-ordinates of the second point of intersection M'. If now the right line be revolved about the point M, until the point M' coincides with M, the secant line will become a tangent; x and y, in equation (2), will become equal to x" and y", and the equation reduce to 2da2y".+ 2b2x" = 0; whence d b a2y" Substituting this value of d in equation (1), we have Y 2_/ Y _,(x - x"), a2y which, since a2y"2 + 62Zx2 = a2b2, reduces to a2yy" q+ b2xx" = a2b2...........,(3), for the equation of a tangent line to the ellipse at a givenpoint. If a = b, the above equation reduces to yy" q+ xx" = a2 for the tangent line to the circle whose radius is a. 152 INDETERMINATE GEOMETRY. 129. If we multiply both members of equation (3), preceding article, by (2), and subtract the result, member by member, from the equation a2y"2 + b2I'I2 _ a2b2, we have a2y"2 - 2a2yy" + b2xI2 - 2b2xx _ a- 2b2. Adding a2y2 + b2x2, to both members, we have a2(yI - y)2 + b2(x, - X)2 = a2y2 + b2X2 - a22. The first member is the sum of two perfect squares, hence a2y2 + b2x2 - a2b2 is positive for all values of x and y, except x = x" and Y = y" All points of the tangent, except the point of contact, are therefore without the ellipse, Art. (109). 130. If in equation (3), Art. (128), we make y = 0, we find Sill2 > a sin a' - b2 cos2 a' > a COS2 a' a2 and the denominator, under the second radical sign, is positive, and the value of y imaginary. In the same way, it may be shown that if y is real, x must be imaginary. Therefore, if one of the conjugate diameters of the hyperbola cuts the curve the other will not, and the converse. If then, we regard the above value of x as real, we may place it equal to a', and the imaginary value of y equal to b' V/ - 1, whence 12. a22 ba 262 a2 sin2 a - b2 cos2 a a2 sin2 a' - b2 COs2 C from which, deducing the values of the denominators, and substituting in equation (1), we have y2 X2 - = - I, or a'2y2 -'2x a2 ( h) ah%12 (A, for the equation of the hyperbola referred to its centre and conjugate diameters, in which, a' and b' are the semi-conjugate diameters. This equation is of the same form as equation (h), Art. (107), and from it we may prove as in Art. (143), that each diameter bisects a system of \A' /jchords parallel to its conju)B,z7<\ A gate, or parallel to the tangent at its vertices, if it have vertices. If a second hyperbola be described upon DE as a transverse axis, having BA for its conjugate, it is said to be conjugate to the first INDETERMINATE GEOMETRY. 175 hyperbola; that is, two hyperbolas are conjugate when the transverse axis of one is the conjugate of the other, and the reverse. The equation of the conjugate hyperbola, obtained by changing x into y, Art. (108), and a into b, in equation (h), is a2y2 - b2x2 = a2b2. 149. The parameter of any diameter of either the ellipse or hyperbola, is a third proportional to the diameter and its conjulgate, the conjugate being the mean. Thus, for the parameter of 2a' - B'A' 2b'2 2a': 2b' 2: 2p; whence 2p - The parameter of the transverse axis, 2b, is also the paraa meter of the curve, Art. (115). For the parameter of the conjugate axis, we have 2a2 b 150. As equations (e') and (h'), Arts. (143), (148), are precisely the same as equations (e) and (h), except that a' and b' enter instead of a and b, it follows that any algebraic expression deduced from the latter, will become the corresponding expression for the former, by changing a into a' and b into b'. Thus the proportions of Arts. (125), (126), become y,2~ ~,,(:' (a —+ x')((a' -') (a' + x")(a,- - X,) y,~ y,,l. (x, + a')(x' - a') (x + a')(x" - a'); the first of which shows that, the squares of the ordinates drawn to any diameter of an ellipse, are to each other as the rectangle of the 1 76 INDETERMINATE GEOMETRY. segments into which the diameter is divided; and the second that, the squares of the ordinates drawn to any diameter of an hyperbola, which intersects the curve, are to each other as the rectantgles of the distances from the foot of each ordinate, to the vertices of the diameter. These properties enable us to construct either curve, having X 1 given two conjugate diameters and the angle formed by them. Thus, let A'B' A and D'E' be two such diameters. Revolve D'E' until it becomes perpen- dicular to A'B'; on the two as axes, describe an ellipse (or hyperbola), in which draw any number of ordinates mp, m'p', &c.; then revolve these until they become parallel to the primitive position of D'E', their extremities will be points of the curve. 151. The equations of the tangent, Arts. (128), (131), become, when referred to conjugate diameters, a'2yy" + b/2cex"/ = 2bI2 a12?YY /~ - bI22XXI/ - a 12k2 a'2yy,, " = b2'2 the first for the ellipse, and the second for the hyperbola. 152. The equations of condition for supplementary chords, Arts. (135), (136), when drawn from the extremities of a diameter 2a', become b'i cc' = / -, for the ellipse............(1), a 2 cc =, for the hyperbola, in which, since the axes of co-ordinates are oblique, c and c' re INDETERMINATE GEOMETRY. 177 present the ratio of the sines of the angles which the chords make with the axes, Art. (20). 153. Likewise, equation (1), Art. (137), will belong to a diameter and tangent at its vertex, when referred to conjugate diaineters, if we change a into a' and b and b', and regard d and d' as the ratio of the sines, &c.; thus, b/2 dd = - -. a/2 Comparing this with equation (1) of the preceding article, we have cc' = dd', and the same for the hyperbola. Hence, if c = d, c' = d' and the converse. Therefore, in either curve, if a chord, drawn from the extremity of a diameter, is parallel to a tangent, its supplement will be parallel to the diameter passing through the point of contact, and the converse. Also, if one of two supplementary chords is parallel to a diameter, the other will be parallel to its conjugate: or, a set of supplementary chords may always be drawn from the extremities of any diameter parallel to a set of conjugate diameters. 154. The properties of supplementary chords, diameters, and tangents, discussed in the preceding article, give the following constructions. First. If either the ellipse or hyperbola is traced upon paper; draw any two parallel chords, nn and n'n', and bisect them by a straight B' line, this will be a diameter, Art. (100). If two diameters be thus constructed, AT their intersection will be the centre of the curve. 12 ] 78 INDETERMINATE GEOMETRY. Second. On any diameter, A'B', found as above, describe a semi-circle and draw two chords from the point M, in which it intersects the curve to the extremities of the diameter; these chords will be supplementary and perpendicular to each other; draw two diameters parallel to these and they will be the axes. And, in general, to construct a set of conjugate diameters making a given angle with each other. Upon any diameter describe an arc capable of containing the given angle; from the point in which it cuts the curve, draw two chords to the extremities of the diameter; through the centre draw two diameters parallel to these chords; they will be the required diameters. Third. If one diameter, as D'E', is given, and it be required to construct its conjugate. From the extremity of any diameter draw a chord Rm, f" g IT'D'parallel to the given diameter; draw the supplement, Orn, of this chord; through the centre draw a diameter CA' parallel to this supplement, it will be the one required. Fourth. To draw a tangent at a given point, as A'. Join this point with the centre; through the extremity O, of any diameter, draw a chord parallel to CAT; draw the supplement of this chord, Rmn; parallel to which, draw a line through the given point; it will be the required tangent. Fifth. To draw a tangent parallel to a given line, as L. Make the same construction as in Art. (137), using any diameter as OR, instead of the transverse axis. Or thus: draw any two chords mnR, n'R', parallel to the given line; bisect them by a straight line; the points A' and B', in which this intersects the curve will be the points of contact, through either of which draw a line parallel to the given line, it will"be the required tangent. INDETERMINATE GEOMETRY. 179 155. Let CB' and CD' be any two semi-conjugate diameters of the ellipse. On the transverse axis AB, describe a semi-circle; through the points gt B' and D' draw two ordinates and produce' them to M and M'; draw the radii CM //, and CM', and denote the angles MCB A P and M'CB by i3 and 3'. The right angled triangles CPB' and CPM, give tang a: tangf:: PB': PM:: b: a, Art. (127); hence b tang a - tang /. a Also, the triangles CP'D' and CP'M', since the angles at the bases are supplements of a' and /3', give tang ac = b tang /'. Multiplying these equations, member by member, we have 62 b2 tang oa tanga = - tang / tang = - a2 a Art. (144); hence tang / tang 3' = - 1, and the two radii CM and CM' are perpendicular to each other, Art. (28); therefore /3' = 90~ + /3, sin/' = cos /, cos/' = - sin /3. 156. From the triangles CPB' and CPM, we have 180 INDETERMINATE GEOMETRY. CP = a' cos a, CP = a cos 3, PB' = a' sin a, PM = a sin /; whence, from the first two, a' cos a = a cos............(1), and from the second a'sina: a sin::3 PB': PM:: b: a, or a' sin a = b sin ~............ (2). In the same way, the triangles CP'D' and CP'M' give b' cos a' = a cos /3' = - a sin/.........(3), b' sin a' = b sin /' = b cos..............(4), after substituting for cos /' and sin /' their values, as deduced in the preceding article. Multiplying equations (1) and (4), member by member, and then, (2) and (3), and subtracting the latter product from the former, we have a'b'(sin.a' cos a - sin a cos o') = ab(cos2 3 + sin2 P), or a'b' sin(a' - a) = ab............(5). Squaring both members of (1) and (3) and adding, we have a'2 COS2a + b'2 cos2 COSa = a. In the same way, from (2) and (4) we obtain a'2 sin2 c' - b62 sin2 a' = b2. Adding the last two equations, member by member, a'2+ b/2 = a2 + b2............(6). INDETERMINATE GEOMETRY'. 181 157. If we unite equation (1) of Art. (144), with (5) and (6) of the preceding article, we have tang a tang a' 6 — -............(1), a2 a'b' sin (a' - a)= ab...........(2), a'2 + b'2 = a2 + b2...........(3), three equations containing six quantities, either three of which being given, the others may be determined. If the angle a' - a, made by the conjugate diameters with each other, is given equal to co, we have a' -- a = w tanga' = tang (a +- a), and this value in equation (1), will give an equation from which tang a may be found, and thus both a and a', become known. Let us resume equation (2), a'b' sin (a' - a) = ab............(2), and at the vertices of any two conjugate diameters A'B' and D'E', draw tangents forming a parallelogram. Also at the vertices of the axes, draw tangents forming a rectangle. From the right angled triangle D'RC, we have D'R - D'CsinD'CR = b'sin(a-' -a); - \ Q, hence, the first member of equation (2) is equal to CB' x D'R, or the area of the parallelogram CQ, while the second member is equal to CB x CD, or the area of the rectangle CS. If these equals be multiplied by 4, we have four times the parallelogram CQ, or the parallelogram QQ', equal to four times the rectangle CS, or the rectangle SS'; that is, the parallelogram constructed 182 INDETERMINATE GEOMETRY. upon any two conjugate diameters of the ellipse, is equivalent to the rectangle upon the axes. Multiplying both members of equation (3), by 4, we have 4a'2 + 4b62 = 4a2 + 4b2 But 4a'2 = (2a')2, 4b12 = (2b/)2 &c.; hence, the sum of the squares upon any two conjugate diameters of the ellipse, is equal to the sum of the squares upon the axes. 158. If in equations (1), (2) and (3), of the preceding article, we change b2 into - b2 and b'2 into - b'2, we have, for the hyperbola, tang a tang a' =............(1), a'b' sin (a' - ) ab............ (2), a'2 - b12 = a2 - b2............(3), from which either three of the quantities may be determined when the others are known. If a' - a =, is given, a and a' may be determined as in the preceding article. The perpendicular D'R is equal to D'C sin D'CR = b' sin (a' - a); hence, the first member of equation (2) is equal to [-/O"~ f-~~ )CA' x D'R, or the area of the parallelogram CQ; while the second member is equal to the rectangle CS. Multiplying these equals by 4, we obtain the INDETERMINATE GEOMETRY. 183 parallelogram constructed upon any two conjugate diameters of the hyperbola equivalent to the rectangle upon the axes. From equation (3), we have 4a'2 - 4b'2 = 4a2 - 4b2, that is, the difference of the squares of any two conjugate diameters of the hyperbola, is equal to the difference of the squares of the axes. 159. If the conjugate diameters of an ellipse are equal to each other, the two expressions for a'2 and 6b2, Art. (143), must be equal, which requires that a2 sin2 t- 62~ COS2 a = a2 sin2a' ~ b2 cos2 a', and every set of values of a and a' which will satisfy' this equation, provided they at the same time satisfy equation (1), Art. (144), will give the position of a set of equal conjugate diameters. Substituting in the above equation sin2 a - 1 - cos2a, sin2' = 1 - cos2 a', it becomes a2 b2) cos2 = (a2 - b2) os2 a'............(1), which, unless a = b, can only be satisfied by making COS2 Cos2 a', or COS osa = cos ao'. The first value cos a = cos a' gives a = a'; hence the two diameters coincide and are not conjugate. The second value cos a = - cos a', satisfies equation (1), Art. (144), and requires the angles to be supplements of each otler, or a' + a = 180~; hence, Art. (135), the diameters must be parallel to the supplementally chords drawn from the extremities of the transverse to 184 INDETERMINATE GEOMETRY. the extremity of the conjugate axis. They will therefore make a greater angle with each other than any other set of conjugate diameters. If a = b, equation (1), will be satisfied for every set of values of a and a'; hence, in the circle, the conjugate diameters are equal to each other. When a' = b', equation (e'), Art. (143), becomes a'2y2 + a'2x2 - a'4, or y2 + x2 = a'2 for the ellipse referred to its equal conjugate diameters. 160. By an examination of equation (3), Art. (158), it will be seen that a' can not be equal to 6', unless a = b; that is, in the hyperbola, there are no equal conjugate diameters, except when the hyperbola is equilateral, in which case each diameter is equal to its conjugate. OF TI-E HYPERBOLA REFERRED TO ITS ASYMPTOTES. 161. If in equation (1) of Art. (143), we put - 62 for b2, it becomes (a2 sin2 a' - b2 Cos2 a')y2 + (a2 sin2a - b2 cos2 c)x2 + 2(a2 sin a sin a' - cosa cosa' )xy - a2b2......(1), for the equation of the hyperbola referred to any set of oblique axes having their origin at the centre. We may assign such values to the arbitrary constants a and a', in this equation, as to cause the coefficients of y2 and x2 to be 0, and thus have a2 sin2a' - b2 cos2 a' = 0, a2 sin2 a - b2 cos2 = 0...... (2) whence by dividing the first by a2 cos2 a', and the second by INDETERMINATE GEOMETRY. 185 a2 cos2 a, and recollecting that sin- tang2 we deduce cos tanga' - i -, tang a -z a a But it is evident that we can not use tang a' = tang a, as in such case, the two new axes of co-ordinates would coincide. If, therefore, we take tang a' b a we must take tang a - -, a and the reverse. These values may be readily constructed, thus: At the vertex A, erect a perpendicular to BA, on which lay off the distances AL and AL', each equal to b and draw the lines CL and CL', these will be the new axes of co-ordinates. For the right angled triangles CAL and CAL' give tang ACL = tang ACL' = b tang a'. a But the tangent of ACL', taken with a contrary sign, is equal to the tangent of 360~ - ACL', and also equal to = tang a; a hence 186 INDETERMINATE GEOMETRY. 3600 - ACL' = -, and CL' is the new axis of X, and CL the new axis of Y, the angles a and a' being as marked on the figure. Since CL = Va2 + b2, the right angled triangle ACL gives b a sin a', cos a' a v a2 b2 Va2 + b2 and since sin cc' - sin a, Cos a' = CS a, we also have - b a smin oc COS a a V a2 + b2 2 b2 Substituting these values, together with conditions (2), inl equation (1), we have - ( )b2 a2b2 2b2 a2s -t+ b2 a2 + b2 whence xy =?1xy =............(), a2 + b2 placing in for a 6. Solving this equation, we have x $ INDETERMINATE GEOMETRY. 1 87 in which, as x increases, y diminishes; when x becomes infinite y becomes 0; and as y can be negative for no positive value of x, it follows that the axis of X, or the line CL', continually approaches the curve and touches it at an infinite distance without cutting it. By solving the equation with reference to x, it may be proved that the line CL enjoys the same property. These two lines are called asymptotes of the hyperbola; and in general, an asymptote of a curve is a line, which continually approaches the curve and becomes tangent to it at an infinite distance. By an inspection of the figure, it is readily seen that the asymptotes of the hyperbola are the diagonals of the rectangle described on the axes. Equation (3) is called the equation of the hyperbola referred to its centre and asymptotes. If the hyperbola is equilateral, a' = 45~, tang a' = 1, the angle LCL' = 900, and the asymptotes are perpendicular to each other. 162. If we take the expression, Art. (132), x - a CT, X/! and make x" - a, the least value it can have for points of the curve, Art. (107), we shall obtain CT - a = CA, which is the greatest value of CT. As x" increases, CT diminishes until x" = oo, when CT = 0, its least value, and the tangent coincides with the asymptote; hence all tangents 188 INDETERMINATE GEOMETRY. drawn to the hyperbola intersect the transverse axis between the centre and the vertex of that branch to which they are drawn; and the asymptotes are the limits of all tangents. 163. If we multiply both members of equation (3), Art. (161), by sin 2a', the sine of the angle LCL' included by the asymptotes, we have xy sin 2a' = m sin 2a'. The second member of this equation is constant, and the first for any point of the curve, as M, is CP x PM sin MPn = CP x Mn, which is the area of the parallelogram CPMR. Hence, the areas of all parallelograms described on the abscissas - ) Z? 1 // and ordinates of points of the curve, referred to the <4 G o<>asymptotes, are equal, each being measured by the expression m sin 2c'. If the point M is placed at the vertex A, the parallelogram becomes the rhombus AOCO', each of its sides being CL = 2 /a2 + b2. 2 2 This rhombus, described on the abscissa and ordinate of the vertex, is called the power of the hyperbola, is equivalent to the parallelogram described on the abscissa and ordinate of any point of the curve, and as is readily seen from the figure, is one eighth of the rectangle described on the axes. INDETERMINATE GEOMETRY. 189 In the equilateral hyperbola 2a' = 900, sin'2od = 90~, sin and the power becomes a square, the area of which is nm. 164. Let X." y" denote the co-ordinates of any point, as M, of the hyperbola. The equation of a right line passing through this point, will be of the form Ff y - yy" = d(x - x")............(1). The equation of the hyperbola being y- n2............(2), the condition that the point M shall be on the curve, will be xIy" = m. Subtracting this from equation (2), we have cy - x"y'I = 0. Combining this with equation (1), by substituting the value of y taken from (1), we obtain Y( - x"I) + dx(x - X') = 0, or (x - x")(y" + dx) = 0. Placing the two factors of the last equation, separately, equal to 0, we obtain for all the common points, Art. (128,) X - X'" 0 OTr x = Xt "r y" + dx= 0 or x =. (3. di 190 INDETERMINATE GEOMETRY. The first value of x is evidently the abscissa of the point M, the second must then be that of M'. Now if the line MM' be revolved about M until the point M' coincides with M, it will become a tangent, and the value of x in equation (3) will become x", whence we have d = - y X/ This value, in equation (1), gives y - yy (x -x), for the equation of the tangent, referred to the asy.mptotes. If in this equation we make y = 0, we have X -." =.'x = PT, that is, the subtangent is equal to the abscissa of the point of contact. And, since PT = CP, we have MT = MS; that is, the palrt of the tangent included between the asymptotes is bisected at the point of contact. If in equation (1) we make y = 0, we find x - xa" = =- CT' -- CP = PT', d the samne value found in equation (3) for the abscissa of M'; hence PT' = M'P', and the two triangles MPT' and M'P'T", having their angles also equal, are equal, and MT' = M'T"; that is, if any straight line be drawn cutting the hyperbola, the parts included between the curve and asymptotes will be equal. This property enables us to construct the hyperbola by points when a single point and the asymptotes are given. Through the INDETERMINATE GEOMETRY. 191 point, as M, draw any right line; from the point in which it cuts one of the asymptotes, lay off the distance TI1M', equal to the distance MT', from the given point to that in which it cuts the other asymptote; the extremity of this distance will be a point of the curve. 165. If a tangent TS be drawn at any point, M, of the hyperbola, and the half tangent MT be denoted by t, the o half diameter MC by a', and the perpendicular Mn be drawn, the two right angled triangles MnC and MnT will give //, a'2 = (x + Pn)2 ~ j2n t2 (x - Pn)2 + Mn2. Subtracting and reducing, we have aL -2 t2 = 4xPn. But the right angled triangle MPn, in which the angle MPn -- 2a', gives Pn = y cos 2a', hence at2 - t2 = 4xy cos 2aC = 4m cos 2a'l. The second member of this equation is constant, and will therefore be the same for any position of the point M. If this point be placed at the vertex A, the half diameter will be CA = a, and the half tangent AL = b; hence 192 INDETERMINATE GEOMETRY. a2 - b2 = 4m cos 2a; whence a'2 t2 a2 - b2. But, Art. (1 58), we have at2 - 6;2 = a2 b2, therefore, we must have t = 61'or 2t - 2b'; that is, if a tangent be drawn at any point of the hyperbola, the part intercepted between the asymptotes will be equal to the conjugate of the diameter passing through the point of contact. Since the line E'D' is equal and parallel to TS = QO, it follows that the figure QOST is a parallelogram, and that the vertices of any parallelogram described on a set of conjugate diameters, will lie on the asymptotes. OF THE POLAR EQUATIONS OF THE ELLIPSE AND HYPERBOLA. 166. If in equation (e), Art. (105), we substitute the values of x and y from formulas (2) of Art. (69), x = a' + r cos v, y = b' + r sin v, we shall obtain after reduction (a2 sin2 v + b2 cos2 v)r2 + 2(a2b' sin v + b2a' cos v)r + a2b'2 + b2a'2 - a2b = 0......(1), for the general polar equation of the ellipse. By changing b2 into - b2, this will become the general polar equation of the hyperbola. INDETERMINATE GEOMETRY. 193 By assigning particular values to a' and b', in the above equation, the pole may be placed at any point in the plane of the curve. -First. If the pole is on the curve, we must have, Art. (108), a2b62 + b2a'2 - a262 = 0, and the equation reduces to [(a2 sin2 v + b2 cos2 v)r- + 2(a2b' sin v + b2a' cos v)]r = 0, which may be satisfied by placing r = 0, or (a2sin2v + 62 cos2v)r + 2(a2b siv + b2a' cosv) = 0......(2). The pole being on the curve, one value of r is necessarily equal to 0, and the other deduced from equation (2), will, for each value of v, give the distance from the pole to the second point, in which the radius vector cuts the curve, Art. (70). If this second value of r becomes 0, the radius vector will become tangent to the curve, and equation (2) will reduce to a2bl sin v + b2C' cos v = 0, or sin v tang2a tang v...., cos v atb' as it should be, Art. (128). For the hyperbola, we shall have the same discussion and result; except that - b2 takes the place of b2. Second. If the pole be placed at the centre, we have a' - 0, b = 0, which reduces equation (1) to (a2 sin2 v + b2 COS2 v)rT2 a2b2 = 0; whence 13 194 INDETERMINATE GEOMETRY. / a2b2 r - -4- (s). Va2 sin2 v - b2 cos2 v The second value of r is negative for all values of v, and therefore gives no point of the curve, Art. (69). The first value is positive for all values of v, and as v varies from 0 to 3600, will give all points of the curve. A_ If v = 0, sinv= 0, cos v = 1, and r reduces to A.- -— c u r't r - a = CB. If v = 90~, sinv == 1, cos v - O. and r = b =CD. If in the first value of r, (3), we put for sin2 v its value 1 - cos2 v, divide both terms of the fraction under the radical sign by a2, and then place a2 b2 a2 - e representing the eccentricity of the ellipse, Art. (118), we shall obtain i -e- e2 cos2v For the hyperbola, equation (3), becomes arb2 a2 sin v - b cos2 v the second value of which is negative for all values of v. The first value is positive but imaginary, unless the denonminator is negative which requires a2 sin2 V- b2COSav < 0, or tangy < =i: b INDETERMINATE GEOMETRY. 195 If v = 0, we have, as above r = a = CA. As v increases from 0, the denominator will be negative until a2 sin2 v = b2 cos2 v, tang = b, when the value of r will be infinite, in which case v = LCA, and the radius vector coincides with the asymptote CL, Art. (161). As v increases beyond this value, a2 sin2 v becomes greater than 62 cos2 v, and r will be imaginary until tangy = - -, a in which case v - ACL" and the radius vector coincides with CL". When v -= 1800, we have r = a = CB, and as v still increases, we shall continue to have real values for'r until it coincides with CL"', when tang v again becomes equal to b, and from this point the values of r will be imaginary until the radius vector coincides with CL', when they again become real and continue so to v = 3600~. The first value of r thus gives all the points in both branches of the hyperbola. By a process similar to that pursued in the ellipse, the first value of r may be reduced to / e2 cos v - 1 96 INDETERMINATE GEOMETRY. in which e represents, the eccentricity of the hyperbola, Art. (119). Third. If the pole be placed at the right hand focus, for which a' - v/a2 - b2 = c, b' = O, equation (1), becomes (a2 sin2 v + b2 cos2 v)r2 + 2b2c cos vr - b4 = 0. If for sin2 v we put its value 1 - cos2 v, and for a2 b2, its value c2, this equation reduces to (a2 - c2 cos2 vr2 +- 2b2c os yr = b, from which - b2c cos V b1 4'b4C2 cos2 v r =: k + a2 - C2 cos2V a2_c2cos2v (a2 - c2 cos2 v)2 or reducing - b2ccosv i ab2 -:: ab2 - b2c cos v a2 - c2 cos2v (a + coS v)(a - c os v) whence, the two values b2 -_ b ~ = b.. (4), r = b —...(5). a + c cosv a- c cosv Since for the ellipse c = /a2 - b2 < a and cos < 1, the second value of r is always negative and must therefore be rejected. As v varies from 0 to 360~, the first value of r will be positive, and give all points of the ellipse. For the hyperbola, expressions (4) and (5) become 2=......(6), r = 62 a + c cosv a - c cos v INDETERMINATE GEOMETRY. 197 The first value of r will be positive whenever the denominator is negative. But this can not be unless cos v is negative and numerically greater than a. Every value of v, beginning with 0, C will then make r negative until a a COSV T - - -_ V' -+6 C 1,a2 + ba when the radius vector will be parallel to the asymptote CL", Art. (161), and r will be infinite. As v now increases, cos v will increase numerically until v = 1800, when cos v = - I, and b2 r = -- C - a which is positive, and gives the vertex B. As v increases from this point, cos v will diminish numerically and r will be positive until we again have a COS = - _, when r = Ao, and the radius vector becomes parallel to the asymptote CL"'. All values of v not included within these limits will make the first value of r negative and give no points of the curve. Thus, it appears that this value of r gives all the points in the left hand branch of the hyperbola, and no others. The second value of r will be positive, when the denominator is positive. Commencing with v = 0 cos v = 1, we have b2 a - c 198 INDETERMINATE GEOMETRY. which is negative. As v increases, cos v diminishes, and r will remain negative until a = c cos v, when a a COS V --- c a2 q+ b2 r reduces to infinity, and the radius vector takes the position FR, parallel to the asymptote CL. As v increases from this point, r will be positive until it takes the position FR' parallel to CL'. When v = 900, cos v = 0 and bI r = - _FM. When v = 180~, cosv - - 1, and b2 r = FA. a + c The second value of r, therefore, gives all the points in the right hand branch, and no others. If in expressions (4), (6) and (7), we put - c for c, the pole, in each case, will be placed at the left hand focus. If the eccentricity of an ellipse be denoted by e, we have, Art. (118), C c2 a2 b2 e = -e = _ = a a2 a2 from which, we deduce c = ae, 62 = a2(1 - e2)............(8). Substituting these values in expression (4), we find r = __a(1 - e2) 1 + e cos v INDETERMINATE GEOMETRY. 199 which expresses the value of r in terms of the eccentricity of the ellipse. For the hyperbola, Art. (119), we have c = ae, - b2 = a2( - e2)..........(9). These values in expressions (6) and (7), give a(1 - e2) a(l - e2) r -=,' — r _ I q- e cosv 1 -- e cos v in terms of the eccentricity of the hyperbola. From equations (8) and (9) we deduce the numerical value a hence, the numerator of each of the above values of r is equal to one half the parameter of the curve, Art. (149); as is also the case in the parabola, Art. (104). DISCUSSION OF THE GENERAL EQUATION OF THE SECOND DEGREE. 167. Every equation of the second degree between two variables, must be a particular case of the most general form ay2 + bxy + cx2 + dy + ex + f 0 = 0............(1), which, by assigning particular values to the constants a, b, c, &c., may be made to represent every line of the second order, Art. (33). Although there are six terms in the above equation, and apparently six arbitrary constants, yet it must be observed that both members of the equation may be divided by the coefficient of either term, as a, thus reducing it to the form y2 +- b'xy + c'xm2 + d'y + e's +f' = 0, 200 INDETERMINATE GEOMETRY. in which there are but five constants, and to which we can assign but five arbitrary conditions. In order then to estimate the number of arbitrary constants in any general equation, or equation given in form only, we divide by the coefficient of one of the terms, and then count the number of different coefficients remaining. This will indicate the number of arbitrary conditions which the given equation may be made to fulfil. In commencing the discussion of equation (1), we may regard the axes of co-ordinates to which the line represented by it, is referred, as at right angles; for if they were oblique, a change of reference might be made by means of formulas (5), Art. (67), and a new equation of precisely the same form would evidently result. 168. By solving equation (1), of the preceding article, with reference to y, we obtain bx d 1 Y. 4=2a - 2a V(b2 _4ac)x2 + 2(bd- 2ae) x+d2 - 4af...(1), from which we may readily construct the line by points as in Art. (22).. Each value of x which will make the quantity under the radical sign positive, will give two real values for'y, and correspond to two points of the curve. These points may be constructed by laying off from A as an origin, the assumed value of x, as AP; at P erect the peipendicular PM, on which lay off PR = bx +d 2a A.' P t X from R lay off RM' in the ASi~l I` positive direction of the ordinates, and RM in the ne-. XI' gative, each equal to the second part of the value of INDETERMINATE GEOMETRY. 2 01 y; PM' will be represented by the first, and PM by the second value of y, and M' and M will be the corresponding points of the curve. Since the point R is midway between the two points M and M', it follows that the chord MM' is bisected at R. But since the points R, r, &c., are constructed by laying off the different values of the expression bx + d 2a it follows that they must all lie on the right line whose equation is bx +- d bx d Y, or y - -- -; 2a 2a 2a hence, this line will bisect all chords drawn parallel to the axis of Y; it is therefore a diameter of the curve, Art. (100), and may at once be constructed by laying off AA' - d, and through 2a A', drawing the line A'R, making with AX an angle whose tanb gent is - -, Art. (26). 2a Hence, if an equation of the second degree be solved with reference to y, the first member placed equal to that part of the second, which does not contain the radical, will give the equation of a diameter bisecting chords parallel to the axis of Y. If the equation be solved with reference to x, a similar discussion will show that, the first member, placed equal to that part of the second which does not contain the radical, will give the equation of a diameter bisecting chords parallel to the axis of X. 169. If in equation (1) of the preceding article, we place bd - 2ae = m, d2 - 4cf = n, it becomes a2 0 2 INDETERMINATE GEOMETRY. bxz d l. Y = 2- (b2 - 4ac)x2 + 2mnx +n......(l). Let us now remove the origin of co-ordinates to the point A', A'R being taken as a new axis of abscissas, the new axis of ordinates being the same as the primitive, A'Y. In the formulas (3), Art. (67), we must then have __b a' = O, b' d a' 900~, tang _ b, 2a 2a and the formulas become x = x' cos a, y d' sin a + y', 2a or by substituting for x' in the second, its value taken from the first, we have I'cosa, - bx +-dcl x -= X cos a, y _ }+ yt 2a Substituting these values of x and y in equation (1), squaring both members and clearing the denominators, we have 4a2y'2 = (b2 - 4ac) cos2 ax'2 + 2mn cos ax' + n............(2), and this may be put under the form 4a2y'2 = (b2 - 4ac) cos2 a + (b _ 4ac) cos n. If to the quantity within the parenthesis, we add m2 (b2 - 4ac)2 cos2 a: it will be a perfect square. To preserve the equality, we must then subtract, without the parenthesis, the same expression multiplied by (b2 - 4ac) cos2 a, or INDETERMINATE GEOMETRY. 203 m2 b2 - 4ac The above equation then takes the form 4a2y 2(b2_4ac)cos2a(x +~ m- m. (3), (62 -4cc) cs 62 -4cc If now we again transfer the origin to a point, as A", on the axis of X', at a distance from A' equal to - m- -........... (3'), (b2 - 4ac) cos a the axis of abscissas remaining the same and the new axis of ordinates being parallel to A'Y, the formulas (2), Art. (67), become $ -- + X my = ty + (b2 - 4ac) cos a Substituting these in equation (3), and transposing, we have,n,2 4a2y"2 - (b2 - 4ac) cos2 ax"2 - n b...........(4), b - 4a~ for the equation of the line referred to the two axes A"X' and A"S. If ba - 4ac is negative; the essential sign of the second term of the first member will be positive. If the second member is also positive, the equation will be of the same form as equation (e'), Art. (143), and will therefore represent an ellipse referred to its centre and conjugate diameters. If the second member is negative, the equation will indicate that the sum of two positive quantities is equal to 0, and can be satisfied by no values of x" and y". The line represented by it is then said to be imaginary, and an imaginary ellipse is a particular case of the ellipse. If the second member of the equation is O, it will indicate that 204 INDETERMINATE GEOMETRY. the sum of two positive quantities is equal to 0, and can be satisfied by no values, except y" = O, x" = 0, which, Art. (16), are the equations of a point, also a particula'r case of the ellipse. If b = 0 and a = c, we have tang a = 0, cos a = 1, and equation (4) will reduce to the form yJ/2 +- x"2 = R2 which is the equation of a circle, Art. (35), another particular case of the ellipse. If b2 - 4ac is positive; and the second member negative, equation (4) will be of the same form as equation (h'), Art. (148). If the second member is positive, the, signs of all the terms may be changed and it will still be of the same form, x" having the place of y, and y" the place of x, Art. (108). In either case it will therefore represent an hyperbola referred to its centre and conjugate diameters. If the second member is 0, the equation may be solved with reference to y"2, and will take the form y1l2 = - r12r12 or yl, =- _- r,X representing two right lines which intersect; a particular case of the hyperbola. If. b = 0 and a = - c, the equation takes the form y -,2 -_ X12 - R2 the equation of an equilateral hyperbola, Art. (112); another particular case of the hyperbola. If b2 - 4ac = O, the expression (3') will be infinite, and the transformation made on equation (3) becomes impossible, but under this supposition equation (2) reduces to INDETERMINATE GEOMETRY. 205 4a2y12 = 2m cos ax' + n............(5). If in this, we transfer the origin to a point on the axis of X', at a distance from A', equal to............. cos(6) 2rn cos a the formulas of Art. (67) will become n x =+ r$"x y = yll 2m cos a and these substituted in equation (5), give 4a2y"'2 = 2m cos ax", or y12 = 2m cos x", 4a2 which is of the same form as equation (7), Art. (99), and therefore represents a parabola. If m = 0, expression (6) will be infinite and the last transformation become impossible, but in this case equation (2) becomes 4a2y'2 = n, or' = /n, x' being indeterminate, and will represent two right lines parallel to the axis of X', when n is positive; one right line which coincides with the axis of X' when n = 0, Art. (21); and two imaginary right lines when n is negative. These are particular cases of the parabola. 170. The above discussion evidently depends upon the fact that the given equation contains the second power of y, or that a is not 0. If a = 0 and c is not, the equation may be solved with reference to x, and the same results deduced in precisely the same manner. 206 INDETERMINATE GEOMETRY. If a = 0 and c = 0, and b is not, the general equation takes the form bxy + dy ex f= 0............(). Let us now, by the aid of the general formulas, Art. (67), x = a' J x', y = b' + y', change the origin of co-ordinates, without changing the direction of the axes. We thus obtain bx'y' ~ (a'b J- d)y' + (b'b - e)x' + ac'b'b 4' b'd + a'e+ f =0....(2). In this equation we have two arbitrary constants, a' and b', and may therefore assign such values to them as to give a'b + d = O b'b + e = 0, or at d. b' e b b Substituting these values in equation (2), it reduces to bxy' d + f = O or xy-'yt 7 d b b2 which, since the axes of co-ordinates are at right angles to each other, is the equation of an equilateral hyperbola referred to its centre and asymptotes, Art. (161). Equation (1) then represents the same hyperbola, referred to two right lines parallel to its asymptotes. If a = O, b = O, c = 0, the equation ceases to be an equation of the second degree. From the previous discussion, we conclude, that every equation of the second degree between two variables represents one of the conic sections, that is, either a parabola, an ellipse or hyperbola, or one of their particular cases. INDETERMINATE GEOMETRY. 207 A parabola when b2 - 4ac = 0. An ellipse when 62 - 4ac < 0. An hyperbola when b2 - 4ac > 0. The parabola when 62 - 4ac = 0. 171. Under this supposition, the value of y, equation (1), Art. (169), reduces to br + d 1.Y a 2i- 2a /: mx + n............ Every value of x, which will make the quantity under the radical sign positive, will give two real values of y and two corresponding points of the curve. The value of x, which makes this quantity 0, will give two equal values of y, the two corresponding points unite, and the ordinate produced is tangent to the curve, Art. (35). Every value of x, which makes the quantity under the radical sign negative, gives imaginary values for y and no points of the curve. If we place 2mx + n = O, we have x n 29n which is the only value of x that will reduce the quantity under the radical sign to 0. Denoting this value by x', the value of y may be written bx + d 1 Y = 2a 2a /2m( - x')............(2)7 since'2mx + n = 2m (x + n 208 INDETERMINATE GEOMETRY. Now, if rn is positive; whether x' be positive or negative, every value of x > x' will give two real and unequal values for y; x = x' will give.two equal values; and every value of x < x' will give imaginary values. Hence, the curve extends indefinitely in the direction of x positive, is tangent to the ordinate PV, which corresponds to the abscissa x', and has no points on the left of this ordinate, as indicated by the full line in the figure. If m is negative, every value of x > x' will give imaginaly values for y; x = x' will give equal values, and x < x' will give two real and unequal values. Hence, the curve is limited in the di-______ N' rection of x positive by the a "~:":\ (? X produced ordinate PV, and extends indefinitely in the direction of x negative, as indicated by the dotted line in the figure. Hence, in order to obtain the limit of the curve in the direction of the axis of abscissas, we solve its equation with reference to y, place the quantity under the radical sign equal to 0, and deduce the value of x, this value will be the abscissa of the limnit, lay it of and through its extremity draw a line parallel to the axis of Y, it will be the limit; and this limit will be tangent to the curve at the vertex of that diameter which bisects the chords parallel to the axis of Y. If the coefficient of x under the radical sign is positive, the curve will lie entirely on the right of this limit; if negative, on the left. By solving the equation with reference to x, we may, in a siinilar way, construct the limit in the direction of the axis of Y. If m = O, the value of y, equation (1), becomes bx + d1 _ 2a 2a or INDETERMINATE GEOMETRY. 209 bx d -/n bx d Vn 2a 2a 2a 2a 2a 2a the equations of two parallel straight lines, Art. (28), when n is positive; which reduce to one straiglht line, when n = 0; and to two imaginary parallels, when n is negative, as seen in Art. (169). Hence, an equation of a parabola being solved with reference to either variable, if the quantity under the radical sign is a positive constant, the equation will represent two parallel straight lines. If this quantity is 0, or the radical disappears, the equation will represent one straight line. If this quantity is a negative constant, the equation will represent two imaginary parallels. It may be remarked, that in the first case, the line whose equation is bx + d 2a bisects all chords, terminated in the two lines and parallel to the axis of Y, and therefore strictly fulfils the condition of a diameter, Art. (100). In the second case, the line represented by the equation is the diameter itself. In the third case, the diameter may be constructed while the lines do not exist. 172. By solving the equation with reference to x, we find for the equation of the diameter which bisects all chords parallel to the axis of X, Art. (168), by +- e 2c e x --; whence y - -2c 2c b b but since b2 - 4ac =- O, we have 14 210 INDETERMINATE GEOMETRY. b 2c 2a b hence the coefficient of x in the above equation is equal to the coefficient of x in the equation bx d 2a 2o and the two diameters represented by these equations are parallel, Art. (28). 173. By an application of the foregoing principles we are enabled to represent on paper, a parabola whose equation is given, without taking the trouble to determine many of its points. First, find the points in which the curve cuts the axes of coordinates, Art. (22); then solve the equation with reference to each variable in succession, and construct the diameters which bisect the chords parallel to the axes, Arts. (168), (26); then construct the limits of the curve in the direction of both axes, Art. (171); and draw a curve tangent to these limits at the points at which they intersect the diameters and through the points first determined, taking care to make it symmetrical with respect to both of the diameters. Examples. First, when m is not 0. 1. 2 2xy + x2 -- y + 2x -1 = 0......(1). By comparing this with the general equation, Art. (167), we see that a = 1, b - 2, c 1, -- 4ac = 4 - 4 = 0; INDETERMINATE GEOMETRY. 211 hence, the curve is a parabola. Making y = 0, we obtain x2 + 2xr- 1 = o; x = - 1 1 X/2. Assuming any line as a unit of measure and laying off AB= - 1 + 72, V AB/ = - 1 - // / we have the points in // — / 4 which the curve cuts / the axis of X. Making X = 0, we find 1 5 2 ] 4 and may thus determine the points C and C' in which the curve cuts the axis of Y. Solving the given equation, first with reference to y, and then with reference to x, we have 2x + 1 1'Y= sx:12 i 42 + 5...... (2), x = y- - y + 2...............(3). The equations of the diameters are 2x ~ 1 y _= 2, x=y —1, which represent the lines DV and D'V'. Placing the quantities under the radical signs (2) and (3), equal to 0, we deduce for the first, 5 212 INDETERMINATE GEOMETRY. for the second, y = 2. Laying off AP = 5 and drawing the line PV, it must be 4 tangent to the curve at V, and since the coefficient of x under the radical is - 4, the curve will lie on the left of this tangent. Laying off AR = 2, and drawing the line RV', it will be the limit in the direction of the axis of Y, and the curve will be represented as in the figure. 2. y2 - 2xy + x2 + y- 2x= O. 3. y2+2xy + xz2-2y - 1= 0. 4. y2-2xy+x2-2y- 2x=Q. 5. y 2xy- +x2+- 2y =O. 6. y - 2xy 2 + x O. Second, when nm = 0 and n positive. 1. y2 - 2xy +-x2 —2y + 2x-1 = 0. 2. y2 - 2xy + x2 - -0. 3. y2 + 4xy + 4x2 + 4 =-0 Third, when n = O, n = 0. 1. y2 2xy + x2 +q 2y - 2x + 1 0 O. 2. y2 -4xy- 4x2 = O. Fourth, when sm = 0, and n negative. y2 9- 2xy +- x2 + 1 = O, 2. y2+y2 + y + 1 = O. INDETERMINATE GEOMETRY. 213 174. If it is required to construct the curve with accuracy; we may first solve itsiequation with reference to y, construct the diameter and determine the limit as in Art. (171). This limit is tangent to the curve at the point in which it intersects, the diameter. Solve the equation with reference to x, construct the diameter and determine the limit in the direction of the axis of Y. This is also tangent to the curve at the point in which it intersects the diameter, Since these tangents are parallel to the co-ordinate axes respectively, they are perpendicular to each other and intersect on the directrix, Art. (97). Through their point of intersection draw a line perpendicular to either diameter, it will be the directrix, Art. (100). Join the two points of tangency by a chord, this will pass througl the focus, Art. (97). With either point of tangency as a centre, and the distance to the directrix as a radius, describe an arce, it will cut the chord in the focus, Art. (88). Through the focus draw a perpendicular to the directrix, it will be the axis, and the curve may then be constructed as in Art. (88). To illustrate, let us recur to example (1) in case first, of the preceding article. Having determined the limits PV and RV', through their point of intersection S, / draw SO perpendicular to DV, it is the direc- //' trix; joins' the. points V and V'; with V'E describe the arc EF cutting VV' in F, F is the focus through which the axis may be drawn parallel to DV. The ellipse when b2 - 4ac is negative. 175. The value of y, equation (1), Art. (168), may be put under the form 214 INDETERMINATE GEOMETRY. - bx + I Y —(b- -c)2 +'-_ + 2a a V - 4ac b2- 4ctcJ Those values of x, which will reduce the radical to 0, and give c-qual values of y, will evidently be obtained, by placing 2mnx n x2 + + - 0. b2 - 4ac b2- 4ac Solving this equation, and denoting the least value of x by x' and the other by x", the value of y, may be put under the form bx+d 1 / =bx + d d- 1 - (b2 - 4ac)(x - x')( - ".... (1). 2a 2a These roots x' and x" may be real and unequal, real and equal, or imaginary. When real and unequal. For every value of x > x" the factors x - x" and x - x' will both be positive, their product also positive, and the quantity under the radical sign negative. The corresponding values of y will therefore be imaginary, and there will be no corresponding points of the curve. For x = x", the quantity under the radical sign is 0, the two values of y equal, and the ordinate produced is tangent to the curve at the vertex of the diameter whose equation is, Art. (168), bx + d y --.- - 2a For every value of x < x" and > x', the two factors x - x" and x - x' will have contrary signs, their product will be negative, and the quantity under the radical sign positive, and there will be two corresponding real values of y and two points of the curve. For x = x', the quantity under the radical sign again becomes 0, and the ordinate will be tangent to the curve at the other vertex of the diameter. INDETERMINATE GEOMETRY. 215 For every value of x < x', the factors x - x" and x — x' will be negative, their product positive, and the values of y imaginary. Therefore, if two distances AP and AP', represented by x' and and x", be laid off on the axis of X, and through their extremities two lines be drawn parallel to the axis of Y, these lines will be tangent to the curve, and no point of the curve can lie without them. Hence, to obtain the limits of the A MP T - curve in the direction of the axis of abscissas; we solve the equation with reference to y, place the quantity under the radical sign equal to 0, and deduce the roots of the equation, these will be the abscissas of the limits; lay off these abscissas, and throu7gh their extremities draw lines parallel to the axis of ordinates, they will be the required limits. These limits will be tangent to the ellipse at the vertices of the diameter which bisects all chords parallel to the axis of Y. By solving the equation with respect to x, we may obtain, in a similar way, the limits in the direction of the axis of Y. If the roots x' and x" are equal, we have (X - X)(X - x) = (X - x)2 and the value of y reduces to bx d x - x' Y 2a - 2a V/2_ 2 4ac, 2a 2a which will evidently be imaginary for every value of x except x = x', and this gives for the corresponding value of y, denoted by y', bxr' + d 2a 216 INDETERMINATE GEOMETRY. y' and x' are then the co-ordinates of a single point, to which the ellipse in this case reduces, Art. (168). If the roots x' and x" are imaginary, the product (x - x') (x - x") will be positive for all values of x *; hence, every value of x, in equation (1), will give imaginary values for y, and there can be no points of the curve, which is said in this case to be imaginary, Art. (168). 176. An equation of an ellipse being given, the curve may be well represented by following the rule laid down in Art. (173). Examples. First, when x' and x" are real and unequal. 1. y2 - 2.y + 2x2 + 2y - X 0, in which b2 - 4ac 4 - 8 -= 4, and POx' AP"- 1, y = x 9F - X + 1I X 1= AP"= * NOTE.-TO prove this, we have only to recollect that imaginary roots always enter an equation in pairs, and must be particular cases of the general form x = a ba/-l, the factors corresponding to which are zx-(a +- byT) and - (a-b ), their product being x262-2ax+a +t b2 (x- a)2 + b2 which is evidently positive for all values of x, since it is the sum of two perfect squares. INDETERMINATE GEOMETRY. 217 x" - AP' - _! 2. y2 - 2zy + 2x2 - 2y- 2x = 0. 3. y2 + 2xy + 22 2x = O. 4. 2y2 - 2xy + 3x2 + 2y + x 1 = 0. Second, when x' and x" are real and equal. 1. y2 _ 2xy + 2x2 - 4x + 4 0. 2. y2 + - 2 - 2x + I = 0. Third, when x' and x" are imayinary. 1. y2 _+ xy + x2 + x + y + 1 = O. 2. y2 q- x2 +q 2x + 2 = 0O. 177. In order to construct the curve accurately; we solve the equation with reference to y, con- y struct the diameter and determine the abscissas of the limits as in A Pt P." p' Art. (175). Substituting these in - either the equation of the curve or diameter, we find for the ordinates of the vertices V and Q, bxI' + d bzx + d 2a 2a. Substituting these in expression (2), Art. (17), we have D 2 2 XI - X,~ D = \/ ) + (x' - 2 + 4a2 VQ. 4a2 2a Since this diameter bisects chords parallel to the axis of Y, its 218 INDETERMINATE GEOMETRY. conjugate will be V'Q', passing through the centre C and parallel to AY, Art. (143). If we denote the abscissa of the point C by z, and substitute it in equation (1), Art. (175), we have for the corresponding values of y, P"V' and P"Q', bx + d _ 2a 2a(b2 -. 4ac)(z - x)(z - a,) The difference of these two values is the length of V'Q'; hence, V'Q' = /(b2 _ 4ac)(z - x')(z - x"), or substituting for z its value, which is evidently X +.Xi/ 2 we have X! - XII V'Q' - V2/4ac - b2. 2a The length and position of these two conjugate diameters being now known, the curve may be constructed as in Art. (150). The angle V'CQ, made by the conjugate diameters, may be readily measured, since the tangent of the angle CDP", in any position of the diameter, will have the same numerical value as tang a, and therefore be equal to - taken with a positive 2a sign; whence, by a reference to a table of natural sines, &c., CDP" becomes known, and since VICV = 900 - CDP", we have V'CQ = 1800 - V'CV = 90~ + CDP". The two conjugate diameters and the angle made by them INDETERMINATE GEOMETRY. 219 being thus known, the curve may be constructed as in Art. (150), or the axes as well as the angles a and a' may be determined from equations (1), (2), and (3), Art. (157). The Hyperbola whlen b2 - 4ac is positive, 178. Let us resume the value of y, equation (1), Art. (175), bx + d 1 ~Y = u - 2 2av (b2 - 4ac) )(x - - x")......(1), in which, we must remember that x' and x" are the values of x obtained by placing the quantity under the radical sign, in the general value of y, equal to zero, and that they will be real and unequal, real and equal, or imaginary. When real and'unequal. For every value of x > x", the factors x - x"' and x - x' will both be positive, and the quantity under the radical sign positive. The corresponding values of y will therefore be real and unequal, and there will be two corresponding points of the curve. For x =- x" the quantity under the radical sign is zero, and the corresponding ordinate produced will be tangent to the curve at the vertex of that diameter which bisects chords parallel to the axis of Y, Art. (168). For every value of x < x"t and > x', the two factors will have contrary signs, their product will be negative, and the corresponding values of y imaginary, and there will be no corresponding points of the curve. For x =- x', the corresponding ordinate produced, again becomes tangent to the curve at the other vertex of the above diameter. For every value of x < x', the factors will both be negative, their product positive, and the corresponding values of y real. - Therefore, if two distances AP and AP', represented by wx and x", be l-aid off on the axis of X, and through their extremities two 220 INDETERMINATE GEOMETRY. lines be drawn parallel to the axis of Y, these lines will be tangent to the curve, -e 17 Z no point of the curve will lie between them, and the curve will extend to infinity in both directions without them. Ience, we obtain the limits of the hyperbola in the direction of either axis of co-ordinates in the same way as. described in Art. (175). If the roots x' and x" are equal, the value of y, equation (1), as in the corresponding case in the ellipse, Art. (175), reduces to bx + d 2x- X' = VPb2 - 4ac, 2a 2a which will evidently be real for every value of x.. This equation then represents two right lines which intersect, and to which the hyperbola in this case reduces. If the roots x' and x" are imaginary, the product (x - x') (z - X") will be positive for all values of z; [see note, Art. (175)], hence every value in equation (1), will give real values for y, and two corresponding points of the curve, and there will be no __, __1 limits in the direction of the axis of X, as was to be expected, since the abscissas of these limits x' and x" are imaginary. It also follows, that the diameter which bisects chords parallel to the axis of Y, has no vertices, or does not intersect the curve, which must be as represented in the figure. 179. An equation of an hyperbola being given, the curve may be well represented by following the rule laid down in Art. (173). INDETERMIN ATE GEOMETRY. 221 Examples. First, when x' and x" are real and unequal. 1. y2 2l xy- 2 + 2- =0. in which b2 - 4ac = - 4 x 1 X - I/ 4 4 =r 4 8, - A _ y - x -'2x2 - 2. 2. y2 -x-2+2x-2y+1=0. 3. Y2 + xy- 2x2 +- x =O. 4. y2 2xy - 2 - 2y + 2x + 3 - 0. Second, when x' and x" are real and equal. 1 y2 _ 22 + 2y + 1 = O. 2. y2 - X2 = 0. 3. y2 + xy 2x2 + 3 x- 1 = 0. Third, when x' and x" are imaginary. 1. y2- 2xy - x2 - 2 =- O. 2. y2 + 2xy - x2 + 2x + 2y 1 = O0. 3. y2 - 2xy - x2 - 2x - 2 =-0. 180. The curve may also be constructed accurately, by first determining the length and position of two conjugate diameters, )precisely as in Art. (177). The expressions for these diameters ~222 INDETERMINATE GEOMETRY. will be the same as those determined for the ellipse. For the one which bisects chords parallel to the axis of Y, Xa - X"/ 2a -/b2 + 4a2; and for its conjugate' _- XSt 2a /4ac - b, the first of which will be real, and the second imaginary, when x' and x" are real, and the reverse when x' and x" are imaginary. In this case, the angle V'CD [see figures in Art. (178)], included between the two conjugate diameters, is always equal to 90~ - CDA. But we know that tang CDA is numerically equal b to tanga a -. We therefore have 2a tang V'CD - cot a, from which the angle may at once be found, and then the curve be constructed as in Art. (150), or the axes, together with a and aC', may be found from equations (1), (2) and (3) of Art. (158). OF CENTRES AND DIAMETERS. 181. The centre of a curve is a point, through which, if any straight line be drawn, terminating in the curve, it will be bisected at this point. It follows from this definition, that for each point, as M, of a curve which has a centre, there will be another corresponding point, as M', on the opposite side of the centre and at the same distance from it. If therefore the origin of co-ordinates be placed at the centre, the co-ordinates of these two points will be equal INDETERMINATE GEOMETRY. 223 with contrary signs; that is, if the co-ordinates of one point are + x' and + y', those of the other will be m, - x' and - y', and the equation of the curve must be satisfied by p the substitution of each of these I sets of co-ordinates. But, this can not be the case, unless all the terms of the equation containing the variables are, of an even degree; for if some are of an odd degree, the signs of these terms will be different when - x' and - y' are substituted, from what they are when + x' and + y' are substituted, while those of an even degree will remain the same. It is evident then, that the equation can not be satisfied, in both cases. In order then to ascertain whether a given curve has a centre, we first examine its equation and see if all its terms are of an even degree with respect to the variables. If they are, the origin of co-ordinates is a centre. If they are not, we substitute for the variables their values taken from the formulas (2), Art. (67), and see if such values can be assigned to the arbitrary constants a' and b' as will cause all the terms of an odd degree to disappear. If so, the curve will have a centre at the new origin, and the values of a' and b' will be its co-ordinates when referred to the primitive system. If no real and finite values can be thus assigned, the curve will have no centre. 182. To apply the above principles to lines of the second order, we resume the general equation ay'2 + bxey + cx2 + dy + ex + f = 0, and substitute for x and y their values taken from the formulas of Art. (67)..x = a' + x', y = b' + y'. 224 INDETERMINATE GEOMETRY. we thus obtain, after reducing, and denoting the sunl of all the terms independent of the variables by f'. ay'2 + bxJ i+ cx'2 + (2 ab' 4- ba r+d)y' +(2ca'+ bb' + e)x' +f'= O. The terms of this equation will all be of an even degree, if 2ab' ba' + d = 0, 2ca' +- b'+ e = 0, which give for a' and b', the values a' - 2ae bd 2cd - be -b2V 4ac' b 4ac These will be real and finite when b2 - 4ac is not zero, from which we conclude that there is always a single centre for each ellipse and hyperbola. When b2 - 4ac = 0, and the numerators are not zero, the above values reduce to infinity; from which we conclude that, in general, the centre of the parabola is at an infinite distance, or that the parabola has no centre. If b2 — 4ac = 0 and 2ae - bd = O, we must also have 2cd - be = 0, 2ae for by substituting in this the value of d =-, taken from the last expression, it becomes 4ace - be be = or (4ac- b2)e = 0; b b hence, in this case the two values of a' and b' both become 0, Oe 0 indeterminate; from which we conclude that there are an infinite number of centres, which was plainly to be anticipated, as in this case the parabola reduces to two parallel right lines, Art. (171), INDETERMINATE GEOMETRY. 225 and any point of the diameter midway between them will fulfil the condition of a centre. 183. A4 diameter of a curve is any straight line which bisects a system of parallel chords drawn in the curve, Art. (100). In lines of the second order, if the axis of X be a diameter and the axis of Y be placed parallel to the chords which this diameter bisects, it is evident that the equation of tthe curve, when referred to these axes, must be of such a form as to give for each single value of x, two values of y, equal with contrary signs. Thus if AX be a diameter, taken as the axis of /'~ X, and AY be parallel to the chords a which AX bisects, then for each value of x as Ap, the two corresponding values of y, pm and pm', must be equal with contrary signs. This can not be the case as long as the equation of the curve contains any term with the first power of y. The reverse is also true; for if the equation contain no term with the first power of y, for each value of x there will be two equal values of y with contrary signs, and these two values taken together will form a chord bisected by the axis of X. This axis will therefore be a diameter. The same reasoning will show that if the axis of Y be a diameter and the axis of X parallel to the chords which it bisects, the equation of the curve can contain no term with the first power of X. 184. Let us now take the general equation of the second degree, Art. (167), and see if by any change of the position of the axes of co-ordinates, we can make either of these axes a diameter. For this purpose, let us substitute for x and y, their values taken 15 226 INDETERMINATE GEOMETRY. from formulas (3), Art. (67). The new equation, leaving out the dashes of the variables, will be of the form, my2 + pxy. + nx2 + qy + rx + s = 0, in which m = (a tang2 a' + b tang a' + c) cos2 a'............(l). n = (a tang2 ua +b tang a + c) cos2 a.............(2). p = (2a tang a tang a' +b(tang a +tang a')+ 2c)cos a cos a'.. (3). = [(2ab' + baE + d) tang a' + (2ca' + bb' + e)]cos a'...(4). r = [(2ab,' + ba' + d) tang a + (2ca' + bb' + e)] cos a...(5). If now the axis of X is a diameter, and the axis of Y parallel to the chords which it bisects, we know from the preceding article, that we must have p _o, O q= o. We have then to assign such values to the arbitrary quantities a, a', a' and b', as will satisfy the equations 2a tang a tang a' + b(tang a + tang a') + 2c = 0............(6), (2ab' + bha + d) tang a' + 2ca' + bb' + e 0.............(7), and whatever the curve is, this can in general be done; for any value assigned to a in equation (6), taken with the corresponding deduced value of a', will of course satisfy this equation. Tang a' being thus fixed, equation (7) can only be satisfied by means of values attributed to a' and b'. But any value of a' taken with the corresponding deduced value of b' will satisfy this equation. In the same way it may be shown that if the axis of Y is a diameter, and the axis of X parallel to the chords which it bisects, we must have p - O, r - 0, and that these equations can always be satisfied. INDETERMINATE GEOMETRY. 227 If both of the axes of co-ordinates are diameters, at the same time, and each parallel to the chords which the other bisects, we must have -= 0, q -- 0, r = 0. We have seen above, that it is always possible to satisfy the equation p = 0,...... (6), by assigning at pleasure a value to either a or a', and deducing the corresponding value of the other. These two angles being determined, a proper direction is given to the new axes of co-ordinates, while the new origin is yet to be fixed, so that we may have at the same time = O, r —0; that is (2ab' + ba' + d) tang a' + (2ca' + bb' + e) = 0, (2ab' + ba' + d) tang a +- (2ca' + bb' + e) = 0. These equations being the same, except that tang a in one, occupies the place of tang a' in the other, it is evident they can not both be satisfied, at the same time, unless we have the terms separately equal to 0, that is, 2ab' + ba' + d = O, 2ca' +r bb' - e = 0, which give for a' and b' the values 2ae- bd 2cd -be b2 4ac b2 - 4ac We recognise these values as the co-ordinates of the centre of the curve, Art. (182), and therefore conclude that the new origin must be at the centre, and that the new axes are conjugate diameters, Art. (143). And since the above values are finite only for the ellipse and hyperbola, and infinite for the parabola, we conclude that both of the co-ordinate axes may be diameters at the same time in the ellipse and hyperbola, but not in the prrabola. 228 INDETERMINATE GEOMETRY. And since there are an infinite number of values of a and at which will fulfil the above conditions, we conclude that in the ellipse and hyperbola, there is an infinite number of conjugate diameters. We have seen above that equation p = 0, being satisfied, the axis of X will be a diameter, if we also have q = 0. If in this equation (7) we regard a' and b' as variables, it will be the equation of a straight line, and any values of a' and b' which are the co-ordinates of a point on this line will satisfy the equation, Art. (23); hence, the new origin may be any where on this line. But this new origin must be on the new axis of X, and may be any where on this axis, (now a diameter of the curve). Hence q = 0, must be the equation of this new axis of X, or diameter, referred to the primitive axes, a' and b' being the variables. If the axis of Y be made a diameter, similar reasoning will show that r = 0 will be the equation of this diameter. The fact that q = 0 is the equation of a diameter, leads to two important conclusions. First. Since by assigning all possible values to a' this equation may be made to represent all possible diameters, and since the coordinates of the centre, Art. (182), when substituted for a' and 6', in this equation, must satisfy it, as they were obtained by placing 2ab' + ba' + d = O, 2ca ~ + bb' - e - O, we conclude that every diameter passes through the centre. Second. If any straight line be drawn through the centre, and the origin of co-ordinates be placed at the centre, and the right line be taken as the axis of X, the values of a' and b' will satisfy the equation q = 0; and the position of'the line being given, a is known, and the corresponding value of a', deduced from the INDETERMINATE GEOMETRY. 229 equation p = 0, will satisfy it also and give a proper direction to the axis of Y. Both of these equations being thus satisfied, we conclude that the right line is a diameter; hence, every right line passing through the centre is a diameter. 185. We have seen in the preceding article, that both axes of co-ordinates can not be diameters in the parabola, but that the axis of X will be a diameter and the axis of Y parallel to the chords which it bisects, when -- 0~, = - 0, and as the equation when referred to these axes is still the equation of the parabola, we must have, Art. (169), p2 - 4mn = 0, and since p = 0, - 4mn must equal 0. But m can not be 0, for if it were, the equation referred to the new axes would reduce to nx2 + rx + s = 0O which is the equation of no curve; hence, we must have n = 0, and the equation will reduce to my2 + rx + s = 0. Hence, in the parabola n = 0 is a condition consequent upon p = 0 and q = 0. This fact may be verified thus: Since in the parabola all diameters are parallel, and make with the axis of X an angle whose tangent is - b, Art. (172), and since the new axis of X is 2a 230 INDETERTMINATE GEOMETRY. Substituting this value in equation (2), Art. (184), we have ab2 b2. b2 b2 - 4ac 0 n = - - - C - +c -- O. 4a2 2a 4a 4a If the axis of Y is a diameter, it may be proved, in the same way, that we must have m = 0, and that the equation of the parabola will take the form nx2 + qy + s O0. It may be further remarked, that any value whatever being assumed either for tang a or tang a' and substituted in equation (6), will, for the parabola, give - for the value of the other. 2a Also, if - be substituted in the same equation for tang a 2a 0 or tang a', the corresponding value of the other will be ~, or indeterminate. This is evidently a consequence of the parallelism of the diameters of the parabola. OF LOCI. 186. The term locus, in Analytical Geometry is applied to the line or surface, in which are to be found all of the positions of a point or line, which changes its position in accordance with some determinate law. Thus, if a point is moved in a plane, so that it shall always be at the same distance from a fixed point, the locus of the point will be the circumference of a circle. Also, a plane tangent to a surface at a given point, is the locus of all right lines drawn tangent to lines of the surface at this point. INDETERMINATE GEOMETRY. 231 187. The determination of the loci of points, which are moved in a given plane subject to certain conditions, gives rise to a great variety of interesting problems, several of which it is proposed to solve and discuss in detail, for the purpose of indicating to the student the general method to be pursued in the solution of all. It should be remarked, that pains should be taken to select the best position for the co-ordinate axes in each problem, as its solution may be thus much simplified. 188. Problemn st. To determine the locus of a point, which in any of its positions is at equal distances from a fixed point and fixed right line. c Let F be the given point and BC the given right line. Through F draw FB perpendicular to BC and denote the known - distance FB by p. At the middle point of FB erect AY perpendicular to it and take AX and AY as the co-ordinate axes. Let M be any position of the moving point, the co-ordinates of which are AP = x, and PM = y. By the conditions of the problem, we must have MF = MC. But MP= _v-P2 +jVp12 y2 + (X 2 and MC = BP = BA + AP = x + P, hence 232 INDETERMINATE GEOMETRY. y2 + (_ m -s =x + P 2 2 Squaring both members and reducing, we obtain y2 = 2px, an equation expressing the relation between x and y for all positions of the point M. It is therefore the equation of the locus, which is a parabola, Art. (88). 189. Problem 2nd. To- find the locus of a point moving in such a way, that the sum of its distances from two given points shall always be equal to a given line. Let F and F' be the two given points, and 2c the distance between them. Let 2a represent the given line. At C, the middle point of FF', erect the perpendicular CD and - n take CF and CD as the co-ordinate axes. Let M be any position of the point and denote its co-ordinates by x and y, and AP' cU P'- denote by r and r' the distances from the point to F and F'. The right angled triangle FMP gives FM2 = MP + -2 or, since CF = c, r2 = y2 + ( - C)2. In the same way the right angled triangle F'MP, gives r,2 = y2 + (X + C)2. Adding these two equations, member by member, we have r2 +.12 = 2(y2 + x2 + c2)............(1), and subtracting them, INDETERMINATE GEOMETRY. 233 r/2 - r2 = 4cx, or (r +') (r - r') 4cx......(2). But by the condition of the problem, r' + r = 2a............(3). Substituting this in equation (2), we have r - r 2cx a Combining this with (3), we deduce r' a + E, r a -...........(4). a a Squaring these values and substituting in (1), we obtain a2 + C X = y2 + 2 4 C2 a2 or a2y2 + (a2 - c2) x2 = a2 (a2 - C2) or putting b2 for a2 -c2, a2 y2 + b2x2 - a2b2, the same as equation (e), Art. (105), and the locus is an ellipse. 190. PProblem 3d. To find the locus of any point of a given right line, which is moved so that its extremities shall be constantly in two other right lines, at right angles to each other. Let AX and AY be the two right lines at right angles, and M any point of the given line CB. Denote the i distance BM by a, and MC by b. Take AX e and AY as the co-ordinate axes and let AP = x, PM =y. Since MP is parallel to AB, we have PC ~ MC:: AP: BM, 234 INDETERMINATE GEOMETRY. or /b2 - y2: b:: x: a; whence b2x2 - a2b2 - a2y2, or a2y2 + b2X2 = a2b2 which is evidently the equation of an ellipse whose semi-transverse axis is BM and semi-conjugate MC. 191. Problemn 4th. To find the locus of the centres of all circles which pass through a given point and are tangent to a given right line. Let M be the given point, and BX the given line. Through M, draw MA perpendicular to BX, and let AX and AM be the axes of coordinates. Denote the ordinate MA by p, the abscissa of this point B P AF X will be 0. Let C be the centre of one of the circles and denote its co-ordinates by x' and y'. The equation of this circle, Art. (34), will be (X - x )2 + (y - y-)2 = 2. But since it passes through the point M, the co-ordinates of this point will satisfy the equation, and give x'2 + (p - y')2 -= R2 and since the circle is tangent to BX, we have R = y'; hence x12 + (p - y)2 = y122 or x'2 - 2py' -- p2 INDETERMINATE GEOMETRY. 235 which expresses the relation between x' and y' for any position of the circle, it is therefore the equation of the locus. If the origin be now transferred to V midway between M and A the formulas (2) of Art. (67) become x' =, y' = y - the substitution of which gives X2 = 2py, the equation of a parabola of which M is the focus and BX the directrix. and this is evidently another method of enunciating and solving problem 1st, Art. (188). 192. Problem 5th. To find the locus of the intersection of right lines, drawn from the extremities of the transverse axis of a given ellipse, to the extremities of chords of the ellipse perpendicular to the transverse axis. Let ABD be the given ellipse and DD' any chord perpendicular to AB. Through D and D' draw the lines AD and BD', it is required to find the locus of M, their point of intersection. Let the equation of the B P given ellipse be a2y2 + b2x2 = a262b and denote the co-ordinates of the point D by x' and y'. The equation of condition that this point shall be on the ellipse will be a2y/2 + b2x'2 = a2b or y'2 - (a2 _ X2)..(1) 23G INDETERMINATE GEOMETRY. The equation of the right line AD, passing through the two points A and D, Art. (31), will be y Y ( + a)............(2), X' -+- a and of the line D'B, Y = ( -a).........(3). X' - a Multiplying these equations, member by member, we have y2 = __V (2 (4), X/2 a2 in which y and x are the co-ordinates of the point of intersection, for the two particular lines AD and D'B. If y' and x' be eliminated from this equation, it is evident that y and x will belong to no particular lines, but will be the co-ordinates of the point of intersection of all the lines which fulfil the required condition; and the resulting equation will be the equation of the required locus. Substituting the value of y'2 taken from equation (1) in equation (4), it reduces to Y2 = b (x2 _ a2), a2 which is the equation of an hyperbola having the same axes as the given ellipse, Art. (105). This method of determining loci, by combining two equations belonging to particular lines, so as to eliminate the arbitrary constants which serve to determine the position of the lines, thus deducing an equation independent of these constants, and therefore belonging to all lines which fulfil the required condition, is of frequent use. 193. Problem 6th. If from the extremity of a diameter of a circle INDETERMINATE GEOMETRY. 237 any straight line, as AR, be drawn until it intersects the tangent BR at the other extremity, and the distance AM be laid off equal to NR, it is required to find the locus of M. Let A. be the origin, and AB and AY the co-ordinate axes. Let A P 1" AB = 2a, AP=- x, PM = y. Then drawing NP' parallel to MP, we have AP PM:: AP': P'N. Also, since AM = NR, AP - P'B, P'N = vP'B x AP' = v(2 - x). The above proportion then becomes x: y:: 2a - x:Vx(2a -x); whence y3 x3 y2 or Y =-\/- _; 2a- x 2a- x for the equation of the locus. The equation being of the third degree, the line is of the third order, Art. (33). All negative values of x give imaginary values for y. x = 0 gives y = i4-O. Each positive value of x < 2a gives two real values of y, equal with contrary signs. x = 2a gives y = i so. All positive values of x > 2a give imaginary values for y, and the curve is as indicated in the figure, the line BR being an asymptote, Art. (161). It is called the Cissoid of Diocles. 194. The following problems may be solved by pursuing methods similar to those indicated in the preceding articles. 238 INDETERMINATE GEOMETRY. 7. To find the locus of a point moving in such a way, that the difference of its distances from two given points shall always be equal to a given line. la D X 8. Given the line AB and the two lines AC DB and AD', to find the locus of M moving U r \ so that MP shall be a mean proportional -A P B between PC and PD. 9. Given the base of a triangle and the difference of the angles at the base, to find the locus of the vertex. 10. Given the base of a triangle, to find the locus of the vertex when one angle at the base is double of the other. 11. To find the locus of the point of intersection of a tangent to an ellipse, with a perpendicular let fall upon it from either focus. 12. Given the semi-circle ASB, to find the locus of the point M, so that we may always have AP: PS AB: PM. l 13. Given the indefinite right line AB, the point C, and the perpendicular CD, to, - find the locus of M so that we may always have MR _ AD. OF SURFACES OF REVOLUTION. 195. A suiface of revolution, is a surface which may be generated by revolving a line about a right line as an axis. By revolving, is to be understood, moving the line in such a manner, that each point of it will generate the circumference of a INDETERMINATE GEOMETRY. 239 circle whose centre is in the axis, and whose plane is perpendicular to the axis. The moving line is called the generatrix. From the definition it follows, that every plane perpendicular to the axis will cut a circle from the surface. Every plane passed through the axis will cut from the surface a meridian curve, or line, and if this be revolved about the axis will generate the surface. 196. In order to obtain the general equation of a surface of revolution, Art. (54), let us take the axis of the surface for the axis of Z, and the co-ordinate planes at right angles. The general equation of the generatrix will then be, Art. (52), X =f(z), y = f(z).......(1), and let r denote the distance of any point of this line from the axis. Since, from the nature of the surface, this point in its revolution must describe a circle whose centre is in the axis of Z, and whose plane is perpendicular to this axis, that is parallel to the plane XY, we must have in every position of the point, x2 + y2 = r2............ (2), and since this point is on the generatrix, the values of x and y taken from equations (1), must fulfil the condition expressed by equation (2), and give 2 2 f(7) + f'(z) r_2 Equating these two values of r2, we have X 2 + y2 = f(z) + f'(z).......(3), an equation expressing the relation between the co-ordinates of the point in all of its positions. It is therefore the equation of the surface, in which f(z) and f'(z), are the values of x and y obtained by solving the equations of the generatrix. 240 INDETERMINATE GEOMETRY. 197. To illustrate, let us find the equation of a surface generated by revolving a right line about an axis not in the same plane with it. The axis of revolution being taken as the axis of Z, we may take for the equations of the generatrix, Art. (44), x = az + a, y = bz + /3, from which, we have f(z) = az + a, f'(z) = bz /3. Substituting these in equation (3), it becomes +2 + y2 = (az + a)2 + (bz + 3)2. If the axis of X be assumed perpendicular to the generatrix and intersecting it, the projection of the generatrix on the plane XZ will be parallel to the axis of Z, and its projection on the plane YZ will pass through the origin of co-ordinates; hence, Art. (45), we have a = O,, = 0, and the above equation becomes x2 - y2 _- 2Z2 = 2.........()(1). If we intersect this surface by a plane parallel to XY, the equation of which, Art. (62), is z c, x and y indeterminate, we shall obtain, Art. (62), x% + y2 = b2c2 + a2, for the equation of the projection of the intersection on the planel XY, which represents a circle whose radius is /Vb2c2 q+ a2, Art. (35); and this circle will be real, whatever be the value of c; and the smallest possible when c = 0, in which case the cut INDETERMINATE GEOMETRY. 2 41 ting plane is the plane XY, Art. (62). And since this projectiol is equal to the intersection itself, we see that every intersection by a plane perpendicular to the axis will be a circle, as we know it should be, from the definition of the surface. If we make y 0= in equation (1), we have -2 b2X2 a2 or b2z2 -_ 2 a2 for the intersection by the plane XZ, Art. (62). If we make x = 0, we have for the intersection by the plane YZ, b22 _ y2 _ _- a2 and these are evidently the equations of two equal hyperbolas, the conjugate axis of each lying on the axis of Z, Art. (105). And since the surface may be generated by revolving either of these meridian curves about the axis, it is called a hyperboloid of revolution of one nappe. Of ole nappe, since, as is readily seen, it forms one uninterrupted surface. 198. If the generatrix is in the plane with the axis of revolution, this plane may be taken for the plane XZ, and as befibre, the axis of revolution for the axis of Z, in which case the equations of the generatrix will be, Art. (52), X = f(z), y'(z) 0, and equation (3) of Art. (196) will reduce to 2 + y2 = f(Z)2............(), in which f(z) is the value of x deduced from the equation of the generatrix. Examples. 1. The equation of a right line in the plane XZ, and passing 16 242 INDETERMINATE GEOMETRY. through a point on the axis of Z, whose co-ordinates are x' 0, zf = c, will be, Art. (29), z x = a(z- c), friom which we have /A X Atf(z) = a(z - c). This substituted in equation (1), gives X2 + y2 = a2(z - c)2, for the equation of the cone generated by revolving the right line about the axis of Z. This equation may be put under the form (X2 + 2) = (Z _ c)2, or denoting the angle made by the generatrix with the base by v, we have - tang v; a whence (x2 + y2) tang2 v (z — )2, the same equation as that deduced in Art. (80). 2. If the axis of a parabola in the plane XZ, coincide with the axis of Z, and its vertex be at the origin of co-ordinates, its equation will be, Art. (84), 92 = 21)z2, from which we have A fzu = / t2nz, g fi 2 = 2pz, which substituted in equation (1), gives INDETERMINATE. GEOMETRY. 243 x2 + y2 = 2pz, for the equation of the surface generated by revolving a parabola about its axis; called a paraboloid of revolution. 3. If the transverse axis of an ellipse, in the plane XZ, lies on the axis of Z, and its centre is at the origin of co-ordinates, its equation will be, Art. (105), a2xX2 + b2z2 = a2b2, whence b2 X (a2 - Z2) = f(z), and this in equation (1), gives 62 X2 + y2 - (a2 _ z2), or a2(x2 + y2) + b2z2 a2b2.. (2) for the equation of a surface generated by revolving an ellipse about its transverse axis. If the conjugate axis of the ellipse lies on the axis of Z, the equation will be, a2 a2z2 + b2x2 = a2b2, whence x2 = l(b2 - Z2) = fz2 and the equation of the surface 62(x2 + y2) + a2z2 a.62b..........(3). These surfaces are called ellipsoids of revolution; or spheroids. The first is the prolate, and the second the oblate spheroid. If in either of equations (2) and (3) we make a = b, the ellipse becomes a circle, and the equation reduces to x2 + y2 + z2 = a2, for the equation of a sphere. 4. If in equations (2) and (3) we change b2 into - b2, we have 244 INDETERMINATE GEOMETRY. a2(x2 + y2) _ b2Z2 - a2b2 and b2( x2 + y2) - a2z2 a2b2. The first represents the surface generated by revolving an hyperbola about its transverse axis, or hyperboloid of revolution of two na2peps. Of two nappes, since it consists of two distinct parts, one being generated by one branch of the hyperbola, and the other by the other branch. The second represents the surface generated by revolving the hyperbola about its conjugate axis. Its equation, after dividing by b2, becomes x2 + y2 _ Z2 = a2 of the same form as equation (1), Art. (197). From which we see that this surface may not only be generated by revolving an hyperbola about its conjugate axis, but also by revolving a right line. asbout another, not in the same plane with it. OF SURFACES OF THE SECOND ORDER. 199. Surfaces, like lines, Art. (33), are classed into orders according to the degree of their equations. We have seen, Art. (57), that the plane is the only surface of the first order. The equation of every surface of the second order must be a particular case of the most general equation of the second degree between three variables, nzx2 + ny2 +- pz2 + m'xy +- n'xz + piyz +m//"x + n"y + p"lz +- = 0........... (1), which, for the same reason as that given in Art. (167), may be INDETERMINATE GEOMETRY. 245 considered as referred to a system of co-ordinate planes at right angles. Points of the surfaces may be determined as in Art. (55), by assigning values to x and y, and deducing the corresponding values of z; but the nature of the surface will, in general, be best ascertained by intersecting it by planes and discussing the curves of intersection thus obtained. 200. If we combine the above equation, with the equation of a plane having any position, Art. (55), and then refer the line of intersection to co-ordinate axes in its own plane, the resulting equation will be of the second degree. For one of the equations being of the first, and the other of the second degree, the result of their combination will necessarily be of the second degree. We therefore conclude, that the line of intersection of any surface of the second order by a plane, is a line of the second order, or one of the conic sections, Art. (170). 201. In the surface represented by the general equation of Art. (199), conceive a system of parallel chords to be drawn. The equations of one of these chords will be of the form, Art. (44), z = az + a, y = bz +.........(1), and these equations may be made to represent any chord of the system, by giving proper values to Mx and 1, a and b remaining unchanged. If equations (1) be combined with the general equation (1), Art. (199), and x and y be eliminated, a result will be obtained of the form 2 s t z + ~ z + _ = 0, r t in which the two values of z will be the ordinates of the points in 246 INDETERMINATE GEOMETRY. which the chord pierces the surface. If x', y' and z' denote the co-ordinates of the middle point of this chord, since z' will equal the half sum of the two values of z, we shall have z - -, 2r or putting for s and r their values, as found by the actual combiniation of the equations, (22nia + m'b +n') + /3(2nb J4 m'a + p') r m"a + n"b + p, 2(nz2 + nb2 + p + m'Cab + n'a + p'b) Since the point (x', y', z') is on the chord, we also have'= az' + a = bz' =' +. If now these three equations be combined, so as to eliminate a and,; x', y' and z' will belong to the middle point of no particular chord, and the resulting equation will therefore represent the locus of the middle points of sall the chords of the system, Art. (192). Combining the equations, by substituting for a and /3, in the first, their values taken from the last, we obtain after reduction, (2ma+ m'b ~ n')x' + (2nb ~r m'a + p')y' /+ml"a + n"b-I-p" 2p + n'a + p'b which is the equation of a plane, Art. (57). We therefore conclude, that every system of parallel chords of the surface nmay be bisected by a plane. In order that this plane shall be perpendicular to the chords which it bisects, we must have the two conditions, Art. (59), 2ma + mn'b + n= 21n) + m'a -t +' 2p + n'a + p'b 2p Jr n'c + p/b and these equations can always be satisfied by at least one set of real values for a and b; for if they be combined and either a or b INDETERMINATE GEOMETRY. 247 eliminated, there will result an equation of the third degree, containing the other, which must have at least one real root, and may have three. Hence, in every surface of the second order, there is at least one plane which is peipendicular to the system of chords wohich it bisects. 202. Let such plane be taken as the co-ordinate plane XY, the axis of Z being perpendicular to it, that is, parallel to the chords. This plane will intersect the surface in a line of the second order, Art. (200), the axis of which may be determined as in Art. (100) or (154). Let this axis be taken as the axis of X and a line, perpendicular to it in the plane XY, as the axis of Y, and suppose the surface to be referred to this new system of co-ordinate planes. Since the plane XY bisects a system of chords parallel to the axis of Z, the equation of the surface must be of such a form, that for every value of x and y, it must give two equal values of z with contrary signs. It can therefore contain no term involving the first power of z, Art. (183). We must then have in the general equation of Art. (199), n'= o, _ p' = o,' = 0............(). And since the axis of X bisects all chords in the plane XY, parallel to the axis of Y, the equation of the surface must also be of such a form that for all values of x, (z being equal to 0), there must be two equal values of y with contrary signs. The equation can then contain no term involving the first power of y. We must therefore have, in addition to the above equations (1), m = 0 n"/ = 0, and the general equation (1), Art. (199), must reduce to the form mzr2 + ny2 + pz2 + m"x + = 0............ (); 2-18 INDETERMINATE GEOMETRY. and as the above transformations are always possible, this equation may be made to represent all surfaces of the second order by assigning proper values to the constants which enter it. 203. To discuss the above equation more fully, let us first transfer the origin of co-ordinates to a point on the axis of X, at a distance from the primitive origin represented by the arbitrary quantity a', the axes remaining parallel to the primitive. The formulas of Art. (72) become X -' a + x', y-y __ Z z. Substituting in the above equation, we obtain mx, t +.2 12 + ny2, + (2 + mi) ~x +,/2 + m"a' + I = o...(1 ). Since a' is arbitrary, we may assign to it such a value as to make 2ma'+ - m" = O, or a' 2min which case the equation, after denoting the absolute term by 1' and omitting the dashes of the variables, reduces to mx2 + ny2 + lPz2 + 1' = 0............(2). If m = 0, this transformation will in general be impossible, as we shall then have at' = =............ (3). In this case we may assign to a' such a value as will malke m"a' + - =1 O, or al - and equation (1) will reduce to ny2 + pz2 + m"x = 0............(4). INDETERMINATE GEOMETRY. 249 If, however, we have at the same time in" = 0, this transformation will be impossible. But in this case, equation (1) will at once reduce to ny2 + pz2 + 1 = 0, x indeterminate............(5), which is evidently the equation of a right cylinder with an elliptical or hyperbolic base, according as n and p have the same or contrary signs, Art. (170), the axis of the cylinder coinciding with the axis of X. Moreover, in this case equation (3) gives a't = indeterminate, 0 and any point of the axis of X will fulfil the required condition. If n = 0, n = 0, equation (3), Art. (202), reduces to pZ2 + mzx ~ 1 = 0, y indeterminate. If m = 0, p = 0, it reducesto ny2 + m"x -+ = 0, z indeterminate; both of which are equations of right cylinders with parabolic baseq, the axis of the first being parallel to the axis of Y, and that of the second parallel to the axis of Z, Art. (76). If Dto" = 0 also, in the last two equations, the first will give Zvhich represents two plx an d y indetenrminate, P which represents two planes parallel to the plane XY, Art. (62); which are real when I and p have contrary signs; become one *wheen = 0; and. are imaginary when I and p have the same sign; and are particular cases of the cylinder, analogous to the l)articular cases of the parabola discussed in Art. (171). In the same way it may be proved, that the second equation will represent two planes parallel to the plane XZ. 250 INDETERMINATE GEOMETRY. If m = O, n - O, p = 0, the equation ceases to be one of the second degree. From this discussion, we see that all surfaces of the second order will belong to one of the three classes represented by the following equations. First, mMx2 + ny2 + pz2 + 1 = 0. Second, ny2 + pz2 + m"x = 0. ny2 ~- ]z2 + 1 = 0 Third, pz2 + m"x + 1 = 0 ny2 + m"x + 1 = 0 J 204. The centre of a surface is a point, through which if any straight line be drawn terminating in the surface, it will be bisected at this point. If the Origin of co-ordinates be placed at the centre, it is evident that for every point on one side of this origin, there must also be another in a directly opposite direction, at the same distance, and having the same co-ordinates with a contrary sign. Hence, the equation. of the surface must be of such a form, that it will not change, when for + x, + y and + z, -, - y and - z are substituted; that is, all of its terms must be of an even degree with respect to the variables. In order then to ascertain whether a given surface has a centre; we see if all the terms of its equation are of an even degree, if so, the origin of co-ordinates is a centre; if they are not, we then see if the origin of co-ordinates can be so placed as to make all the terms of the transformed equation, of an even degree. If this is possible, the surface will have a centre, which will be at the new origin. If it is not possible, the surface will have no centre. 205. By applying the above principles to surfaces of the second INDETERMINATE GEOMETRY. 251 order, we see that all of the first class have centres. That none of the second have centres. That the cylinders represented by the first equation of the third class have an infinite number of centres, each point of the axis fulfilling the required condition. That those represented by the second and third equations have no centres. 206. Any plane which bisects a system of parallel chords of a. surface, is called a diametral plane; and if the chords are perpendicular to the plane, it is a principal diametral plane, or simply a principal plane. Two diametral planes intersect in a diameter common to the two curves cut fiom the surface by these planes, and this intersection is also a diameter of the surface; and two principal planes intersect in an axis of the surface. A diametral plane may be constructed, by drawing three parallel chords of the surface, not in the same plane, and bisecting them by a plane. By constructing t\vo planes in this way, we determine a diameter, and the middle point of this diameter will evidently be the centre. 207. The co-ordinate planes being at right angles to each other, we see that each of them, in surfaces of the second order of the first class, is a principal plane. For, if equation (2), Art. (203), be solved with reference to either variable, we shall have two equal values with contrary signs, and these two values taken together; will form a chord, perpendicular' to the co-ordinate plane of the other two variables, and bisected by it. From this, it also follows that the axes of co-ordinates are axes of the surface, Art. (206). In the second class, equation (4), Art. (203), the co-ordinate planes ZX and YX, are also principal planes, and the axis of X is an axis of the surface. 2,52 INDETERMINATE GEOMETRY. In the cylinders represented by the first equation of the third class,, the planes ZX and YX are principal planes, and the axis of X is the axis of the cylinder. In the cylinders represented by the second, the plane XY is the ornly principal plane, and there is no axis. In those represented by the third, the plane ZX is the only principal plane, and there is no axis. DISCUSSION OF THE VARIETIES OF SURFACES OF THE SECOND ORDER. 208. All the varieties of the first class of surfaces of the second order, or those which have a single centre, may be obtained by making in their equation, Art. (203). First. mn, n and p all positive, 1 being negative or positive. Second. Either two positive and the other negative, I being positive. Third. One positive and the other two negative, 1 being positive. For if all are negative, the signs of both members of the equation may be changed, giving the first case. If two are negative, the other positive and I negative, the signs may be changed, giving the second case. If one is negative, the others positive, and 1 negative, the signs may be changed, giving the third case. First, m,, n and p positive, I negative or positive. 209. Supposing 1 to be negative, the equation of the first class, Art. (203), may be put under the form mx2 +- ny2 - pZ2 = 1............(1) Let us intersect this surface by planes parallel, respectively, to INDETERMINATE GEOMETRY. 253 the co-ordinate planes ZY, ZX and XY. The equations of the cutting planes, Art. (62), will be x - h, y = k, z = g. Combining these with the equation of the surface, Art. (62), we obtain ny2 -+ pz2 = - mh2; mx2 + pz2 = I - n2;.........(2), rnx2 +- ny2 = 1 - p?2; for the equations of the projections of the several intersections on the co-ordinate planes; and since the curves are parallel to the planes on which they are projected, the projections are equal to the curves themselves. Each of these equations represents an ellipse, Art. (169), and these ellipses will be real when the second members of the equations are positive, or h < ~-/1, g < m n P Tf 1, =./, =, v- +,, m. n p the above equations reduce to ny2 + pz2 = 0, mx2 + pz2 = 0, mx2 + ny2 O0, and the first members of each being the sum of two positive quantities, they can only be satisfied by making y =O, z = 0; x O, z = 0; = O, y - 0, which are the equatiolns of points. 254 INDETERMINATE GEOMETRY. If h > ~k > X > it X m n p the second members of the above equations'(2) will be negative, and they can be satisfied by no values of the variables, and the ellipses will be imaginary, that is, the planes will not intersect the surface. If h = O, k - O g = 0O equations (2),'become ny 2 Jr pZ2 - 1, mx2 + pZ2 -- 1 m nx2 + ny2 _=, which are the equations of the principal sections, and each of these sections is evidently larger than any other made by a parallel plane. From this discussion we conclude that if the surface, represented by equation (1), be intersected by a system of planes parallel,'spectively, /,,/'- —,-.~ to the co-ordinate planes, the curves of intersection will all be ellipses, and Al'.. Do:. /,, _:I ~/ these ellipses will diminish as the dis"/ tance of the cutting plane from the centre, on either side, is increased, until they reduce to points; after which there will be no intersection and no points of the surface. The surface is, then limited in all directions, as in the figure, and is called an Ellipsoid. If we make y = 0, z = 0, in equation (1), we have mx2 or x = _ CB or CA. In a similar way we find INDETERMINATE GEOMETRY. 255 i - CE or CE', -- CD or CD'. n 1P Placing the expressions for these semi-axes, respectively equal to a, b and c, we have a, b, - c, rn whence 1n 11 m = 2 -, X= =-, and substituting these in equation (1), we obtain x2 y2 z2 + - +- = a2 b2 C2 or b2C2X2 + a2c2y2 + a2b2z2 = a2b2c2, an equation for the ellipsoid, referred to its centre and axes, analogous to equation (e), Art. (105). 210. If m = n, equation (1) of the preceding, article may be put under the form Zn which is the equation of a surface of revolution, the axis of Z being the axis of revolution, Art. (198). But since m = n, we have a = b or CA = CE, and the surface is generated by revolving the ellipse BDA about its conjugate axis, and is the oblate spheroid, Art. (198). Likewise if i = p, equation (1), becomes 2 - mx2 2 X2 3- 22 =f(X 256 INDETERMINATE GEOMETRY. which is the equation of theprolate spheroid. If n = n = p, we obtain x2 + y2 + z2 1 which is the equation of the sphere, Art. (198). If I = 0, equation (1) becomes mx2 + ny2 + pz2 = 0, which, since the first member is the sum of three positive quantities, can only be satisfied by making x = 0, y -= z -= 0, which are the equations of a point, Art. (41). If I is positive, equation (2), Art. (203), takes the form mnx2 + 2ny2 +- pz2 = - l, which can be satisfied forno values of x, y and z, and therfore represents no surface, or an imaginary surface. From this discussion we see that the particular cases of the Ellipsoid are, the ellipsoid of revolution, the sphere, the point, and the imaginary surface. Second, m and n positive, 1) negative and 1 positive. 211. In this case equation (2), Art. (203), takes the form?mx2 + ny2 - pZ2 = - 1............(1). Intersecting the surface by planes as in Art. (209), we have, for the equations of the projections of the curves of intersection, ny2 _ pz2 = - I - mh2; mx2 _ pz2 = - 1 - 2;...... (2) mx2 + ny2 = - I + pg92 Each of the first two of these equations represents an hbyperbo -INDETERMINATE GEOMETRY. 25; la, whose transverse axis coincides with the axis of Z, Art. (105). and which increases in length, indefinitely as h and k increase. The third equation represents an ellipse, Art. (105), which is real when pg2 > i, or g > +-, p and which increases as g increases. This ellipse becomes a point when pg2 - I, or g = i yWL..-b (3), and imaginary, or there is no curve, when py2 < or Y < orp If h = 0, k = 0, g = 0, Art. (62), equations (2), become ny2- _pz2 = -1, rax2 pZ2 — = - i, mx2 + ny2 = - 1I which are the equations of the principal sections. The first two represent hyperbolas, whose transverse axes are less than those of any of the parallel hyperbolas. The third equation call be satisfied by no values of x and y, fiom which it appears that the plane XY does not intersect the surface. From this discussion, we conclude, that if the surface represented by equation (1) be intersected by a system of planes, parallel respectively to /c the co-ordinate planes, the sections plarallel to ZX and ZY, will be hy- / perbolas having their transverse axes // parallel to the axis of Z, while the 7 I sections parallel to XY, will be ellip17 258 INDETERMINATE GEOMETRY. ses when at a greater distance from the origin, above or below, than the value of g in equation (3). Hence, the surface extends to infinity in all directions from the centre, and consists of two distinct and equal parts or nappes, as in the figure. It is therefore called, an Hyperboloid of two nappes. If we make y = O z = O, in equation (1), we have -nX - I x = ~\/-" in which is imaginary, and the surface does not cut the axis of X. In a-similar way, we find f~y2= - 1, Yl,_ and pzX =, z = i = CA or CB. Placing = I /- 1, \,/ 4.Ct, we have mn, n = _, p=_ 62 b2 and these in equation (I), give x y2 z2 1 _+ - ~2 b2 a2 or a2b2z92 + CC2y2 22 - bc22 - a,262c INDETERMINATE GEOMETRY. 259 for the equation of the hyperboloid of two nappes, referred to its centre and axes. 212. If m = n, equation (1) of the preceding article may be put under the form X2 + y2 _ i + z = f (z)2 which is the equation of a surface of revolution, Art. (198), evidently generated by revolving the hyperbola about its transverse axis BA, or the hyperboloid of'revolutior of two nappes. If 1 = 0, equation (1) reduces to rnx2 + ny2 - pz2 = 0O If this surface be intersected by any plane parallel to XY, we have for the projection of the intersection mx2 + ny?2 = pg92 which is the equation of an ellipse always real, whether g be positive or negative. If g = 0, we have mx2 -+ ny2 = 0, which can only be satisfied by y = 0, x - O, which are the equations of a point. If we make first x - 0, and then y = 0, we obtain for the intersections by the co-ordinate planes YZ and XZ, the equations ny2 2 pz2 = O; mx2 _ pz2 = 0 or y zit th z n n each of which evidently represents two right lines passing through 260 INDETERMINATE GEOMETRY. the orioin, Art. (169), and the surface can only be a cone having its vertex at the origin, and axis coinciding with the axis of Z. The particular cases of the Hyperboloid of two nappes are, therefore, the hyperboloid of revolution of two nappes, and the cone. Third, n, positive, n and p negative, I positive. 213. In this case, equation (2), Art (203), takes the form mx2 - ny2 -- pz2 -............(1). Intersecting by planes, as in Art. (209), we obtain ny2 - pz2 = 1 + mh2. mx~s2 - pz2 = - 1 + nk2............(2). mx2 - ny2 = - + pg2 The first of these equations represents an ellipse, which is always real, and increases as h increases in either direction, from the origin. The second represents an hyperbola, whose transverse axis coineides with the axis of Z when the second member is negative, or nk2 < 1, and k < n and with the axis of X, when k > 4The third is also the equation of an hyperbola, whose transverse axis coincides with the axis of Y, when and with the axis of X, when INDETERMINATE GEOMETRY, 261, sg,> ~- z. If in the last two of equations (2), we make ~=:kI, g _ we have mx pz2 - 0, z mn2 - ty2 = O, X z \/ each of which represents two right lines. If h = O, k = 0, g = 0, equations (2) become ny2 +t pz2 = 1, mx2 - pz- = -, rnx 2 - ny2 -1, for the equations of the principal sections. The first represents an ellipse, which is smaller than any par-. allel section, and is called the ellipse of the gorge. The other two represent hyperbolas. We therefore conclude that, if the surface be intersected by planes parallel iz respectively to the co-ordinate planes, the sections parallel to Z X and YX are hyperbolas; while A those parallel to YZ are ellipses,,/ always real, whatever be their dis- i tances on either side of the centre. The surface then extends to infinity in all directions from the centre, without being separated into two parts. It is called an hy2perboloid of one nappe. If we make y = 0, z = 0, in equation (1), we have 262 INDETERMINATE GEOMETRY. mx2 = -, = / 1 rn7 which is imaginary. In a similar way, we find CD, CA -, both of which are real. Placing = c/_, b, a, we deduce m _, - = b, p = 2- - d2 62 T2 and these in equation (1), give a2b2X2 - a2C2y2 b2c2z2 =- a2b2c2, for the equation of the hyperboloid of one nappe, referred to its centre and axes. 214. If n = p, equation (1) of the preceding article may be written y2 z2= + x2 f(-)2 n which is the equation of a surface of revolution, Art. (198), evidently generated by revolving the hyperbola about its conjugate axis, or the hyperboloid of revolution of one nappe. If 1 = 0, equation (1) reduces to mxz2 - y2 - pZ2 - 0, which may be shown, as in Art. (212), to be the equation of a cone INDETERMINATE GEOMETRY. 203 having its vertex at the origin, and its axis coinciding with the axis of X. The particular cases of the Hyperboloid of one nappe are, therefore, the hyperboloid of revolution of one nappe, and the cone. 215. All the varieties of the second class of surfaces of the second order, or those which have no centre, may be obtained by making in equation (4), Art. (203): First, n and p positive, in" being positive or negative: Second, n positive and p negative, mn" being positive or negative. For, if n and p are negative, the signs of both members of the equation may be changed giving the first case. If n is negative and p positive, the signs may be changed giving the second case. First, n and p positive, in" positive or negative. 216. If mn" is negative, equation (4), Art. (203), may be put under the form ny2 -+ p2 m"tx.........(1). Intersecting the surface as in Art. (200), we have for the projections of the several curves on the co-ordinate planes, ny2 + pz2 = m"h, pz2 = n"x -n'2, ny2 = en"x -- p2. The first represents an ellipse, which is real as long as h is positi;e, and increases indefinitely as h is increased, becomes a point when h = 0, and is imaginary for all negative values of h. The other two represent parabolas, the axes of which coincide with the axis of X, Art. (84). And since the parameters of these parabolas are, respectively, - and _, whatever be the 1P z1 20 4 INDETERMINATE GEOMETRY. values of k and g, it follows that all the parallel sections are equal to each other. By making h = O, k = O, = O, we have for the prlincipal sections ny~2 + po2z = O, Pz2: r x ny2 = nlx. The first represents a point, the origin of co-ordinates, and each of the others a parabola, having its vertex at the origin. From this it appears that the surface extends to infinity in the positive direction of the axis of X, but does not extend at all to the left of the origin; that the intersections by one system of planes are _A 1 ellipses, and by the other two, para/ < + ____ ibolas. It is therefore called an elliptical paraboloid. If 1n" is positive, equation (4), Art. (203), takes the form ny2 + pz2 = -- x, in which, if we change x into - x, we shall have equation (1). But the only effect of this change is to estiiiate the abscissas from A to the left. The equation will then represent the same surtface revolved 1800 about the axis of Y. 217. If n = F, equation (1) of the preceding article may be written y2 + z2 X —- = 7 2 which is the equation of a parcaboloid of revolution, generated by,revolving the parabola about its axis, and this is the only particular case of the elliptical paraboloid. INDETERMINATE GEOMETRY. 2 65 Second, n positive and p negative, mn" positive or negative. 218. It will only be necessary to discuss the case where m" is negative; for, if nt" is positive, it may be shown, as in Art. (216), that the equation will represent the same surface revolved 180~ about the axis of Y. This being the case, equation (4), Art. (203), takes the form ny2 - pZ2 = mrx............(1). Intersecting the surface, as in Art. (216), we have ny2 -- pz2 = m"h... (2), pz2 - - m"x +- nk2, ny2 = m/"x - pg2. The first is the equation of an hyperbola always real, and having its transverse axis on the axis of Y when h is positive, and on the axis of Z when h is negative, Art. (105). The other two are the equations of parabolas, the first extending indefinitely in the dirction of the negative abscissas, and the second in the direction of the positive abscissas, Art. (171). By making h = 0, k = 0, g = 0, we have for the princi-pal sections ny2 - p2 = 0....(3), pz = - m"x, ny2 = m"x. The first may be put under the form ny2 = pz2, or y - - z n wrhich represents two right lines passing through the origin. The other two represent parabolas each equal to those cut out by the corresponding parallel planes. From this, it appears that the surface is unlimited in all directions; that the sections by one system of planes are hyperbolas, and by the other two, parabolas. It is therefore called a hyperbolic paraboloid. It has no particular case. 266 INDETERMINATE GEOMETRY. We have seen above that the plane YZ intersects the surface in two right lines represented'by equation (3), and that any plane par\ \ """- allel to YZ, intersects the surface > -% in an hyperbola, the projection of - which is represented by equation (2). If we denote the ordinate of any point of one of these right lines by y', to distinguish it from the ordinate of a point of the curve corresponding to the same value of z, we shall have ny'2 - pz2 - 0. Subtracting this equation, member by member, from equation (2), we have ny2 - ny'2 = nm"h; whence y - y' = n(y 4- y') Now as z is increased, y and y' are both increased, and y -y' becomes smaller and smaller, and when y and y' become infinite, y - y' becomes 0,'or the two points coincide; that is, the right line continually approaches the curve and touches it at an infnite distance, or is an asymptote, Art. (161). Heace, the two right lines represented by equation (3), will be the asymptotes of the projections of the hyperbolas cut out by the planes parallel to YZ. Or, if two planes be passed through these lines and the axis of X, the plane which cuts from the surface an hyperbola, will cut from these planes, lines which will be the asymptotes of the hyperbola. OF THE INTERSECTION OF SURFACES OF THE SECOND.ORDER BY PLANES. 219. It has been proved, Art. (200), that every intersection of a surface of the second order, by a plane, is a line of the second INDETERMINATE GEOMETRY. 2G7 order. The discussion of the nature of these sections, except when they are parallel to one of the co-ordinate planes, is much simplified by referring them to axes at right angles, in their own planes. For the purpose of this discussion, let us resume the general equation, Art. (202), mnx2 +'ny2 + p2 + m1"x + I - 0............(1), in which the origin is at some point A, on the line AX, this being the intersection of two principal planes, Art. (206). Let any plane be passed intersecting the surface, and let A'X' be its trace on the plane A' XY, making an angle / with the axis of X, and let 8 denote the angle made by this plane with the plane X1 XY. For any point of the curve of intersection, as M, we shall then have x = AP', y = PP', z MP. Let this point be- now referred to the two axes A'X' and A'Y', at right angles to each other and in the plane of the curve. Through P draw PN perpendicular to A'X', and PO parallel to AX; also draw NS perpendicular to AX. Join M and N, then the angle MNP = 0. Denote the distances AA' by a', A'N by x', MN by y', and we shall have x = a' + A'S + OP, y = NS -NO. 2 8 INDETERMINATE GEOMETRY. The right angled triangles MPN, A'SN, and PON, give z = y' sin O, NP = y' cos 8, A'S = x' cos/3, NS = x' sin/3, NO = NP cos /, PO = NP sin /3. Substituting these values in the preceding equations, we obtain x = a' + x' cos/ + y' cos d sin/3, y =x' sin/ - y' cos cos/3. If these values, with'the value, z = y' sin 8, be substituted in equation (1), the result can only belong to points common to the plane and surface, and will therefore represent the line of intersection. Making the substitution and reducing, we obtain (mn cos2 3 - +n sin2 /)x'2 - [cos2 0(1m sin2 ]- n cos2 /) + p 1sin2 0] y/2 + 2(m — n) sin 3 cos /3 cos 8 x'y'+ cos /3(2na'+- mn")x' + cos 8 sin /(2a'm + m")y'+ ma'2 + m,"a'+ 1 =.0...(2). By assigning proper values to a', we may always cause the plane to intersect the surface, and by assigning proper values to,3 and 0, we may cause the above equation to represent the several varieties of lines of the second order. 220. For instance, let it be required that the intersection shall be a right line or lines. If it is possible to cut a right line from the surface by a plane in any position, the same right line may be cut out by a plane perpendicular to the plane XY. For it is only necessary that the cutting plane should occupy the position of the plane which projects the line on the co-ordinate plane XY. We may therefore regard 0 in the above equation as equal to 900, which gives cos 0 = 0, sin8 = 1, and see if it is possible to give such values to a' and /3, as will make the equation represent one or more right lines. INDETERMINATE GEOMETRY. 269 221. For those surfaces which have a centre, we may also regard in" = 0, Art. (203). Substituting this value with the above, for cos 0 and sin 0, in equation (2), Art. (210), and omitting the dashes of x and yj it reduces to (in cos2 f + n sin2 r)X2 + py2 + 2a'm cos 3x + mna'2 + 1 = 0. Solving this with reference to y, we have y = ~t / _- [(m Cos2,3 +nsin2)x+ z2+2macos, +, 2+i.... (1) In order that this represent one or more real right lines, it is necessary that _ 1 shall be positive, and that the factor within the parenthesis shall be a perfect square, Art. (178), which requires P< 0.... (2), and (m cos2 o + n sin2 /3)(ma2' + 1) = 2a'2 cos2............(3). Deducing the value of a' from the last condition, we obtain a /: I(n_ oo R + n )m........... 4). mn sin2 Since p is positive in the ellipsoid, Art. (209), condition (2) can not be fulfilled; whence the conclusion, that no right line can be cut from this surface. Since m%, n and 1 are positive in the hyperboloid of two nappes, Art. (211), the value of a' will be imaginary for all values of /3. Condition (3) can not then be fulfilled, and no right line can be cut fromn this surface. Since in and I are positive and n and p negative in the hyperboloid of one nappe, Art. (213), condition (2) will be fulfilled, and the values of a' will be real for all values of /3 which give 270 INDETERMINATE GEOMETRY. n sin2 f < me cos2 /, or tangu2 3 < -n n and equation (1) will then represent two real right lines which intersect, Art. (178). Hence, an infinite number of right lines may be cut from the surface of the hyperboloid of one nappe by planes. 222. If we take the value of V/ma'2 + 1 from condition (3) of the preceding article, and substitute it in equation (1), extract the square root of the factor within the parenthesis, and substitute in the result the value of a', from equation (4), we shall obtain t ~(_ Sj /zMoos2s + nsin2,C + c ml> which may be put under the form c sinot + n + cot )...(1), and will represent the two right lines cut out by any plane making the angle / with the plane XZ. By changing /3 into /,', we shall obtain at once the equations of two other lines cut out by another plane. The lines cut out by two different planes are, V'x x not parallel; for the citting planes, which are also their projecting planes, are not parallel. Neither can they intersect, for if they intersect at all, it must be in the perpendicular to the plane XY, at the point P; and if we substitute A'P for x in the equation of INDETERMINATE GEOMETRY. 271 the first set of lines, and A"P for x in the equation of the second set, we must obtain the same value for y in each case. But denoting the distance PO by d, we have d d A'P = d A - d sin /3 sin i' and these values being substituted for x, each in equation (1), xvill give values which are unequal. 223. For those surfaces which have no centre, we may regard m and I as equal to 0, Art. (203). Substituting these values with cos = 0, sin d - 1, in equation (2), Art. (219), and omitting the dashes of the variables x and y, it reduces to n sin2 fix2 + py2 + In" cos f3x + mla' = 0. Solving this with reference to y, we have y, = i\/1 _(n sin2,X2 + mi" cos /3s + mllal)........... (1)..p In order that this shall represent one or more real right lines, we must have, as in Art. (221), p < o.........(2), and m"12 Cos2 I - n sin2 / m"a'...........(3); whence' / Cos02 n sin2 /3 In the elliptical paraboloid, n and p are both positive, Art. (216), condition (2) can not be fulfilled, and no right line can be cut from the surface. 272 INDETERMINATE GEOMETRY. In the hyperbolic paraboloid, n is positive and p negative, Art. (218); condition (2) is fulfilled, the value of a' will be real, and fulfil condition (3) for all values of /3; and an infinite numbel of right lines may be cut from this surface by planes. Substituting the value of a' in equation (1), and extracting the square root of the quantity within the parenthesis, we obtain -~' + mu' cos/3) -(/ n p s n sin which will represent the two right lines cut out by any plane making the angle /3 with the plane XZ. By changing / into i3', we shall obtain the equations of two other right lines cut out by another plane. It may be proved, as in the preceding article, that the lines cut out by two different planes are not parallel, and do not intersect. 224. The preceding discussion of the rectilineal sections of surfaces of the second order, enables us to classify these surfaces as they are classed in Descriptive Geometry. This classification is: 1. Plane suifaces, which may be generated by a right line moving along another right line and parallel to its first position. 2. Single curved surfaces,'which may be generated by a right line, moving so that its consecutive positions shall be in the same plane. 3. Double curved surfaces, which can only be generated!,y curves. 4. WVaped surfaces, which may be generated by a right lia:, moving so that its consecutive positions shall not be in the same plane. The cylindrical and conical surfaces are sivgyle czurved, as the consecutive elements of the first are parallel, Art. (74), and those of the second intersect, Art. (77); that is, are in the same plane. INDETERMINATE GEOMETRY. 2 73 The ellipsoid, hyperboloid of two nappes, and elliptical paraboloid, are double curved; since no right line can be cut from them, Arts. (221), (223); that is, no right line can be so placed as to lie wholly in either surface. The hyperboloid of one nappe, and hyperbolic paraboloid, are waiped; since the right lines cut from the surfaces by consecutive planes are not parallel, neithler do they intersect, Arts. (222), (223), and therefore can not lie in the same plane. 225. If it be required that the intersection represented by equation (2), Art. (219), shall be a circle, it is necessary that the coefficient of x'y' be equal to 0, and that the coefficients of x'1 and y'2 be equal to each other, Art. (169). This requires 2(m - n) sinf cos/, cds -- 0............(1), m cos2 /3 + n si2 / = cos2 (m sin2 / +ncos2 ) + p sin+.....(2). The condition (1), (mn and n being in general unequal), may be satisfied by making either sin 3 -= 0, cos / = 0, Cos -= 0. Sin/3 = 0 substituted in condition (2), gives M = ncos2 0 + p Sin12 0 re(sin2 0 + cos2 8), since sin2 8 + cos2 0 = 1. From this we deduce m tang2O r m = n + p tang2, or tang = / I............0 (3). In a similar way we find ccs: = o, tan3 (4); p - n 18 274 INDETERMINATE GEOMETRY. cos = 0, tang 13 = i_ V/.......() In the first case, the cutting plane is parallel to the axis of X; in the second, parallel to the axis of Y; and in the third, parallel to the axis of Z. It may be remarked that if any position of the cutting plane be found to give a circle, every parallel, plane intersecting the surface will also give a circle. For if the angles /3 and 6 remain the same, a' may be changed at pleasure, without affecting the equality of the coefficients of x'2 and y'2. 226. In the ellipsoid, in which m, n and p are positive, Art. (209), in order that the first set of values of tang 6, (equation (3), preceding article), may be real, we must have n > A > pI, or p > n > n. In order that the second set may be real, we must have p~ > ~n > mn, or m > n > p. In order that the values of tang f may be real, we must have n >Fp > en, or m > > p> n. It is evident that no two of these conditions can be fulfilled at the same time. If either of the first is fulfilled, we shall have, [see expressions for a, b, _/ and c, Art. (209)],'\x t x/ CE> CB > CD, or CE < CB CE > CD. or CB < CE < CD, and for either of the third CB > CD > CE, or CB < CD < CE. Hence, a cutting plane passed parallel to the mean axis of the surface may have two positions, such that the sections shall be circles, these positions being determined by the two proper values of tang 0 or tang 8f; and in no other position can the section be a circle. If m = n, both sets of values of tang 0 become 0, and tang i3 becomes imaginary. Hence the two cutting planes unite in one, parallel to XY, or perpendicular to the axis of Z; as should be the case, since the surface becomes an ellipsoid of revolution, its axis lying on the axis of Z, Art. (210). If n = p, the first set of values of tang 0 become imaginary, while the second and those of tang f3 become infinite, and the cutting plane is perpendicular to the axis of X, Art. (210). If n'= n = p, the values of tang 8 and tang /3 become -, indeterminate, and every position of the cutting plane gives a circle, as it should, since the surface becomes a sphere. 227. In the hyperboloid of two nappes, in which en and n are positive and p negative, the values of Art. (225), after giving to the letters their proper signs, become tang 8 — -/ m tang0 = + - n q- y tang p= -+ n + p The values of tang i3 are imaginary. 276 INDETERMINATE GEOMETRY. If m < n, the first set of values of tang d is real, and the second imaginary. If m > n, the reverse is the case. But, if m < n, we have c > b; andif m > n, we have c < b, Art. (211). Hence, in this surface, the cutting plane must be parallel to the longest of the two axes which do not intersect the surface. If m = n, the values of tang O./ A ibecome 0, and the cutting planes'- / I'. unite in one perpendicular to the axis of Z; as they should, since in this case, we have the hyperboloid of revolution of two nappes, Art. (212). Since the above values of tang 0 do not depend upon 1, they will remain the same when I = O, Art. (212), that is, in a cone with an elliptical base, it is always possible to pass planes in two different directions so as to cut circles. These are,called subcontrary sections. If one of them be regarded as the base of the cone, the other will be sub-contrary to the base; that is, in a scalene cone with a circular base, it is always possible to pass a system of,)lanes not parallel to the base, which shall cut out circles. If the cone is a right cone with a circular base, it is a surface of revolution, and the sub-contrary sections unite in one, perpendicular to the axis or parallel to the base. 228. In the hyperboloid of one nappe, in which m is positive, n and p negative, we have tang/ =n + m ta n + =m tanugB-+-~' nu+gm m + P n - -. tang p =:yi /p +- +nm n -p INDETERMINATE GEOMETRY. 277 The first are imaginary. If in < p, the second will be real and the third imaginary, and the reverse when n > p. X If n < p, we have b = CD > a= CA, and if n > p, we have / z CD < CA. x Hence, the cutting plane is parallel to the greatest of the two axes which pierce the surface. If n =- p, the above real values of tang 0 and tang f become infinite and the two planes unite in one, perpendicular to the axis of X, Art. (214). When the surface becomes a cone, the discussion is similar to that in the preceding article. ~229. In the elliptical paraboloid, in which m = 0, n and p positive, the values of tang 0 and tang j3 become tang 8 _ /- 0 tang ~/__ n P p — n tang -- P n - p The first are imaginary. If n < p, the second are real and the third imaginary. If n > p, the reverse will be the case. Hence, the cutting plane must be parallel to the greater axes of the elliptical sections, Art. (216). If n- = p, the above real values become infinite and the cutting planes unite in one perpendicular to the axis of X, Art. (217). 230. In the hyperbolic paraboloid, in which m = O, n positive and p negative, we have 278 INDETERMINATE GEOMETRY. tang d = i /, tang 8 -4tang/ = /- P 2+ ~n The second and third are imaginary, and the first real, and the position of the cutting plane will be given by the equations sin / 0, tango = _/n But these values with the value rn = 0, substituted in con(lition (2), Art. (225), make the coefficients of,,2 and y/2 both equal to 0, and equation (2) pf Art. (219), takes the form ex -+ f = 0, y indeteiminate, which represents a right line. Since any plane parallel to either of the planes determined by the above values of sin 3 and tang 8' will also cut a right line from the surface, we see that there are two different systems of right line elements, each of which is parallel to a given plane. We conclude, also, that no circle can be cut from the hyperbolic paraboloid. 231. The intersection of any two surfaces of the second order may be found as in Art. (62); but as their equations are of the second degree, the result of their combination, so as to eliminate one of the variables, will be of the fourth degree; hence, in general, the projections of the lines of intersection will be lines of the fourth order, the discussion of which will be complicated and of little interest. If, however, it is known that two such surfaces. intersect in a line of the second order, it will, in general, be fbund that they will also INDETERMINATE GEOMETRY. 279 intersect in another line of the second order;:that is, if one surface enters the other in a line of the second order, it will leave it in a line of the same order. To prove this, let us take the most general equations of the two surfaces, mx2 + ny2 + pz2 + n'xy + n'xz + 1p'yz + WnS"x + n"y + plz +- = 0......(1), qx2 + ry2 +,s2 + q'xy + r'xz + s'yZ +__ q" + -//"y -+ szt + i = 0......(2), and let the plane of the curve in which it is known the two surfaces intersect be taken as the plane XY. If we make z = 0, in each of the above equations, we shall have mx2 + ny2 + in'xy + mn"x + n"y + I = 0..........(3); qx2 + ry2 + q'xy + qx l'+ /y + 1' = 0..........(4); each of which must represent the known curve of intersection of the surfaces. These equations must then be the same, which can only be the case when the corresponding coefficients are equal, or when those of the first equation are equal to those of the second multiplied by a constant factor, as k. If we now multiply equation (2) by k and subtract from equation (1), we obtain (p - k)Z2 + (no - r.')xz + (p' - ls')yz + (p" - s")z = 0, which equation must be satisfied for all values of x, y and z, belonging to points cornmmon t o the tw urfaces. Since is a common factor, it may be satisfied by placing z =0 or, (p - s)z + (n' - kI')x, + (p' - ks')y + (p"ll - s") =, 280 INDETERMINATE GEOMETRY. z = 0 evidently belongs to the plane XY, in which the known line of intersection lies. The other is the equation of a plane, Art. (57), which by its combination with either (1) or (2), will give another line common to both surfaces, and this line must, of course, be one of the second order, Art. (200). 232. Let x", y" and z" be the co-ordinates of a given point on a surface of the second order. These when substituted for the variables in the general equation mx2 +- ny + pZ2 - mx I = 0............(1), must satisfy it, and give the equation of condition mzx"2 + ny"2 +- pz"2 + m"x" - I = 0...........(2). Subtracting this, member by member, from equation (1), and factoring the terms, we have m(x - x")(x + x,) + n(y - y")(y + y") + p(z - z)(z ~ z',) + m + "(- x") = 0...........(3), which is the equation of the surface, with the condition introduced, that the point x", y'", z" shall be on the surface. The equations of any right line passing through the given point, are (x-'") = a(z - z"), y- i " = b(z - z)............(4). If these equations be combined with equation (3), we shall obtain (z - z")[ma(x + x") + nb(y + y") +p(z + z") + l =......(5), in which x, y and z must denote the co-ordinates of all points coinmon to the line and surface, Art. (58). Since this is an equation of the second degree, there are but two such points; and these may be determined by placing the factors of (5) separately equal to 0. INDETERMINATE GEOMETRY. 281 z - z" = O, gives z-= z, y y", x = x'; which evidently belong to the given point. Placing ma(x + x") +. nb(y + y") + p(z +- z") + m"a = 0.........(6), x, y and z in this must represent the co-ordinates of the second point in which the line pierces the surface. If now any plane be passed through this right line, it will cut from the surface a line which will contain both of the points; and if the second point be moved along this line until it coincides with the first, the right line will become tangent to the line cut from the surface, and the values of x, y and z in equation (6), will become equal to x'", y" and z". Substituting these values in equation (6), it becomes 2mzax" ~ 2nby" + 2pz" + mr"a = 0............ (), an equation which shows the relation that must exist between a and b, in order that the right line represented by equations (4) may be tangent to a line of the surface at the given point; and since a and b in this equation are indeterminate, it follows that an infinite number of right lines may be drawn, each tangent to a line of the surface at the given point. If now ill equation (7), we substitute for a and b their values taken from equations (4), we-obtain (21,.zx" +?-,- ")(x - x") + 2ny"(y - y") + 2pz"(z - z") = 0, ol, since from equation (1), - 2mx"2 - 2ny"2 - 229z"2 = 2m"x" + 21, we have finally, 2nnx,,x + 2vny"y + 21JZ "z +',"(x + "',) +.21 = 0...........(8), an equation which expresses the relation between x, y and z for all points of the tangent line in all of its positions. The surface represented by it, is then the locus of all right lines drawn tangent to 282 INDETERMINATE GEOMETRY. lines of the surface, at the given point, or point of contact. This equation being of the first degree between three variables, is the equation of a plane. This plane is said to be tangent to the surface, at the given point; and in general, a plane is tangent to a sujface, when it has at least one point in common with it, through wVhich if any plane be passed, the sections made in the surface and plane will be tangent to each, other. For those surfaces which have a centre, the origin of co-ordinates being placed at this centre, we have m" = 0, Art. (203), and equation (8) reduces to mnxx" q-.ny' + pzz" - 1 = 0............(9). If m = n = p, equation (9) becomes XZx + yy" +" +- __ = R2, for the equation of a tangent plane to a sphere, Art. (210). For those surfaces which have no centre, m,- = 0, 1 = 0, Art. (203), and equation (8) reduces to 2nyy" + 2pzz" + m"(x + x") = 0. 233. Let x', y' and z' be the co-ordinates of a fixed point without a surface of the second order. If it be required that the tangent plane to the'surface shall pass through this point, its coordinates must satisfy equation (8), of the preceding article, and give the equation of condition 2.mx,"x' + 2ny"y' + 2z 2 "(X + X"( ) + 21 = 0.........(1). In this equation x", y" and z" are unknown, but since the point w0hich they represent must lie on the surface, we must also have the condition mZ/x2 + ny"2 + pz"2 + i2"" + I = 0............(2), INDETERMINATE GEOMETRY. 2 83 and these two equations are all the means which we have of determining the values. 6f x", y" and z"; and since we thus have three unknown quantities, and but two equations, it follows that the unknown quantities are indeterminate. Hence we conclude that, in general, an infinite number of planes can be drawn from a point without a surface of the second order tangent to the surface. If straight lines be drawn, from the different points of contact of these planes, to the fixed ppint, they will evidently form a cone which will be tangent to the surface, in the line formed by joining the points of contact. But since the co-ordinates of these points must all satisfy equation (1), when substituted for x", y" and z", the points must lie in the plane which will be represented by this equation when x", y" and z" are regarled as variables. This curve of contact must then be a plane curve, and since it lies on the surface at the same time, it must be a line of the second order, Art. (200). We therefore conclude,that, in general, the line of contact of a tangent cone and surface of the second order, is a line of the second order. And the same will be true of a tangent cylinder, inasmuch, as the cone becomes a cylinder, when its vertex is removed to an infinite distance. 234. If it be required that the tangent plane pass through a second given pdint x"'; y"', z"', without the surface, or contain the right line joining these two points, we shall also have the equation of condition 2mx"Xl" + 2nytly"' Jr 2pz"z"' -- mI"(x"' + xI") + 2 = 0, and this united with equations (1) and (2) of the preceding article, will give three equations involving three unknown quantities, and since two of these equations are of the first. and the other of the second degree, there will in general be two sets of values for x", yi" and z". Hence we conclude that, in general, two planes may 284 INDETERMINATE GEOMETRY. be passed through a right line tangent to a surface of the second order, and only'two. 235. A right line, or a plane, is normal to a surface when it is perpendicular to a tangent plane, at the point of contact. There evidently can be but one normal line to a surface at a given point; but, since every plane containing a normal will be perpendicular to the tangent plane, there will be an infinite number of normal planes. 236. The equations of a normal line, to a surface of the second order, will be of the form, Art. (50), X-x", = a(z- Z), y - yt = b(z - z")...........), in which it is necessary to determine the values of a and b on con-, dition that the line shall be perpendicular to the tangent plane represented by equation (8), Art. (232). The equations of condition, Art. (59), a = - c, b= -d, give mnx" /.4+ mni ny ~ and these, in equations (1), give z - zIIy,, mx" + (z - z"), y - Y" ( -")...(2), pz"/PZ for the equations of a normal line to any surface of the second order. By supposing m'n" = O, we shall have the particular equations for those surfaces which have a centre; anld by makingm Mn = O, we have them for those surfaces which have no centre. INDETERMINATE GEOMETRY. 285 If n = p, equation (2) reduces to yzII YIz = 0, which, having no absolute term, shows that the projection of the normal on the plane YZ passes through the origin of co-ordinates; hence the normal intersects the axis of X. But when n = p, the surface becomes one of revolution, the axis of X being the axis of revolution, Arts. (210), (214), (217).' We therefore conclude that all the normals to a surface of revolution of the second order intersect the axis of revolution; and that the meridian plane, passing through the point of contact of: a tangent plane, is a normal plane: Or, a tangent plane to a sucface of revolution of the second order is perpendicular to the meridian plane passing through the point of contact. PRACTICAL EXAMPLES. 237. Although examples have been occasionally given, in immediate connection with the articles which they are intended to illustrate, it is believed to be advantageous to add, in this place, a number of others, a portion of which the teacher may give out with each lesson; or may defer them until the subject has been completed, when their solution will serve as a general review of the principles of the course. Each example should be carefully constructed, on the black board, in proper proportion, a unit of convenient length being first assumed; or, when it can be done, should be accurately drawn on paper, with mathematical instruments. By this exercise, the principles of the subject will be strongly impressed upon the mind of the pupil, while, at the same time, a good test of his knowledge will be afforded to his teacher. The axes of co-ordinates are supposed to be at right angles, unless otherwise mentioned. 28 6 INDETERMINATE GEOMETRY. The teacher may multiply the examples to an unlimited extent, by simply substituting, for the numbers used, any others which mlay occur to him. 1. Construct the points whose equations are, Art. (16), x = 2, y =-; x= -1, 1 y = 4; x =-3, y - - 2; x - 3, y = - 5. 2. Find the expressions for the distances between the points, whose equations are, Art. (17), x'= 1, y' 3; x" 0, y"- - 2; XI:- 3_ Y' 4; Xl 2X y" - 1. 3. Construct the points whose polar equations are, Art. (18), v = 201, r = 5; v = 190~ r = 2. 4. Construct the right lines whose equations are, Art. (26), 2y - 3x + 1 = 0; 3y - x = 0. 5. Find the point of intersection of the right lines, whose equations are as in the last example, Art. (27). 6. Find the expression for the tangent of the angle, included by the same lines, Art. (28).'7. Ascertain if the lines represented by the equations 2y- 5x - 1 = 0, y = 3x- 2, are parallel, or perpendicular to. each other, Art. (28). 8. Find the equation of a right line, passing through the point x' = 2, y' - - 4, and parallel to the line whose equation is, Art. (30), 3y + 2x -- 1 0. INDETERMINATE GEOMETRY. 287 9. Find the equation of the right line passing through the same point and perpendicular to the same line, Art. (30); also, the length of the perpendicular, Art. (17). 10. Find the equation of a right line passing through the two points, x= 3, y' - 4; x!= - 2, y/ = - 1. 11. Find and discuss the equation of a circle, the co-ordinates of whose centre are x' = 3, y' = - 2; and whose radius is 3, Art. (34). Also, when the co-ordinates of the centre are XI = - 2, y' = 0; and the radius 4. 12. Find the intersection of the last circle with the right line whose equation is, Art. (27). y -- - 3x - 1. 13. Ascertain if the point x = 1, y = - 2, is on, without, or within the circle whose equation is, Art. (37), x2 +- y2 = 9 14. Find the equation of a circle which shall pass through the point x = 3, y = - 2; the origin being at the centre. Also of one passing through the point x = 4, y - 2, the origin being at the left hand extremity of the diameter, Art. (38). 15. Construct the points, in space, Art. (40); x = 1, y= 2, z - 3; x - - 2, y - 3, z - 4. 16. Find the expression for the distance between the two points given in last example, Art. (42). 17. Construct the point whose polar co-ordinates are, Art. (43), 288 INDETERMINATE GEOMETRY. U = 35~0, v = 70~, = 4. 18. Construct the right line, in space, whose equations are, Art. (44), x = 2z +- 3, y z 2, and cfind the equation of its third projection. 19. Find the point of intersection of the two lines, in space, whose equations are, Art. (47), x = - 2z- +3, y = z - 2; x = 3z - 1, 5y = -- 10z - 2. 20. Find the expression for the cosine of the angle included between the lines given in the last example, Art. (49). 21. Ascertain if the lines whose equations are, x = 2z + I, y =3z- +4; = -2z 3, y = z -- 2; are parallel, or perpendicular, Art. (49). 22. Find the equations of a right line which shall pass through the point x' =- 3, y' = 2, z' = - 1, and be parallel to the line whose equations are, Art. (49), x - 3z- 3, y = 4z +- 3. 23. Find the equations of a right line which shall pass throujgh the same point and be perpendicular to the same line, as in tle last example, Art. (49). 24. Find the equations of a right line which shall pass tllhroughl the two points, Art. (51), - 1, y' = 2, z'= 0; x"= 3, y=O, z" - 2. INDETERMINATE GEOMETRY. 289 25. Find the intersection of the two lines whose equations are, Art. (53), x2 + z2 - 5 = 0, + y - 3 = 0; x - 3z + 5 = 0, z2 + 4y2- 8y = 0. 26. Find the equation of a plane whose directrix is represented by'4y - 3x + 1 = 0, the projections of the generatrix making angles with the axis of Z, whose tangents are 2 and - 3, Art (55). 27. Find the equations of the traces of the plane represented by 2z - 3y + x + 4 - 0, and the points in which it cuts the co-ordinate axes, Art. (56). 28. Find the point in which the right line, whose equations are x 2z + 2 0 2. y + 3z -1 = 0, pierces the plane given in the last example, Art. (58). 29. Ascertain if the same line and plane are perpendicular to each other, Art. (59). 30. Find the equations of a right line passing through the point x2 =- 1, y' = - 3, z' - 0, and perpendicular to the plane represented by 3x - 4y + z- I = 0; also the point in which the line pierces the plane, and the length of the perpendicular, Art. (60). 31. Find the expression for the cosine of the angle made by the line whose equations are 19 290 INDETERMINATE GEOMETRY. x = 3z + 5, = - + 1, with the plane given in the last example, Art. (61). 32. Find the intersection of the two planes whose equations are, Art. (62), 3x- 5y + z = 0, x y 3z 1 - 0. Also, the expression for the cosine of the angle included by the same planes, Art. (63). 33. Find the equation of a plane passing through the origin of co-ordinates, and the two points, Art. (65), xi = - 1, yI = 2, z' = 3; "=- O, - 2, Z - 1. 34. The equation of a circle being X2 + y2 = 9; find its equation referred to a system of co-ordinate axes, making an angle of 450 with each other, the new axis of X being parallel to the primitive, and the new origin being at the upper extremity of the vertical diameter, Art. (67). 35. Find the general equation of the circle referred to any set of oblique co-ordinate axes, Art. (67). 36. Find the general polar equation of the right line, Art. (69). 37. Find the equation of a cylinder, the equation of the directrix being, Art. (75), y2 2x -X2, and the elements being parallel to the line, x - 2z + 4, y = - 3z- +1. INDETERMINATE GEOMETRY. 291 38. Find the intersection of the cylinder of the preceding example by the plane whose equation is, Art. (62), 3x - 2y - 3z + 2 = 0. 39. Find the general equation of a cylinder, with an elliptical base, the origin of co-ordinates being at the centre of the base, Art. (75). 40. Find the equation of a cone, the co-ordinates of the vertex being x' = 1, y' = 2, z' = - 3, and the equation of the directrix, Art. (77), y2 = xz. 41. Find the intersection of the same cone by the plane whose equation is, Art. (62), x + 2y - 3z - 0. 42. Find the equation of a right cone, the equation of the directrix being x2 + y2 = 9, the altitude being 5, Art. (77). 43. Intersect the same cone by a plane passing through the axis of Y and making an angle of 45~ with the base, and find the equation of the intersection in its own plane, Art. (81). 44. Find the general equation of a cone with a hyperbolic base, the origin of co-ordinates being at the centre of the base, Art. (77). 45. Construct the parabolas whose equations are, Arts. (85), (86), 2 = 4X; y2 = - 3x; x2 9y. 46. Ascertain whether the point x' = - 3, y' = 3, is 292 INDETERMINATE GEOMETRY. without, on. or within each of the parabolas given in the preceding example, Art. (87). 47. Find the equation of a parabola which shall pass through the point x' = 3, y' = 5, Art. (29). 48. Find the intersection of the circle and parabola whose equations are, Art. (27), x2 +- y+ = 6, y2 = 2x. 49. Find the equation of a tangent to the parabola y2 = - 2x, at the point y' = 4, x' = 8, Art. (90). Find also the equation of a normal at the same point, Art. (98). 50. Find the equation of a tangent to the parabola y2 = 4x, and parallel to the right line whose equation is, Arts. (30), (90), 2y = 3x + 5. 51. Find the equations of the two tangents to the parabola represented by y2 = 6x, which shall pass through the point ~_ = 1, y' 4, Art. (93). 52. Find the equation of the polar line to the point c - 2, d = 1, for the parabola represented by y2 = 3x, Art. (95). 53. The equation of the polar line to the same parabola being y = x + 2, find the co-ordinates of the pole, Art. (95), 54. The equation of a parabola being y2 = 4x, find its equation when referred to a diameter and tangent at its vertex, the tangent making an angle of 450 with the axis, Art. (99). 55. Determine the axes, and construct the ellipses, whose equations are, Art. (106), 2y2 + 32 =. 4 4 4y2- x2 = 9. INDETERMINATE GEOMETRY. 203 56. Determine the axes.and construct the hyperbolas, whose equations are, Art. (107), y2 _ 3x2 = - 5; 2y2 - 4x2 = 4. 57. Ascertain whether the point x' = 2, y' = 3, is without, on, or within each of the curves given in the last two examples, Arts. (109), (110). 58. Find the equation of an ellipse which shall pass thlough the point x' = 3, y' = 2, the origin of co-ordinates being at the centre, and the semi-transverse axis equal to 4, Art. (125). 59. Find the equation of an hyperbola which shall pass through the point x' = - 3, y' = - 2, the origin being at the centre, and the semi-conjugate axis equal to 2, Art. (126). 60. Find the intersection of the ellipse and parabola, whose equations are, Art. (27), 2y2 + 4x2 = 8, y2 = - 5x. 61. Find the intersection of the ellipse and hyperbola, whose -equations are, Art. (27), 3y2 - X2 = 3, 2y2 - 3x2 - 6. 62. Find the equations of a tangent and normal, to the ellipse represented by 4y2 +' x2 = 9, at the point x" = 1; y" = V, Art. (128). 63. Find the equations of a tangent and normal to the hyperbola 4y2 - 2x2 - 8, at the point x" = v/~, y" = V/2, Arts. (131), (30). 294 INDETERMINATE GEOMETRY. 64. Find the equation of a tangent to the ellipse 4y2 + (39x2 = 36, and making the angle 450 with the axis, Arts. (30), (128). 65. Find the equations of the two tangents to the ellipse re1)resented by 4y2 + 3x2 - 12, wvhich shall pass through the point x' - 1, y' = 4, Art. (133). 66. Find the equations of the two tangents to the hyperbola represented by y2 _ 3x2 = _ 57 which shall pass through the point x' = 2, y' = 3, Art. (134). 67. The equations of anr ellipse and its polar line being 4y2 + 22 - =8; y = 2x. + 6, find the co-ordinates of the pole, Art. (139). 68. The equation of an hyperbola being 3y2 - 2x2 = - 6, find the equation of the polar line of the pole c = 4, d = 0, Art. (140). 69. Construct an ellipse, the two conjugate diameters of which are 6 and 4, making an angle of 120~; also an hyperbola having the same conjug~ate diameters, Art. (150). 70. Find the position and length of the equal conjugate diameters of the ellipse, whose equation is, Art. (159), 4y2 - 3x2 = 12. 71. Construct the asymptotes of the hyperbola, INDETERMINATE GEOMETRY, 295 4y2 2x2 = - 8, and find its equation when referred to them, Art. (161). 72. Construct the hyperbola whose equation is, Art. (170), 2xy + 3y + x - 1 =0. 73. For examples illustrating the discussion of the general equation of the second degree between two variables, see Arts. (1'7), (176), (179). 74. Ascertain if the line represented by the equation y4 _ -2 - 2x - 3 - 0, has a centre, and determine its co-ordinates, Art. (181). 75. For examples relating to loci, see Art. (194). 76. Find the equation of the surface generated by revolving the right line whose equations are 4x = 3z + 2, 2y = -z +:6, about the axis of Z, Art. (196). 77. Find the equation of the paraboloid of revolution generated by the parabola represented by, Art. (198), y2 - _ 3x. 78. Find the equations of the spheroids, generated by the ellipse represented by 4y2 + -2 = 4. 79. Find the equations of the hyperboloids, generated by the hyperbola represented by 9y2 - 4x2 = - 36. 296 INDETERMINATE GEOMETRY. 80. Find the equation of the surface generated by revolving the parabola represented by 2y2 = x + 1, about the axis of Y. 81. Find the equations of the surfaces generated by revolving the lines represented by Y2 _ y = 2x2S about the axis of Y. Also the surface generated by revolving the first line about the axis of X. 82. Find the position of the planes which will make circular sections, Art. (226), in the ellipsoid whose equation is'2x2 + 3y2 + 4z2 = 1. 83. Find the position of the planes which will make circular sections, Arts. (227), (228), in the hyperboloids whose equations are x2 + 2 -y2- z2= -- 3; 4x2 -_ y2 _ 32 = _ 2. 84. Find the position of the planes which will make circular sections, Art. (229), in the paraboloid whose equation is 2y2 + 3z2 - 4x = 0. 85. Find the equation of a tangent plane, Art. (232), to the ellipsoid, whose equation is 4x2 + 2y2 + z2 = 10, at the point whose co-ordinates are x" = 1, y" = - 1, Z = 2. INDETERMINATE GEOMETRY. 2 97 Also the equation of a normal line, Art. (236), at the same point. 86. Find the equations of the tangent planes, Art. (232), and normal lines Art. (236), to the hyperboloids whose equations are 2x2 + y2 3z2 = _ 8; 3Z2 2/2 _ _87; at the point of the first, represented by x" = 2, y = - 1, z" = 3; and at the point of the second, represented by x" = 2, y" = - 3; z" = 1. 87. Find the equations of the tangent planes, Art. (232), and normal lines, Art. (236), to the paraboloids whose equations are 2y2 + 3zx = 4x; 4y2 - z2 = 5x; at the point of the first, represented by x" = 5, y" = 2, z" - - 2; and at the point of the second, represented by., = 4, y = - 3, z" = - E 4. THE END.