fJomM'^^.S/^^ be ''laken ^ 'i'-,. r-,: ^ fel .,r-.r, ■■■7',.'/^o AaimtSi:^'" ^m. ^ —.Z' fyvmll Utifemitg pilrt;atig THE GIFT OF ,p:.a,.fJ.A ./.. A3./''//S.f... 6896^2 the integral c 3 1924 031 264 769 Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031264769 AN ELEMENTARY TREATISE THE INTEGRAL CALCULUS. /-^ORNELL^ DUBLIN : PRINTED AT THE UNIVERSITY PRESS BY PONSONBY AND ■WELDBICK. PREFACE. This Book has been written as a companion volume to my Treatise on the Differential Calculus, and in its construction I have endeavoured to carry out the same general plan on which that hook was composed. I have, accordingly, studied simplicity so far as was consistent with rigour of demonstra- tion, and have tried to make the suhject as attractive to the heginner as the nature of the Calculus would permit. I have, as far as possible, confined my attention to the general principles of Integration, and have endeavoured to arrange the successive portions of the subject in the order best suited for the Student. I have paid considerable attention to the geometrical ap- plications of the Calculus, and have introduced a number of the leading fundamental properties of the more important curves and surfaces, so far as they are coimected with the Integral Calculus. This has led me to give many remarkable results, such as Steiner's general theorems on the connexion of pedals and roulettes, Amsler's planimeter, Kempe's theorem, Landen's theorems on the rectification of the hyperbola, Genocchi's theorem on the rectification of the Cartesian oval, and others which have not been usually included in text- books on the Integral Calculus. A Chapter has been devoted to the discussion of Integrals of Inertia. For the methods adopted, and a great part of the vi Preface. details in this Chapter, I am indehted to the kindness of the late Professor Townsend. My friend, Professor Crofton, has laid me under very deep obligations by contributing a Chapter on Mean Value and Probability. I am glad to be able to lay this Chapter before the Student, as an in- troduction to this branch of the subject by a Mathematician whose original and admirable Papers, in the Philosophical Transactions, 1868-69, and elsewhere, have so largely contri- buted to the recent extension of this important application of the Integral Calculus. In this Edition I have introduced a brief account of the application of the Integral Calculus to Spherical Harmonics, and also a short Chapter on Fourier's Theorem, which I hope will be found useful additions to the Book. Trinity College, February, 1888. TABLE OF CONTENTS. CHAPTER I. ELEMENTARY FORMS OF INTEGRATION. PAGE Integration, . .1 Elementary Integrals, 2 Integration by Substitution, . . . . . 5 Integration of ■ -, ... 8 Euler'a expressions for sin 9 and cos fl deduced by Integration, . .10 Integration of . =^, .14 {x — p)ya + 2bx + cx^ dx (a + 2bx + cx'')^ de 16 de , de 7- and - — -, . . . 17 cos 9 sm 9 19 " " fl + ioos9' ■ ■ ■ ■ Different Methods of Integration, . . 20 Formula of Integration by Parts, . . .20 Integration by Rationalization, . . . . 23 Rationalization by Trigonometrical Transformations, ... 26 Observations on Fundamental Forms, ... .29 Definite Integrals, .... . . . . 30 Examples, . 36 "VIU Contents, CHAPTEE II. INTEGRATION OF BATIONAL FBACTIONS. Eational Fractions, Decomposition into Partial Fractions, Case of Beal and tJnequal Soots, Multiple Seal Boots, Imaginary Boots, . Multiple Imaginary Boots, Integration of Examples, {x-a)'»{x- -J)"' dx ffi" - l' sc™-! dx 60 61 CHAPTEE III. INTEGEATION BY SUCCESSIVE BEDUCTION. Cases in which sin™ 9 cos"6j S e'" cos'^xdx, „ „ j DOS'" X BIO. nxdx, Beduction by Differentiation, 63 66 72 73 75 76 77 78 79 79 80 Contents. is PAGE Eeduction of -. , ... 81 J (a + ex)" " J (ffi + 2te+ m, 1 -a^n dx, Differentiation of Definite Integrals, Integrals deduced by Differentiation, .... Diflferentiation of a Definite Integral when the Limits vary, Integration under the Sign of Integration, { r"dx Page lU 115 119 120 123 126 127 128 128 128 130 131 132 134 135 138 139 143 144 147 148 . 161 Contents. XI cos mx dx + «" d9. cos Jo I I log {sin 9) Theorem of Frullani,* . „ , . f" tau-'aa; - tan-'^ic , V alue of dx, Jo X Remainder in Lagrange's Series, . Gamma-Functions, . . . , Proof of Equation Sim, n) = iMll^ r{m + n)' „ T {n) r {I - n) = -r^ Table of Log r(i)), Examples, ... . . PAGE 153 154 155 157 158 159 161 162 164 169 171 CHAPTEE VII. ABEAS OF PLANE CUBVES. Areas in Cartesian Coordinates, The Circle, .... The EUipse, The Parabola, The Hyperbola, . ... Hyperbolic Sines and Cosines, The Catenaiy, Form for Area of a Closed Curve, The Cycloid, . 176 . 178 . 179 . 180 . 181 182 . 183 . 188 . 189 • This theorem was communicated by Frullani to Plana in 1821, and published after- wards in Mem. delSoc, Ital., 1828. » xu Contents. PACE Areas in Polar Coordinates, . 190 Spiral of Arcliimedes, . 194 Elliptic Sector — Lamtert's Theorem, . 196 Area of a Pedal Curve, .... . 19& Area of Pedal of Ellipse for any Origin, . 201 Steiner's, Theorem on Areas of Pedals, . 20J „ on Eoulettes, . 203 Theorem of Holditch, . . . . . 206 Kempe's Theorem, . 210 Areas hy Approximation, 211 Amsler's Planimeter, ... . 214 Examples, ... . . . 218 CHAPTEE VIII. LENGTHS OF CUEVES. Rectification in Rectangular Coordinates, The Parahola — The Catenary, The Semi-cubical Parabola — ^Evolutes, The Ellipse Legendre's Theorem on Rectification, Fagnani's Theorem, The Hyperbola, . Landen's Theorem, Graves's Theorem, Difference between Infinite Branch and Asymptote for Hyperbola, The Iiima9on — ^the Epitrochoid, Steiner's Theorem on Rectification of Roulettes, Genocchi's Theorem on Oval of Descartes, Rectification of Curves of Double Curvature, Examples, 222 223 224 226 228 229 231 232 234 236 237 238 239 243 246 Contents. xiu CHAPTEE IX. VOLUMES AND SUEFAOES OF SOLIDS. Tke Prism and Cylinder, ... ... . 260 The Pyramid and Cone, . . . . . . . 251 Surface and Volume of Sphere, . 252 Surfaces of Eevolution, . . 254 Paraboloid of Eevolution, . 256 Oblate and Prolate Spheroids, . . ... . 257 Surface of Spheroid, .... ... . 257 Annular Solids, . 261 Guldin's Theorems, . 262 Volume of Elliptic Paraboloid, ... . . . 265 Volume of the Ellipsoid, . 266 Volume by Double.Integration, . .... . 269 Double Integrals, . 273 Quadrature on the Sphere, . 276 Quadrature of Surfaces, . 279 Quadrature of the Paraboloid, . 280 Quadrature of the Ellipsoid, . 282 Integration over a Closed Surface, . 284 Examples, . 288 CHAPTEE X. INTEGKAIiS OF INERTIA, Moments and Products of Inertia, .... Moments of Inertia for Parallel Axes, or Planes, Eadius of Gyration, Uniform Eod and Eectangular Lamina, . 291 292 293 294 xiv Contents. PAGE Eectangulflr Parallelepiped ^^" Circular Plate and Cylinder, . . 295 Right Cone, 296 Elliptic Lamina, 296 Sphere '297 Ellipsoid 29& Moments of Inertia of a Lamina, • 299 Momental EUipse, ■ • • 300 Products of Inertia of a Lamina, • 301 Triangular Lamina and Prism, ... . . 302. Momental Ellipse of a Triangular Lamina, . .... 304 Tetrahedron, 304 SoUdEing, . . 305 Principal Axes, ....... . 307 Ellipsoid of Gyration, . . . . 309 Momental Ellipsoid, 30& Equimomental Cone, . .... .310 Examples, . . . ... . 311 CHAPTER XI. MULTIPLE INTEGRALS. Double Integration, . . . . . . . . .313 Change in Order of Integration, . . . 314 Diriohlet's Theorem, _ _ 3jg, Transformation of Multiple Integrals, . . .... 321 Transformation of Element of a Surface, 324 General Transformation for n Variables, 325 Green's Theorems, 326 Application to Spherical Harmonics, 332 Laplace's Theorem on Expansion by Spherical Harmonics, . . 334 Examples, 3^2 Contents. xv CHAPTER XII. ON MEAN VALUE AUD PEOBABILITY. FAGB Mean Yaluea, 346 Case of one Independent Variable, . . 347 Case of two or more Independent Variables, . . 350 Probabilities, ' . . 352 Buffon's Problem, ... ... .365 Curve of Frequency . ... 369 Errors of Observation, . . 374 Lines drawn at Eaudom, .... .... 381 Application of Probability to the Determination of Definite Integrals, . 384 Examples, 387 CHAPTER XIII. ON FOUBIBB S THEOBBM. Fourier's Tbeorem, ... 392 Examples, ... 399 Miscellaneous Examples, ... ... 401 The beginner is recommended to omit the following portions on the first reading :— Arts. 46, 49, 50, 72-76, 79-81, 8g, 96-125, 132, 140, 142-147, i49, 158-167, 178, 180, 182, 189-193, Chapters x., xi., xii., xiii. INTEGRAL CALCULIIS. CHAPTER I. ELEMENTARY FORMS OF INTEGRATION. I. Integration. — The Integral Calculus is the inverse of the Differential. In the more simple case to ■which this treatise is principally limited, the object of the Integral Calculus is to find a function of a single variable when its differential is Jtnoimi. Let the differential be represented hj F{x)dx, then the function whose differential is F(x) dm is called its integral, and is represented by the notation F{sci) dx. Thus, since in the notation of the Differential Calculus we have df{x)=f'{x)dx, the integral oif'{x) dx is denoted hjf(x) ; i.e. |/(«) dx =f{x). Moreover, as f{x) and /(«) + C (where C is any arbitrary quantity that does not vary with x) have the same differen- tial, it follows, that to find the general form of the integral of fix) dx it is necessary to add an arbitrary constant to f{x) ; hence we obtain, as the general expression for the integral in question, {f'{x)dx=f{x) + C. (i) M 2 Elementary Forms of Integration: In the subsequent integrals the constant Cwill he omitted, as it can always he supplied when necessary. In the appli- cations of the Integral Calculus the value of the constant is determined in each case hy the data of the problem, as will be more fully explained subsequently. The process of finding the primitive function or the inte- gral of any given differential is called integration. The expression F{x) dx under the sign of integration is called an element of the integral ; it is also, in the limit, the increment of the primitive function when x is changed into X + dx (Diff. Calc, Art. 7) ; accordingly, the process of inte- gration may be regarded as the finding the sum* of an infinite number of such elements. We shall postpone the consideration of Integration from this point of view, and shall commence with the treatment of Integration regarded as being the inverse of Differentiation. 2. Elementary Integrals. — A very slight acquaint- ance with the Differential Calculus will at once suggest the integrals of many differentials. We commence with the simplest cases, an arbitrary constant being in all cases under- stood. On referring to the elementary forms of differentiation established in Chapter I. Diff. Calc. we may write down at once the following integrals : — x'^dx ■■ m + 1 fdx_ - I J a;™ {m- 1)*'"-^" ^"' ^ = log(.). (J) X f . cos ma; f sm mx sm mxdx = , cos mx dx = . (c) J m ' ] m ^ ' f dx , f dx — r- = tan x, -t-r- = - cot x. (d\ J cos'* J sm^a; ^ ' * It was in this aspect ttat the process of integration was treated by Leib- nitz, the symbol of integration J being regarded as the initial letter of the word stmt, in the same way as the symbol of di£Eeieatiation d ia the initial letter in the word difference. Fundamental Forms. 3 dx X , . X' a a' + x' a 1 j^dx = ^; |«^^^~. {9) These, together with two or three additional forms which shall be afterwards supplied, are called the fundamental* or elementary integrals, to which aU other forms,t that admit of integration in a finite number of terms, are ultimately re- ducible. Many integrals are immediately reducible to one or other of these forms : a few simple examples are given for exercise. Examples. 3- 4- S- 6. fdx tan^(?ar. „ -log (oosit) yx, fxdx 7fr7»- ,,-^l-rXh I I e<"dx. Bec0. I J " a - e"' * The fundamental integrals are denoted in this ohapteiby the letters a, t, e, &a. ; the other formulae by numerals i, 2, 3, &c. t Byintegrahle forms are here understood those contained in the elementary portion of the Integral Calculus as inyolving the ordinary transcendental func- tions only, and exduding what are styled EUiptlc and Hyper- Elliptic functiona. [1*] ' „-- IX' " 10. I ^^ . „ JJic". Elementary Fi/mis of Integration. fdx J r dx I - II. [— ^. „ log («-«). i X — a 3. Integral of a Sum. — It follows immediately from Art. 1 2, DifE. Calc, that the integral of the sum of any number of differentials is the sum of the integrals of each taken sepa- rately. For example — \{Aiif +33^"+ Cif + &o.) dx=A /a!" dx + B^ af'dx + Clardx + &c. ^«'»" .B««" Ccf*"- „ + &o. (2) m+ 1 n + I r + I Hence we can write down immediately the integral of any function which is reducible to a finite number of terms con- sisting of powers of x multiplied by constant coefficients. Again, to find the integrals of cos^xdx and sm'xdx; here f , , f I + cos 2x , a; sin 2* , , 00s xdx = dx = - + , (3) Bin^xdx - I - cos 2X , X Bimx dx=- — . (4> 2 24 A few examples are added for practice. Examples. c{i-x^)'dx ^ , , ! I . . Ans. log a; - a;' + - J a; ° , clx-2) dx /- 4 J X's/x VX 3. J \&'s^»Ax = /(sec'a - i) dx. „ tan x-x. Integration hy Substitution. sin (m + n)x sin (»» — «) as 4. f cos mx cos nxdx. Ana. — } — H -, r— • ■' 2{m + n) 2(m~n) sin(»j - »)« Bmtm + n)x I !. \smmxsni.nxax. ,, — } ; ) r-. ' ■> J " 2{m-n) x(m + n) €. I . / dx. ,, a sin-i v^a^ - (c'^. 3\a-x a Multiply the numerator and denominator by ya + x. 7. J » yx + adx. ' Ans. - {x + a) a {x + a) . «. 1-7= — 7-- .. ^(('^ + «') -^ )• Multiply the numerator and denominator by the complementary Biird V a; + a — y x. Ia + bx , . ix ab' — ia' , , , ,, . - — j;- dx. Am. — + — — — log (» + S'a;). a \ hx . b i i _ « + d« J fflS' — ba' Here ^_^ = + . o' + 4'» b' V {of + 4' a;) ' 4. Integration by Substitution. — The integration of many expressions is immediately reducible to the elementary forms in Art. 2, hy the substitution of a new variahle. For example, to integrate {a + hx)'"' dx, we substitute z for a + IX-, then dz = hdx, and ^ ' J 6 {n+ i)b {n+ i)h Again, to find 1 x^dx {a + te}"' as befoK [ {z-afdz we substitute z f or a + 5a?, as before, when the integral be- comes I ["(z - cC)'^dz 2a or h^\{n- 3) s"-= (« - 2) s"-' (» - i) z» 6 Elementary Forms of Integration. On replacing % hj a + hx the reqtdred integral oan be ex- pressed in terms of x. The more general integral f x"^dx J-f {a + &«)"' teger, 1 (z - a)" where »» is any positive integer, by a like substitution be- comes Expanding by the binomial theorem and integrating each term separately the required integral can be immediately obtained. Again, to £nd r dx ]»"•(« + &«)"' we substitute a for - + b, and it becomes X (s - h)" which is integrable, as before, whenever m + « is a positive integer greater than unity. Thus, for example, we have r dx I / X \ }x{a + bx) a \a + bxj It may be observed that all fractional expressions in which the numerator is the differential of the denominator can be immediately integrated. For we obviously have, from (6), Integration of dx 9? - 0? Examples. J a + sin X dx -1 f i^dx • ^ / • - 5. 6 b cos ic' a^dx 3. *^jlog^-. 4. f'^^^^. J a; log it; f , x' I ji«». — log (a + i cos a;) log a , x'dx }^ f t^g J a;' (a + *»)' !a;c?a; (a + te)l" (a + te)i' J X' - em ' I - 1 . 4 \«/ -aoga;)''. log (log a;). log (a + hx) 3«2 + 4a}a! 43 '*' 2l^{a\hxf' 2? a + to ff+ lix a? X <^x (* + 4a;)' 2 (a + 4a;)* 2a (a + hx)i 3P 42 ■ 3 (a + 4a;)* 3a (a + 4a;)t 5i' - tan"' lia; - a x^iax — ci^ Assume 2aa; — b' we have dx I , , ex + b taii->-— == (7) a + 2bx + ex'' y^ZTb^ ^ac - 6»" Integration of- ' - a + 2bx + cx^ If ac < b\ f ± = 1,^ cx + h-^/F^ ]a+2bx + cx' z^W-ac ex + b + -/b^ - ao This latter form can he also immediately obtained from (6) . In the particular case when ao = b', the value of the inte- gral is - I 7. Integration of ex + b' (p + qx) dx a + zbx + ex^' This can at once he written in the form q {b + ex) dx pc - qb dx c a + 2bx + ex' c a + zbx + ex'' The integral of the first term is evidently — log (a + 2bx + ec^), 2e ° ' while the integral of the second is obtained by the preceding Article. For example, let it be proposed to integrate {x COS0 - \)dx (i^-2xca%Q-\- 1' The expression becomes in this case cos 6 (x - cos 6) dx sin' 6dx 1^ - 2;8 cos + I {x ~ cos QY + sin^ ' hence f (« C0S9 - !)«?» COS0, ,„ . . -^ 4 = log (»" - 2x cos 0+1) J »'' — 2a; cos y + I 2 ° ^ ' . rt , , a; - cos . - sm tf tan"' — ^ — ^r— . (0) sm0 ^ ' 10 Elementary Forms of Integration. When the roots oi a + zbx + ex' are real, it -will he found simpler to integrate the expression loj its decomposition into partial fractions. A general discussion of this method will' he given in the next chapter. EXAUFLES. f ^^ -^ In,. f 2x-r+ -/s f <& , , I — — . „ taii"'(a!+ 2). fdx I - 2a; + 2a:2 ' * ' 8. Exponential Talne for sin dand cos Q. — By com- paring the fundamental formulae (/) and (A) the well-known exponential forms for sin Q and cos can he immediately deduced, as follows : Suhstitute z v^- i for x in hoth sides of the equation { dx I ^ fi + x\ ]r^ = -2^°^[7^x)^''"'^*-' and we get dz ^ , /i + zy^\ — i = / — log 7= + const.; Exponential Forms of sin and cos 6. 11 or, by (/), tan-' 2 = — ^ log- ( ^- — .— ) + const. 2t/- I ° \l -Z\/- 1/ Now, let s = tan 0, and this becomes . I 1 A + -/- I tan 0\ , a = — ^=1 log I -;= — — I + const. 2^/- I \i - •/- I tan 0/ When = o, this reduces to o = const. XT 2;.^/l COS + >;/- I sin . . / . ,,., Hence e'«v-' = -;=; = (cos 9 + -/- i sin 0)% cos - '/- I sin ■r^^g ^ ^.,. .^ - ' or e*^ = cos + -v/- i sin 0, e-«V=i = cos - -/^ sin 0. 9. Integration of V^a;^ + a* Assume* ^/i^~±^ =& - x, then we get + a^ = s'' - 2a:z, hence (s — x)ds = zdx, or = — : ' s - X % •■• y^r^^ = J 7 = log« = log {x + yx' ± a'). {1} This is to he regarded as another fundamental form. By aid of this and of form (e) it is evident that all ex- pressions of the shape dx */a + 26a; + c^ * The student will better understand the propriety of this assumption after reading a subsequent chapter, in whieh a general transformation, of which the aboye is a particular case, will be given. 12 Elementary Forms of Integration. can be immediately integrated ; a, b, c, being any constants, positive or negative. The preceding integration evidently depends on formula {i), or (e), according as the coefficient of «* is positive or negative. Thus, we have 1 , , ■ = ~7^ log \cx + b+ \/c(a + 2bx+ cx^) ), (lo) r dx \ . ,( ex -h \ , . . = = —= sm-M , (1 1) J -\/a + 2te - coi? ^/c \\/ac + bv cbeiag regarded as positive in both integrals. When the factors in the quadratic a + zbx + cx^ are real, and given, the preceding integral can be exhibited in a simpler form by the method of the two next Articles. 10. Integration of y/{x-a){x-fi) Assume x — a = z^, then dx = 2zdz dx = zdz ; hence '/x — a dx zdz ■) As before, assume x - a = z', and we get do) 2ds -/(«-a)(j3-a;) y^-a-z'' Hence, by (e), f dm . Iso - a ,. ' = 2 Sin \- — . Otherwise, tlius : assume x = a cos^ fl + /3 sin' 9, then /3-« = O-a)co8'0, x - a = [^ - a) sia" 9, (i3> / and dx= 2{j3- a) sin cos rffl ; . ,-' hence y- ^ r = 2d9; \ ^/ {x - a){(5 - x) V,,^^ ... f '^^ = 20 = 2 sin-^ /^ }^/{x-a){j5-x) V/S-a" 12, Again, as in Art. 7, the expression (p + ga) (fe ya + 2bx + ex' can he transformed into q ( J + ca?) dx pc — qb dx ** y^a + 2 Ja; + ft?;* ^ V a + 2hx + ca^ and is, accordingly, immediately integrahle hy aid of the- preceding formulae. 14 Elementary Forms of Integration. Examples. f dx Ans. 2 log (•;/«; + '/x - a). ' '^^ — ax ■' Y ax - x' 3- I y " -• „ 2 Biii->\/a; - i. J V 3* — a:' — 2 4- f ^ „ log {2X + I + 2 v/rTaT'a^. J ^/ I + I + a;^ 5" / ; J (a + cx'p a{a + cx'')i ^ ^' 16. To find the integral of dx (a + 2bx-v co^)^' This can be written in the form <^dx which is reduced to the preceding on making Ci» + J = s.. Integration of ■ sin 0' Hence, we get { dx b + ex } {a + 2bx + cx')^ (etc - ¥) {a + zbx + cxy^' Again, if we substitute - for x, xdx . - ch becomes (15) (a + 2bx + cx^)^ (as' + 25s + cf and, accordingly, we have f xdx a+bx J (a + 26a! + c*')* {ac - b''){a + 25a! + ca?y Comtining these two results, we get f {p + qx) dx bp - aq + (cp - bq) x J (a + 26a! + cx^)^ {ac - 6") (a + 26a; + cx^)^' ^ <^9 ^ de 17. Integration of -. — « and ;,. sm a cos (16) It will he shown in a subsequent chapter that the integra- tion of a numerous class of expressions is reducible either to that of -; — n, or of 2i '• '^® accordingly propose to inves- tigate their values here. For this purpose we shall first find the integral of -r-^ 7,. ° sm cos ft dO ^ dQ cos' 9 ditanO) Here consequently sin 9 cos 9 tan 9 tan 9 d9 sin 9 cos 9 [2] = log (tan 0). (17) 18 Elementary Forms of Integration. Next, to find tlie integral of d9 sin ff This can be written in the form dd . d 0' 2 sm-cos- 2 2 and, by the preceding, we have -^— T, = log(tan-). sin ^ \ 2/ Again, to determine the integral of 3 ih for fl. a.nd the t ^xTireasion becomes -. — ~ (18) we substitute ., , . : the integral 2 " ' "■ sm0 ^ of this, by (18), is - log f tan I j, or log f cot | ), or log jcot P - -j| . Accordingly, we have 1.4»--H(i-!)l--h(r-DI- <-) This integral can also be easily obtained otherwise, as follows : — r dQ r cosOrfg _ r b, -^—. = ^^= t-^-' I /— tan ^1 (20) «i + 6cos0 y«-^_6^ \4a + b^^ 2]' ^^°^ (2) when a < i, by formula {h), log ^ i \.[2l) ^Q J I -v/j + a + v^ 6 - a tan a + 6 cos ^^2 _ ^2 ° "^i 1^^^ Q .' -v/j + a - v/S - a tan - If we assume a = J cos a, we deduce immediately from the latter integral J cos a + cos sm a ° r a -6'] cos- L f J The integral in (20) can be transformed into \ f '^^ = ' cos- i^JiS^^S^] ' I J a + 6 COS vV~*'- (a +6 cos 0)' V^' 20 Elementary Forms of Integration. In a subsequent chapter a more general class of integrals ■which depend on the preceding will be discussed. 19. Methods of Integration. — The reduction of the integration of functions to one or other of the fundamental formulae is usually effected by one of the following methods : — (i). Transformation by the introduction of a new va- riable. (2). Integration by parts. (3). Integration by rationalization. (4). Successive reduction. (5). Decomposition into partial fractions. Two or more of these methods can often be combined with advantage. It may also be observed that these different methods are not essentially distinct : thus the method of rationalization is a case of the first method, as it is always effected by the substitution of a new variable. We proceed to illustrate these processes by a few ele- mentary examples, reserving their fuller treatment for sub- sequent consideration. 20. Integration by Transformation. — ^Examples of this method have been already given in Arts. 4, 10, &c. One or two more cases are here added. Ex. I. To find the integral of sin'* cios^xdx. Let sin » = y, and the transformed integral is p'(i - 2/Vy =yfdy - y^dy- ^ f ± a") dx. Here I log {x + v^«' + «') rfa; = « log (^?^!^±a^. r xdx = a; log (a; + \/a;' ± «') - y^x' + c^. (26) Integration by Rationalization. /^23 Examples. I. \ X^lo^xdx. Ans. ^—- (log a; —] . n + i \ ° n + ij 2. 1 tan-'xrf«. „ X tem-^x log (i + »'•'). 3- X tan^ xdx. ,, a; tan x + log (oosa;) . 4- BYDr^XdX (l-x^f I ll^^H /w A^_\ " / ; + ,'^''SU ^ J- Let X = sin y, and the integral becomes dy — ^ = p"i (tan y) = (/ tan j/ + log (cos y). ,, «*(»- — 2x + 2). e' »* i&. 22. Integration by Rationalization. — Ey a proper assumption of a new variable we can, in many cases, change an irrational expression into a rational one, and thus inte- grate it. An instance of this method has been given in Art. 9- The simplest ease is where the quantity under the radical sign is of the form a + bx : such expressions admit of being easily integrated. For example, let the expression be of the form {a + bx)i' where w is a positive integer. Suppose a + bx = s', then , 2zds , z' - a ax = — T— , and x = — ; — : . o ^^ ^^ making these substitutions, the expression becomes 2(3'' - «)"& 24 Elementary Forms of Integration. Expanding by the Binomial Theorem and integrating the terms separately, the required integral can be immediately C^ CvX found. It is also evident that the expression ^ can (a + bx)t be integrated by a similar substitution. 23. Integration of j^:^^^, where «i is a positive integer. Let a + cs" = s" : then o, the integral is easily seen to be ( ^/A(a + cs^) -^ x^/Ao - Ca\ , ., °^ K^Aifl + C3^) - x>/Ac - Ca) z-Za {Ac - Ca) (2). If — < o, the value of the integral is ,i ' { I , ,X\/Ca - Ac , , \ , tan-' ^. . (29) - yA ( Ca - Ac) '/A{a + ex'') Examples. jt^ f ^^ ' . , ( S" \ ^ I . 2^/3 + 4g'-' + 5^ (4-3»='^)(3 + 4*')»' " ^ °^2>/iT^-s^' 26. Rationalization by Trigonometrical Trans- formation. — It can be easily seen, as in Art. 6, that the irrational expression \/a + zbx + cx^ can be always trans- formed into one or other of the following shapes: (i) (a' -.=)*, (2)(a^ + 2^)4, (3) [f-a^)\; neglecting a constant multiplier in each case. Accordingly, any algebraic expression in x which con- tains one, and but one, surd of a quadratic form, is capable of being rationalized by a trigonometrical transformation : the first of the forms, by making 2 = a sin ; the second, by s = a tan % ; and the third, by s = a sec 0. Rationalization hy Trigonometrical Transformation. 27 For, (i) ■when s = a sin Q, we have (a* - s^)^ = a cos Q, and dz = a cos OcW. (2). When s = a tan 0, .... (a' + 2')^ = a sec B, and cos^y (3). When 2 = a sec0, . . . . {s^ - c^Y = a tan d, and ffe = a tan Q seo Odd. A. ntunber of integrations can he performed hy aid of one or other of these transformations. In a subsequent place this class of transformations will he again considered. For the present we shall merely illustrate the method by a few ex- amples. Examples. Let X = tan B, and the integral beoomea 100s ede _c 0 6. I . sill? xdx. la t^dx i P dx i\/x — a){^ -x) See Art. 11. Jo ... f Jo 13. I ; .where a >i. J„ « + cos X (^ dx " 8 4 r Jo ji, J 1 +«+»■-" n I eos'a; (?j:. f^ xdx I J 2 f + a;- ' 2 ^ 3-/ 3 2_-_4 3-5' " -/«nri I — 2» cosa;+ a- I — a' COSECANT. 33 56. Differential coefficient of cosec x. Let y = cosec x ; proceed as in the last example, and we find dy _ cos X dx sin* X ' 57. Since tan x, cot x, sec x, and cosec x are all fractional forms, we may deduce tlie differential coefficient of each of these functions by Art. 31 from those of sin x and cos x. Thus, let • sin X y = tan x = , ^ cosaj dsmx . dcosx , cosa; — ; sma; — j — ^, - ay dx ■ dx . ^ „^ therefore -^ = „ , Art. 31, dx cos X cos' X + sin' X , = 5 , Arts. 51 and 52, cos' X 1 cos a; Similarly we may proceed with cot x, sec x, and cosec x. Since vers a; = 1 — cos x, the differential coefficient of vers x by Arts. 27 and 33 = — differential coefficient of cos x = sin X. T. D. C. ( 34 \ CHAPTER IV. DIFFERENTIAL COEFFICIENTS OF THE INVEESE TRIGONOME- TRICAL FUNCTIONS AND OF COMPLEX FUNCTIONS. 58. Let y = ^ [x), so that y is a known function of x ; it follows from this that x must be some function of y, although we may not be able to express that function in any simple form. The best mode for the reader to convince himself of this will be to recur to algebraical geometry and suppose x and y to be the co-ordinates of a point in a curve the equation to which is y = (x). Fot every value of x there will be generally one or more values of y, positive or negative, as the case may be. So for any value of y there will be generally one or more definite values of x, which, as they really exist, may be made the subjects of our investigations, even ' although our present powers of mathematical expres- sion may not always furnish us with simple modes of repre- senting them. 59, A simple example will be given in the equation y = x'-2x+l (1). Solve this equation with respect to x, and we have x=l±y^ (2). Here (2) shews that. if any value bo assigned to y we must have for x one of two definite values. Now in (1), X being the independent variable and y the dependent variable, we have by Arts. 28, 33, and 44, l|=2-2 («)• Definite Integrals. 35 In like manner, with the same condition, we have aos^ nxdx = -. (37) Jo 2 Again, to find the value of fP \/{x-a) {^-x)dx. Assume, as in Art. 1 1, « = a cos' + /3 sin' ; then, when fl = o, we have x = a; and when = -, a; = |3. Hence, as in the article referred to, we have It 1 ^ ^(x-a){!^-x)dx = 2 (j3 - a)M \vD? e cos' ed9. Jso 2 ['sin'6lcos'6lcZfl = i [W'2 flrffl Jo Jo = i|^sin'^«f0 = ^; .■.^^y{x-a){p-x)dx=l{(i-ay. (38) [Sa] S6 Examples. i Examples. J {« + Bin a;)» " 2 (a; + sin a;)"" [ajsinada!. i, sin »- a; cos a:. f '-^^ dx. „ 2 log (I + a:) - x. J i + a; I(o + Ja;") '»■'■' f "^'^^ _ ? I ^" J(«»+»»)r " 3 (»' + =»')»■ t dx . , jx+ I f a:Var 1 , /'^-'\ o. f ^ . 1. —7 tan-1 ( — tan a: ) . ^ Jo»cos«a; + A«sm«a! " o* \« / .0. f^?^. „ -^^los(«cos»« + «sin».). J a + i tan^a; i (4 - «; icos (log x) dx' . ,, , i— *— i — . „ Bin (log a). dx 1 2. Show that the integral of — can he ohtained from that of le^dx. ;gm+l _ (im*-l Write the integral of a;"rfa: in the form ; and, hy the method of indeterminate forms, Es. j, Ch. It. Diff. Calc., it can easily he seen that the true value of the fraction when m+ l=o is log ( - J , or log x, omitting the arbitrary constant. 13. /e''»sin»ia;cos«a!i&. This is immediately reducible to the integral given in formula (23). 14. 1 : — . Ans. - tan-' I — =^— — • 'J5 + 4Bm« 3 \3/ Examples. J (i + «« Ans. 3(« + g)^(4«'-3«) 16. I i{a + »)!<&. Let « + »«« = 2'. ,8 f (?_+ ff^o^ '")''* [ KP + q J a + i - i cos :); This is equivalent to :qdx ^ pi- ga t die Cqdx pi -ga r 1 b ^ b ]a+ b cos x' and accordingly can be integrated by Art. 18. xo'dx Ixe'' e* (i + »)'■ ■ I + a:" xdx 2 fxdx I f dx I / y/ 1 + a' - A J a:-\/a»TT " 3 Xv^l +aiS + 1/ let a;' + I = s', ^3. f-^. ,. I,ogfi^^±^^\ 24. Integrate %y aid of the assumption x a + * cos fl i + a cos 9 a + i cos fl The expression transforms into dx \/(a»-«»)(i-a:»)' integi ^^TZli log {x + v^a^ - I), &c. accordingly, when a>b, its integral is _ sin-'a; : and when « < J, it ia 38 Examples. »5. Deduce Gregory's expansion for tan"'* from formula (/). When * < I, we haye : I - it' + »* - a;' + &e. ; I + »■= t dx 3? sfi x' . .: tan-' x = \ 7 = a + + &o. J I + *' 3 5 7 No constant is added since tan~' x vanishes with x. 26. Deduce in a similar manner the expansions of log (i + x), and sin''!-^ Ja 27. Find the integral of a + 4 cos fl + c sin-fl This can be reduced to the form in Art. 18, hj assuming - = cot a, &e. dx Ua + l Ana. log } ) -/ 02 + 42 (4-«x4'\/(a» + *2)(i + «»))" This can be integrated either by the method of Art. 13 or by that of Art. 2J.. »9- 3°- 31- 3* 1 — . . Am. - sec ' I x ]. J x^x" - I « V / IT 14 sin xdx I , • 11 log (l + \/2' J I, cos a! w s \ ' V . dx I f« di (4 + 3^)5' " 8" 33. j„ v/«=-a:Var. irffl' 4 4 34. 1 K versiuM - J 2, , > 3, / , f I* <^* 3 >/,',/ . -r-. :• » - log (« + 3 + 7 log (." - •)• fdx I , » - I I , -7 . „ - log tau-ia;. X* - 1 4 ° X + I 2 f dx J a* + 5*2 + 4 fxdx I - tan-'a: — ^ tan-' - 3 6 : »■= - I 4 »■' + I 6. [(Si^-ajifa: i, fx/-2\ ^ , (1^— ^1 \^ + a^-to - " 6log*+2log{«-2) + -log(». + 3). Here the denomiaator is equal to a; (a: — 2) (k + 3) ; and we have • x^+ X — I _-^i -^2 -^3 . a;(»- 2)(a; + 3) x x— 2 x + i' 42 Integration of Rational Fractions. ^euoe x^ \x-i= Ai{x'> + a; - 6) + AiX(x + 3) + --a3).. (sc-an) +Ai{x-ai){a;-as) .. {a!-a„} + &C. +A„{x-ai){x-ai) . . {x-aa.i) J and since, by hypothesis, hoth sides of this equation are identical for aU values of x, we may substitute oi for » throughout; this gives /(oi) = Ai{ai - ai){ai - 03) . . . (ui - a„), In like manner, we have A -li^ A -l^ A - /("") (,\ i> (aj) (as) (a„) Hence, when all the roots are unequal, we have — 7~\ "77 — ? ■• 77 — T <" **C. + —77 — L , ^(x) [(ii)x-ai ^ {02} X - Oi i){an)x~a„ Accordingly, in this case The preceding investigation shows that to any root (a), which is not a multiple roof, corresponds a single term in the^ integral, viz. (4> •44 Integration of Rational Fractions. ■one wMcli can always be found, whether the remaining roots are known or not ; and whether they are real or imaginary. 38. Case where STunierator is of higher Degree than Denominator. — It should also be observed that even when the degree of a; in the numerator is greater than, or equal to, that in the denominator, the partial fraction cor- Tesponding to any root (a) in the denominator is still of the form found above. For let ^; =«.-#-,. where Q and R denote the quotient and remainder, and let A R . . be the partial fraction of —p- corresponding to a single root a ; then, by multiplying by 0(«) and substituting a in- stead of cc, it is easily seen, as before, that we get For, example, let it be proposed to integrate the ex- pression a^ - zse^ - 5X+ 6 Here the factors of the denominator are easily seen to be CD - I, x+ 2, and a; - 3 ; accordingly, we may assume = 0^ + 00; + ^+ + + a? - 23^ - 5x + 6 X - 1 X + 2 X - ^ To find a and /3, we equate the coefficients of «* and a? to •zero, after clearing from fractions : this gives, immediately, ■a = 2, and /3 = 9. Again, since f{x)=a^- 2x^ - 50? + 6, we have 0'(a;) = 2,3? - AH! - 5- Real and Unequal Roots. 45- Accordingly, substituting 1,-2, and 3, successively for x in the fraction a' ^x' - 4a! - 5 we get 6 15' 10 and hence a^ J ___i 32 243 ar" - 20;' - 5* + 6 6(i»-i) i5(a;+2) io(a!-3)'' «'cfo a;' „ . log(a!- i) -T r 7 = - + «'+ 9a! 2_i^ -' HD^ - 2iiir - 5x + o 3 6 ^2 24 ^ - — log(a! + 2)+— log(a:-3). 39. Case of SlTen Pofvers. — If the numerator and denominator contain x in even povrers only, the process can generally be simplified ; for, on substituting z for a;", the- fraction becomes of the form Accordingly, whenever the roots of (j>{z) are real and unequal, the fraction can be decomposed into partial fractions, and to any root (a) corresponds a fraction of the form 0'(«) z - a' The corresponding term in the integral of is obviously represented by /(n) f dx 0'(a) ]x^-a 46 Integration of Rational Fractions. This is of the form (/) or (A), according as a is a positive or negative root. The case of imaginary roots in ^(z) will be considered in a subsequent part of the chapter. It may be observed that the integrals treated of in Art. 3 are simple cases of the method of partial fractions discussed in this Article. EXAMPIES, r [ 2x + 3) J a;' + a' - [2x + 3) dx zx' Here the factors of the denominator evidently are x, J fe^ =-^°S « + f ^"S (^ - I) - 6 log (X + 2). __i !_ /_[ L_V (a;2 + a^){x^ + b^) a' - b^ \x'^ + 4' »» + a^} ' fdx Here hence the value of the required integral is I K.-.){^-(i)4--(3t- (xdx (xi + «)(ii;« + b)' Multiple Real Roots, 47 Substitute z for x'' and the transformed integral is (I ds 2 (a + a) (a + 4)' Consec^uently tlie value of the required integral ia fZ-jjIpB _ I Wig -4— '—-. ^M. 3a; +11 log (a; -2) -2 log (a; -I), a:* — 3a; + 2 (x^ - ■i)dx I , , , • , , > 3 ^a-7. + 6 - « -log(a:-i) + -log(a;-2) + l;log(^ + 3)- *• l.TTTKJTl)- » - log a; + log (.+ !)- -log (a: 4 2). f (?a:(g' + 4'a^) J a;''*i(a + Ja;») ' Let !!!» = -. 2 40. Multiple Real Roots. — Suppose ^(a;) has r roots each equal to a, then the fraction can be written in the shape In this case we may assume fix) _ Ifi , M^ ^ _irr_ P {x - aY4>{x) {x - af ix-ay-^ ' " x-a VC*)' where the last term arises from the remaining roots. For, when the expression is cleared from fractions, it is readily seen that, on equating the coefficients of like powers at both sides, we have as many equations as there are unknown quantities, and accordingly the assumption is a legitimate one. 48 Integration of Rational Fractions. In order to determine the coefficients, Mi, M^, &c. . • . Mr, clear from fractions, and we get /(«) = M4[x) +Mi{x - amx) + Mzix - ay^{x) + &o. , . . (6) This gives, when o is substituted for x, f{a)=M4{a),0TMi=-^y (7> Next, differentiate with respect to x, and substitute a instead of « in the resulting equation, and we get /(a) = Jfif (a) + M4{a) ; (8) which determines M^ By a second differentiation. Ma can be determined ; and so on. It can be readily seen, that the series of equationfl thus arrived at may be written as follows — /(a) = M4{a), f{a)=M4'{a) +l.M4{a), f'{a) = M,xP"{a) + 2.M4'ia) + I.2.M4{a), r{a) = M4"\a) + 3 . M4"{a] + 2.3. M4'{a) +1.2.3. ^4{«)y /'(a) = M,-^\a) + ^.M4"'{a) + 3 . 4 . i!f3^"(a) + 2.3.^.M4'{a) 4 l.2.3.4.M4{a), in which the law of formation is obvious, and the coefficients can be obtained in succession. The corresponding part of the integral of \ f{x) dx {x - ay-ijj {x) evidently is j|/Jog(«.-a)-^-^--^^^,-...-^^3^y^^— p,. (9) If ^{x) have a second set of multiple roots, the cor- responding terms in the integral can be obtained in like manner. Imaginary Roots. 49 41. Imaginary Roots. — The results arrived at in Axt. 37 apply to the case of imaginary, as well as to real roots ; however, as the corresponding partial fractions appear in this case nnder an imaginary form, it is desirable to show that conjugate imaginaries give rise to groups in which the coefficients are all real. Suppose a + h ^/- i and a-h v - i to he a pair of con- jugate roots in the equation ^{x) = o ; then the corresponding quadratic factor is (\ ^ $,+> &'>'' t^"* expression becomes f dx J l,x - i){x + if [x^ + i)' Assume x = -, and tbe transformed expression is evidently z f a6^ J(j;-l)(s+i)2(««+i)' The quotient is easily seen to be z - i ; and, by the method of Art. 38, we may assume 26 _ A B Zi + M Hence (Arts. 37, 40), we have ^ = 1= --r Next, Z and Jf are found by mating s^ = — i, in the equation ifi=(Zz + M){z-i){z+i)'; .: i=2{Zz + M){z + I) = 2 {Zz^ + (X + M)z + M], which gives 2 4 4 In order to find the remaining coefficient G, we make z = o, when we get o = -i-A + S-^ C+M; .-. C=|. 5 Multiple Imaginary Roots. 53 hence we hare z' I I Q a - I («-!)(« + 1)2(2'+ I) ■^8(z-i) 4(3 + 1)'^ 8(2+ I) 4(«"+i)' + I log (a + i) - 5 log (a* + I) + - tan- 'a. 004 ■Hence dx I I * I , I -it' , «+ 1 I r dx I I * I , I -it' , «+ 1 I ^ , I 8. /" ,- / ■,, Am. - log — J (»-i)2(a:+3) 2 °a; + 3 a; - I 43. Multiple Imaginary Roots. — To complete the discussion of the decomposition of the fraction —j-x, suppose the denominator ^{x) to contain r pairs of equal and imaginary roots, i. e. let the denominator contain a factor of the form { {x - ay + b'}''; and suppose 0(«) = ( (» - a)' + J')'' ^i{x) In this case we assume f{x) LiX + Ml Ltx + Mi [{x-ay+V'Y,pi{x) [{x-aY + V']'- {(»-«)'' + 6'j'-' LrX + Mr P + . .. + -, T-„ — - + ■ {x-af+b'' i,i{x)' the remaining partial fractions heing obtained from the other - roots. There is no difSculty in seeing that we shall stiU have as many equations as unknown quantities, Li, Mi, Li, M2, . . . when the coefficients of like powers of x are equated on both sides. To determine Li, Mi, Li, &c. ; let the factor {x - a)" + J* be represented by X, and multiply up by X'', when we get ^ = lix + Mi+ {LiX + Mi)X + &c. + (LrX + Mr) X-1 + ~- (l l) 0iW «i(«) ^ ' 54 Integration of Rational Fractions. The coefficients Li and Mi axe determined as in Art. 41. To find Li and M^ ; difEerentiate with respect to x, and sub- stitute a + h«/ - I for X in the result, when it becomes = Li + 2{xa - a){LiXo + Mi), where Xo = a.+ h-/ - i. Hence, equating real and imaginary parts, we get two equations for the determination of Lt and M^. By a second differentiation, L^ and M^ can be determined, and so on. It is unnecessary to go into further detail, as sufficient has been stated to show that the decomposition into partial frac- tions is possible in all cases, when the roots of <^{x) = are known. The practical application is often simplified by transfor- mation to a new variable. 44. The preceding investigation shows that the integra- tion of rational fractions is in all cases reducible to that of one or more fractions of the following forms: dx dx {A + B)dx {Lx + M)dx x-a' {x-af (x-af + b^' {{x-ay+bY The methods of integrating the first three forms have been given already. We proceed to show the mode of dealing with the last. 45. In the first place it can be divided into two others, L{x-a)dx {La + M)dx {{x - of + Vy ^ {{x-ay^b'Y' The integral of the first part is evidently -L z{r- \]\{x-af ^Vy To determine the integral of the other part, we substitute s for a; - a, and, omitting the constant coefficient, it becomes (s^+jy Multiple Imaginary Roots. 55 Again But we get by integration by parts C z^ds _ r zdz _ I f / i_ ]{&^ + h'-Y~r'{z^ + bY~ 2{r-i)]^ \{z^ + ¥) (f + by-'' 2{r-i){z^ + b°-)'^'^ 2(r-i). Substituting in the preceding, we obtain f dz 2r-3 r dz a , . ]{s' + b'Y ~ 2{r-i)b'}{z' + bY-' "*" 2{r-i)b' (f + b^y-'' ^^^' This formula reduces the integral to another of the same shape, in which the exponent r is replaced by r - i . By successive repetitions of this formula the integral can be re- dz duced to depend on that of 7-; — r-r. ■^ (s" + b") The preceding is a case of the method of integration by successive reduction, referred to in Art. 19. Other examples of this method will be found in the next Chapter. The preceding integral can often be found more expedi- tiously by the following transformation : — Substitute 6 tan dz for z, and the expression -^ — j^ becomes, obviously, ^i\°°^"" OdB. The discussion of this class of integrals will be found in the next Chapter. 46. We shall next return to the integration of '^ , 0(« ) which has been already considered in Art. 39 in the case 56 Integration of Rational Fractions. where the roots of ^(s) are real. To a pair of imaginary roots, a + h + e J «*-2»^ccos 20 + c" ~ 8 COS0 ci °^ \^a? + 2* v/c"cos + c, 4 sin 0^ dx tan-'f£fA/c^m0 47. Integration of r— . , =— . ^' * (a!-fl)'»(a!- J)" This expression can be easily transformed into a shape Integration of -. r — ; rr- . 67 "whlcli is immediately integrate, by the following sutstitu- tion : — Assume » - a = (» - 6) s ; then a-h% (a- b)z , a - b , (a-b)dz 4C = ; .•.x-a = - '—, x-b = , ax = -, h— ; I -s I - a I -s (i -s)'' and the expression transforms into (i - z)'"*^'dz {a - 6)'»+»-ia»* Expand the numerator by the Binomial Theorem, and the integral can be immediately obtained. (Compare Art. 4.) For example, take the integral dx {x - aY{x- by Here the transformed expression is •(i -zYdz or J{«- 4<,j' J^] (J - f + 3 -^)^^ = (^ j^- 3^ + 3 log. + Jj. Substituting for z, the integral can be expressed in terms of x. . 48. Integration of 7 {a + cx'Y where m and n are integers. Let a + ca;" = 2, and the expression becomes (s - «)"*& a form which is immediately integrable by aid of the Bino- mial Theorem. 58 Integration of Rational Fractions. It is evident that the expression is made integrable by the same transformation when n is either a fraotional or a nega- tive index. It may be also observed that the more general expression -. — .^^ can be integrated by the same transformation, ■where f[3?) denotes an integral algebraio function of a;'. x^dx "■ J (a + ex'^y f _ J(i + x^dx x'} .2\3- Examples. Am. 0* X^ ,2 ,, + - + «' log («'-»»). I a » 4 >j 3;2 + i 4(*Hip'» "^^^ "'• dx 49. Integratioa of where » is a positive integer. Suppose a an imaginary root oiaf - 1 =0, then it is evi- dent that a"' is the conjugate root : also, by (3), the partial fraction corresponding to the root a is I a or na'^^{x - a)' n{x- a)' If to this the fraction arising from the root a"' be added, we get M « I °" ) or'^ ^(g + «-' )- 2 W (a; - a « - a"')' w (a!* - (a + a"') « + I But, by the theory of equations, a is of the form 2kTr /—^ . 2kir cos — + V- I sm , Integration of . 59» a;" - I where k is any integer ; 2kTr .'. a + a^ = 2 COS ■ n Hence, if be substituted for — , the preceding fraction. becomes 2 a; cos - I n »' - 2x cos + i' The integral of this, by Axt. 7, is COS0, „ ,. 2sin0. Vaj-cosflX log (I -2XQ0sB + x') tan~M — -v— ?> — . » ° ' n V sin y There are two cases to be considered, according, as n is. even or odd. (i). Let n= zr: in this case the equation »"■ - i = o has- two real roots, viz., + i and - i ; and it is easily seen that [da I , CD- I I kw^ , At ,, = — log h — S cos — log f I - 2x cos — + »') J as"" -I zr "x+i 2r r '° '' " r , I m -003 — \ -issin^tan- — ^ , (13) \ ^^"^ / where the summation represented by S extends to all integer values of k from i to r - i. (2). Let w = 2r + I, we obtain f dx log(a!-i) I ^ zkir , ( zkir -^rr. = -+ SCOS log I-2a!C0S +X' \x^*^-i zr+i zr+i 2r+i °\ 2r+i 2kT ,7. /«-cos 2 .„ . 2*77 , _, / zr + I ^'^^z-;t^^-H- ^hr-^ , (i4> zr + I zr + 1 I . 2^77 sm zr+ 1 n\a! - a 60 Integration of Rational M-actions. where the summation represented by S extends to all integer values of k from i up to r. 50. Integration of , 'wbere m is less than n + 1. As before, let a be a root, and the corresponding partial iraction is ——-. r or —. r ; hence the partial fraction Wa""' {^-a) n{^ - a) arising from the conjugate roots, a and a"^ is X - a'^J n «" - (g + g"') X + l 2 xcosmO - cos(m - 1)6 n aj" - 2® cos 4- I ' where is of the same form as before. The corresponding term in the proposed integral is easily ■seen, by Art. 7, to be -|cos»«61og(a!'-2a!cos0+ i) -zsiumfltan"' — =~fl~ • ('S) By giving to k all values from i to — i, when n is even, and ft T from I to when n is odd, the integral required can be written down as in the preceding Article. Samples. 61 EXAMPIES. . J «» + 6a; + 8 2 *= V» + 4/ j8 _ ^ _ ^ - >. 2 log (a; -2) + log (» + !). r(^ + Ba:2)(?a: ^ J»-^i, , , ,, 3- J <« + te') - " -log:i; + ^^^log(«+ia;^). f g^'^Jg I a; - I v'l , / a; \ ■"^ +' 4y^2 a;2-a>/2 + l 2^/2 \i-icV f (2g-5)i?a; 7 , " , f^ + ^\ ■ J (»+ 3)(»+ I)"" " 2 (a; + I) + 7 ^^ l^n j • f in which m and n are supposed to have positive* signs. sin" Q By this formula the integral of — z-ndd is made to de- ■^ ^ cos" d pend on another in which the indices of sin 9 and cos 9 are each diminished by two. The same method is applicable to the new integral, and so on. If m be an odd integer, the expression is integrahle im- mediately by Art. 51. If w be even, and n even and greater than m, the method of Art. 5 2 is applicable ; ii m = n, the expression becomes / tan^O d9, which will be treated subse- quently ; ii n m, the integral reduces to cos"-"e ' * The formulae of reduction employed in practice are indicated by the capital letters A, B, &c. ; and in them the indices m and n are supposed to have always positive signs. By this means the formulae will be more easily apprehended and applied by the student. [5 a] 68 Integration by Successive Reduction. and if n < m, it reduces to t, — . The mode of find- J cos 6> ing these latter integrals wiU be considered subsequently. Again, if the index of sin be negative, we get, by changing the sign of ot in (2), [• cos"g „ cos"-'e w-i f cos"-'e Jsin^ei (»M-i)sin'»-'6l"^^JsS^ ^ ' "We shall next consider the case where the indices are both positive. 55. Indices both Positive.— If sin^fl (i - cos'ft) be written instead of sin"" d in formula (2), it becomes I n'- I f . si »j+ I J • ™n „« 7a cos»-'0sin"'+'0 sin'"0 cos" Odd = sin^e (cos'-^e - cos" 6) dO = m + 1 cos"-' sin™" d m + I fl — 1 C 7t — \ c + sin^fl cos"^0rf0 sin^e oos"edd: m+ I J m+ 1 } hence, transposing the latter integral to the other side, and dividing by , we get f sin^e cos«e dO = "°^""'^ '"''"' - + ??— ? f sin^ecos"-^^^^ (C) J m+n OT+wJ ^ ' In like manner, from (3), we get f sin-" e cos" ede = ^^— ^ f sin-™-^ e cos" Odd- «^"'""'^<'°s""fl _ J m + nj m + n ^ ' By aid of these formulae the integral of sin™ cos" Odd is made to depend on another in which the index of either sin 6, or of cos 9, is reduced by two. By successive appU- cation of these formulae the complete integral can always be foui^d when the indices are integers. Indices both Negative. 69 56. Formulae of Reduction for sin." B dO and cos" These integrals are evidently cases of the general formulse (C) and (2)) ; however, they are so frequently employed that we give the formulse of reduction separately in their case, f „n ,n sin cos"-' « - I f „ , n ,„ , . cos" ed9 = + cos"-' 6 dd. (4) f • ^ n 7/1 COS Q sin"-' «-if-„,/i,/, ,\ sm" ddB = + sm"-'' Q dO. (5) J n n J The former gives, when n is even, f cos" 0rf0 = — fcos"-' e + ^^-=-i cos'-^ Q J n \ n - 2 -. ^'^ -';;"- 3) cos"-»e+&c. n {n - 2){n — Of) . . . 2 ' ^ ' A similar expression is readily ohtained for the latter integral. Examples, f . , , sinfl cosfl/ . . 3\ ■? I. ein*9(?fl. Ans. (siii29 + -l + |8. f. . sin e cose/sin^9 sin-fl i\ 9 2 V 3 12 %j i6 f . sinfl oos'8/ „ 5\ S / . \ 3. C03«fl(?9. ,, T lcos2fl + ij + -^lsin8oosfl+eJ. 57. Indices botli Negative. — It remains to consider the case where the indices of sin Q and cos B are both negative. Writing - m and - n instead of m and n, ia formula (C), it becomes r dB _ -i_ n + I (• dB j sin^O cos"0 ~ {m + n) qos'^'B sin^-'fl ''' m + n] sin"0 CO8""0 ' 70 Integration hy Successive Reduction. m + n or, transposing and multiplying by n + I dQ I m+n dO sin'"0co8"O" sin^e cos"^=0 (» + I ) cos"'-'0 sin"'-'0 w + i J i Again, if we substitute n for » + 2 in this, it becomes r dd ^ I Jsin'"0cos"0 ~ {n - i)cos"-'0sin'»-'0 m + n- 2{ dO -1 Jsin"'0cos»-^0" ^ ' Making alike transformation* in formula (D), it becomes [ dO - I I Bin'"flcos"0 {m- 1) 8in™-'0 oos"-'0 m + n- 2 + m ■ •" w sia'»-'6/ cos»0' In eacb of these, one of the indices is reduced by two degrees, and consequently, by successive applications of the formulae, the integrals are reducible ultimately to those of dO dO one or other of the forms — r. or -: — 7, : these have been cos sm a already integrated in Art. 17. The formulae of reduction for -: — ^ and 7: are so sm"0 cos"0 important that they are added independently, a& follows :— * It may be observed that formulae {B), (i)), and (F) can be immediately obtained irom {A), (C), and {£), by interchanging the letters m and n, and substituting ^ instead of 8. For, in this case, sin 9, cos d, and dS, transform into cos ip, sin (j), and — d^ji, respectively. Application of Method of Differentiation . 7 1 f d9 sing n -z f d9 . . Jcos"0 ~ [n - i) co8"-'0 "*" w - I Jcos"-*0' ^'^' f dQ -cos6> n-2 f dd - . Jsiii»e~ (w- i)siii"-'0 "^w - Jsin'-'fl' ^' It may be here observed that, since sin'0 + cos^fl = i, we have immediately f dQ _ [■ dQ f dO _ , J sin™© cos"0 ~ J siii"'-''0 cos"0 "*" J sin^fl coa'^'9 ' ^^' and a similar process is applicable to the latter integrals. This method is often useful in elementary cases. Examples. fde rsine 64. Redaction of J «*" (log «)"(&. Let y - log X, and the integral reduces to that discussed, in the last Article. The formula of reduction is «"» (log x)" dx = ^—^—!- a'" (log x)''-'dx. ( 1 7), again 78 Integration by Successive Reduction. EXAMFLBS. fgax I 5 *52 ^2ll x^e"dx. Am. — } a;' - ^ «■ + — '— a; - i-1-— 1— j . ^' J *i ■ » ~3(a^ 57■ + 2ri+iT7TTJ^~• 65. Redaction of I of* cos aajrfa;. Here «" cos fla-rfa; = a;""' sin aajrfa; : J o aj f „ , . , «"-' cos ra « - 1 r „ , , a""' sin axdx = + »""' cos aar aa', J a a ] hence •f „ , a;""' (aa? sin aa! + M cos oa;) n[n~i)t , \af^: dx (m - i) a;*""' {a + cx^) + ex'" {a + cx')i (m - i) ax'"'^ mcx™^ = '- 1 • {a + cx^)^ (a + ca;'*)*' hence, transposing and integrating, we obtain x'^dx x"^^(a + cx')i (m - i)a x^-'^dx [a + cx'')^ mc mc J (a + ca?)i (20) By this formula the integral is reduced to one or more dimensions ; and by repetition of the same process the ex- pression can be always integrated when m is a positive integer. The formula (20) evidently holds whether m be positive deduction ^■^ } {a + cxY 81 or negative ; accordingly, if we change m into - (m - 2), we obtain, after transposing and dividing, da; f dx (a + ea;')* (m - 2) c J af^{a + cx'Y ~ {m-i) ax'^'- {m- i) a, 69. more generally, fve bave Hence a;^' {a + cx')i' . (21) — [x'^^ {a + ca;')") = (w - i ) a!™-' (0 + ca;'')" + 2wca!'" (a + ca;')""' = (a + ca;")"-' { (m - i ) a a;"-'' + (m + 2« - i ) ca;'" j . a!*"-' (a + ex')" x"' {a + ca^y-^dx = ' (m- i)a {m + 211— i)c_ {m+ 2n- 1)0 x'"-'^{a + cx''Y"'dx. (22) Consequently, when m is positive the integral can be reduced to one lower by two degrees. If m be negative, the formula can be transformed as in the preceding Article, and the integration reduced two degrees. We next proceed to consider the case where n is negative. 70. Redaction of SB^dx {a + oxY' m and n being both positive. „ f x'"dx Jiere 7 57- ' }{a + ex')" xdx {a + cx^' Let x'^'^ = u, and f xdx J (a + cxy '^ ^' or and we get r x'"dx ]{a + cx')" 2 (« - i)c(a + ca;'')""' = «, 2 (m- i)c (a f caj")"-' "^ 2(w - i)cj {a + cxy-'' ^^^' 82 Integration hy Successive Reduction. By successive applications of thisfonn the integral admits of being reduced to another of a simpler shape. We are not able, however, to find the complete integral by this formula, unless when n is either an integer, or is of the form -, where r is an integer. r s^ diX 71. Redaction of -. = ^.. J (a+ 20x + cx^)* By differentiation, we have — [x'"-\a + 2lx + ca?)^ = {m- i)x'^' (a + 2bx + c^')i »"""' (J + cx) {m- 1) ax"''^ + (zOT - I ) J*""' + mea;" [a + 2hx+ cx'')i {a+ 2bx + cx'^)i ' , r x"'dx x'^^(a+ 2bx + cx')i hence -. 7 rrr = =^ - \ [a + 20X + cx^)i mc {2m-i)bC x^^'^dx (m- i)ar af^'^dx mo ]{a+ 2bx + cx'^)i mc ] {a + 2bx + cai^)^' ^ This furnishes the formula of reduction for this case : by successive applications of it the integral depends ultimately on those of xdx , dx and {a + zbx + c*")* {a-¥ 2bx+ cx'ji' These have been determined already in Arts. 9 and 12. Again, the integral of -—. —- can be reduteed to * ^ x'"{a+2bx + cx')i the preceding form by making x = -. z 72. Tbe more general integral r x^'dx J (a + zbx + ex')" admits of being treated in lilie manner. Reduction of rr-. 83 J (a + 20X + car)" "For if a+ zhx + ci? be represented by T, we have, by difierentiation, ^ /^'"""'N _ H- 0«""' _ 2 (w - i)!?""-' (6 + ca) (»» - i) a!*^" (a + 2 to + ca;') - 2 (w - i) a?*"'' (5 + ca;) _(m-\) aa?"-'^ 2l{m - n) «*""' {m -m - i)e!>f^ jfn 2^ "jln " Hence, we get the formula of reduction rss^dx _ -x'"-^ 2{m-n)b (xF-^dsc J ~T" " {2n-m-i)cT'^' "^ (2w-m-i)cJ I" {m - i)a ^ai^'^dx ."'" {211- m- i)c] r» ■ ^^^' By aid of this, the integral of , -when ?» is a positive integer, is made to depend on those of -=^ and -=j. Again, it is easily seen that the integral of -^ is reduced to that of dx ra;rfa! i (• (5 + a J 2^ = -J fi cx)dx h c dx h \ dx - I oiax 2(«-i)cr«-' cj T"" ^ ' [6 a] 84 Integration by Successive Reduction. 73. Reduction of ; ; J (a + 20a; + cx^^Y' In order to reduce ^, we have d fh + cx\ _ c 2W (6 + cxY dx { T" J ~T" T^' Hence . ^. c 2n{ae-b^) 2«e 2n{ae-¥) (2w-i)e rfit; 5 + ca; (2« - i)c f (30) J (i + cos a cos a')" sm*"'aj •where tan - = tan - tan -. 212 2 Hence, if we take o and — as limits for x, we have 2 dx J ( I + cos a cos «)" Bin'""'a 76. Integration of (l - COSO C03t/)"~^dl/. f{x) dx + /3') -/a + 2Bz + Cz"' where L, M, A, B, C, are constants. To integrate this form ; assume* s = /3 tan (6 + 7), where * For this simple method of determining the integral in question. I am indebted to 'H.t. Gathcart. 88 Integration by Successive Reduction. fl is a new variable, and 7 an arbitrary constant, and the transformed expression is {Z/3 sin (fl + 7) + If cos (0 + ■y)\dB /3 y A cos' (0 + 7) + 2^/3 cos (6» + 7) sin (0 + 7) + 0^' sin' (6> + 7) " Again, tlie expression under the square root is easily transformed into ^\A+ C/3' + (^- C/3') cos 2(0 + 7) + 2^j3 sin 2(6 + 7)) ^ + Cj3' + cos 20 {(^ - C/S") cos 27 + 2^^ sin 27} + sin 20 {2jj3 cos 27 - (^ - Cj3'-) sin 27) . Moreover, since 7 is perfectly arbitrary, it may be assumed so as to satisfy the equation 2S[i cos 27 - (^ - 0/3') sin 27 = o, or tan 27 = . _ „^; : and consequently the proposed expression is reducible to the form (Z'co3 + If^sin0))p*i [ (p + i)tx<' -a} > — j3(8ia!i!-cosa!) + cos'a(3sina;-oosa;)[. If f- J sin ^+-^Isk d9 sin"' 9 cos" 9 sin™-' 9 cos""' 9 J sin"'"^9cos"9' determine the values of A and £ by differentiation. ■ («' - a')dx f 8in'9(f8 '■ J (1 + cos 9)2' .4«s. 2 tan — ( !sin"*9 d9 f 7- transforms into i""-"*' 1 (i + cos9)« J where 9 = 2^. sin"0 dif) cos^""*"^* 90 Examples. r dx -J sin a ia ^ i(/«-*\'i ") e /tan -\ icos 9 (?9 . ? sin 9 8 i. 1 1 5 I ; -r^. Ans. - . tan M — — I. (j + 4 COS fl)» 9 5 + 4 003 9 27 \ 3 / 15. I (sin-'a;)* (fo; = a {(sin-^ic)' -4 . 3 . (sin-'a:)'' + 4. 3 . z.l} + 4v^i - a;* sin"'« {(sin'^a;)' -3.2}. . , - f{cosx)dx . 16. Proye ty Art. 74, that any expression 01 the form -. j r- is ■^ ' ^ "^ (a H- cos iuj" capable of being integrated when / (cos x) consists of integral powers of cos x. t^. Show, in like manner, that the expression /(cos a;, Binx)dx {a + b cos »)» can be integrated when /(cos x sin x) consists only of integral powers of cos » and sin X. {A + Sx+ Cx' ) dx = M rug ^» X P*J T «Ai 18. If f--5£±££±^)-^ P log (« + ,3:.) + Qlog(<. + 4» + «»;''), J (ff + fix){a ■^ bx + ex^) „ f V 3 ' ■92 Integration hy nationalization. 78. Rationalization of i^(a!, ^/a + zbx + cx')dx. It has been observed (Art. 28) that the integration, in a finite form of irrational expressions containing powers of x beyond the second, is in general impossible without introducing new transcendental functions. We shall accordingly restrict our investigation to the case of an algebraic function containing a single radical of the form y^a + zbx + cx^, where a, I, c are any constants, positive or negative. Integrals of this form have been already treated by the method of Eeduetion (Art. 76). We shall discuss them here by the method of rationalization. The expression* '—V-r can be made ra- ^ W -/a + 26a! + ex' tional in several ways, which we propose to consider in order : — (i). Assume y^a + zbx+cx^ = z- x -\/c, (i) Then a + 26a; = s' - zxz -v/c ; .". Mx = zdz - ^/c {xdz + zdx), or dx[h + z ./c) = dz{z-x v/c) = dz ./a + 2bx + ca?', dx dz (2) y^a^- 2bx + cx' b + z^o 2{b + z^/c) This substitution obviously renders the proposed ex- pression rational ; and its integration is reducible to that of the class considered in Chapter II. * It will be shown subsequently that the integration of all expressions of ■the form F(x, y a + ibx + ex') dx is reducible to that of the above when J is a rational algebraic function. It may also be observed that, in general, the most expeditious method of in- tegration in practice is that of successive Eeduetion (Arts. 71, 72, 76). Rationalization of F{x, ^/a + zbx + ex') dx. 93. When 5 = o, we get dx d% , a' - ffl , . , s =, and X = — —- (see Art. 9). V « + ex' Z^/o 2Zy/c By aid of the preceding substitution the expression dx transforms into (» -ip) -v/a + 26a! + C3? dz (Art. 13) For example, to find s* — 2%p -yc - a — 2'^b dx {p + qx) \/i + a? TT 2' - I , dx 2 Mere x = , and 22 ' (p + qx) v^i + a^ qz' + 2pz - q' J (^ + qx) -s/ \ -V x' »/p' + f \q% +p + ^/p' +q'J When the coefficient c is negative the preceding method introduces imaginaries : we proceed to other transformations in which they are avoided. (2). Assume* \/a + zbx + ex' = ^/a + xz. (4). Squaring both sides, we get immediately zh + ex= 2z v^ + xz' ; .: dx{o - s") = 2dz{Ya + xz) = zdz \/a + 2bx + ci?. TT dx zdz , , Hence — = . (5). /ya + 2bx + cx' c - s' * This is reducible to the preceding, by changing x into -, and then em- ploying the former transformation. 94 Integration hy Rationalization. And X = -^^-^ — z — '-. (6) This substitution also evidently renders the proposed expression rational, provided a be positive. For example, to find f dx i X\/\ - «' Assume -v/ 1 - a? = \ - x%, and we get dx r^« /t _ y/7 ■ l;7r^=jT = ^°^^ = ^°< — -X — > (3). Again, when the roots of a + 2 te + cx^ are real, there is another method of transformation. For, let a and /3 be the roots, and the radical becomes of the form 4. f a:™ { (a' + a;'i)i + a;}" (to. JIalung the same assumption as in £z. 2, the tranafoimed expression is which is immediately integrable when »j is a positive integer. f dx [(r + a:')i + x]"^^ [(l + x>)i t- aj]"-' ^' J {(i+a;2)l-a;}»' "*' 2(»+i) 2 (» - i) Ti ^£ V^ff + 2ia; + ca:^ (v^* + 2*a; + ca;^ ± xy cj Let \/« + iJa; + ca;' + a; v c = ^i then, as in Art. 78, we get dx dz \/ a -k^ 2bx + cx"^ h +z y' c hence the proposed expression transforms into de S» (J + 1! V «) General Investigation, 97 79. Creneral Investigation. — The following more general investigation may he worthy of the notice of the student. Let It denote the quadratic expression a + ibx + ca?; then, since the even powers of v^-B are rational, and the odd contain '/M as a factor, any rational algehraio function of x and of vB can evidently he reduced to the form P+ Q^R F+qyiL where P, Q, P', Q' are rational algebraic functions of sc. On multiplying the numerator and denominator of this fraction by the complementary surd P' - Q' yS, the deno- minator becomes rational, and the resulting expression may be written in the form where M and If are rational functions. The integration of Mdx is eiiected by the methods of Chapter II, ,_ tJsritdic Also which is of the form -/B /(jg) dx (j) (x) '/a + zhx + c;^ Let, as before, -/a + zbx + ca^ = vc (x - a){x - /3), and substitute w ^ r-; instead of x, when the radical becomes \ +2fiZ + vs' a' + 2n'z + v'z'. (9) Again, if the quadratic factors under this radical be made each a perfect square, the expression obviously becomes rational. [7] 98 Integration by EaUonalization. The simplest method of fulfilling these conditions is by reducing one factor to a constant, and the other to the term containing z^. Accordingly, let X — aX = O, fi — afi = O, n — J3/jl' - o, V — j3v = o ; or fi = o, fi = o, X = nX', V = /3y'. On making these suhstitutions the expression (9) heoomes X + y'fs' X + V'Z' In order that v- cX'v' should he real, X'and v must have opposite signs when c is positive, and the same sign when c is negative. It is also easily seen that without loss* of generality we may assume X' = i, and v' = + i. o a Hence, when c is positive, we get x = — ^-^, and when a + /3s' c IS negative, x = . ° 1 + z^ These agree with the third transformation in the' preced- ing Article. More generally, when the factors in (9) are each squares, we must have (ji - a/j.')' - (X - aX) (v - av) = O, or ;u"-Xv+(Xv' + i;X'-2/i/)a+(ju''-XV)a' = 0, (lo) and a similar equation with j3 instead of a. Moreover, by hypothesis, a satisfies the equation a + iba + Co? = o. • For the sulistitutiou of y^ for — ;- transforms A. a^ + Pv'z^ . . a + Py' , — ; 7-r- mto ^ ; .•. &c. General Investigation. 99 Accordingly (lo) is satisfied if we assume the constants A, fi, &c., so as to satisfy the equations fi^-\v = a, X'v ^\v-zixfi=2b, fi^-X'v' = c. (ii) Again, solving for z from the equation X (X' + 2/i's + vV) =\ + 2fiS + vz^, (12) we obtain j , a:' - I + v^I + a;* » » I , v/i +a;' + a;v' 2 /-l°g „ ' -v/ 2 I - a;'' This and the preceding were given by Euler (Cah. Int., torn. 4) : the connexion, however, of their solution with the method of recurring ec^uations does not appear to have been pointed out by him. f (a:* - x)dx ^ 'i/»*T'^"+~I 5- -• ^«». J a;2. Let 3? + -: = z, &c ar ■^y/i^+x^ + i ■ z, &c. 6. f (x^-i)dx J x^/{x^ + aa; + 1) (a;» + iSa; + i)* , -v/a;' + ax + t + v/a;'+|8j+ I ^»«. 2 log — ■' f (i- ^K.,_ . ^^.sin-(^V Examples. 103 8. f^.. ^„,,3{?^iZ_3f)(„ + i,,,. J {a+bx)i lob' Ji - «' V' I + a' + a* " \/3 I - a* Assume r = (i +V)i ein fl, &c. "• f ^- — I — " '^'i — ^V J (' + «'"){(' + it'")" - «'}» M' +«'^)'"/ (I + x)i + (i + »)»• AsEume t + x = z'. '^' J (l-a^)(i+a;*)r . I , (i + a^)J + 3:^/2 I , (i + a:*)* ' -4««. — ^ log i — ^ = tan ' ^ -~. 14- 1 ^ —r. Ans. . 5- J i-a;* ' Ans. — ^ log — H -:;taii-' — . 7../1 \ I - x' I 21/ z ■\/l+a' ix 16 \—LZ.^— ^ ia;'' + 2»a:^ + «* Ans ' , a;\/2(c-a) + .v/l + 2(!a:' +a2 3;» ■^"'* /-;^ r log -'^^ — ^ -—. , when oa. I . Ix\/%{a-e)\ , „ — , sm"' I —^ — —z — 1 , when a ><;. ( 104 ) CHAPTER V. MISCELLANEOUS EXAMPLES OF INTEGRATION. _ . (Aoosas+ B sin X + C)dx 82. Integration of ^ ; — ; — . aoosx + Biace + c Let a cos* + b sin x + c = u, then -a sinx + b cosa; = -;-, ax Next assume A oosx + B sinx + C = Xu + /i — + V, and, equating coefficients, we have A = Xa + fib, B = Xb - /la, C = Xc + v. Solving for A, /x, v, we get _ Aa + Bb _ Ab - Ba _ (Aa + Bb) c ^~ a' + b" ^~ a' + b' ' ''~^~ a? + b' ' _. f (^ cos « + 5 sin a; + C) dx Hence ^^ -1-^ — ^ — J a cos « + sm a; + c (Aa + Bb) X Ab - Ba , , , . = -^ — I — 1~~ + — r — 7^ log {a cosa:+ 6 sin a; + c) a^ + ¥ a^ + b' ° ' (a' + b')C-{Aa + B b)c ^ a'^ + b^ r dx J a cos a; + J sin a; + c' The latter integral can he readily found ; for, if we make a = r cosa, b = r sin a, we get o cos a; + J sin a; = r (cosar cos o +Bina! sin a) = rcoB{x - a). Integration of Ji'^osx, smx)dx ^ ^^^^ a cosa; + o smx + c On making x~ a = B, the integral reduces to the form con- sidered in Art. i8. As a simple example, let us take ({A + B tan oe) dx J a-vh tan x T-r A ^^ B tan x ^ cos a; + 5 sin a; Mere a + i tan x a cos a; + 6 sin a; ' and we evidently have f{A-^B\&-D.oc)dx (Aa + Bb)x ~^Ah-Ba, , , . ^ a + 6tana. = "a" 6" ^ ^^T^logC^cos^ . Asma;). 83. Integratioa of /(«°««^. «^^^)^^ . a cos K H- sm a; + c where /is a rational algebraic function, not involving frac- tions. As in the preceding Article, assume x = Q + a, and the expression heeomes of the form ^ (cos Q, sin Q) dO Acosd + B Again, since sin' 6=1- cos' 6, any integral function of sin 9 and cos 6 can be transformed into another of the form 0j (cos 6) + sin 6 02 (cos 6). Accordingly, the proposed expression is reducible to 01 ( cos 0) rfS 2 (cos 0) sin 6 rffl Acosd + B Acosd + B The latter is immediately integrable, by assimiing ^ cos + jB = 3. To integrate the former, we divide by ^ cos + B, and integrate each term separately. 106 Miscellaneous Examples of Integration. 84. Integration of /{cosx)dx (ffli + Si cos x) (fla + h COS x). .. .{a„ + b„ COS x) where/, as before, denotes a rational algebraic function. Substitute z for cos x and decompose m («! + 61 2) {a^ + bzz) . . . . (fl» + b„z) by the method of partial fractions : then the expression to be integrated reduces to the sum of a number of terms of the form dx A + £ cosa;' each of which can be immediately integrated. EXAICPLES. f —^ ;. An., -i log ( L±E^) - A tau- te). J cos a; (5 + 3 cos ») 10 \i-sina!/ lo \ 2 / I . ■■ ■ 1 r, when a>i. sin'^x (a -t- « cos x) J — a cos a J' , li + a coix\ „js-' ( I . {a^ — b'^)amx (a' - 4')* Va + icosa;' Idx . tana; i , , /ir a;\ i* f dx :; — ; ; r. AtU. ; lOg tan I - H H \ -. . cos''a:(a + icofla;) a a^ V4 ^7 o»Ja + eoosai 85. Tntegration of [f{x) +/(x)}e'dx. The expression e^Pdx is immediately integrable whenever P can be divided into the sum of two functions, one of which is the derived of the other. For, let P = f{x)+f{:x), then le'Pdx = l^f{x)dx + ld'f'{x)dx. Differentiation under the Sign of Integration. 107 Again, integrating by parts, we have / e'/(x) dx =f{x) e - J ^f{x) dx. Accordingly, \[Ax)^f(x)\e^dx = e'f{x). For instance, to find V(^ X ^xf dx. X I I I + a; Here . . , - , ,, , (i + ar)- \ + X (i + xy consequently the value of the proposed integral is Examples. I. I e* (cos it + sin ») <&. Ans. e^im^x. fi + « log a! , , «« ^—dx. „ e'logx. X Ia;' + I , X - \ «* ; rs dx. ,, e" . (x+lf " X+I 4. e' I ^^ ) dx. „ ;. J \i + x^J " i + x- 86. Differentiation ander the Sign of Integra- tion. — The integral of any expression of the form {<«^<^) ^ da ] da ' In other words, if M = J (a;, a) dx. du t dij) then da -\t-- provided a be independent of « ; in which case, accordingly:, it is permitted to differentiate tinder the sign of integration. By continuing the same process of reasoning we obviously , get „-J da- '^' ^'^ ■where «« = J ^{x, a) dx, a being independent of x. For example, if the equation f e*" ^dx = — Integration under the Sign of Integration. 109 be differentiated w times with respect to a, we get \----m) ="^'"^1TG (See Art. 49, Diff. Calc). Again, in Art. 2 1 we have seen that J e" (a sin mx-m cos mx) e" sm jwa; ax = — ^ '-. mr + a Accordingly, f . , f d\'f^{a&mmx-mwcosa; + rw) = 2 logw + 2 — + r- - ao. . Consequently, when m> i, we have f, , 5\j 1 /sina; sin 2a! sin 3a! \ log(i+2»icosa;+m')aa;=2a;logOT+2 ;— ^+ , „ —■•• I- J o\ y ° \ »i z^m' ym^ J From the ahoTe it is easily seen that the integral Jlog(i + acosa;)&! can be exhibited in the form of an infinite series when a is less than unity : for making a = ; we have "" ° I +m' log (i + a cosa;) = log (i + 2m cos x + m') - log (i + w'). The relation between m and a admits of being exhibited in a simple form ; for let a = sin a, and we get m = tan -. Making this substitution in (4), we get log(i + sinaCOsa;)£?a; = 2a;log(cos- A « • , , a sin 2a! . N - . + 2 ( tan - sm a- - tan' — + &c. J. (5) I Examples. 1 13 EXUIFI.ES. I2D0SX+ ismii!)dx . i3« 5, , ... 3 cos a; + 2 sin as 13 13 — . „ -taii9 + ^-^tan-i(tan9v/2). -Biii''9 2 z^ 1 e'( a:° + g + T.)ix d'x l+iC" ^- [^i^^^^g = |log(i + 1 f 1 7. Prove that f e«» , . f, „ , «S+a; «' (3 + a;)« ) and determine when the series is convergent, and when divergent. 8. Prove that f e + e . , sm between the limits X and Xo ; suppose also that X - Xo is divided into n intervals Xi - «o, «2 - Xi, Xs- x-i, . . . X - x„.i ; then by definition (Diff . Calc, Art. 6), we have in the limit when ici = sjo ; accordingly we have 0(a!,) - ^{Xo) = {Xi - Xo){(l>'{Xo) + to), Limits of Integration. 115 where to becomes infinitely small along with Xi - x^. Hence we may write ^ (»i) - ^ («o) = (»i - a'o) {^'(a'o) + £o) , * ('"a) - (*«) = ('^'s - %) {0'(«j) + £2), (X) - (a;^i) = (X - «:„_,) {^'C^ft-O + s«-i}> where to, ti . . . eb-i become evanescent when the intervals are taken as infinitely small. By addition, we have {X) - f (Xo) = {Xi - Xo) '{xo) + {Xt- Xi) ^'{x,) + . . , + (X - X„.i) ^'(a-n-i) + {Xi - Xo) £0 + («2 - «i) £1 + . . . + (X - X,i.j) £„-i. Now if I) denote the greatest of the quantities eoj eu . • • £n-i> the latter portion of the right-hand side is evidently less than (X - »(,) r/ ; and accordingly becomes evanescent ulti- mately (compare Diff. Calc, Art. 39). Hence i> (X) - (xo) = limit of [(»i - Xo) ^'{xo) + {x^- Xi) f'{xi) + . . . + (X-aj„_0^'(«„-i)], (1) when n is increased indefinitely. This result can also be written in the form ^ (X) - (p («o) = '2(j)'{x)dx, where the sign of summation S is supposed to extend through aU values of x between the limits Xo and X. 91. Definite Integrals, Iiimits of Integration. — The result just arrived at, as already stated in Art. 31, is written in the form"" f{X)-f{xo) = \^f{x)dx, ^ (2) where X is called the superior, and Xo the inferior limit of the integral. [8 a] 116 Definite Integrals. Again, the expression "0 is called the definite integral of ^{x)da) between the limits Xa and X, and represents the Umit of the sum of the infinitely small elements ^ (a) dx, taken between the proposed limits. From equation (i) we see that the limit of (xi-x„)f'{x^) + {Xi - Xi)f{xi) + . . . + (X - a;„.i)/(a;„_i), when Xi - Xo,Xi- Xi, . . . X- x„.i become evanescent, is got by finding the integral of /'(«) dx (i. e. the function of which /'(«) is the derived), and substituting the limits Xo, X for x in it, and subtracting the value for the lower limit from that for the upper. If we write x instead of X in (2) we have f{x)-f{x,) = r f{x)dx. (3) in which the upper limit* x may be regarded as variable. Again, as the lower limit a!o maybe assumed arbitrarily, /(a;o) may have any value, and may be regarded as an arbitrary constant. This agrees with the results hitherto arrived at. In contradistinction, the name indefinite integrals is often applied to integrals such as have been considered in the pre- vious chapters, in which the form of the function is merely taken into account, without regard to any assigned limits. As already observed, the definite integral of any expres- sion between assigned limits can be at once found whenever the indefinite integral is known. A few easy examples are added for illustration. • The student should ohserve that in {3) the letter x -which rtands for the superior limit and the x in the element f'(x) dx must he considered as being entirely distinct. The want of attention to this distinction often causes much confusion in the mind of the beginner. i: Elementary ExampkB, 117 Examples. ulitdx. Am. ; . « + I f 1 sin e de -_ Jo cos^fl " ^ f ^r 4 /_ ,_ r* dx T Jo a- + a*' " 2»' I f" dx ■6. I e""* + «'' " sin ^* 1* m e-i" sin JMa; » -I) (^ Jo «(«+ l) ....(» + OT- l) This formula, combined with the equation a;"-' (i -«)"•-' rfa; = f a;'"-' (i - a;)""' rfa;, shows that when either m or ra is an integer the definite integral J 1 a!''-'(i -xy-^dx can be easily evaluated. When m and « are both fractional, the preceding is one of the most important definite integrals in analysis. We purpose in a subsequent part of the Chapter to give an investigation of some of its simplest properties. ExiltPLES. '• 1 a?{i-xfdx. Am. . •0 3.7. II. 13 r««(i-ai)» dx. ,, . 5.7-9. 13-17 Elementary Examples. 119 ir IT 93. Talues of sin.''«rf'a;and ooa'^xdx. One of the simplest and most Tiseful applications of definite integration is to the case of the circular integrals considered in the commencement of Chapter III. We hegin with the simple case of sr sivPxdx. If in the equation (Ait. 56) r . „ , cos « sin"-'« » - I f . „ , , J n n ] we take o and - for limits, the term — — vanishes 2 n for toth limits, and we have IT IT f ^ • , « - 1 f *' ■ . , syD!'xdx = em^'^xdx. Jo n Jo , Now, if « be an integer, the definite integral can be easily obtained ; its form, however, depends on whether the index n is even or odd. (i). Suppose the index even, and represented by 2m, then IT IT fs 2m — 1 (^ srD?'"xdx = sm"^'a!cfe. Jo 2m Jo Similarly, n IT six^-^xdx= sai}^^xdx; Jo 2m-2j„ and by successive application of the formula, we get \\in-'xdx = '■^■j ^^"'-'^ . ^. (5) Jo 2 . 4 . . . . . 2m 2 ^'" 120 Definite Integrals. [2). Suppose the index odd, and represented by 2to + i, then Jo 2OT + iJo Hence, it is easily seen that rsin"""^ca. = ^-^-^---- ^^ , . (6) Jo 3 • 5 • 7 (2OT+I) ^ ' Again, it is evident from (35)> Art. 32, that cos"xdx = sin"a!rfi», and consequently (5) and (6) hold when cos x is substituted for sin x. 94. InTestigation of sin^^cos^^flfc, Jo From Art. 55, when m and n are positive, we have f 2 . , W - I f 2 . sm'^xcoa^xax = ■ sin'"a!COs""'i!;rf'ir, Jo m + njo «■ IT f 2 . , tn — 1 [^ , and Bin"** cos"a; aa; = sin"*"'ar cos"a! cS». Jo m + nj„ Hence, when one of the indices is an odd integer, the value of the definite* integral is easily found. * The result in this case follows also immediately from Art. 92, by making cos2 x = z; for this substitutioiL transforms the integral into ifi ^ i)l»2 2 dz. I fl Elementary Examples. For, writing 2m + i instead of m, we have zm 121 sin""*'a;cos"a;c?a; ■■ 2m + n + I sin"*"' (BOOS" «!«&;. Eence sin""" a; cos" a; (fe 2»l(2OT - 2) {2m + n+ i)(2ffj + w- i) . . . . (w + 3). 2.4.6 ... (2m) In like manner, (?» + l)(» + 3) . . . (« + 2»l +.1) sin a; co^xdx (7) 1: 8in""a;cos"'a;£fo! 2W- I fa , = —. r SI 2(m + «)Jo sin"" a; cos"*"''*^*. Hence dx f'sin""a!cos'^a;l+l(i|; _ 2 .4. . 6 . . . 2« V'l -a' 3 • 5 • 7 • • ■ (2» + I) 7. Deduce 'Wallis's value for ir ty aid of the two preceding definite integrals. xfdx . 2 . 4 . 6 ...(»- i) g . Ans. . > lia+ix^f'' 3-5. 7.... » a/o*"*' when » is an odd integer. 9. f x'{2ax-a'')^dx. Jo Ehmentary Examples. 123 95. Talue of I e-* ^ dx, when n is a positive integer. In Art. 63 we have seen that e"^ «" cfe = - e^a!" + n e^x^'-^dx. Again, the expression — vanishes when x= o, and also when a; = 00 (DifE. Calc, Art. 94, Ex. 2). Hence e^i^dx = n\ e-^x"-'^dx. (10) Consequently e"**!" £& = i . 2 . 3 . . . «. (11) Many other forms are immediately reducihle to the pre- ceding definite integral. For example, if we make a; = as we get in which a is supposed to be positive. Again, to find x?" Qjog x)" dx ; let x = e"", and the in- tegral becomes • ■ Jo (m+i)«*' Since log x = - log ( - J, this result may be written in the form f * X- (log -Y dx = '•/•^V:^ ( 1 3> 124 Definite Integrals. The definite integral e^ »"■■ uv > Bv ; when V is negative, Av B{X- x^ and < ,4 (X - a;„). 97. Taylor's Vheorem. — The method of definite inte- gration combined with that of integration by parts furnishes a simple proof of Taylor's series. For, if in the equation (X\h \dx cx\h /(X + A)-/(Z)=j^ /H, we assume a; = Jf + A - z, we get dx = - dz, and also f{x)dx = \ f\X+h-%)d%\ .-. /{X + h) -f[X) = ['/'(X + h- z)dz. Again, integrating by parts, we have {f'{X + h-z)dz = zf{X+h-z) + ( zf"{X + k-z)d Hence, substituting the limits, we have £/'(X + h-z)dz = hf[X) + r zf"{X + h-z) dz. Taylor's Theorem. 127 In like manner, I s/'(Z ^h-z)d%= ^/"{X + h-z) +^jf"iX+ h -<■ z)dz, whicli gives f * zf"{X + h-z)dz = -f"(X) + f * -f"'{X + h-z)dz; Jo 2 J 2 and so on. Accordingly, we have finally f{X+h) =/(Z) + ^/(Z) + ^/'(Z) + . . . + ^/^"-'K^) .{/(..(X..-.)'* (.6) r Jo * This is Taylor's well-known expansion.' 98. Remainder in Taylor's Theorem expressed as a Definite Integral. — ^Let It„ represent the remainder after n terms in Taylor's series, then by the preceding Article we have i?„=[/N(X+A-.)^. (17) There is no difficulty in deducing Lagrange's form for the remainder from this result. For, by Art. 96, we have J ol . 2 .3 .. . («-l) I .2 ... W where U lies between the greatest and least values which /W(X+ h - z) assumes while s varies between o and h. * The student will observe that it is essential for the validity of this proof (Art. 90), that the successive derived functions, /'{«),/" (*)i ^"-i should be finite and continuous for all values of x between the liroits X and X + h. Compare Articles 54 and 75, Diff. Calc. 128 Definite Integrals. Hence, as in Art. 75, Diff. Calc. (since any value of z between o and h may be represented by (i - 0) A, where > o and < i) ; we have i2„= /(»)(X+0/O I . 2 . . . « ' where is some quantity between the limits zero and unity. 99. Bernoulli's Series. — If we apply the method of integration by parts to the expression /(a;) rfb we get I f{x) dx = xf{x)- \xf{x)dx ; .: f f(x) dx = Xf{X) - [ f'{x)xdx. Jo Jo In like manner, \y'{x)xdx=^nX) -[\f"{x) "^ Jo 1,2 Jo 1*2 Jo-' ''''1.2 I . 2.3"^ ^ '' Jo -^ ^ ■' I . 2.3* and so on. Hence, we get finally \^f{x)dx =^/(X) - ^f{X) +_^/'(X)-&c. . . . (18) Jo I 1.2 1.2.3 Compare Art 66, Diff. Calc, where the result was obtained directly from Taylor's expansion. 100. Exceptional Cases in Definite Integrals. — In the foregoing discussion of definite integrals we have sup- posed that the function f{x), under the sign of integration, has a finite value for aU values of x between the limits. We have also supposed that the limits are finite. We purpose now to give a short discussion of the exceptional cases.* They may * The Complete iavestigation of definitb integrals in these exceptional cases is due to Cauchy. For a more general discussion the student is referred t» M. Moigno's Cakul Integral, as also to those of M. Serret and M. Bertrand. Exceptional Cases in Definite Integrals. 129 be classed as follows : — (i). 'Wheii/(a!) becomes infinite at one of the limits of integration. {2). Wben/(a;) becomes infinite for one or more values of x between the limits of integration, (3). "When one or both of the limits become infinite. X In these cases, the integral f{x)dx may still have a finite valije, or it may be infinite, or indeterminate : depend- ing on the form of the function/(a;) in each particular case. The following investigation wiU be found to comprise the cases which usually arise. loi. Case in ^vlnebf(x) becomes infinite at one of the liimits. — Suppose that f{x) is finite for all values of x between Xo and X, but that it becomes infinite when x = X. The case that most commonly arises is where /(«) is of the form ,4.' . , in which i//(«) is finite for all values {X - a;)"' ^^ ' between the limits, and n is a positive index. Let a be assumed so that if{oo) preserves the same sign between the limits a and X; then \,^{x-xr-\{x-xr''\^{x- 4/{x) dx The former of the integrals at the right-hand side is finite by Art. 96. The consideration of the latter resolves into two cases, according as n is less or greater than unity. (i). Let 11 < 1, and also let A and B be the greatest and least values of ^p{x) between the limits a and X : then, by Art. 96, the integral ■ L. -^ lies between A y= r- and B y= — . Moreover, since n < i, we have evidently p ^ dx _ {X - a)'-» Ja(J:-a:)''~ i-n ' and consequently, in this case, the proposed integral has a finite value. [9] 130 Definite Integrals. (2). Let n > I, and, as before, suppose A and B the greatest and least values of ip{x) between a and X; then r^^ ^'' ^^^^^^^ A]j0w^ ^^^^{"c dx [x-xr Again, we have f «?» _ I J {X-x)" ~ {n- i)(X -«)"-'■ Now 7= 7-— becomes infinite when x = X, but has a {X - ij)"-' ' finite value when x = a; consequently the definite integral proposed has an infinite value in this case. f dx When w = I, t^Z — > ^ ~ ^°^ i^ ~ ^)- This becomes infinite when a = X ; and consequently in this case also the proposed integral becomes infinite. The investigation when f{x) becomes infinite for x = Xn follows from the preceding "bj interchanging the limits. 102. Case fvtaere f{x) becomes infinite between the IJimits. — Suppose f{x) becomes infinite when x = a, where a lies betweeji the limits x^ and X; then since f{x)dx=\ f{x)dx+\ f{x)dx, the investigation is reduced to two integrals, each of which may be treated as in the preceding Article. Hence, if we suppose f{x) = . ^^ ' , it follows, as in the last Article, that f{x)dx has a finite or an infinite value according as n is less or not less than unity. The case in which /(«) becomes infinite for two or more values between the limits is treated in a similar manner. , Case of Infinite Limits. 131 For example, if /(ffii) = oo, f{a^ = 00, . . . /(«„) = 00, where a^a^ . . . an lie between the Umits X and a;,, ; then [ f{x)dx= \ ^ f{x)dx+\ ^f{x)dx + &c. + \ f{x)dx, each of which can be treated separately. 103. Case of Infinite liimits. — Suppose the superior limit H to be infinite, and, as in the preceding discussion, let f{x) be of the form . , where i/'(a;) is finite for all values of X. As before, we have f{x)dx= f{x)dx + \ /{x)dx, J Xq J Xq J a The integral between the finite limits x„ and a has a finite value as before. The investigation of the other integral con- sists again of two cases. (i). Let n> I, and let A be the greatest value of ^{x) between the limits a and 00, then f J is less than A dx But r-i^.=_L_r_i ' 1. The latter term becomes evanescent when X= 00 : accord- ingly in this case the proposed iategral has a finite value. _ In like manner it is easily seen that if n be not greater than unity, the definite integral f" dx [9 a] 132 Definite Integrals. has an infinite value ; and consequently f * i^[x) dx Ja («;-«)" is also infinite, provided i//(a;) does not become evanescent for infinite values of x. Hence, the definite integral J^{x)dai r„ {x - a)" has, in general, a finite or an infinite value according as n is greater or not greater than unity : \li{x) being supposed finite, and a-o being greater than a. If X become - oo, a similar investigation is applicable; for on changing x into - x, we have f{x)dx = -\ f{-x)dx, in which the superior limit becomes od. 104. Principal and Cfeneral Talues of a Definite Integral. — We shall conclude this discussion with a short account of Cauchy's* method of investigation. Suppose /(«) to be infinite when x = a, where a lies be- tween the limits «„ and X; then the integral f{ m. Let a be a root of ;»■" + i =0, and, hj Art. 37, we have Again, by the theory of equations, o is of the form (2/i;+i)7r / — . [2lc+ i)7r cos — + -v/- I sin i —, zn 2% in which h is either zero or a positive integer less than n ; .-. a""" = cos (2A + i) + '/^i sin [ih + i) 0, (2m + i)7r where ■ zn Hence B = — ^ — ; and accordingly we have zn 2?i + ^2 + . . . + 5„ = — {sin + sin 30 + . . . + sin (2W - 1) 0) . zn To find this sum, let jS = sin + sin 30 + . . . + sin {zn - i) ; then 2>S'sin = 2 Bin'0 + 2 sin sin 30 + . . . + 2 sin sin [zn - i)0 = I - cos 20+ cos 20- cos 40+ . . . + cos(2?j - 2)0 - cos 2n0 = i-cos2?z0=2sin'n0=2sin'(2m+ i) - = 2 ; B= ' sinO ■ {2m + i)7r' zn Value of\ Accordingly, we have rdx. 139 wsm (2TO+ l) 2ft Hence, by (19), Jol + «"" ~ 2J..I +»"" ~ 2/8 . {zm+lJTr' (22) sm- 2W We now proceed to consider the analogous integral —, where m and n, as before, are positive integers, Jo I ~ * and n> m. log. Investigation of We commence by showing that dx -dx. f. = o. This is easily seen as follows : dx C^ dx It i dx Now, transform the latter integral, by making x = -, and we get •• dx _ r dz 1 I - «' ~ J 1 1 - : dz dx r* dx Jo I -*■ Again, proceeding to the integral " x^'^dx 140 Definite Integrals. we otserve that i + x and i - x are the only real factors of I - a;'", and that the corresponding partial quadratic fraction in the decomposition of :is I -X'" « ( I - a') ' Consequently, the part of the definite integral which corre- sponds to the real roots disappears. Moreover, it is easUy seen that the method of Arts. 107 and 108 applies to the fractions arising from the n - 1 pairs of imaginary roots, and accordingly f" x^™dx ^= 277(^1 + ^8 + . . . + .B».i), J-» I - X ■where J3i, Bt, . . . Bn-i have the same signification as before. Again, since the roots of «"" - i = o are of the form ktr J — . Ut: cos — + a/- I sm — , n n it follows, as in Art. 108, that Bi-^ Bi + . . . + B„.i = — [sin 20+ sin 40 + .. . + sia2(M- i)0], where 6 = ^ —, as before. 2W Proceeding as in the former case, it is easily seen that sin 20 + sin 40 + ... + sin 2 [n - i) COS0 -C0S(2M- l)0 , 2OT + I '— = cot TT. Hence 2sin0 o^'^dx IT ,2m+i = — cot — TT ; 2W ' x''"'dx w . 2m+ I , . cot T. (23) , I - X"' 211 2n Exampks. 141 Again, if we transform (22) and (23) by making a'" ■ , 2m + I and a = , we get 2» smaTT 's^-'^ = TT cot arr. (24) The conditions imposed on m and « require that a should be positive and less than unity. Moreover, since the results in (24) hold for all integer values of m and n, provided n > m, we assume, by the law of continuity, that they hold for all values of a, so long as it is positive and less than unity. 1 10. The definite integrals discussed in the two preceding Articles admit of several important transformations, of which we proceed to add a few. For example, on making m = s" in (24), we get f' du air f* du , 1 = — ; j^ = air cot J 1 + M<» sin «Tr J 1 - M» air. If - = r, these become a f" du _ IT r" J 1 + w*" ~ . tt' Jo du ir , IT - — cot -, (^5) where r is positive and greater than unity. Again r stfda; _ p x^dx f" x"dx Joi +«» ~J„ I +a^ Jii +«'■ Now, if in the latter integral we make a; = -, we get f aT'dx 1 + X' " a^dx _ r» z-^di J 1 1 + «" ~ J 1 I + s r x"dx f «" + «-» , , ,^ 142 Definite Integrals. Moreover, from (22), when n is less than unity, we have f afdx 2 cos 2 Accordingly 2C0S- nir In like manner, it is easily seen that f ' «" - ar" -ff2 We add a few examples for illustration. EziUFLES. " dx ■^ ° («» - *»)» ^" J (»:» + »»)(»» + - Ans. " 2ab(a+b)' 4. tan"fl m. Ans. Jo »"+ ar» »' r 1 \'^ r e J\^ ~ ^ I J mv 2n cos — a b 2 003 - COS - («"»+ «-"«)(«'» + r5») 2 2 »a?. ,, , , ^x + e-uai 003 a + COS 4 Jo er* - e-ira: C03»+C0SA It siould te otserved, that in these we must have a + i < tt. 8. Hence, when i < ir, prove that ghx + e-5« \ / 2 cos aa; B» = , i: ' — «"** , sin 5 003 axax = - ffnx _ g-,ra: e« + 2 COS + < Jo «"- Sin axdx=— ■ ■ e-^" 2 6" + z cos 4 + 6-" Q. I (fo. -^KS. IT cot flir . Jo I -z a III. Differentiation of Definite Integrals. — It is plain from Art. 86 that the method of differentiation under the sign of integration applies to definite as well as to in- definite integrals, provided the limits of integration are independent of the quantity TOth respect to which we dif- ferentiate. On account of the importance of this principle we add an independent proof, as follows : — Suppose u to denote the definite integral in question, i.e. let •i u = if>{x, a)dXf J a, where a and 5 are independent of a. To find 3- let A«« denote the change in u arising from the da change Aa in a ; then, since the limits are unaltered, 144 Definite Integrals. Am = [^(a;, a + Aa) - ^(«, a)j«&; At« r* ^{x, a + Aa) - (j> {x, a) " ' Aa Ja Aa Hence, on passing to the limit,* we have du p d^ [x, a) da Jo da Also, if we differentiate n times in succession, we oh- viously have ^i _ p d"{x, a) d^" ~ ]a da" The importance of this method will be best exhibited by a few elementary examples. 112. Integrals deduced by Differentiation. — If the equation i be differentiated n times with respect to a, we get e'^dx = - a \y e-^^ = iLlll^, as in Art. 95. Again, from the equation C" dx _ n- Jcjip' + a" 2 I ah' we get, after n differentiations with respect to a, r" dx TT I .3 . 5...(2» - i) I Jo («' + a)"*^ ^22.4.6... 2n a^^i' which agrees with Art. 94. * Eoi exceptions to this general result the student is refcired to Sertrand's CaUulJntigral, p. 181. Differentiation under the Sign of Integration. 145 Again, if we take o and oo for limits in the integrals (23) and (24) of Art. 21, we get $-"'■ 005 mxdx = — ;, e-^" wa. mx dx = — -,. (31) Jo «*+wi' Jo «"+»»' ^'^ ' Now, differentiate each of these n times with respect to a, and we get f e-«* «" cos mxdx=(-iY f-^X ( ■ - "' \ Jo ^ ' \daj \a' + m') _ \n .cos(m + i)0 {a^ + m?) ^ f° -ar ■n • J Iw. sin(m+i)0 e "* a;" sm mx dm = L_ "^ ' , , Jo "iT^' (32) {a^ + ni") » where w = « tan 0. (See Ex. 17, 18, Diff. Oalc, pp. 58, 59.) Next, from (24) we have = IT cot Jo i-x air. Accordingly, if we differentiate with respect to a, we have m^^ log x^ it^ ) \- X sin'' air Again, if the equation 1; [r'iv'l be transformed, by making y= —, it evidently gives Jj (a + te)"" ~ Ha6»' [10] 146 Definite Integrals. Now, difEerentiating ■witli respect to a, we have r a^'-'dcc {a + hxY*^ n{n + 1)0,' b"' If we proceed to differentiate m- i times witli regard to a, we have af^^ dx 1 .2 . ^ .. .{m-i) I Jo (a + fe)*"*" w.(?j+ i)(?i + 2) . .. (w + OT - i) ' al^b"' 113. By aid of the preceding method the determination of a definite integral can often be reduced to a known integral. We shall illustrate this statement by one or two examples. Ex. I. To find |'''log(i + sin a cosir) Jo cos « Denote the definite integral by u, and differentiate with respect to a ; then [" cos adx ,, . , „, = -. = n (by Art. 18). J I + sm a cos (B da Hence, we get ■ dx log (i + sin o cosa) cos a; 1: No constant is added since the integral evidently vanishes along with o. _, I (r"" sm mx , £jX. 2. M = dx. In this case a fT"" cos mx dx = du f dm Jo C dm , , fnC\ .-. u = a\ , = tan-» — . ]a +mr \aj No constant is added since u vanishes with m. Case tchere the Limits are Variable. 147 Ex. 3. Next suppose ^P°f ■v-^.. log(i + «"«') Here du C" 200^ dx la^]^ {7TaV)(iT6^) I V V° 2adx p zadx "1 " «'-i'Ljo i+6V~ Jo FTfflVj I fa \ "■ . "=6 {x, a)dxda J a„ Jo is here taken as an abbreviation of ^{a:,a)dx\da, Ja„LJa J m -which the definite integral between the brackets is sup- posed, to be first determined, and the result afterwards integrated with respect to a, between the limits oo and oi. The principle* established above may be otherwise stated, thus : In the determination of the integral of the expression \mj \mj Compare Ex. 2, Art. 113. If we make oo = o and oi = 00 in the latter result, we obtain f sin t. Jo X mx jT dx = -. Value of e~^'dx. Ex. 4. To find the value of 151 e-'^'dx. Denoting the proposed integral by k, and substituting ax for X, we obviously have f e-''"'''adx = k; Jo Hence But ''C"''')a.Sa 2 I + a"' I I + iB^ 4 1 f rfa; 2 10 Hence e-^'dx = k = ~'/^. Jo 2 (36) This definite integral is of considerable importance, and several others are readily deduced from it. 117. For example, to find (A) u = Here e "'dx. du da -X'-- dx ' a —. X' Again, let 'z = -, and we get e "'ds = u ; 152 Definite Integrals. Jo «' Jo .•. — - = - 2M : hence u = Ce~^". da To determine C, let a = o, and, by the preceding example, M becomes "^^-^ — . 2 Consequently J/ -&=^e-=''. (37) Again, to find (B) u = [ ^'''^"cos zbxdx. Jo Here ^ = - 2 I e^'*" a; sin 25«f&, But, integrating by parts, we have e^'^" sin,2?/a;, ,26 .a have + '£_ fef-o^^'cosaSarefo; Hence 25 6-"°'^'' sin 2bxxdx = — | e"*"*" cos 2ixdx. du 2hu du 2b db or — =- db a' ' "" u a' Hence u = Ce "'. Also, when b = o, u becomes — '-, 2a e-''°^°cos2J^&; = :^^c''"'. (38) Exami)les, 153 Again, if we differentiate n times, with respect to a, the equation e-'^'dx ■■ 2^/a and afterwards make a = i, we get (C) I ^■^A . ■■3.5.^.(^.- _L) y- Next, to find P) We ohviously have r cos ma Jo I + ° cos mzdx ae-'^'i'^^lda-- I + a' -i: a 6'°-" ('■^^°) cos mx dx da ' cos mxdx 1 + x"- ' But, by (38), we have e''^''^' cos mxdx = ~ — e 4a= ' -v/i 6'°-"' ia.' da P cosm Jo .1 + mxdx Hence, hy (37), we have " cos mxdx TT _„ = -e-*". I + a;^ 2 (39) Again, differentiating with respect to m, we obtain p xsinmxdx t _^ J I + a;^ 2 " ''^ •' 154 Definite Integrals. Examples. I. xiix. ^'- e-"' Ans* isec=?. 4 2 2. J2 i+x x' >i log^tan^j, ahen a ; > o and < i. 3- r 1 I" + ;-» - 2 di Jo I - s log z' i> ° \sm. cmj 4- IT i~L dx log (i + cos fl cosa;) . ° ^ 'cos X •5 - — flM . 2 \4 / C, cos3:log(-= Jo \a;' + - dx. '' T r«-{bx) dx = 1^(00) - 0(0) I log (46) Eor example, let (j>(ax) = tan"'(aa;) then we have ^(p) = o,. and 4>(oa) = -. ' 2 Accordingly we have tan"' ax - tan"' bx 1: dx ■■ -r 2 j& h ^ dx 'Mt * JEdurational Times, iS'iS- The student will find some remartdble exten- sions of the formulae, given ahove, to Multiple Definite Integrals, by Mr. Elliott, in the I'roceedinffs of the London Mathematical Society, 1876, 1877. Also by Mr. Lendesdorf, in the same Journal, 1878. 158 Definite Integrak. 119a. Remainder in liagrange's Series. — ^We next proceed to show that the remainder in Lagrange's series {Diff. Calc, Art. 125) admits of heing represented by a definite integral. This result, I believe, was first given by M. Popoff (Comptes Rendus, 1861, pp. 795-8). The following proof, which at the same time affords a demonstration of the series, of a simple character, is due to M. Zolotareff :— Let z = X + y ip{z) ; and consider the definite integral Sn = [y<^{u) +■■» - u}"F'{u)du. DijBferentiating this with respect to x, we get, by (33), Art. 114, iJx ^ = ns^,-rr[f{x)}»F'{x). , (47) If in this we make w = i, we get So = yi>{a;)F'{x) + -^; but s„ = F{s) - F{x) ; .■.F{z)^F{x)+y,t,{x)F'{x) + ^. (48) In like manner, making « = 2, we have 2S. = y'[i,[x)rV{x) + '2' dsi dx I .2rfa: L J 1 . 2 dx^ Substituting in (48) it becomes Again, I d'si 1.2 dx^ Gamma Functions. ' d^ dx' w)m<») 150 I d^Sa 1.2.3 ^*^ ' «s«-i = y"{^(«)}"J^(a;) + d'^'s^ d"- .n-i daf^ I .2 .. .11 dx"'^ Hence we get finally dx' {{x)rVix) I d"s„ 1.2 . . . n dx" ' mVF'{x) + &C. . 2 . . . H \dxj [y ^(m) + « - M]"i?"(M) du. (49) Consequently the remainder in Lagrange's series is always represented by a definite integral. "We nest proceed to consider a general class of Definite Integrals first introduced into analysis by Euler. 120. Gaiunia Functions. — It may be observed that there is no branch of analysis which has occupied the atten- tion of mathematicians more than that which treats of Definite Integrals, both single and multiple ; nor in which the results arrived at are of greater elegance and interest. It would be manifestly impossible in the limits of an elementary treatise to give more than a sketch of the results arrived at. At the same time the Gamma or Bulerian Integrals hold so fundamental a ' place, that no treatise, however elementary, would be complete without giving at least an outline of their properties. "With such an outline we propose to conclude this Chapter. The definitions of the Eulerian Integrals, both First and Second, have been given already in Art. 95. The First Eulerian Integral, viz., I «'»-'(! -x)"-'dx, is evidently a function of its two parameters, m and w ; it is usually represented by the notation £(m, n) . 160 Definite Integrals. Thus, we have by definition \ a:*"-' (i - x)"-'clx = B{m, n). [ (50) The constants m, n, are supposed j90s«%e in all cases. It is evident that the result in equation (14), Art. 95, still holds when p is of fractional form. Hence, we have in all cases T{p+i)-pr{p). (51) This may he regarded as the fundamental property of Gamma Fmictions, and by aid of it the calculations of all such functions can be reduced to those for which the para- meter^ is comprised between any two consecutive integers. For this purpose the values of r(p), or rather of log r{p), have been tabulated by Legendre* to 1 2 decimal places, for all values of p (between i and 2) to 3 decimal places. The student will find Tables to 6 decimal places at the end of this chapter. By aid of such Tables we can readily calculate the approximate values of all definite integrals which are re- ducible to Gamma Functions. It may be remarked that we have r(i) = i, r(o)=oo, r{-p)=oo, p being any integer. For negative values of p which are not integer the function has a finite value. Again, if we substitute zx instead of «, where z is a con- stant with respect to x, we obviously have [ e-i^-^clx=^. (53) * See TraiU des Fonctiona EUiptiqiiea, Tome 2, Int. Euler, chap. 1 6. Expression for B{m, n). 161 With respect to the Fj'rst Eulerian Integral, we hare already seen (Art. 92) that [ a!™-'(i - 3;)"-'rfa! = | «»-' (i - x)'"-^dx; .'. B {m, n) = B [n, m). Hence, the interchange of the constants m and n does not alter the value of the integral. Again, if we substitute for x, we get [ar^Hi-x)'^^dx = r , ^""'f^, ■ U ^ ' Jo (i+yr™ ^^"""^ jo (7T^ = "^^"'''')- ^53) "We now proceed to express B {m, n) in terms of Gamma Functions. 121. To prove that B [m, n] = „, ' — ^'. From equation (52) we have r [»») = [ e-^g^a^-'cte. J" Hence r (m) e-" g"-! = f e-' ("^) z™*""' «•»-' dx ; .-. r (ot) f e-" 2"-' rfis = [ I j e-'C") 3'"*"-' ds \ «•"-' (/.p. 162 Definite Integrals. But, if s (i + «) = y, we get I r , , rCm + n) r a;'"-' r(OT) r(«) = rim + n) ;^ — ^ ' ^ ' ^ '' Jo (i +: j\mM»' Accordingly, by (53), we have „, , Vim) Tin) , . B[m, n) = — V^ — -T' (54) ^ ' ' Tim + n) ' Again, if w = i - w, we get, by (24), „/ \ „/ 1 f" x^'^dx w , . Till) r(i - w) = = -. . (55) If in this w = -, we get r(i) = ^. This agrees with (36), for if we make 0^ = z, we get t{'-\ e-^'dx = - e-' z-i dz = — iSZ. (5 6) Jo 2 Jo 2 Again, if we suppose in the double integral x'^^if-^dxdy X and y extended to all positive values, sub]'ect to the condi- tion that X + y h not greater than unity ; then, integrating with respect to y, between the limits o and 1 - x, the integral becomes ip ™,/ s«,j iT{m)Tin+ 1) , , , - ar-^{\ -xYdx = - —^ — 5^ XT, by (54) ; wjo ' n r(OT + w+i) ' ^ ^^^' ff «-, ^,^ ^ T{m)T{n) , , JJ " " T{m + n+i)' ■ ^^" in which x and y are always positive, and subject to the cofl-" dition x + y < 1. Gamma Functions. 1^3 12 2. By aid of the relation in (54) a number of definite integrals are reducible to Gamma Functions. For instance, we have Jo (TT^^ ~ J (I + 2/)""" Ji (I + 2')"""" Now, substituting- for y in the last integral, we get Ji(i +2/r»^Jo(i+^)" ji(,i + y; Jo vi -r-- Hence ^ ^ , ^ r 1 a;'"-' + «"-' r (w) r(?t) 1 aa, - i^!!i±yii, (58) Next, if wo make a; = — , we get ^m+n ' J„(i +.'e)"'^» Jo(a2/ + *)' y'^-'^dy _ r{m) r(w) ]o (ay + &)'""' ~ a"'bT{m + n) Again,* let x = sin'0, and we get 1: [' »»•-■ (i - «)"-' rfaj = 2 [' sin'"^' d cos'"-' 0c?0 ; I This result may also be written as follows ; rf^V^? (59) W-0cos-e.^0 = I^'-?)i^. (60) , 2 r(m + w) sinJ"-' e cos^-' 6d9 = V ~r- (61) * These results may he regarded as generalizations of the fomralEe given in Arts. 93, 94, to wMoh the student can readily see that they are reducible wheo the indices are integers. 164 Definite Integrals If we make q= i, we get sitf-' ede = ^^ y , . (62) 2 jp + 1\ Again, if ^ = y in (61) it becomes ' ^7} = I 'sin?-' fl cosP-' 0d9 = 4-r f 'sia?"' 20rf0. 2r(i?) Jo 2P-'\„ Let 20 = z, and we have IT IT rsin''-'29rf0 = i sin''-'3«?z = Tsin^'-'acfz Jo 2 Jo Jo ^ y^ Kzl If we substitute 2m iorp, this becomes rwr(m + i)=^r(2m). (63) Again, make p = tan'S in (59), and we get n 8in'"^'ecos°"-'0rfg r{m)r{n) J (a sin" 6/ + * cos' 0)'»«' " 2a"' 6" r{OT + «)' ^ "^^ 123. To find the Value* of \n) \nj \nj ' ' ' \ n )' n being any integer. * Tliis important theorem is due to Etder, by ■whom, as already noticed, the Gamma Functions were first investigated. r.,«.^r(i)r(i)r(l).,.r(l^). 165 Multiply the expression by itself, reversing tW order of the factors, and we get its square under the form that is, by (55), sm — sm — sin — ... sin n n n n To calculate this expression, we have by the theory of equations 1 - of IT ,V 2n- A / (w-l)7r ^, I -2a;cos-+a;'' i -2a;cos — +x- )...( i -2a:cos^ '—+X'- 1. n J\ n Making successively in this, » = i,- and x = - i, and re- placing the first member by its true value n, we get / . ttV/ . 27rV ( . {n- Ott V w = 2 sin — 2 sin - — ■ I . . . 2 sm -^ — , \ 2nj \ 211) \ 2n J TrVf 2,rY / (n-i)7r- w = 2 cos — 2 cos — 1 ... 2 cos 2Hj \ 2nJ \ 211 whence, multiplying and extracting the square root, „ , . 77 . 277 . (m - 1)77 n = 2""' sm — sm — ... sm ^—. n n n Hence, it follows that 166 Definite Integrals. 124. To find the values of e'^'^cos Ja;«"'"'fl&, and e''"'sva.hxx"^^dx. If in (52) a - b'^- i be substituted* for z the equation becomes [a -h v^- I (a" + h'Y Let a = (a' + 6°) cos Q, then 6 = (a' + 6^) sin 0, and the preceding result becomes [ e^*0 cos hx + y/- I sin hx)x^'^dx T{m) (cos + y- I sin 0)" = ^—-4; (cos mQ + -y- 1 sm mv). {a- + h'-y Hence, equating real and imaginary parts, we have r (m) e^" COS bxx^''^clx ■■ e-^'siabxxf^'^dx cos md (a' + bj T{m) > (66) sin mO 1 {a' + b'f J in which = tan~M - If we make a = o,B becomes -, and these formulae beoomo 2 * For a rigorous proof of the Tiilidity of tHa transformation the student is referred to Serrett's Calc. Int., p. 1 94. Gamma Functions. 167 cos Ja!®""' dx = , ' cos — , 1 Jo 6'" 2 ' I I • I „ , 7 T{m) . mir \ I sm hxx"^^ dx = ,„ sm — ! r (w) . mjT It may be observed that these latter integrals can be ar- rived at in another manner, as follows : — Erom (52) we have COS hz i *" r (n) — ;^ = e~'"' a;"-' cos bz dx ; „ , . f °° cos J2 & I " r " .-. r(w) ;; — = e-"' COS bzx'^'dxdz. Jo K Jo Jo But, by (32), we have cos bz dz = I g-xs _ Jo — — 6» + a^' f " cos Js & I C " IB" rfiK Jo 2" "T>)Jo s^T^" r(«) WTT 1„/ „x ^ '' 2C0S— , by (27), in which n must be positive and < i. In like manner we find sin bz dz 6""' r in) . WTT ^ ' 2 sm — 2 The results in (67) follow from these by aid of the relation contained in equation (55). 168 Definite Integrals. EZAHFLES. r(,4.)r(— ) 1 a!«-i(i _ a;)»-i(&; r(»j) r(«) (a + a:) rove that f' x^dx [■! ^ Jo(i _ a;4)J Jo(i + g^)\ ly/j IOD 1^ COS (is") ♦'' " o» (I + a)'»r(ffi + n) 3. Prove that fi x^dx r^ dx IT P — JoVi f smbx '■ 1 r(» + i)cos ( M - 1 " f, • dx Y IT \n) )) * » /I i\ dx. X 123. STumerical Calculation of Cramma Func- tions. — The following Table gives the values of log T{p), to 'six decimal places, for all values of p between i and 2 (taken to three decimal places). It may be observed that we have r(ij = r(2) = i, and that for all values of^ between i and 2, t(p) is positive and less than unity ; and hence the values of log T {p) are negative for all such values. Consequently, as in ordinary trigono- metrical logarithmic Tables, the Tabular logarithm is obtained by adding 10 to the natural logarithm. The method of calculating these Tables is too complicated for insertion in an elementary Treatise. JLog ra.) • V 1 2 3 4 5 6 7 8 »l I.OO 9750 9500 9251 9003 8755 8509 8263 8017 7773 I.OI 9-997529 7285 7043 6801 6320 6080 S841 5602 5365 1.02 5128 4892 4656 4421 4<|7 3953 3721 3489 3257 3026 1.03 2796 2567 2338 2110 1883 1656 1430 1205 0981 0775 1.04 0533 0311 0089 9868 «t647 9427 9208 8989 8772 8554 1.05 9-988338 8122 7907 .7692 7478 7265 7052 6841 6629 6419 1. 06 6209 6000 5791 5583 5378 5169 4963 4758 4553 4349 1.07 4145 3943 3741 3539 3338 3138 2939 2740 2541 2344 1.08 2147 195 1 1755 1560 1365 1172 0978 0786 9594 0403 1.09 0212 0022 9833 9644 9456 9269 9082 8900 8710 8525 l.IO 9-978341 8157 7974 7791 7610 7428 7248 7068 6888 6709 I. II 6531 6354 6177 (juco 5825 5^50 5475 5301 5128 4955 1. 12 4783 4612 4441 4271 4101 3932 3764 3596 3429 3262 I-I3 3096 2931 2766 2602 2438 2275 2113 ,1951 1790 1629 1. 14 1469 1309 1 150 0992 0835 0677 0521 0365 0210 0055 I-I5 9.9C9901 9747 9594 9442 9290 9139 8988 8838 8688 8539 1. 16 8390 8243 8096 7949 7803 7658 7513 7369 7225 7082 1.17 6939 6797 6655 '^h'^^ 6374 6234 6095 5957 5818 5681 1. 18 5544 5408 5272 5137 5002 4868 4734 4601 4469 4337 1. 19 4205 4075 3944 3815 3686 3557 3429 3302 3175 3048 1.20 2922 2797 2672 2548 2425 2302 2179 2057 1936 1815 1. 21 1695 1575 1456 1337 1219 IIOI 0984 0867 0751 0636 1.22 0521 0407 0293 0180 0067 9955 9843 9732 9621 95" 1-23 9.959401 9292 9184 9076 8968 8861 8755 8649 8544 8439 1.24 8335 8231 8128 8025 7923 7821 7720 7620 7520 7420 I-2S 7321 7223 7125 7027 6930 6834 6738 6642 6547 6453 1.26 6359 6267 6173 6081 5989 5898 5807 5716 5627 5537 1.27 5449 5360 5273 5185 5099 5013 4927 4842 4757 4673 1.28 4589 4506 4423 4341 4259 4178 4097 4017 3938 3858 1.29 3780 3702 3624 3547 3470 3394 3318 3243 3168 3094 1.30 3020 2947 2874 2802 2730 2659 2588 2518 2448 2379 I-3I 2310 2242 2174 2106 2040 1973 1007 1842 1777 1712 1-32 1648 1585 1522 '459 1397 1336 "75 1214 "54 1094 1-33 1035 0977 0918 0861 0803 0747 0690 0634 0579 0524 1-34 0470 0416 0362 0309 0257 0205 <>iS3 0102 0051 0001 I-3S 9-949951 9902 9853 9805 9757 9710 9663 9617 9571 9525 1.35 9480 9435 9391 9348 9304 9262 9219 9178 9136 9095 1-37 9054 901S 897s 8936 8898 8859 8822 8785 8748 8711 1.38 8676 8640 8605 8571 8537 8503 8470 8437 8405 8373 1-39 8342 8311 8280 8250 8221 8192 8163 813s 8107 8080 1.40 8053 8026 8000 7975 7950 7925 7901 7877 7854 7831 1.41 7808 7786 7765 7744 7723 7703 7683 7664 7645 7626 1.42 7608 7590 7573 7556 7540 7524 7509 7494 7479 7465 1-43 7451 7438 7425 7413 7401 7389 7378 7368 7357 7348 1.44 7338 7329 7321 7312 7305 7298 7291 7284 7278 7273 I -45 726/ 7263 7259 7255 7251 7248 7246 7244 7242 7241 1.46 7240 7239 7239 7240 7240 7242 7243 7245 7248 7251 1.47 7254 7258 7262 7266 7271 7277 7282 7289 7295, 7302 1.48 7310 7317 7326 7334 7343 7353 7363 7373 7384 7395 1.49 7407 7419 7431 7444 7457 7471 7485 7499 7514 7529 Logr(p). Examples. Examples. |„ / • Ans. z\/a. Jo Y a-a; ^ 2. If /(a!) =/(» + *) for all values of x, prove that I»a ra f{x)dx = n I f{x)dx, where « is an integer. dx ■ i: ■ i: yax - 3;* dx x-y x^ — I 5. I sin-' a; «?ar. J •'»(i+a;)v/l + 2a;-a;2 " i,^'2 f " 0. „ . J-i (a-*a;)A/l-ic2 -/ a^ - h^ {a — bx) »y \ — x^ dx 15- - , i<^-/(x-a)(P-x) , , f sin (7a; COS te , ir ,. ^, , 17. Show that 1 dx = -, or o, aocording as o > or< *; and io X 2 that when = i the value of tiie integral is -. 4 r*i (?x , > 1 , A + V'aA 18. «i<«»■—- — log —.]■ J.i\/(i-2«a; + «2)(l-2ix + A«)' y«4 \i-'/ab) n 19. r tan' a; (St. ,> 2\S'~2)' I + cos^a; 4 y/j 21. If every infinitesimal element of the side e of any triangle be divided hy its distance from the opposite angle G, and the sum taken, show that its value is logf cot— cot — j. 22. Being given the base of a triangle ; if the sum of every element of the base multiplied by the square of the distance from the vertex be constant, show that the locus of the vertex is a circle. r^ cos' fl sine or < i . Ans. a < I, its value is o. „ B > I, its value is ztt log «. 29. lif(^x) can be expanded in a series of the form flo + «i cos a; + «2 cos 2a; + . . . + a„ cos «a: + . . . , show tbat any coefScient after Oq can be ezHbited in the form of a definite integral. 2 fir Ans. 8n = — I /Mcosnxdx. 30. Find the analogous theorem when/(a:) can be expanded in a series of sines of multiples of x ; and apply the method to prove the relation (sin 2a; sin 3a; , \ sin a; + &o. 1 , as/' when X lies between + jr. 31. Prove the relation "V^sinfl 32. Express the definite integral f? de n . /-:—-^\.Ysmede = 7r. •'0.%/ smfl JO ' fa" de •''' V I — K-sin *$\ 174 Definite Integrals. ri Iog(i + cosacosa;)rfg ^^ Iful^aA ^^' Jo cosa; ' "2X4 / I" rt' — 4' -i!e-"»cos terfa;, where a > o. „ . „ .j r^- 36. ("log (a' cos' 9 + J8« sin' 6) (?9. „ ir log ^-— . Jo " ■ J (i _ aS)*' " 3 i: 39- (I - x")' « sm - f coa rxdx ■"«■'' I — 2BC0sa + a' 41. Find the sum of the series n n n I + __« . _9 + -.9. , -2 + • • ■ + ~~2* •when n is increased indefinitely. This is evidently represented hy the definite integral !^ dX IT J I + »- 4 42. Find the limit of the sum I I , - + . + , ■(■■..+ . "when » = oc, ^ns. — . Examples, 175 43. Prove that IT IT fa" , m(m- r) [2 „ _ 1 cos™* cos nx dx = —5 ^ I cos"'-" x cos nxax ; Jo m^ - »M and hence, deduce the values of the integrals I" cos2»'a; cos(2« + i)a;(?j;, and I co3'"'+' a; cos 2«3; <&;, when m and n are integers. 44. I log(i - 2a oosfl + a') cos«fl«!9, when a'' < I. Am. Jo f" '^^^ 45. 1 cos — a^. „ 1. J -« 2 47. Prove the following equation : ; -„ ^ = , Tr-\ (i - 2a cose + a'')''-'rf9. Jo (I - 2a cos 8 + «»)» (i - ffi2)»-i Jo ' 48. Prove the more general equation fir sin^flrffl _ ' ["■ Jo (i - 2a cos 9 + a^)" ~ U - a^)*"-"*-! J o (i - : sva.'"9dB (i - 2a cos 9 + a^)n (i - b2)2»-">-i Jo (i - 2a cos 9 + a2)i*">-»' in which »j + i is positive. ( 176 ) CHAPTEE VII. AKEAS OF PLANE CURVES. 126. Areas of Curves. — The simplest method of regarding the area of a curve is to suppose it referred to rectangular axes of co-ordinates; then, the area included between the curve, the axis of x, and the two ordinates corresponding to the values x^ and Xi of x, is represented by the definite integral r ydx. For, let the area in question be represented by the space ABVT, and suppose 5 F divided into w equal intervals, and the corresponding ordinates drawn, as in the accompanying figure. Then the area of the portion PMNQ is less than the rectangle pMNQ, and greater than PMNq. Hence the entire area AB VT is less than the sum of the rectangles represented hypMNQ, and greater than the sum of the rectangles P3INq ; but the difference be- tween these latter sums is the sum of the rectangles Pp Qg, or (since the rectangles have equal bases) the rectangle under MJV and the difference between TV and AB. Now, by supposing the number n increased indefinitely, MJV can be made indefinitely small, and hence the rectangle MIf {TV - AB) also becomes infinitely small. Consequently the difference between the area ABVT and the sum of the rectangles PMNq becomes evanescent at the same time. Areas of Curves. 177 If now the co-ordinates of P be denoted by x and t/, and M]}f by A;», it follows that the area AS VT is the limiting value* of S(y Ax) when the increment A* becomes infinitely small ; pi or area ABVT = ydx; where «! = OV,Xd= OB. Ja;o It should be observed that this result requires that y continue finite, and of the same sign, between the limits of integration. . If y change its sign between the limits, i.e. if the curve cut the axis of x, the preceding definite integral represents the difference of the areas at opposite sides of the axis of x. In such cases it is preferable to consider each area sepa- rately, by dividing the integral into two parts, separated by the value of x for which y vanishes. The preceding mode of proof obviously applies also to the case where the co-ordinate axes are oblique ; in which case the area is represented by f»i sm w y dx, where oi represents the angle between the axes. In applying these formulse the value of y is found in terms of x by means of the equation of the curve : thus, if y =/{x) be this equation, the area is represented by \/{s!)dx, taken between suitable limits. Conversely, the value of any definite integral, such as - jyix)dx, may be represented geometrically by the area of a definite portion of the curve represented by the equation y ==/W- * This demonstration is substantially that giyen by Newton (see Prineipia, Lib. I., Sect, i., lemma 2) ; and is the geometrical representation of the result establi^ed in Art. go. The modification in the proof when the elements of SV axe considered unequal, but each infinitely small, is easily seen. It may be remarked that the result here given is but a particular case of the general principle laid down in Arta. 38, 39, Dif. Cale. [12] 178 Areas o/Flane Curves. On account of this property the process of integration was called, by Newton and the early writers on the Calculus, the method of quadratures. Again, it is plain that the area between the curve, the axis of y, and two ordinates to that axis, is represented by ^xdy, taken between the proper limits : the co-ordinate axes being supposed rectangular. We proceed to illustrate this method of determining areas by a few applications, commencing with the simplest examples. 127. The Circle. — ^Taking the equation of a circle in the form x^ + y'' = a^, we get y = ^/c? - »% and the area is represented by c? taken between proper limits. For instance, to find the area of the portion represented by APBE in the accompanying figure. Let x = a cos, 0, then the area in ques- tion plainly is represented by Fis. a" sin' = — (o - sin a cos a) ; where a = L DO A. This result is also evident from geometry ; for the area BPAE is the difference between DPAC and BCE, or is a' a fis'sma cos a The area of the quadrant ACB is got by making a = - ; and accordingly is — : hence the entire area of the circle is jra'. The Ellipse. 179 128'. The Ellipse. — From the equation of the ellipse I + 1 = I, we get'y = - y^f:^J^^ and tte element of area is b . h but this is - times the area of the corresponding element of a the circle whose radius is a : consequently the area of any portion of the ellipse is - times that of the corresponding part d of the circle. This is also evident from geometry. The area of the entire ellipse is irab. — Again, if the equation of an ellipse he given in the form ttC A^ + By^ = C,. its area is evidently As an application of ohlique axes, let it he proposed to find the area of the segment of an ellipse cut off by any chord Djy. Draw the diameter AA', con- jugate to the chord, and B^ parallel to it. Then, C being the centre, let CA' = a', CB' = b',ACB' = o,, and the equation of the ellipse is -^ + ^ = i ; hence the area DA'iy is represented by h' . fc^' /- 2 — sin w va'^ - a?dx- a'b' sin w (a - sin o cos a), a i CE CE where cos a = -r—,. CA Again, a' J' sin w = ab, by an elementary property of the ellipse, a and 6 being the semiaxes. Hence the area of the se^ent in question is ab{a - sin a cos a). [12 a] 180 Areas of Plane Curces. This restilt can also be deduced immediately from the circle by the method of orthogonal projection. It may be observed that if we denote the area of an elliptic sector, measured from the axis major to a point ■whose co- ordinates are x, y, by S, we may write X ZS ?/ . 2(S - = cos —r = cos a, a ab b ab sma. 129, The Parabola. — Taking the equation of the parabola in the form y* = px, we get y = vpx. Hence the area of the portion APN is r 2,2 p^ a^dx, OT- pix^fi.e. — oBi/. Consequently, the area of the seg- ment PAF, cut ofE by a chord perpen- dicular to the axis, is \ of the rectangle TMM'F. It is easily seen that a similar relation holds for the seg- ment cut off by any chord. More generally, let the equation of the curve be y = oai", where n is positive. Fig 4. Here ydx = a x^dx ■■ ax" + const. « + I If the area be counted from the origin, the constant vanishes, and the expression for the area becomes aa^*'^ xy --— , or -^—. » + I w + I Hence, the area is in a constant ratio to the rectangle under the co-ordiuates. A corresponding result holds for oblique axes. The discussion, when n is negative, is left to the student. Example. Express the area of a segment of a parabda cut ofl ty any focal chord in terms of I, the length of the chord, and^, the parameter of the parabola. Am. -g-. The Hyperbola. 181 130. The Hyperbola. — The simplest form of the equation of a hyperbola is where the asymptotes are taken for 00-ordinate axes ; in this case its equation is of the form xy = , the area between the curve and an asymptote is denoted by & sin a> — , or & sin w log ( — ), where Xi and jSo are the abscissse of the limiting points. If the curve be referred to its axes, its equation is X' y and the element of area ydx becomes - '/(^ - aj^dx. a Hence the area is represented by - '/x' - c^dx, taken between proper limits. v'a Wa^ - Also, integrating by parts, we have v^a;" - d'dx = x ^a? -d^ -\ — . Adding, and dividing by 2, we get 1 '/i^ — a^dx = -/«'- dx Yx' -a' •/x X-J X' ffl «% , /— log (« + v' iB - a'). 182 ^ Areas of Plane Curves. Accordingly, if we suppose the area counted from the summit A, we have ^Pi\r = A ^ y^rzi^ _ ^ log f^±^^?I?) 2a ^ 2 ^ V « ^ = *l^_^^ogf%f' 2 2 °\a b Again, since the triangle CFIf = ^a^, it follows that sector ^^P = ^ log g+|\ For a geometrical method of finding the area of a hyper- holic sector, see Salmon's Conies, Art. 395. 130(a). Hyperbolic Sine and Cosine. — If 8 repre- sent the sector ACP, the final equation of the preceding Article becomes which may also be written « '/ - - + T = « ' a introducing a single letter v to denote the quantity , /« y\ 28 ^'^[a-'D^vr'- Hence, by the equation of the hyperbola, we get __ ;L — p-^ 1. ~ " ' a Thus, in analogy with the last result of Art. 128, calling the following functions the hyperbolic cosine and hyperbolic sine of v, and for brevity writing them cosh v, and sinh v, e* + e"" = 2 coshi), e* - r" = 2 sinht;, (2) the co-ordinates of any point on the curve are - = cosh V = cosh — r, T- = sinh v = sinh —r. a ab ab The Catenarrj. 183 We might have treated the matter differently By intro- ducing the angle defined by the equation » = a see , and therefore y = b tan (for the geometric meaning of this transformation, see Salmon's Conks, Art. 232) ; whence (i) may be written* 28 . . (^ —r=v= log tan - + ab ° \4 and we see that the hyperbolic cosine of a real quantity is the secant, and the hyperbolic sine the tangent of the same real angle. Also, since sinh V I.I cosh v sind» = — ^ — , cos i4 = — ^ — , cot d» = —:—, — , cosec 4) = -;-T — , cosh V cosh V smh v "^ smh v we can obviously extend the names of the other trigonometrical functions likewise. Again, putting in (2) for v, uy - i, or iu, they become, by Art. 8, cos u = cosh iu, i sin u = sinh iu. 131. The Catenary. — If an inelastic string of uniform' density be allowed to hang freely from two fixed points, the curve which it assumes is called the Catenaiy. Its equation can be easily arrived at from elementary mechanics, as fol- lows : — Let V be the lowest point on the curve; then any portion VP of the string must be in equilibrium under the action of the tensions at its ex- tremities, and its own weight, W. Let A be the tension at F; T that at P, which acts along PB, the tangent at P; lPRM = 0. Then, by the property of the triangle of force, we have W:A^PM:RM; * Wten ^ is lelated to « by this equation, (j> is what Professor Cayley {Elliptic Functions, p. 56) calls the gudermannian of v, after Professor Guder- mann, and writes the inverse equation tj> — gdv. 184 Areas of Plane Curves. Again, if s be the length of VF, and a that of the portion of the string whose weight is A, we have, since the string is uniform, W^a'-; .'. s = a tan0. This is the intrinsic equation of the catenary. Calc, Art.' 242 (a).) *. Its equation in Cartesian co- . V ( ordinates can be easily arrived at. ^ For, on the vertical through V take VO = a, and draw OX in the horizontal direction, and assume OX and OP" as axes of co-ordi- nates. Let (Diff. then PN=y, OIf = x, dx = tan (p, rig. 7. di/ = sm 0, d-x ds = cos ; sin rf> dx Hence dy dy dt d(p ds d^ cos'0' dcj) cos(j>' y = a seotp, x = a log (sec + tan (f). (3) No constant is added to either integral, since y = a, and X = o, when = 0. From the latter equation we get also sec

df, the integral of which gives the same result as before. In like manner, to find the area of the curve (0 .;^f = - Let x = a sin=0, then y = b cos'^, and ydx becomes 3aisin^0 cos* (j>d(^ : The Cycloid. hence the entire area of the curve is represented by 189 Jo \zab\ aiTx'(j> COS* fdf = -irab.. Examples. Find the whole area of the evolute of the ellipse ^ + 75 = 1- Ans. 37r(a2 - «2)2 2. Find the whole area of the curve 0^"^^-(l) 2 2llH 2.4.6 2{m + n+ 1) 134. The Cycloid. — In the cycloid, we have (Diff. Oalc, Art. 272), a; = a (S - sin 6), y = a (i - cos 0) ; (i - cos0)^rf9 = 4a^ [ ■ .0 ydx = a" sm* ■de. Taking B between o and tr, we get 3n-a' for the entire area between the cycloid and its base. The area of the cycloid admits also of an elementary geometrical deduction, as follows : — It is obviously sufficient to find the area between the semicircle BPB and the semi-cycloid BpA. To determine this, let points P and P' be taken on the semicirolo such that arc BP = arc JDP' : draw MPp and M'P'p' perpendicular to BB. Take MN and M'N' of equal length, and draw Nq and N'4i also perpendicular to BI): then, by the fundamen- tal property of the cycloid, the line Pp = ^re BP, and P'p = arc BP" : :. Pp -v P'p' = semicircle = ttb. 190 Areas of Plane Curves. Now, if the interval MN be regarded as indefinitely small, the sum of the elementary areas PpqQ dio.i.P'p'q'Q' is equal to the rectangle under MN and the sum of Pp and P'p', or to ira X MN. Again, if the entire figure be supposed divided in like manner, it is obvious that the whole area between the semi- circle and the cycloid is equal to Tta multiplied by the sum of the elements MN, taken from B to the centre 0, i.e. equal to jra". Consequently the whole area of the cycloid is Srf, as before. The area of a prolate or curtate cycloid can be obtained in like manner. 135. Areas in Polar Co-ordinates. — Suppose the curve APB to be referred to polar co-ordinates, being the pole, and let OP, OQ, OJS represent consecutive radii vectores, and PL, QM, arcs of circles described with as centre. Then the area OPQ = OPL + PLQ; but PLQ becomes evanescent in com- parison with OPL when P and Q are infinitely near points; conse- quently, in the limit the elemen- r''dS tary area OPQ = area OPL = ; r and being the polar co-ordi- nates of P. Hence the sectorial area AOB is represented by r'de. where a and |3 are the values of corresponding to the limit- ing points A and B. 136. Area of Pedals of Ellipse and Hyperbola. — For example, let it be proposed to find the area of the locus of the foot of the perpendicular from the centre on a tangent to an ellipse. Writing the equation of the ellipse in the form the equation of the locus in question is obviously r' = fl^cos^fl + ¥ ain'O. a^ ¥ = 1, Area of Pedals of Ellipse and Hyperbola. 191 Hence its area is cos'ede + - siu'edd = =^-^ e + —^ sine cos 0. 2 J 44 _ 2 The entire area of the locus is 2 ^ ' The equation of the corresponding locus for the hyperbola is }•» = a' eos'O - W ein'(?. In finding its area, since r must he real, we must have a'coa'd - 6" sin" positive : accordingly, the limits for are o and tan"^T. b Integrating hetween these limits, and multiplying by 4, we get for the entire area ab + Uf - 6") tan"' 7. In this case, if we had at once integrated between = o and 6 = 27r, we should have found for the area (a' - 6") -. 2 This anomaly would arise from our having integrated through an interval for which r^ is negative, and for which, therefore, the ;oorresponding part of the curve is imaginary. The expression for the area of the pedal of an ellipse with respect to any origin will be given in a subse(iuent Article. Examples. 1. Show tliat the entire area of the Lenmiscate r' = a'^ cos 26 is a^ 2. In the hyperhoUc spiral re = a, proye that the area bounded by any two radii veetores is proportional to the difference between their lengths. 3. Find the area of a loop of the curye »•' = b' cos nB. Am, —. n 192 Examples. 4. Find the area of tie loop of the Folium of Descartes, whose ejuatioa ia a? + yi = ^axy. Transfoiming to polar co-oidinates, ve have 3» cos fl sin 9 »• = . Bin's + cos' 9 Again, the limiting values of 9 are o and - ; •. Area _ 9®' f ^ 9«' f 2 sin' 9 cos' 9 tf9 j„ (sin' 9 + cos' 9)'' Let tan 9 = », and this expression becomes <)d? f " u^du _ 30^ ~]a. In this case the curve consists of one loop, and its area ia I ("2^ / - (^\ [a cos 9^+ J)2rf9= (42 + -J IT. When i = a, the curve becomes a Cardioid, and the area . (2). Let h ^'^^ *'^® entire area is f"(«cos9-i)V9 Jo = f («" cos' 9 - 2aJ cos 9 + b'^Jde . pig. j j, — + J' J a H sin o cos o - 2ah sin a = (^,,2)eos->^-|,,/,-ar^. Area of a Closed Curve hy Polar Co-ordinates. 193 It is easily seen that the sum of the areas of the two loops is ottained by in- tegrating between the limits o and 2x, and accordingly is (!-)■ as in the former case. 137. Area of a Closed Cnrve by Polar Co-ordi- nates. — In finding the whole area of a closed curve hy polar co-ordinates we distinguish between two cases. Wlien the origin is outside, we sup- pose tangents OT, OT', drawn from 0, and vectors OT, OQ, &o., drawn to cut the curve ; then, if these lines intersect it in but two points each, the element of area PpqQ is the difference between the areas POQ and^O^'; or, in the limit, is ^ (n" - r^') dO, where OP = n, Op = r,. Hence, the expression if{r^-r,')dd, Fig. 14. taken between the limits corresponding to the tangents OT and OT', represents the entire included area. If the origin lie inside the curve, its whole area is in ge- neral represented by ^Kri' + ri)dO, taken between the limits 0=0, and 6 = TT. We shall illustrate these results by applyiag them to the circle r* - 2rc cos + c" = a'. If the origin be outside, we have c>a, and ri + ri = 2c cos 9, ■and TiT^ = c'^ - a' ; • r, - j-j = 2 '^/a' - c' sin^ 0. Hence (n* - r/jdO = 4c cos fly^a^ - c* sin' tidO; and the limiting values of 6 are ± sia"'-. Hence the whole area is cos v^a" - [13] c" BUT' Odd. 194 Areas of Plane Curves. Let c sin = a sin ; then PN= PL + AN', ('-= ds + od(^. Fig- n- QT Amsler^s Planimeter, 215 Now, if we suppose AB after a complete circuit of the curve to return to its original position, we have oTjTiously S {d^) = o ; and therefore S {ds') = 2 {ds),i.e. the whole length registered by the revolving wheel at P is the same as if it were placed at A. Next, let X and y he the co-ordinates of B with respect to rectangular axes drawn through G, and let ^C= a, AB = b, L ACX= d; and suppose the angle which BA produced makes with the axis of co ; then we shall have X = a COS + b COS ^, i/ = a sind + b sin ). But + (p = 2d-{9-) = 2bds - ab cos (9 ~ ). Consequently xdy-ydx = a^dO-\- b'd(j)+2bds-ab cos [6 -(l>)d {6 -(j>). But, hy Art. 139, the area traced out by 5 in a complete revolution is represented by -J {xdy-ydx) taken around the entire curve. Also, since AO and AB return to their original positions, the integrals of the terms a^dQ, b'^dcp and ab cos (0-^)d{0- i' and that the whole area of the locus is IT o'S ^'/dU^l^ + t 18. Apply the three methods of approximation of Art. 148 to the calculation 1^ dx I , adopting — as the common I +ic 12 interval in each case. Ans. (i), .693669. (2), .693266. (3), .693224. The 7-eal value of the integral being log 2, or .693147, to the same number of decimal places. 1 9. Prove that the sectorial area bounded by two focal vectors r and r' of a parabola is represented by 3" where c is the chord of the arc, and the semiparameter of the parabola. (('-4^r-{'-^')')' Examples. 221 20. Show that the whole area of the inverse of the ellipse -„- + -jr, = i ia- a' 0' represented by I _f!_^'V(«= *2 \o^ b'-j w 4-/)' where a, p, are the co-ordinates of the origin of inversion, and k is the radius of the circle of inversion. 51. A given arc of a plane curve turns through a given angle round a fixed point in its plane ; what is the area described f 22. Given the base of a triangle, prove that the polar equation of the locus, of its vertex, when the vertical angle is double one of its base angles is a{2 cos 39 + l) 2 cos B Hence show that the entire area of the loop of the curve is 3«_V_3_ 4 23. is a point withiu a closed oval curve, F any poiut on the curve, QPQ' a straight line drawn in a given direction such that QP = FQ' = PO ; prove that as P moves round the curve, Q, Q', trace out two closed loops the sum of whose areas is twice the area of the original curve. Camh. Trip. Exam., 1S74. 24, Prove that the area of the pedal of the cardioid r = a(l - cos fl) taken, with respect to an internal point at the distance c from the pole is —■ (Sa8 - 2ac + 202). {Ibid., 1876.) 25. The co-ordiiates of a point are expressed as follows : y + i' " fl3 + I ' -/v 4- ^ - 3 A ^ find the equation of the curve described by the poiut, and the area of the portion of the plane iaolosed thereby. ( 222 ) CHAPTEE VIII. LENGTHS OF CURVES. 150. IJength of Curves refierred to Rectangular Axes. The usual mode of eonsideriiig the length of a curve is by treating it as the Hmit of a polygon when each of its sides is infinitely small. If the curve he referred to rectangular axes of co-ordinates, the length of the chord joining the points {x, y) and {x+ dx, y + dy) is ^dx^ + dy"r, and, consequently, if s represent the length of the curve measured from a fixed ipoint on it, we shall have ds = ^/daP' + dy', or, integrating, taken between suitable limits. The value of — in terms of x is to be got from the equa- tion of the curve, and thus the finding of s is reducible to a question of integration. The determination of the length of an arc of a cm-ve is called its rectification. It is evident that if y be taken for the independent variable we shall have J Again, when x and y are given functions of a single va- riable (p, we have s = l(S)MI)T* In each case the form of the equation of the curve deter- mines which of these formulae should be employed. The Catenary. 223 The curves whose lengths can be obtained in finite terms (compare Art. 2) are very Hmited in number. We proceed to consider some of the simplest cases. 151. The Parabola. — ^Writing the equation of the parabola in the form «* = zmx, we get -r- = — . ° ay m Hence s = ■ Vy'' +m'^dy. The value of this integral can be obtained from that of the area of a hyperbola (Art. 130), by substituting y for x, and ni^ for - a^. Thus we have y^y'^ m' m fy + ^y'^ + ni' zm 2 °\ m (2) the arc being measured from the vertex of the curve. 152. The Catenary. — The equation of the catenary (Art. 131)' is y=l(f-,i^y ^^- Hence dx 2^ f . dy-\i I dx \ dx s =- 2 i = - e" + e e''dx + - 2 e'^\ If s be measured from the vertex V, we have a I the same result as already arrived at in Art. 131. Again, since PL = P V, and NL is constant, it follows that the catenary is the evolute of the tractrix (see Ex. 9, p. 219). 224 Lengths of Curves. 153. Semi-cubical Parabola. — The equation of this curve is of the form ay- = «'. , x^ dy 3fx\i cIs f g^Y I a^'' ' ' dx z \aj ' dx \ /^a) ' 1+^-] dx= — i+— + const. i\aj 27 \ ^aj If the arc he measured from the vertex, we get 27 (V 4«/ The semi-cuhical parahola is the first curve whose length was determined. This result was discovered by William Neil, in 1660. 154. Rectification of Evolntes. — It may he noted that the rectification of the semi-cubical parabola is an immediate consequence of its being the evolute of the ordinary parabola (see Diff. Calc, Art. 239). In like manner the length of any curve can be found if it be the evolute of a known curve, from the property that any portion of the arc of the evolute is the difference between the two corresponding radii of curvature of the curve of which it is the evolute. For example, we get by this means the lengths of the cycloid, the epicycloid and the hypocycloid. Again, since the equation of the evolute of an ellipse is {ax)l + {hy)l = {a' - b% the length of any arc of this curve can be at once found. This can also be readily got otherwise ; for, writing the equation in the form and making x = a sin'0, we get y = (5 cos^j and ds = {dx' + dy^)i = 3 sin cos (p{a^ sin'0 + j3- eosi')* If the arc be measured from the point x ~- o, y = ^, we get the constant - /3' , _ (a'sin'0 + ^'cos'^)^-ffl ~ IF^^' ^""^ ' " ■ ^^"^ • If a = /3, the expression for ds becomes 3a sin cos ^ hence we get s= - a sin'^, the arc being measured from the same point as above. Examples. 1. Find the length of the logarithmic curve y — ca". Here log « = a!log« + logo; . •. -7- = -, where 4 = , . dy y' loga Hence ,_ f (^' + i^')i^y _ f ydy f V>dy J y J(4' + 2^')J J^(4' + J^')t 2. Find the length of the traotrix. Here, hy definition (see fig. 26), we have FT= a ; . „»,,, y , ds a .: sin FT2f = -, hence — = - - ; a dy y .: s = - a\ — =— a log y + const. If the arc he measured from the vertex A, we get arc AP = a log ( - J • 3. Find in what cases the curves represented by a»'y» = x"*« are rectifiaUe. Here we have [15] 226 Lengths of Curves. (m + «)'' ^ Substituting b for ^ — ^J , and making I + bx" = z-, this becomes — I \ S"! z^dz. This expression is immediately integrable when — is a positive integer. w Hence, if — = r, we see that curves of the form ay^ = a''^' are rectifiable. 2»» Again, if ^ — be a negative integer, the expression under the integral sign becomes rational, and can accordingly be integrated. This leads to tiiie form y'i' = ax''~^. Accordingly, all curves comprised in the equation ay" = it™" are rectifiable, m being any integer. (Compare Art. 62). 155. The Ellipse. — The simplest expression for the arc of an ellipse is obtained by taking « = a sin 0, whence y = b 003 (j>, and ds = (as' cos'^ + 6" sin'^)* dip ; .-. s = («' cos'0 + 5" Bixi?^)id^. It is often more convenient to write this in the form {1 -e'^ bid} ^)id, (3) : = «|(i e being the eccentricity of the ellipse. It may be observed that ^ is the complement of the eccen- tric angle belonging to the point («, y). The length of an eUiptio quadrant is represented by the definite integral n (i - e" sbx' (j>)i d(j>. We postpone the further consideration of elliptic arcs to a subsequent part of the Chapter. 156. Rectification in Polar Co-ordinates. — If the curve be referred to polar co-ordinates we plainly have (Diff. Oalc, Art. 180) rfs" = dr^ + r^dO'^ ; hence we get I r' + -^J d9, or s = I ( I + -^j dr. (4) Rectification in Polar Co-ordinates. 227 For example, the lengtli of the spiral of Archimedes, r = aO, is given by the equation •=-[{r^ + a^)hdr. Comparing this ■with the formula (2) for the parabola, it follows that the length of any are of the spiral, measured from its pole, is equal to that of a parabola measured from its vertex. EZAIIFZES. 1. Cardioid, »• = «(i + cos 9). dr Here — = — a sin 8, and hence 6 6 * = a J" {{I + cosfl)' + Bii>?e}ide = 23 f cos - (?« = 4a sin - + constant. 2 2 The constant hecomes zero if -we measure « from the point for which B = 0. 2. Logarithmic spiral, r = d9. Here, if i = = , -we get log a' rde fi — = J; •■'' = j^ (I +*»)!"» = ffi"* cos mB. Taking the logarithmic differentials, we get —^ = — tan )»9 ; ds ,'. -^r: = sec me. rdd Hence Or, writing ^ for niB, I (C0SJB9)"' dB. (cos ^) dip. This is readily integrated when — is an integer (see Art. 56). [15 a] 228 ' Lengths of Cimes, Whatever be the value of m, we can express the complete length of a loop of the curve in Gamma Functions. For if we integrate between o aaid -, we ob- viously get the length of half the loop. Hence the lengtii of the loop (Art. 122) is i 157. Formula of liegendre on Rectification. — Another formula* of considerable utility in rectification fol- lows immediately from the result obtained in Art. 192, Diff. Calo. For, if this result he written in the form -^ — - =p,-weg6ts-t= Ipdw. (5) Consequently, the total increment oi s -t between any two points on a curve is equal to ^pdu) taken between the same two points. For example, in the parabola we have p = , and ^ ^ COS(i> hence s ~t = a = a log tan ( - + - 1 + const. J COS (L« ° \4 2 y If we measure the arc from the vertex of the curve, and observe that t = -r, this gives aw a smcu s= r— +fl cos w log tan (^.h^). The student can without difficulty identify this result with that given in Ait. 151. * This theorem is due to Legendre. See Traiti des Fonctima ElUptiques, tomeii., p. 58S. FagnanVs Theorem. 229 It should be observed tliat when the curve is closed, its whole length is, in general, represented by r2ir pdw. Equation (5) furnishes a simple method of expressing the intrinsic equation of a curve, when we are given its equation in terms of ^ and w. For, if ^ =/(w) we have dp d(o pdco =f'{bi) + /(a») dw, (6) taken between suitable limits. 158. A^plieation to Ellipse. Fagnani's Theorem. In the ellipse we have p- = cf cos'w + V' sin^'w. Hence, measuring the arc from the vertex A, and observ- ing that in this case FN\i> to be taken with a negative sign, we have arc ^P + PiV = Fig. 28. (ffl" cos'w + J" sin'wjMw, where o = lACN. But, in Art. 155, we have found that if be measured from the vertex B, the arc is represented by (a' cos"^ + V- sin''^)4c?0. Consequently, if we make L BCQ = a = I ACN, and draw QM perpendicular to the axis major meeting the curve in P', we shall have arc BP' = arc AP + PN, or, taking away the common arc PP, BP-AP' = PjV. (7) 230 Lengths of Curves. This remarkable result is known as Fagnani's Theorem*, and shows that we can in an indefinite number of ways find two arcs of an ellipse whose difference is expressible by a right line. We add a few properties connecting the points P and P' in this construction. Examples. 1. If (a:, y) and (»', y') Is the co-ordinates of P and F', respectively; prove the following : — (i). PJf = — , (j). PN= FN', (sr. CN. CN- = CA . CB, (4). cp2 + car'2 = CA^ + CB2 = cp'^ + gn^. 2. Divide an elliptic quadrant into two parts whose difierenoe shall be equal to the difierence of the semiazes. This takes place when F and F' coincide ; in which case CN= f/ab, and PN=a-b. ■We shaU designate the point so determined on the elliptic quadrant as Fag- nani's point. 3. Show that if a tangent he drawn at Fagnani's point, the intercepts between its point of contact and its points of intersection with the axes are respectively equal in length to the semi-axes of the ellipse. 4. If the lines PJTand P'N' be produced to meet, show that they intersect on the conf ocal hyperbola which passes through the points of intersection of the tangents to the ellipse at its vertices. Show also that this hyperbola cuts the ellipse in Fagnani's point, * Fagnani, Giomale de' Letterati d' Italia, 17 16, reprinted in his Prodmioni Matematiche, 1750. It may he noted that if we integrate the equation of Art. 1 16, Diff. Calc, taking the angle C as obtuse, and adopting zero for the lowest limit in each integral, we obtain •\/i - A* sin'^a*?* + 1 \/ 1 - Ifi sin'''* dh — \ -v/i - &^sin' is measured helow the axis. PN-AP = (a' cos' to - 6''sin'tu)'^(?w, where a = I. ACN. As we proceed along the hyperbola the perpendicular p diminishes, and vanishes whea the tangent becomes the asymptote. Moreover, as the limit of w in this case becomes tan"' -r, it follows that the difference between the asymptote and the infinite hyperbolic arc, measured from the vertex, is represented by the definite integral ftan-i *(o''cos'(j - Vsi.v?w)idu). Examples. 1. If « > i, prove that J(a + Acos0)W(/> is represented by an elliptic arc, and that the semiases of the ellipse are the greatest and least values of (a + i cos ^)". 2. If a < i, prove that J (a + i cos ^)i d<^ is represented by the difference between a right line and a hyperbolic arc. 232 Lengths of Curves. 1 60. Iianden's Theorem on a nyperbolie Arc. — We next proceed to establish an important theorem, due to Landen ;* namely, that any arc of a hyperbola can he expressed in terms of the arcs of two ellipses. This can be easily seen as follows : — ^In any triangle, adopting the usual notation, we have c = aco&B + bcosA. Now, representing by C the external angle at the vertex C, we have C == A + £, and hence cdC = {acosB + b cos A) dA + (a cos5 + 6 cos^) dB. Consequently, supposing the sides a and h constant, and the remaining parts variable, we have \cdC = aoosBdA + b cos AdB + zasinB + const., or {'/a' + 6' + 2a6cos CdO= l^a'-b^ sin' A dA+Wb' - a^sin^B dB + lasinB + const. (9) Now, if we suppose a>b, ^/a' - V sin^A dA represents (Art. 155) the are of an ellipse, of axis major 2a and eccen- tricity -. Also -yb^ - a' sm^BdB represents (Art. 159) the difference between a right line and the arc of a hyperbola, whose axis major is b and eccentricity 7. Again, y^a^ + 6^ + zah cosC = J{a- bysiD!'-+ {a + b)'' cos^— , * Landen, Fhilosophical Transactions, i']TS i siso, Mathematical Memoirs, 1780. Landen's Theorem on a HyperloKc Arc. 233 and consequently the integral y^a^ + V + lab cos CdG represents an arc of the ellipse whose semiases are a + 5 and a — h. Hence, Landen's theorem follows immediately. It should be noted that the limiting values of A, B and C are connected hy the relations a sinB = b mxA, and = A + S. Again, if we suppose the angle A to increase from o to it, the external angle will increase at the same time from o to IT, while B will commence by increasing from o to a, and afterwards diminish from a to o (where a = sin"'- I . Moreover, in the latter stage b cos A is negative, and dB also negative, consequently the term b cos A dB is positive through- out the entire integration ; and the total value of ■v/fi^ - a'sin'BdB is represented by 2 yb'-a^ein^BdB. (J Hence, substituting ^ for — , and integrating between the limits indicated, we get, after dividing by 2, IT r {(a + bysiT^<^ + {a- hYco&^^]^d^ Jo = ["(a" - 5^ 5m^A)^dA + ["(5'' - a" sm^B)^ dB. (10) Accordingly, the difference between the length of the asymp- tote and of the infinite arc of a hyperbola is equal to the differ- ence between two elliptic quadrants. This result is also due to Landen. We next proceed to two important theorems, which may be regarded as extensions of Fagnani's theorem. 234 Lengths of Curves. i6i. Theorem* of Dr. Crraves. — If from any point P on the exterior of two confocal ellipses, tangents PT and PT' be drawn to the in- terior, then the difference (Pr+Pr-rr) between the sum of the tangents and the arc between their points of contact is con- stant. For, draw the tangents Q8 and QS' from a point Q, regarded as infinitely near to P, and drop the perpendiculars PN and Pig- 3°- QN' ; th«n, since the conies are confocal, we have z PQN = L QPN' ; .-. PN' = QW. Also, PT=TR + RN= TR + RS+ SN=TS+SN = TS+ SQ- QK In like manner PT'^PN'+S'Q-rS'; .'. PT + PT' =Q8+ Q8' + TS- T'8', or PT + PT' - TT' = Q8 + Q8' - 88'. Hence, PT + PT' - TT' does not change in passing to the consecutive point Q ; which proves that PT + PT' - TT' has a constant value. * This elegant theorem was arrived at by Dr. Graves, now Bishop of Limerick, for the more general case of sphericalconics, from the reciprocal theorem, viz. : — If two spherical conies have the same cycUo arcs, then any arc touching the inner will cut from the outer a segment of constant area. (See Graves' trsmsla- tion of Chasles on Cones and Spherical Conies, p. 77, Dublin, 1841.) It should be remarked that the theorems of this and of the following article were investigated independently by M. Chasles. The student will find in the Comptea Bendm, 1843, 1844, a nimiber of beautiful applications by that great geometrician of these theorems, as well to properties of confocal conies, as also to the addition of elliptic functions of the first species. Theorem of Br. Graves. 235 TMs value can be readily expressed by taking the point at ^, one of the extremities of the minor axis of the exterior ellipse. Let D be the point of contact of the tangent drawn from B, and drop DM, and DN perpen- dicular to CA and CB, respectively. Let CA = a, CB = b, CA'-d, GB'=b', e the eccen- tricity of interior ellipse. -^^S- 3'- Then, by Art. 155, the length of arc BB = a where Again, (i - e^Ein''0)M^, cos a : BK CN CB b hence CB CB CB b" B'B^ = FN^ + BN^ =:{b'-b cos aY + «= sin"a B'B = yy/b"-b' = a'sma. (i -e'sin^^o'^, Consequently we have B'B-BB = a'siaa-a Hence, in general, PT+ FT'- TT' = 2a' sin a - 20 ["(i - e' &in^^fd• = fl cos + 5, we get -r- = - a siu 0, and hence ds = (a' + 6* + 2ah cos Q)idd ; .-.«=[ {{a + hy cos= ^ + (« - hy sm^^dO. Accordingly, the rectification of the lima9on depends on that of the ellipse whose semiaxes are a + h and a-b. 164. Tbe Epitroehoid and Hypotrochoid. — The epitroehoid is represented by the equations (see DifE. Calc, Art. 284) a; = (a + &) cos - c cos — - — 9, y = [a + 0) smo - c sm — — U. Hence dx - (fl! + S) Isin - - sin —7— , (a + b) jcos - 7 cos —7— 0! . Squaring and adding we get dp dO' a + b i' + c'-25ccosyj*(/0. Hence, suhstitutiag — - for 0, we get s = ^^^~- [ {(* + cf sin^^ +{b- cy cos-?>}« 238 Lengths of Curves. Consequently the length of an arc of the epitroohoid is equal to that of an ellipse. The corresponding form for the hypotrochoid is obtained by changing the sign of b. 165. Steiner's Theorem on Rectification of Roulettes. — If any curve roll on a right line, the length of the arc of the roulette described by any point is equal to that of the corresponding arc of the pedal, taken with respect to the generating point as origin. For (see fig. 20, Axt. 145), the element OCof the roulette is equal to OPdw. Again, to find the element of the pedal. Since the angles at N and N' are right, the quadrilateral NN'TO is inscri- bable in a circle, and consequently NN' = OT sin HON'. But, in the limit, NN' becomes the ele- ment of the pedal, and OTbecomcs OP : hence the element of pedal is OPdw ; consequently the ele- ment of the pedal is eqijal to the corresponding element of the Fig. 34. roulette; .*. &c. We proceed to point out a few elementary examples of this principle. In the first place it follows that the length of an arc of the cycloid is the same as that of the cardioid ; and the length of the trochoid as that of the lima9on. Again, if an ellipse roll on a right line, the length of the roulette described by either focus is equal to the corresponding arc of the auxiliary circle. Moreover, it is easily seen, as in Art. 146, that, if one curve roll on another, the elements ds and ds', of the roiilette, and of the corresponding pedal are connected by the. relation ds = ds'fi + ^\ In the case of one circle rolling on another, this relation shows that the arcs of Epicycloids and of epitrochoids are proportional to the arcs of cardioids and of limacons, which agrees with the results established already. Oval of Descartes. 239 1 66. Oval of Descartes.— We next proceed to the rectification of the Ovals of Descartes, some _ properties of ■which curves we have given in chapter xx.j Diff. Oalo. The curve is de- fined as the locus of a point whose dis- tances, rand /,froni two fixed points are connected by the equation mr + Ir' = d, where /, m, d are constants. For convenience we shall write the equation in the form mr + Ir = nc, (12) where e is the dis- tance between the fixed points l?jg- 35- The polar equation of the curve is easily got. For, let F and Fi be the fixed points, and L F^FP = d, then we have /2 = r^ + c^ - zrc cos 6 ; also from (12), hence the polar equation of the locus is readily seen to be OTw - P cos r - zrc ■ + c' = o. (13) m' -r ' " m^ -P For simplicity we shall write this in the form r^ - 2rQ + C = o. (14) Solving this equation for r, we get »' = Q±v/i2'-G, oTFPi = Q + '/QF^,FP^Q.-.s/Qr^. It can be seen without difficulty that, so long as I, m, n are real afid unequal, the curve consists of two ovals, one lying inside the other, as in the figure. ? 240 Lengths of Curves. Again we get from (14), by differentiation [r -Q,)dr= rQ,'dO, where Q' = -^ ; dr CI' Q' , ds ^/q.'+Q,"-0 ■. — - = = — , ; hence — rn = — — . rdf) r-Q ^Q^-G '"^^^ ^Q?-C Or ds = Qv/Q'j^Q^ ^^g ^ yQ^H-Q''- CdO, (15) the upper sign corresponding to the outer oval, and the lower to the inner. Hence the difference between the two corresponding elementary arcs is equal to 2 v/Q' +Q,'^~G dd, or, 2 v/a' + 2a6 cos + 6' - Cd9, (writing Q, in the form a + 5 cos 0) ; this plainly represents the element of an ellipse. Consequently, the difference between two corresponding arcs of the ovals can be repre- sented by the arc of an ellipse. This remarkable theorem is due to Mr. W. Eoberts (Liouville, 1847, P- i95)- Some years after its publication it was shown by Professor Genocchi (Tortolini, 1864, p. g?), that the arc* of a Cartesian is ex- pressible in terms of three elliptic arcs. In order to establish this result we commence by proving one or two elementary properties of the curve. Suppose a circle described through F, Fi, and P ; and let PQ be the normal at P to the oval, meeting the circle in Q, and join FQ, and F^Q ; then let L FPQ = u,, and F,PQ = u> ; , , dr jd/ 1, 7 • / and since m-z- + I —r = o> 'we nave i sm u) = m sm w : ds ds .: FQ : F^Q^l: m. * For the proof of this theorem giyen in the text I am indebted to Mr. Panton. , The Cartesian Oval. 241 Also, since mr + l/ = nc; and (by Ptolemy's theorem) '' FP . F,Q + F,P . FQ = FF, . PQ, we have FQ^FQ^PQ 7 I m 71 ' Henoe, denoting the common value of these fractions by w, we have FQ = lu, FiQ = mu, PQ = mi. Again dr Q' ya' - C tan (X) = — ^r. = — ■ ■ ; .'. cos (u = — ~. Hence the first term in the expression for c?s in (15) is equal to Qdd e mn- P cos 6 ,» cos tx) m^ - I cos u> Again, let l FPFi = ;/-, z PF,C = ^, and we have the two following relations between the angles 0, 0, ;/- : ^ = 6 + ^, ^ sin + m sin = « sin 1^. (16) Hence dd^ = n cos^dxp; .-. {mn - P cos 9)d9 = m(n + lcoB^)dij> -n{m+ I cos \p)dip, or mn - /^ cos ,,, » + ^ cos A , w + ^ cos ti , , , , dO = m ^d Jcosft> ni' - P] fin r J 5— j5U/^'' + »»''+2/»tC0St/irfl/'. (18) Each of these latter integrals is represented by the arc of an ellipse, and, accordingly, the arc of a Cartesian Oval is expressible in the required manner. It should be noted that the limiting values of 0, 0, and ^ are connected by the relations given in (16). Again, it can be shown without difficulty that the axes of the ellipses are the lines {AB,'CI)), {AG, BB), and {AD, BO), respectively/ : a result also given by Signor Gre nocchi. Fir st, with respect to the ellipse whose element is ^Q,^ + Q'^ - C'iB, it is plain that its axes are the greatest and least values of 2 y/Q' + Q,'' - C, or of z-Za^ + b" + zah cos - C ; but these are 2 ,/'{a + by - C and 2 ^{a - by - G, which are plainly the same as the greatest and least values of PPi ; and, con- sequently, are AB and CD. Again, from the equation mr + Ir' = nc, we get mFB + UFB + c) = nc; ■•. FB = '^^-ZUl, ^ l + m In like manner, {n+ l)e FC = l + m Again, since we get the points on the outer oval by changing the sign of I, we have m-l ' m- I ' Reetifieation of Curves of Double Curvature. 243 and, consequently, AD = ^, BC^.^''' m- ■I' l + m' -, 2mc{n + l) 2mc{n-l) ^ but these are readily seen to be the values for the axes of the ellipses in(i8). It should be noted that if we substitute in (15) the values for a and b, the expression for the element ds becomes of the following symmetrical form : ds=—z — Tz^P+n^+2lncQB(j>ddt — ; — -,'yP+m'+2lmcoB\Ld\L mr-P ^ ^ m'-r ^ Ic m^ - P ^/ni^ + ff - 2mn cos ddd. (19) "We shall conclude the Chapter with a brief account of the rectification of curves of double curvature. 167. Rectification of Curves of Double Curvature. If the points in a curve be not situated in the same plane, the curve is said to be one of double curvature. The expression for its length is obtained in an analogous manner to that adopted for plane curves ; for, if we refer the curve to a system of rectangular axes in space, and denote the co-ordi- nates of two consecutive points by {x, y,z),[x+dx,y+ dy, s + rfa), we get for the element of length, ds, the value ds = ^/d:^ + dy"^ + d^. The curve is commonly supposed to be determined by the intersection of two cylindrical surfaces, whose equations are of the form f(x, y) = o, = o, which is of the form r' + 2r{a cos 8 - /3) + (a - j8)* = o. Hence, by (15), the difference between two corresponding elementary arcs is /— e 4V a;3 cos - do. Consequently, if Bi and 62 be the values of 9 for the two transversals in question, we get the difference of the corresponding arcs „ /-^ / . 92 . 9A ■■ oy' ap I sm sm— I . Also, it can be readily seen that the distance between the vertices of the lima9on is 4 y a^ ; . . &c. 248 Lengths of Curves. 1 8. Show that the length of an arc of the ellipse -5 + j^ = i is represented by the integral an^\ («''cos2e + i2sin2fl)3 we have ds = prffl, anc 19. Show, in like manner, that the length of a hyperholio arc is represented This result is easily seen, for we have ds = prffl, and p = — j- ; . . &c. by anA '1 J (a«cos=e-i«6in»e)* 20. Hence prove that the integral fdx (o - J:i;=)9(«'-iV)* is represented by an elliptic arc when aV > ha', and by a hyperbolic arc when ah' < h(C, 21. Prove that the differential of the arc of the curve found by cutting in the ratio « : i the normals to the cycloid y = a + 4 cos «, a; = «« + i sin «, ^(« + «i nVf' + ^nab so? - du. 22. Each element of the periphery of an ellipse is divided by the diameter parallel to it : find the sum of aU the elementary quotients extended to the entire ellipse. Ait%. ti. 23. In the figure of Art. 158, if o = Z -4CiV', and iS = / :BCN, prove that tan a tan ^ a b 24. Find the length, measured from the origin, of the curve y a;2 = «2(i - c). Ans. » = a log ( I - «■ ° \a-xj 25. Find the length, measured from

) cos - d

'00s'-rf* Jo ^ ; 2 [17 a] 260 Volumes and Surfaces of Solids. Fig. 4«- 3. Find the volume and the surfece of the solid generated by the revolution of the tractiix round its axis. (i). Here we have y*dx=- (a» - y^iydy ; hence the volume generated by the portion AP is J» 3 The volume generated by the entire tractrix is — a'; i. e. half 3 the volume of the sphere whose radius is OA. (2). The surface generated by AF is 27r I yds = 2ira I dy (see Ex. 2, Art. 154) = 2jra(a - y). Hence the entire surface generated is 2ira^ ; i. e. half the surface of the sphere of radius OA. 4. Find the volume, and also the surface, generated by the revolution of the catenary around the axis of x. (i). Here the volume of the solid gene- X rated by TF is represented by / - - = — (ys + ax), where a = FV. (2). Again, since we have ■ Fig. 42. air I yds = — I y'^di. Surfaces of Eevolution. 261 Confiequently the surface generated by PV in a complete revolution is - X the Tolume generated ; i.e. = ir[ys + ax). 5. In the same curve to find the surface generated by its revolution round the axis V. Here 8-- Affain f = 27r I xds = It I xe" (fa: + ir I xe "dx. xe''dx= axe"— a\ e'dx^ a{xe^ - ae" + a) Jo Also the value of I* - xe' a dx is obtained by changing the sign of a in the last result. Hence • axe " — a'e "; fx ^^ xe » (&; = «' .*. j8' = ir jjffl' + as (««-«"» J - 8'[e» + e"»] [ = 27r(a'-* + a:s — fly). 177. Annular Solids. — If a closed curve, -which is symmetrical •with respect to a right line, he made to revolve round a parallel line, then the superficial area generated in a complete revolution is equal to the product of the length of the moving curve into the circumference of the circle -whose radius is the distance bet-ween the parallel lines. This is easily proved: for let APBP' he any curve, symmetrical -with respect to AB, and suppose OX to he the azis of revolution ; and draw PN, QM t-wo indefinitely near lines perpendicular to the axis. It is evi- ' dent that PQ = P'Q'. Again, let PJST^ y, P'N=y', PQ, = P'^ = ds, BN = h ; then the sum of the elementary zones described by PQ and P'Q' in a complete revolution is represented by 2Tr{y + y') ds = 4nbds. Fig- 43- 262 Volumes and Surfaces of Solids, Conseqaently the surface generated by the entire curve is ZTrbS, where S denotes the whole length of the curve. A similar theorem holds for the volume of the solid ge- nerated : viz., the volume generated is equal to the product of the area of the revolving curve into the circumference of the san~e circle as hefore. For the volume of this solid is plainly represented hy or by n- {y-y'){y + '!/)dx= 2TrJ {y-y')dx. But the area of the curve is represented by {y -f)dx: 1' consequently, denoting this area by A, and the volume by V, we have F= 2Trb X A. In these results the axis of revolution is supposed not to intersect the curve; if it does, the expression zirb x A represents the differsnce between the volumes of the surfaces generated by the portions of the curve lying at opposite sides of the axis of revolution ; as is readily seen. A similar alte- ration must be made in the former theorem in this case. If a circle revolve round any external axis situated in its plane, the surface generated is called a spherical ring. From the preceding it follows that the entire surface of such a ring is 47r''«J ; where a is the radius of the circle, and b the dis- tance of its centre from the axis of revolution. In like manner the volume of the ring is 2Tr^ a^b. It would be easy to add other applications of these theorems. •''178. Cruldin's* Theorems. — The results established in the preceding Article are but particular cases of two general * Guldin, Centrobaryica, sen de centra gravitatis trium speeierum quantitatii continues, 1635. Giildm arrived at liis principle ty induction from a small num- 1)er of elementary cases, 1]ut his attempt at a general demonstration yraa an eminent failure. See Montuola Bist. dea Math., torn. ii. p. 34. Montucla has sho-wn, tom. ii. p. 92, that Guldin's theorems can he estahlished from geome- trical considerations, without recourse to the Calculus. Guldin's Theorems. 263 propositions, usually called Guldin's Theorems, but originally enunciated by Pappus (see Walton's Mechanical Problems, p. 42, third Edition). They may be stated as foUows : — (i). If a plane curve revolve round any external axis, situated in its plane, the area of the surface generated is equfil to the product of the perimeter of the revolving curve by the length of the path described, during the revolution, by the centre of gravity of that perimeter. (2). Under the same circumstances, the volume of the solid generated is equal to the product of the area of the generating curve into the path described by the centre of gravity of the re- volving area. To prove the former, let s denote the whole length of the curve, X, y, the co-ordinates of one of its points, x, y, those of the centre of gravity of the curve ; then, from the defi- nition of these latter, we have [yds .'. 2Trys = 277 1 yds, i. e. the surface generated by revolution round the axis of x is equal to the product of 8, the length of the generating curve, into 2 Try, the path described by the centre of gravity. To prove the second proposition ; let A denote the area of the generating curve, and dA the element of area corre- sponding to any point x, y. Also let x, y be the co-ordinates of the centre of gravity of the area, then _ "ZydA Wydxdy , ,,.,,. , , » , .> y = — ^— = . (substituting dx dy for dA) ; .•. 2TryA = ztt jjydxdy = TTJ y^dx; where the integral is supposed taken for every point round the perimeter of the curve: but, from Art. 171, the integral at the right-hand side represents the volume of the solid gene- rated ; hence the proposition in question follows. For example, the volume of the ring generated by the revolution of an ellipse around any exterior line situated in its plane is at once 2Tr'abc, where a and b are the semiaxes 264 Volumes and Surfaces of Solids. of the ellipse, and c is the distance of its centre from the axis of revolution. It may be noted that these results still hold if we suppose the curve, instead of making a complete revolution, to turn round the axis through any angle. For, let 6 be the circular measure of the angle of rotation, and in the former case we have flys = J yds. But fly is the length of the path described by the centre of gravity, and 9 j yds is the area of the surface generated by the cu7ve ; .'. &c. In like manner the second proposition can be shown to hold. Again, Ghildia's theorems are still true if we suppose the rotation to take place around a number of different axes in succession ; in which case the centre of gravity, instead of describing a single circle, would describe a number of arcs of circles consecutively ; and the whole area of the surface ge- nerated will still be measured by the product of the length of the generating curve into the path of its centre of gravity ; for this result holds for the part of the surface corresponding to each axis of revolution separately, and therefore holds for the sum. .Again, in the limit, when we suppose each separate rota- tion indefinitely small, we deduce the following theorem. If any plane curve move so that the path of its centre of gravity is at each instant perpendicular to the moving plane, then the surface generated by the curve is equal to the length of the curve into the path described by its centre of gravity. The corresponding theorem holds for the volume of the surface generated. These extensions of Guldin's theorems were given by Leibnitz {Act, Erud. Lips., 1695). 179. Expression for ITolume of any Solid. — The method given in Art. 1 7 1 of investigating the volume bounded by a surface of revolution can be readily extended to a solid bounded in any manner. For, if we suppose the volume divided into slices by a system of parallel planes, the entire volume may, as before, be regarded as the limit of the sum Volume of Elliptic Paraboloid, 265 of a num'ber of infinitely thin cylindrical plates. Thus, if we suppose a system of rectangular co-ordinate axes taken, and the cutting planes drawn parallel to that of a;y ; then, if A^ represent the area of the section made hy a plane drawn at the distance z from the origin, the entire volume is denoted ■by taken hetween proper limits. The area Az is to be determined in each case as a function of s from the conditions of the bounding surface. For example, to find the volume of the portion of a cone cut ofE by any plane ; we take the origin at the vertex, and the axis of 2 perpendicular to the cutting plane ; then, if B denote the area of the base, and h the height of the cone, it is easily seen that wo have A^: B= z" : ¥, or ^^ = ^-- ; z^dz = - £ X h; as in Art. 169. Jo 3 If the cutting planes be parallel to that of pz, the volume is denoted by jA^dx; where A^ denotes the area of the sec- tion at the distance m from the origin. 180. Volume of EUiptic Paraboloid. — Let it be proposed to find the volume of the portion of the elliptic paraboloid - + — = 2Z, P 1 cut off by a plane drawn perpendicular to the axis of the sur- face. Here, considering z as constant, the area of the ellipse - + - = 2z, by Art. 128, is 27rsy^. Hence, denoting by c the distance of the bounding plane from the vertex of the surface, we have 'c F= ZTT'/pq zdz = nc^ -v/p?' 266 Volumes and Surfaces of Solids. This result admits of being exhibited in another form ; for if £ be the area of the elliptic section made by the bounding plane, we have B = 2TrC'>y/pq. Hence F = 5 circumscribing cylinder, as in paraboloid of re- Tolution. i8i. Tlie Ellipsoid. — Next, to find the volume of the ellipsoid «" y^ z' a' 0^ c^ The section of the surface at the distance z from the origin is the ellipse ?! r _ _ £_' a» "^ 6' ~ ' ?'• the area of this ellipse is Trf I --jab, i.e. A, =ir(i --^]ab. Hence, denoting the entire volume by V, we have V=2Trab i I - — ]dz =-Trabc. J 3 182. Case of Obligne Axes. — It is sometimes more convenient to refer the surface to a system of oblique axes. In this case, if, as before, we take the cutting planes parallel to that of xy, and if w be the angle the axis of z makes with the plane of xy, the expression for the volume becomes sin a» j Azdz, taken between proper limits, where A^ represents the area of the section, as in the former case. For example, let us seek the volume of the portion of an ellipsoid cut off hj any plane. Case of Oblique Axes. 267 Suppose BEjyE' to represent the section made by the plane, and ABA'B' the parallel central section. Take OA, OB, the axes of this section as axes of X and y respectively ; and the conju- gate diameter 0(7 as axis of z. Then the equation of the surface is x" %f s= Fig. 44. where OA = a', 0B = h',0O= c'. It will now be convenient to transfer the ongm to the point C, without altering the directions of the axes, when the equation of the surface becomes x^ y^ 2s s- The area At of the section, by Art. 128, is T«'6'ft-i^; (3) hence, denoting C'N by h, the volume cut off by the plane BEB' is represented by Tta'b' sin &> Jo 2S S' or ,-, . (K' h'\ wa sm (0-7 7, . V 30 y But, by a well-known theorem,* we have db'c' sin to = abc, ~i where a, b, c, are the principal semiaxes of the surface. Hence the expression for the volume V in question be- comes * Salmon's Geometry of Three Dimensions, Art. 96. 268 Volumes and Surfaces of Solids. or, denoting ^^ by 7c, V=Trabck'fi-^. (5) This result shows that the volume cut off is constant for all sections for which Jc has the same value. Again, since OiV -pr— -, = 1- k, the locus of iVis a similar ellipsoid ; and we infer (JO that if a plane cut a constant volume from an ellipsoid, the locus of the centre of the section is a similar and similarly situated ellipsoid. 183. Elliptic Paraboloid. — The corresponding results for the elliptic paraboloid can he deduced from the preceding by adopting the usual method of such derivation : viz., by taking a^ =pc, V^ = qc, and afterwards making c infinite ; observing that in this case the ratio - becomes unity. Making these substitutions in (4), it becomes F = TT ^/pqt? ( I -A or -aW 'i/pq. since c = 03, Hence, if a constant length be measured on any diameter of an elliptic paraboloid and a conjugate plane drawn, then the volume* of the segment out from the paraboloid by the plane is constant. Again, the area of an elliptic section by (3) is Y2A A'\ TT TTflsS — -— J, or — \C C^J CB 2h h' c sm. w\c c • For a more direct investigatioa the student is referred to a memoir " On some Properties of the Paraboloid," Quarterly Journal of Mathematics, June, 1874, by Professor Allman. Elliptic Paraboloid. 269 On making the same substitutions, this becomes for the paraboloid sin•' = a' cos'' 61 + h" sin^'e, and r=\ (a' - Wf J sin' B dO. Double Integration. 275 TT If this be integrated between the limits o and - we get 2 the -yth of the entire volume ; hence the entire volume 9 Examples. 1. A sphere is cut by a right cylinder, the radius of ■whose hase is half that of the sphere, and one of whose edges passes through the centre of the sphere ; find the yolume conunon to both suriaces. Am. ■ , « "being the radius of the sphere. 3 9 2. If the tase of the cylinder he the complete curre represented by the equation r = a cos »9, where « is any integer, &id the vdume of the solid be- tween the surface of the sphere and the external surface of the cylinder. 187. It is readUy seen, as in Art. 141, that the volume in- cluded within the surface represented by the equation \a cj is abc X the volume of the surface F{x, y, z) = o. For, let - = «', ^ = «/, - = z, and we shall have ' a ' b "' c ' zdxdy - abcz' dxf dy' , and .*. J ^ zdxdy = abc l\z'dx'di/ ; which proves the theorem. Ilence, for example, the determination of the volume of an ellipsoid is reduced to that of a sphere. Again, if the point [x, y, 2) move along a plane, the cor- responding point {xf, y', z') will describe another plane. From this property the expression for the volume of an ellipsoidal cap (Art. 1 82) can be immediately deduced from that of a spherical cap (Axt. 17b). [18 a] 276 Volumes and Surfaces of Solids. In like manner the volume included between a coiie en- veloping an ellipsoid and the surface of the ellipsoid is reducible to the corresponding volume for a sphere. 1 88. Quadrature on the Sphere. — We next propose to give a brief discussion of quadrature on a sphere, and commence with the results on the subject usually given in treatises on Spherical Trigonometry. In the first place, since the area of a lune is to that of the entire sphere as the angle of the lune to four right angles, the area of a luno of angle A is represented by zlf'A ; where It is the radius of the sphere, and A is expressed in circular measure. Again, the area of a spherical triangle ABC is expressed hy £,' {A + B + C - tt) ; for, the sum of the three lunes exceeds the hemisphere by twice the area of the triangle, aa is easily seen from a figure. Hence, it readily follows that the area S of a spherical polygon of « sides is represented by ■S, = B''{A + B+ C + &0.- (M-2)n-); A, B, C, &c., being the angles of the polygon. This result admits of being expressed in terms of the sides of the polar polygon ; for, representing these sides by «', b', c', &c., we have A = IT - a', B = TT - b', &c., and consequently S = B^ZTT - {a' + 6' + c' +&c.)). Or, denoting the perimeter of the polar figure by 8, •2 + BS= 2TrB\ (6) This proof is perfectly general, and holds in the limit, when the polygon becomes any curve ; and, accordingly, the area bounded by any closed spherical curve is connected with the perimeter of its polar curve by the relation (6). Again, the spherical area bounded by a lesser circle (Art. 170) admits of a simple expression. If p denote the circular radius of the circle, or the arc from its pole to its circumference, the area in question is represented by 27rjB'(i - cos/o) ; Quadrature on the Sphere. 277 for (see fig. Art. 1 70) we have AN='AC-CN=R{i- cosp). This result also follows immediately as a simple case of equation 1(6). Again, the area hounded hy the lesser circle and hy two arcs drawn to its pole is plainly represented by ^^"0(1 - COSjo), where a is the circular measure of the angle between the arcs. We can now find an expression for the area bounded by any closed curve on a sphere ; for the position of any point P on the surface can be expressed by means of the arc OP (fcawn to a fixed point, and of the angle FOX between this arc and a fixed arc through 0. These are called the polar co-ordinates of the point, and are analogous to ordinary polar co-ordinates on a plane. pj 5 Now, let OP = p, and POX = w ; then any curve on the sphere maybe supposed to be expressed by a relation between p and w. Again, suppose OQ to represent an infinitely near vector, and draw PH perpendicular to OP; then, neglecting in the limit the area PQB, the elementary area OPQ, by the preceding is represented by jB^(i - cos p)d(o. Hence the area bounded by two vectors from ip expressed by the integral ii* (i - coap)dii), taken between suitable limits. If the curve be closed, the entire superficial area becomes P?\ (l - COSjo), and ds = Sda ; w .'. area in question = 21^ I cos utfn) = ^IP ; i.e. the square of the diameter of Jo the sphere, 2. When the diameter is less than the radius of the sphere, 2 ids = 2a v' JJ2 — a- sin'oii^tii = 2a£ y' I — ic'sin^ai dca ; .". &c. 189. ttnadratnre of Surfaces. — In seeking the area of a portion of any surface we regard it as the limit of a number of infinitely small elements, each of which is con- sidered as a portion of a plane which is ultimately a tangent plane to the surface. Now let dS denote such an element of the superficial area, and d;((«' sin'v + c'cos'v)^6'8in'i/ + c'cos'v)*) Quadrature of the Ellipsoid. 283 Now, if 8 denote the superficial area* between two curves corresponding to " = a and v = a, after one or two reductions, it is easily seen that S = -^a'b'c'{I+r), (9) , 7- f " sin V dv ~ J„ (6' sin^ V + c" cos" v)= (a* sin^ v + c^cos' v)^' [■»' sin V c?i» "]„(«" sin' V + c" cos" I/) * (6" sin" v + c" cos " v) * ' It is easily shown that the former of these integrals is represented hy an arc of an ellipse, and the latter by an arc of a hyperbola ; it being assumed that a>h> c. For, assuming c? - e', the former integral represents an arc of an ellipse, and the latter an arc of a hyperbola. (See Ex. 19, p. 249). * This form for the quadrature of an ellipsoid ia given by Mr. Jellett in the memoir already referred to. He has also shown that the ellipse and the hyperbola in question are the focal conies of the reciprocal ellipsoid ; a result wMch can be easily arrived at from the forms of I and /' given above. For application to the hyperboloid, and further development of these results, the student is referred to Mr. Jellett's memoir. 284 Volumes and Surfaces of Solids. 192. Integration over a Closed Surface. — We shall conclude this Chapter with the consideration of some general formulaB in double integration relative to any closed surface. We commence by adopting the same notation as in Art. 189, where A, ju, v are taken as the angles which the exterior normal at the element dS makes with the positive directions of the axes of x, y, s, respectively. Again, let each element of the surface be projected on the plane of xy, and suppose* for simplicity that each 2 ordi- nate meets the surface in but two points : then, if the indefi- nitely small cylinder standing on any element dA in the plane of xy intersects the surface in the two elementary por- tions dSi and dSt (where dSi is the upper, and dS^ the lower element), and if vi and vj be the corresponding values of v, it is plain that vi is an acute, and Vi an obtuse angle, and we have dA = cos vidSi = - cos VidS^. Hence, if we take into account all the elements of the surface, attending to the sign of cos v, we shall have J/cos vc^/S = o. In like manner we get \lQOs\dS = o, and J/cos/i£^S= o; the integrals extending in each case over the whole of the closed curve These formulae are comprised in the equation II (a cos A + /3 cos ;« + 7 COS v) dS = o. (10) Again, if s, and Sj be the values of z corresponding to the clement dA, then, denoting by rfFthe element of volume standing on dA and intercepted by the surface, we plainly have dV = (si - Zt}dA = sidSi cos vi + z^dSz cos v^, * It it easily seen that tlis and the following demonstrations are perfectly -general, inasmuch as each ordinate must meet a closed surface in an eyen number ■of points, which may te considered in pairs. Integration over a Closed Surface. 285 and the sum of all such elements, that is, the whole volume, is evidently represented by ^\z cos vdS. Hence, denoting the whole volume hy V, we have V = \\x cos\dS = jjy cos nd8 = jj z cos vdS ; the integrals, as before, being extended over the entire surface. Again, it is easily seen that we have jj X COS vd8 =0, jj y cos vdS = o, J/ a; cos fidS = o, l\ycoa\d8=o, jj s co3\dS= o, jj z cos fidS = o. For, as in the first case, it readily appears that the elements are equal and opposite in pairs in each of these integrals. These results are comprised in the equation jj{ax + Pi/ + yz) {a cos X + /3' cos ;u + y cos v) d8 = (aa' + |3i3' + 77') r. (11) Por a like reason, we have jjxi/ cos vd8 = o, j j zx cos nd8 = o, j j i/z cosXd8 = o. Also //»' cos vd8 = o, // »' coSfjid8 = o, &c. Next, let us consider the integral jjxz cos vd8. This integral is equivalent to H xdV; consequently, if S, y, i, be the co-ordinates of the centre of gravity of the enclosed volume V, we get l\ as cos vd8 = l\xdV = xV; in like manner \\ xz cos \d8 = zV. Again, the integral \\ z^ cos vd8 consists of elements of the form (si' - z^) dA ; but (Si* - Z^) dA = (Zi + Sa) (Si - Za) dA = (zi + Zi)dV. 286 Volumes and Surfaces of Solids. '^■dr=2zV. But the z ordinate of the centre of gravity of (?F is plainly ~ -", and consequently s'cos vd8= 2 - — In like manner it can be shown that //«' cQsXdS = zxV, jj y^ cos fidS = 2yV. Accordingly we have Vx - iJJ«' fX)^\dS = \\xy oos /xdS = ^^xzcoavdS, Vy = liyx coaXdS = iJly^coa/jid8=SSyzoosvd8, Vs = Ij zx cos \d8 = \\zy oosfidS = ^jjz'^ cos vdS. 193. Expression for Volume of a Closed Surface. — ^Next, if we suppose a cone described with its vertex at the origin 0, and standing on the elementary base dS, its volume is represented (Art. 169) by ^pd8, where _p is the length of the perpendicular drawn from to the tangent plane at the point. Also, if r be the distance of from the point, and y the angle which r makes with the internal normal, we have "jj = r cos 7. Hence the elementary volume is equal to -J-r cosy(?;S,and it is easily seen that if we integrate over the entire surface, the enclosed volume is represented by \l\r cosydS. 1 94. Again, if we suppose a sphere of unit radius described with as centre, and if diD represent the superficial portion of this sphere intercepted by the elementary cone standing on dS, then it is easily seen that cos ydS ^r'du; COSydS .: db) = '- — . r^ Now if be inside the closed surface, and the integral be extended over the entire surface, it is plain that // rfw = 4!r, being the surface of the sphere of radius unity ; \cosyd8 1 —i 4;r. ir Esq>ression for Volume of a Closed Surface. 287 Again, if te outside the surface, the cone will cut the surface in an even number of elements, for which the values of cos 7 win be alternately positive and negative, and, the corresponding elements of the integral being equal but with opposite signs, their sum is equal to zero, and we shall have cos 7 dS -. — = o. If be situated on the surface, it follows in like manner that w — -^ dS = iir. T Hence, we conclude that 'cos 7 W'' dS = 47r, 2ir, or o, (12) according as the origin is inside, on, or outside the surface- The multiple integrals introduced into this and the two preceding Articles are principally due to Gauss. The student will find some important applications of this method in Bertrand's Gale. Int., §§ 437, 455, 456, 476, &c. 288 Examples. Examples. 1 . A sphere of 15 feet radius is cut by two parallel planes at distances of 3 and 7 feet from its centre ; find the superficial area of the portion of the sur- face included between the planes approidmately. Ans, 376.990855. feet. 2. Being given the slant height of a right cone, find the cosine of half its vertical angle when its volume is a maximum. ^ i Ans. . 3. Prove that the volume of a truncated cone of height h is represented hy irh — (i2^ + iJr + r'), where E and r are the radii of its two bases. 4. A cone is circumscribed to a sphere of radius if, the vertex of the cone being at the distance -D from the centre ; find the ratio of the superficial area of the cone to that of the sphere. If — IP Ans. — =rT— •. 4D£ 5. Two spheres, A and S, have for radii 9 feet and 40 feet ; the superficial area of a third sphere C is equal to the sum of the areas of A and £ ; calculate the excess, in cubic feet, of the volume of C over the sum of the volumes of A and 5. Ans. 17558. 6. If any arc of a plane curve revolve successively round two parallel axes, show that the difference of the surfaces generated is equal to the product of the length of the arc into the circumference of the circle described by any point on either axis turning round the other. If the axes of revolution lie at opposite sides of the cuito, the sum of the surfaces must be taken instead of the difference. 7. Find, in teims of the sides, the volume of the solid generated by the complete revolution of a triangle round its side e. Air s(a - a)(s—i)(s- c) Ana. 5 -i-. — '. 3 c 8. Apply Guldin's theorem to determine the distance, from the centre, of .the centre of gravity, (i) of a semicircular area ; (2) of a semicircular arc. Ans. (i)±., (2)-. V ^ 9. If a triangle revolve round any external axis, lying in its plane, find an expression for the area of the surface generated in a complete revolution. 10. Prove that the volume cut from the surface a" = Ax^ + By^ E xy, is — — th pi w + I the plane section, and terminated by the plane of xy. by any plane parallel to that of xy, is th part of the cylinder standing on Examples. 289 11. A cone is oiroumsoribed to a sphere of 23 feet radius, the vertex of the cone being 265 feet distant from the centre of the sphere ; find the ratio of the superficial area of the cone to that of the sphere. 12. The axis of a right circular cylinder passes through the centre of a sphere ; find the volume of the solid included between the concave surface of the sphere and the convex surface of the cylinder. Ans, — , where is the length of the portion of any edge of the cylinder intercepted by the sphere. This question is the same as that of finding the volume of the solid generated by the segment of a cu'cle cut off by any chord, in a revolution round the diameter parallel to the chord. 13. Find the volume of the solid generated by the revolution of an arc of a circle round its chord. Ans. zira j Co ! , where a = radius, c = distance of chord from centre, and 003 a = -. a In this we suppose the arc less than a semicircle : the modification when it is greater is easily seen. 14. If the ellipsoid of revolution, and the hyperboloid ii;« + s^ + i2 " «% x' + z^- 42 f: be cut by two planes perpendicular to the axis of revolution, prove that the zones intercepted on the two surfaces are of equal area. 15. Find the entire volume bounded by the positive sides of the three co- ordinate planes, and ©MDM;) = I, Ans. — . 90 16. Find the volume of the surface generated by the revolution of an arc of a parabola round its chord ; the chord being perpendicular to the axis of the curve. Q Am. — iri'c, where is the length of the cliord, and h the intercept made by it on the diameter of the parabola passing through the middle point of the chord. 1 7. A sphere of radius r is cut by a plane at distance d from the centre ; find the difference of the volumes of the two cones having as a common base the circle in which the plane cuts the sphere, and whose vertices are the opposite ends of the diameter pei'pendicular to the cutting plane. Ans. %Ttd (1-2 - rf2). [19] 290 Examples. 1 8. Find the area of a spherical triangle ; and prove that if a curve traced on a sphere have for its equation sin K = f(l), K denoting latitude, and I lon^- tude, the area between the curve and the equator = lf(l)dl. 19. ShowUiat the volume contained between the surface of a hyperboloid of one sheet, its asymptotic cone, and two planes parallel to that of the real axes, ia proportional to the distance between those planes. 20. Find the entire volume of the surface e)^(r)^e)'-- ^-?^. 21. The vertex of a cone of the second degree is in the surface of a sphere, and its internal axis is the diameter passing through its vertex ; find the volume of the portion of the sphere intercepted within the cone. 22. Prove that the volume of the portion of a cylinder intercepted between any two planes is equal to the product of the area of a perpendicular section into the distance between the centres of gravity of the areas of the bounding sectious. 2 J. If ^ be the area of the section of any surface made by the plane of xij, prove, as in Ait. 192, that ^ I COS vdS, the portion 1 24. If a right cone stand on an ellipse, prove that its volume is represented the integral being extended through the portion of the surface which lies above the plsne of xy. by - (0^.0^')8sin'acosa; where is the vertex of the cone, A and A' the extremities of the major axis of the ellipse, and a is the semi-angle of the cone. 25. In the same case prove that the superficial area of the cone is ^ {OA + OA") [OA . OA')i sin a. ( 291 )f—-^ ^ CHAPTER X. I . INTEGRALS OF INERTIA. : .U- f 195. Integrals of Inertia. — The following integrals are of STicli frequent occurrence in mechanical investigations, that it is proposed to give a hrief discussion of them in this Chapter. If each element of the mass of any solid hody be supposed to he multiplied hy the square of its distance from any fixed right line, and the sum extended throughout every element of the body, the quantity thus obtained is called the moment of inertia of the hody with respect to the fixed line or axis. Hence, denoting the element of mass by dm, its distance from the axis by p, and the moment of inertia by /, we have I='2p^dm. (i) In like manner, if each element of mass of a body be multiplied by the square of its distance from a plane, the sum of such products is called the moment of inertia of the body relative to the plane. If the system be referred to rectangular axes of co- ordinates, then the expression for the moment of inertia relative to the axis of s is obviously represented by S {x^ + 'if') dm. Similarly, the moments of inertia relative to the axes of X tad y are represented by S (y' + s") dm and S (ar* + z^) dm, respectively. Again, the quantities ^x'dm, ^y^dm, "Ez^dm, are the moments of inertia of the body with respect to the planes of yz, xz, and xy, respectively. Also the quantities "Sixydm, ."SiSxdm, "Siyzdm, are called the products of inertia relative tO the same system of co-ordinate axes. In like manner the moment of inertia of the hody with reference to a point is ^r^dm, where r denotes the distance of the element dm from the point. Thus the moment of inertia relative to the origin is S (»' + y" + z') dm. TlQal 292 Integrals of Inertia. 196. noments of Inertia relative to Parallel Axes, or Planes. — The following result is of fundamental importance : — The moment of inertia of a body with respect to any axis exceeds its moment of inertia toith respect to a parallel axis drawn through its centre of gravity, by the product of the mass of the body into the square of the distance between the parallel axes. For, let I be the moment of inertia relative to the axis through the centre of gravity, 1' that for the parallel axis, M the mass of the body, and a the distance between the axes. Then, taking the centre of gravity as origin, the fixed axis through it as the axis of z, and the plane through the parallel axes for that of zx, we shall have I='2,{a? + f)dm, I' = •2,{{x + a)' + y'']dm. Hence J' - J= za'Saidm + a^'2dn^ = d?M, since "Sixdm = o as the centre of gravity is at the origin ; .-./' = 7+ a'Jf. (2) Consequently, the moment of inertia of a body relative to any axis can be foimdwhen that for the parallel axis through its centre of gravity is known. Also, the moments of inertia of a body are the same for all parallel axes situated at the same distance from its centre of gravity. Again, it may be observed that of all parallel axes that which passes through the centre of gravity of a body has the least moment of inertia. It is also apparent that the same theorem holds if the moments of inertia be taken with respect to parallel planes, instead of parallel axes. A similar property also connects the moment of inertia relative to any point with that relative to the centre of gravity of the body. In finding the moment of inertia of a body relative to any axis, we usually suppose the body divided into a system of indefinitely thin plates, or lamince, by a system of planes perpendicular to the axis ; then, when the moment of inertia is determined for a lamina, we seek by integration to find that of the entire body. / Radius of Gyration. 293 197. Radius of ©yration. — If k denote tlie distance from an axis at which the entire mass of a body should be concentrated that its moment of inertia relative to the axis may remain unaltered, we shall have me = I=^fdm. (3) The length h is called the radius of gyration of the body with respect to the fixed axis. In homogeneous bodies, which shall be here treated of principally, since the mass of any part varies directly as its volume, the preceding equation may be written in the form where d V denotes the element of volume, and V the entire volume of the body. Hence, in homogeneous bodies, the value of k is indepen- dent of the density of the body, and depends only on its form. We shall in our investigations represent the moment of inertia in the form j-_ -jij-ji. and, it is plain that in its determination for homogeneous bodies xve may take the element of volume for the element of mass, and the total volume of the body instead of its mass. Also, in finding the moment of inertia of a lamina, since its radius of gyration is independent of the thickness of the lamina, we may take the element of area instead of the element of mass, and the total area of the lamina instead of its mass. 1 98. If ^ and B be the moments of inertia of an infi- nitely thin plate, or lamina, with respect to two rectangular axes OX, OY, lying in its plane, and if G be the moment of inertia relative to OZ drawn perpendicular to the plane, we ^^^^ C = A + B. (4) For, we have in this case A = 'S^y'^dm, B = 'S.x'dm, and C= ^{x'' + y'')dm. Again, for every two rectangular axes in the plane of the lamina, at any point, we have ^x^dm + ^y'dm = const. Hence, if one be a maximum, the other is a minimum, and vice versd. We shall, in all investigations concerning laminse, take C for the moment of inertia relative to a line perpendicular to the lamina. 294 Integrals of Inertia. igg. Uniform Rod, Rectangular IJamina. — We commence with the simple case of a rod, the axis being perpen- dicular to its length, and passing through either extremity. Let X be the distance of any element dm of the rod from the extremity ; then, since the rod is uniform, dm is propor- tional to dx, and we may assume dm = fxdx: hence, the moment of inertia Zis represented by n'SiX^dx, or by ^I'.V., -^ where I is the length of the rod Hence 7 = "-- = If -. 3 3 If the axis be drawn through the middle point of the rod, perpendicular to its length, the moment of inertia is plainly the same for each half of the rod, and we shall have in this case I=M-. 12 Next, let us take a rectangular lamina, and suppose the axis drawn through its centre, parallel to one of its sides Here, it is evident that the lamina may be regarded as made up of an infinite number of parallel rods of equal length, perpendicular to the axis, each having the same radius of gyration, and consequently the radius of gyration of the lamina is the same as that of one of the rods. Accordingly, we have, denoting the lengths of the sides of the rectangle by 2a and ib, and the moments of inertia round axes through the centre parallel to the sides, by A and B, respectively, A=-Mb\ B=-Ma\ (5) 3 3 Hence also, by (4), the moment of inertia round an axis through the centre of gravity and perpendicular to the plane of the lamina, is - M{a' + 6'). (6) By applying the principle of Art. 196 we can now find its moments of inertia with respect to any right line either lying in, or perpendicular to, the plane of the lamina. Circular Plate, Cylinder. 295 200. Rectangular Parallelepiped. — Since a parallel- epiped may be conceived as consisting of an infinite number of laminse, each of which has the same radius of gyration relative to an axis drawn perpendicular to their planes, it follows that the radius of gyration of the parallelepiped is the same as that of one of the laminte. Hence, if the length of the sides of the parallelepiped be 2a, 2b, and 2c, respectively ; and, if A, B, C be respectively the moments of inertia relative to three axes drawn through the centre of gravity, parallel to the edges of the parallel- epiped, we have, by the last, A = -M{h^ + c'), S = -M{c' + a'), C = -M{a'+b'). (7) 201. Circular Plate, Cylinder. — If the axis be drawn through its centre, perpendicular to the plane of a circular ring of infinitely small breadth, since each point of the ring may be regarded as at the same distance *• from the axis, its moment of inertia is r'dm, where dm represents its mass. Hence, considering each ring as an element of a circular plate, and observing that dm = fi2Trrdr,-w6 get for C, the moment of inertia of the circular plate of radius a, C=2 TTfi r'dr= -'—- = M- Oonsequently, the moment of inertia of a ring whose outer and inner radii are a and b, remeetively, with respect to the same axis, is /^ a' -¥ ^^a^ + V irdr = TTfi = M . '""^i J» 2 Again, by (4), the moment of inertia of a circular plate about any diameter is M—, since the moments of inertia are 4 obviously the same respecting all diameters. In like manner, the moment of inertia of a ring relative to any diameter is ^^a' + J' 296 Integrals of Inertia. Also, the moment of inertia of a. rigkt cylinder about its axis of figure is M-, a being the radius of the section of the cylinder. Again, the moment of inertia relative to any edge of tho cylinder is - Ma^. 202. Right Cone. — To find the moment of inertia of a right cone relative to its axis, we conceive it divided into an infinite number of circular plates, whose centres lie along the axis ; and, denoting by as the distance of the centre of any section from the vertex of the cone, and by a the semi-angle of the cone, we have 2 Jo 10 ' where h is the height of the cone, and b the radius of its base. Hence, since by Art. 169 the volume of the cone is - J'A, we have (8) i=^m\ 10 203. Elliptic Plate. — Next let us suppose the lamina an ellipse, of semi-axes a and h ; and let A and B be the moments of inertia relative to these axes, respectively. Describe a circle with the axis minor for diameter, and suppose the lamina divided into rods by sections perpendicular to this axis. Let B^ be the moment of inertia for the circle round its diameter. Then, denoting by dB and dB^ the moments of inertia of corresponding rods, we have dB-.dB^ {npY : (np'Y = {pay : {oby = a' : 6» ; .-. B:B'=a^:b\ Fig- 47. Sphere, f 297 But B', by Art. 201, is ; ^ M'a" M / 46 4 ilf Similarly, A= — V. Hence the moment C round a line tkrough the centre of the ellipse, perpendicular to its plane, is M f («' + i')- (9) It is plain, as hefore, that the expression for the moment of inertia of an elliptical cylinder relative to its axis is of the same form. 204. Sphere. — If we suppose a'sphere divided into an infinite number of concentric spherical shells, the moment of inertia of each shell is plainly the same for all diameters ; and accordingly, representing the mass of any element of a shell by dm, and by x, y, z any point on it, we have 'S.x'dm = "Siy^dm = '^z'dm. But 2(«' + y' + z')t?OT = Sr^^w; 2 .*. S («" + y') dm = -'2, r^dm. Hence, (a) the moment of inertia of a shell whose radius 2 is r with respect to any diameter is - mr'^, where m repre- sents the mass of the shell. Again, (6) for a solid sphere of radius H, since the volume ^ of an indefinitely thin shell of radius r is ^irr^dr, we get^ ^^- ; T- K r« ^r'^dv = 47r T' _ Z)5 _ -J T7-D2 i 5 5 "When this is substituted, the moment of inertia of a solid homogeneous sphere relative to any diameter is found to be -MR\ (10) 298 Integrals of Inertia. 205. Ellipsoid. — Let the equation of an ellipsoid be «' w" z' and suppose A, B, C to be the moments of inertia relative to the axes a, b, e, respectively ; then C = fi-^ix^ + f)dr = n\\\{x' + f) dxdyds. Now, let and we get C = fialc I {a^x'^ + 6V') dafdi/dz', where the integrals are extended to all points within the sphere x" + i/^ + z" = I. But, by the last example we have x'^'dx'dy'dz' = y"dsifdi/dz' = 15 •. C = -^ TTfiahc (a' + b') = — (a' + b'). 15 ^ 5 ^ ' (II) 1:1 like manner. M M 5 ' 5 It should be remarked that the moments of inertia of the ellipsoid with respect to its ^&q principal planes are — ffl% — 6% — c% respectively. Moments of Inertia of a Lamina. 299' 206. moments of Inertia of a IJaniina. — Suppose that any plane lamina is referred to two rectangular axes drawn through any origin 0, and that a is the angle which any right line through 0, lying in the plane, makes with the axis of » ; then, if I be the moment of inertia of the lamina relative to this line, we have / = 'Sip'dm = 2 (y cos a - a; sin aYdm = cos^a '^y'dm + sin^aS«'rfOT - 2 siu a cos a "S^xydm = « cos'a + 5 sin'a - 2A sina cosa; (12) where a and h represent the moments of inertia relative to the axes of x and y, respectively ; and h is the product of inertia relative to the same axes. Again, supposing X andFto be the co-ordinates of a point taken on the same line at a distance R, from the origin, we get cos o = -5-, sin a = -5- ; and, consequently, IR' = aX' + bY' - 2h XT. Accordingly, if an ellipse be constructed whose equation is aX' + bY' - 2AXZ= const., (13) we have IR' = const. ; and, consequently, the moment of inertia relative to any line drawn through the origin varies inversely as the square of the corresponding radius vector of this ellipse. The form and position of this ellipse are evidently inde- pendent of the particular axes assumed ; but its equation is more simple if the axes, major and minor, of the ellipse had been assumed as the axes of co-ordinates. Again, since in this case the coefBcient oi XY disappears from the equa- tion of the curve, we see that there exists at every point in a body one pair of rectangular axes for which the quantity h, or "Sixydm = o. This pair of axes is called the principal axes at the point ; and the corresponding moments of inertia are called the principal moments of inertia of the lamina relative to the point. 300 Integrals of Inertia. Again, if A and £ represent the principal moments of inertia, equation (12) becomes J= ^ COS'a + P sin'a. (14) Hence, for a lamina, the moment of inertia relative to any axis through a point can be found when the principal moments relative to the point are determined. The equation of the ellipse (13) becomes, when referred to the principal axes, AX" + BY" = const. 207. momental Ellipse. — Since the moments of inertia for all axes are determined when those relative to the centre ■of gravity are known, it is sufficient to consider the case where the origin is at the centre of gravity. With reference to this case, the ellipse AX'' + BY'' = const. (15) is called the momental ellipse of the lamina. Again, if two different distributions of matter in the same plane have a common centre of gravity, and have the same principal axes and principal moments of inertia, at that point, they have the same moments of inertia relative to all axes. This is an immediate consequence of (14). Hence it is easily seen that the moments of inertia for any lamina are M the same as for the system of four equal masses, each — , placed on the two central principal axes, at the four dis- tances + a and + h, from the centre of gravity, where a and b are determined by the equations A = -Mh\ B=-Ma\ 2 2 Again, if two systems of the same total mass, in a plane, have a common centre of gravity, and have equal moments of inertia relative to any three axes, through their common ■centre of gravity, they have the same moments of inertia for all axes. Momental Ellipse. 301 This follows immediately since an ellipse is determined when its centre and three points on its circumference are given. Again, it may he observed that the houndary of an elliptical lamina may be regarded as the momental ellipse of the lamina. For, if / he the moment of inertia relative to any diameter making the angle a with the axis major, we have But, hy Art. 203, I = A coa'a + B sin'a. A = ^l^ B = ^a^; 4 4 M .\ 1= — (6" cos'a + 0' sin' a) 4 ^ ^ M ,,,/cos'o sin" a' 4 V a* b ~ 4 r' ' Hence the moment of inertia varies inversely as the square of the semi-diameter r ; and, consequently, the ellipse may he regarded as its own momental ellipse. ' 208. Products of Inertia of IJainina. — Suppose the lamina referred to its principal axes at a point ; and let p and q be the distances of any element dm from two axes, which make the angles a and (5 with the axis of « ; then we have ^pqdm = 2 («/ cos a - a; sin a) (y cos /3 - « sin j3) dm = eosa cos j3 ^y'dm + sin a sinj3 ^ai^dm - sin (a + (5) "SiXydm = A Gosacos^ + B sin a sin j3, since A = ^y^dm, B = '2,a?dm, and "Sixydm = o. Hence, if '2pqdm = o, we have A cos a cos jS + 5 sin a sin]3 = o, 302 Iniegrak of Inertia. and accordingly the axes are a pair of conjugate diameters of tlie momental ellipse AX^ + SY^ = const. Hence, if two laminae in the same plane have for any point two pairs of axes for which ^pqdm = o and ^p'^dm' = o, they have the same principal axes at the point. Tlus follows from the easily established property, that if two ellipses have two pairs of conjugate diameters in common, they must be similar and coaxal. 2og. Triangular JLamina and Prism. — Suppose a triangular lamina, whose sides are a, b, c, to be divided into a system of rods parallel to a Side a ; j^ and let A represent the moment of inertia relative to a line parallel to the side a, and drawn through the opposite vertex; also let p be the perpendicular of the triangle on the side a, and x the distance of an olementary rod from the vertex; then we have, since the mass dm of the ■elementary rod may be represented by /x — dx^ Jr dec A = Jix'dm = u.'SiK' — dx P y^ X \ y^ p \ Fig. 48. a?dx = u — 4 M In like manner, let B and C be the moments of inertia relative to lines drawn through the other vertices parallel to h and c ; and let y, r be the corresponding perpendiculars of the triangle, and we have i>=¥f, C = M Again, if A^, B^, Co, represent the moments of inertia Triangular Lamina and Prism. 303 relative to three parallels to tlie sides, drawn through the centre of gravity of the lamina, we have, by (2), ^0 = -^ Mp\ 5„ = ^ Mi\ Co = ^^MrK . (16) Also, if Ai, Bi, Ci, he the moments of inertia relative to the sides a, b, c, respectively, it follows, in like manner, from (2), that A.-^^Mp', B,= ^-Mq\ C, = ^Mr\ (17) Again, it is readily seen that the values of A, An, Ai, &c., are the same as if the whole mass M were divided into three equal masses, placed respectively at the middle points of the sides of the lamina. Consequently, by Art. 207, the moments of inertia of the triangular lamina relative to all axes are the same as for M three masses, each — , placed at the middle points of the sides of the triangle. Hence, if I be the moment of inertia of a triangular lamina with respect to the perpendicular to its plane drawn through its centre of gravity, we have 7=^ilf(«^ + 6' + 0. (18) This expression also holds for the moment of inertia of a right triangular prism with respect to its axis* In like manner the moments of inertia of the triangular lamina relative to the three perpendiculars to its plane, drawn through its vertices, are i*(5H.-f), '-m{,.,'JS), iif(.-.»--f)i and the same expressions hold for a triangular prism relative to its edges. • By the axis of a prism is understood the right line drawn through its centre of gravity parallel to its edges. 304 Integrals of Inertia. 2IO. Momental Ellipse of a Triangle. — It can be shown witliout difficulty that the ellipse which touches at the middle points of the sides may he taken for the mo- mental ellipse of the triangle. For, let X, y, z be the middle points of the sides, and it is easily seen that o is the centre of this ellipse ; also, if I, It, Is be the moments of inertia of the ^^' "*'' lamina relative to the lines ax, hy, cz, respectively, it can be readily shown from (17), that we have / I: I,: J3 {axy ' {byf ' {czy {oxy • {oyf • {ozy Accordingly, by Art. 207, the ellipse xyz may be taken for the momental ellipse of the lamina. 211. TTetrahedron. — If a solid tetrahedron be supposed divided into thin laminae parallel to one of its faces, and if A, B, C, I) represent its moments of inertia with regard to the four planes drawn respectively through its vertices parallel to its faces ; then, denoting the areas of the corre- sponding faces by a, b, c, d, and the corresponding perpen- diculars of the tetrahedron by p, q, r, s, respectively, it is easily seen, as in Art. 209, that we shall have ^2 a rp A = '2x^dm= u.'2,a?a—dx = u.-i\ x*dx In like manner we have V Solid Ring. 305 Again, if Ao, Bo, Go, Do te the corresponding moments of inertia relative to the parallel planes drawn through the centre of gravity of the tetrahedron, we have, hy (2), Ao = -^Mp\ Bo 80 Mq\ Co=i-^Mr% Bo ■.^MsK (19) Also, if Ai, Bi, Ci, Di be the moments of inertia relative to the four faces of the tetrahedron, we have A, = — Mp\ B,= — Mq\ 10 10 Ci = — Mr\ Di = — M^. (20) 10 10 ^ ' 212. Solid Ring.* — If a plane closed curve, which is symmetrical with respect to an axis AB, he made to revolve round a parallel axis, lying in its plane, hut not intersecting the curve, to prove that the moment of inertia I of the generated solid, taken with respect to the axis of revolution, is represented by Mih' + 3F), where M is the mass of the solid, h the distance between the parallel axes, and It the radius of gyration of the generating area relative to its axis. For, if the axis of revolution be taken as the axis of x, and, if y, Y be the distances of any point P within the generating area from AB, and from OX, respectively ; and, if dA be the corresponding element of the area, then the volmne of the elementary ring generated by dA is 2n- YdA, and its mass 27rju TdA ; hence the moment of inertia of this elementary ring, relative to the axis of X, is zwfiY^dA. Accordingly, we have 1= 2iTpL'2,Y^dA = 2irn'2{h + yYdA = 27r/iS (A3 + 2,h^y + iJi'f + y^) dA. * The theorems of this Article were given by Professor Townsend in the Quarterly Journal of Mathematics, 1869. [20] 306 Integrak of Inertia. Moreover, since the curve is symmetrical with respect to the axis AB, it is easily seen that we have "SiydA = o, ^y'dA = o. Also, by definition, ^y'dA = AF. Hence 1= 2iriihA{h^ + z¥). Again, by Art. 177, M= zirfihA ; .: I=M{h'' + 3k'). (21) This leads immediately to some important cases. Thus, for example, the moment of inertia of a circular ring, of radius a, round its axis is mU' + ^J). Again, if a square of side a revolve round any line in its plane, situated at the distance k from its centre, we have There is no difficulty in adding other examples. 213. Creneral Expression for Products of Inertia. — We shall conclude this Chapter with a short discussion of the general case of the moments and products of inertia, for any body, or system. Let us suppose the system referred to three rectangular planes, and let p, q, r represent the respective distances of any element dm from the three planes « cos a + y cos/3 + s cos 7 = o, X cos a + y cos |3' + s cos y = o, ai cos a" + y cos j3" + 2 cos 7" = o. Then Sfij'£?»w=S (ascosa+ycosjS+zcosy) (aJCOso'+ycos/S'+zcosy')*^ = cosacoso'2ar'cfm + cos|3cosj3'S2''*^''* + cos7 00S7'Sz'rfOT + (cos a cos |3' + cos |3 cos a) "Sixydm + (cos 7 cos a + cos a cos 7') ^zxdm + (cos /3 cos 7' + cos 7 cos /3') ^yzdm ; and we get similar expressions for ^prdm and 'S.qrdm. Principal Axes. 307 Now, suppose that we take ^x'dm = a, '^y^dm = 5, "^^dm = c, "Lt/zdm^f, "Sixzdm = g, "Sixydm = A ; then the preceding equation may be written ^pqdm = cos a {a cos a + h cos fi' + g cos 7') + cos /3 (A cos a' + 6 cos |3' + / cos 7') + cos 7 {g cos a'+/ cos |3' + c cos 7') ; {22) along with similar expressions for ^rpdm and 'Sqrdm. 214. Principal Axes. — Next, let us suppose that the planes are so assumed as to satisfy the equations ^pqdm = o, 'Srpdm = o, ^qrdm = o ; then it is easily seen* that these planes are a system of con- jugate diametral planes in the ellipsoid represented by the equation aX^ + bY^ + cZ' + 2fYZ + zg!^ + zhXY = const. (23) Hence it follows that at any point there exists one system of rectangular planes for which the corresponding products of inertia, for any body, vanish : viz., the principal planes of the preceding ellipsoid.f These three planes are called the principal planes of the body relative to the point, and the right lines in which they intersect are called the principal axes for the point. Again, every two solids have for every point at least one common system of planes for which "Sipqdm = o, ^rpdm = o, 'Sqrdm = o, ^p'q'dm' = 0, 'Sr'p'dm' = o, ^q'r'dm' = o; where the unaccented letters refer to the elements of one solid, and the accented to those of the other. This is obvious from the property that every two con- centric ellipsoids have one common system of diametral planes. *. Salmon's Geometry of Three Dimensions, Art. 72. + The exceptional cases when the eUipBoid is of revolution, or is a sphere, will be considered suhsequently. [20 a] 308 Integrals of Inertia. Again, if two solids have for any point more than one system of planes for which the foregoing six products of inertia vanish, they must have the same principal planes at the point. This follows since the two ellipsoids in that case must be similar and coaxal. 215. Principal moments of Inertia. — Let us now suppose the co-ordinate planes to be the principal planes of the body for the orig^, then the moment of inertia relative to the plane a; cos a + y ops /3 + z cos 7 = o is 'S.p^dm = S (» cos o + y cos /3 + z cos yYdm = oos^ a'S.a? dm + oos'/3 ^y^dm + cos'y 'Si^dm, (24) since in this case we have ^xydm = o, ^zxdm - o, "Zyzdm - o. Again, let I be the moment of inertia of the body relative to the line through the origin whose direction angles are "> i3> 7 ; then we have I + ^p''dm = 'Sr'dm = 'Si{x' + y' + z') dm ; .'. / = cos" o S (y' + s') dm + cos" j3 S (z" + a^) dm + cos"7S(a^ + ^")cfm; or 1= A cos' a + B cos'jS + C cos''7, (25) where A, B, C are the moments of inertia of the body relative to its three principal axes. A, B, Care called the three principal moments of inertia of the body relative to the origin. If the centre of gravity be taken as the origin, the corresponding values of A, B, C are called the principal moments of inertia of the body. We suppose, in general, that A is the greatest, and the least of the three principal moments. It follows from (25) that the moment of inertia of a body relative to any line passing through a given point is known, whenever the angles which the line makes with the principal axes are known, as also the moments of inertia relative to these axes. Momental Ellipsoid. 309 2 1 6. Ellipsoid of Cryration. — Suppose, as before, the solid referred to its three principal axes at any point, and let a, b, c be the corresponding radii of gyration, i.e. let A = Ma\ B = Mb\ C = Mc\ and I = M¥; then equation (25) becomes F = a" cos' a + 6' cos'jS + c'cos^'y. (26) Now, if we suppose an ellipsoid described having the principal axes for the directions, and a, b, c for the lengths of its corresponding semi-axes ; then (26) shows that the radius of gyration of the body, relative to the perpendicular from the origin on any tangent plane to this ellipsoid, is equal in length to this perpendicular. (Salmon's Geometry of Three Dimensions, Art. 89.) The foregoing ellipsoid is called the ellipsoid of gyration relative to the point. It should, however, be observed that by the ellipsoid of gyration of a body is meant the ellipsoid in the particular case where the origin is at the centre of gravity of the body. 217. Momental Ellipsoid. — If X, Y, Z be the co- ordinates of a point H taken on the right line through the origin 0, whose direction angles are a, /3, y, we have X=ORcQ%a, Y=OE co&fi, Z=OB cosy. Substituting the values of cos «, cos j3, cos y, deduced from these equations, in (25), it becomes 1 . 0B? = AX^ + BT' + CZ\ Suppose, now, that the point R lies on the ellipsoid AX^ + BY'' + CZ^ = const., (27) and we get I . OR' = X, denoting the constant by X ; OR'' (28) Hence the moment of inertia relative to any axis, drawn through the origin, varies inversely as the square of the cor- responding diameter of the ellipsoid {2"]) . 310 Integrals of Inertia. From this property the ellipsoid is called the momenta! ellipsoid at the point. When the origin is taken at the centre of gravity of the body, this ellipsoid is called the central ellipsoid of the body. If two of the principal moments of inertia relative to any point be equal, the momental ellipsoid becomes one of re- volution, and in this case all diameters perpendicular to its axis of revolution are principal axes relative to the point. If the three principal moments at any point be equal, the ellipsoid becomes a sphere, and the moments of inertia for all axes drawn through the point are equal. Every such axis is a principal axis at the point. For example, it is plain that the three principal moments for the centre of a cube are equal, and, consequently, its moments of inertia for all axes, through its centre, are equal. 218. £quinioinental Cone. — Again, since cos'a + cos'/B + cos'7 = I, equation (25) may be written in the form {A - I) cos'a +(-»-/) cos»/3 + (C- I) cos'y = o ; hence the equation {A - I) X^ + [B -I)Y'+{0-I)Z' = o (29) represents a cone such that the moment of inertia is the same for each of its edges. Such a cone is called an equimomental cone of the body. Again, the three axes of any equimomental cone, for any solid, are the principal axes of the solid relative to the vertex of the cone. When 1= B, the cone breaks up into two planes; viz., the cyclic sections of the momental ellipsoid. For a more complete discussion of the general theory of moments of inertia and principal axes, the student is referred to Eouth's Rigid Dynamics, chapters i. and 11. ; as also to Professor Townsend's papers in the Camb. and Bub. Math. Journal, 1846, 1847. Examples. 311 Examples. Find the expressions for tie moments of inertia in the following, the hodies being supposed homogeneous in all cases : — 1. A parallelogram, of sides a, *, and angle 8, with respect to its sides. M M Ana. — i' sin' fl, — a? sin' 8. 3 3 2. A rod, of length a, with respect to an axis perpendicular to the rod and at a distance d from its middle point. Jns. M(- + dA. J. An equilateral triangle, of side a, relative to a line in its plane at the distance rf from its centre of gravity. Ani. JSff'^ + dA. 4. A right-angled triangle, of hypothenuse c, relative to a perpendicular to Its plane passing through the right angle. Ans. M—. 5. A hollow circular cylinder, relative to its axis. f2 ^ y'S Ans. M , where r and r' are the radii of the hounding circles. 2 6. A truncated cone with reference to its axis. Ans. - — TT ;;, where i and b' are the radii of its bases. 10 42 - 4 ' 7. A right cone with respect to an axis drawn through its vertex perpen- dicular to its axis. ■iM I S'\ Ans. - — I A'' H — I , where h denotes the altitude of the cone, 5 \ 4/ and h the radius of its base. 8. An ellipsoid with respect to a diameter making angles a, ;8, 7 with its axes. Ans. — (a'sin'o + i'sin'jS + c'sin'7) . g. Area hounded by two rectangles having a common centre, and whose sides are respectively parallel, with respect to an axis through their centre perpendicular to the plane. Ans. —- , ,,, • . 12 ab — a b 312 Examples. 10. A square, of side a, relatiYe to any line in its plane, passing through its centre. Ans. M—. 12 11. A regular polygon, or prism, with respect to its axis. Ans. ~[lt^i-2rA, where Jt and r are the radii of the circles cireumsciibed, and inscribed to the polygon. 12. Prove that a parallelogram and its maximum inscribed ellipse hare the same principal axes at their common centre of figure. 13. Prove that the moments and products of inertia of any triangular M lamina, of mass M, are the same as for three masses, each — , placed at the 3 " • three vertices of the triangle, combined with a mass - M placed at its centre of 4 gravity. 14. Prove that the moments and products of inertia of any tetrahedron are M the same as for four masses, each — , placed at the vertices of the tetrahedron, 4 ^° combined with a mass - M placed at its centre of gravity. 15. If a system of equimomental axes, for any solid, all lie in a principal plane passing through its centre of gravity, prove that they envelop a conic, having that point for centre, and the principal axes in the plane for axes. 1 6. Prove also that the ellipses obtained by varying the magnitude of the moment of inertia form a confocal system. 17. Prove that the sum of the moments of inertia of a body relative to any three rectangular aixes drawn through the same point is constant. 18. Prove that a principal axis belonging to the centre of gravity of a body ia also a principal axis with respect to every point on its length. 19. Prove that the envelope of a plane for which the moment of inertia of a body is constant is an ellipsoid, confocal with the ellipsoid of gyration of the body. 20. If a system of equimomental planes pass through a point, prove that they envelop a cone of the second degree. 2 1 . For different values of the constant moment the several enveloped cones are confocal ? 22. The common axes of this system of cones are the three principal axes of the body for the point ? 23. The three principal axes at any point are the normals to the three sur- faces confocal to the ellipsoid of gyration, which pass through the point. (M. Binet, Jour, de I'Ec. Tob/. 18 13.) ( 313 ) CHAPTEE XT. MULTIPLE INTEGRALS. 219. Double Integration. — In the preceding Chapters we have considered several cases of double and triple integra- tion in the determination of volumes and other problems connected with surfaces. We now proceed to a short treat- ment of the general problem of Multiple Integration, com- mencing with double integrals. The general form of a double integral may be written r.y C7 f{x,y)dxdy, in which we suppose the integration first taken with respect to y, regarding a; as constant. In this case, Y, y^, the limits of y, are, in general, functions of x ; and the limits of x are ■constants. For example, let us take the integral U- ■a (X ,1-1 4,m-i / x'-^y'"-^dxdy, in which I is supposed greater than m. a' therefore l7= — \ «'-'-—-«'"«&;= ,, It should be observed that in many cases the variables are to be taken so as to include all values limited by a certain •condition, which can be expressed by an inequality : for in- stance, to find U= gi-l ym-\ (Ixdy, extended to aU positive values of x and y subject to the con- dition X + y < h. 314 Multiple Integrals. Here the limits for y are o and h-x; and the subsequent limits of X are o and h. Hence U= h-x iff'^y'^-^dxdy I m x^^ {h - x)'"dx. Let X = hu ; then h P- U= u^U\-uYdu = h^<«r{i) r(m) T{l + m + l) ' (0 by Art. 121. 220. Change of Order of Integration. — We have seen (Art. 115) that when the limits of x and y are constants we may change the order of integration, the limits remaining unaltered. But when the limits of y are functions of x, u the order of integration be changed, it is necessary to find the new limits for x as functions of y. This is usually best obtained from geometrical considerations. For example, in the integral U- f{x, y) dxdy, the limits for y are given by the right line y = x and the hyperbola xy = a^; and the integral extends to all points in the space bounded by the axis of y, the hy- perbola AL, and the right line OA, where A is the vertex of the hyper- bola. Draw AB perpendicular to the axis of y. Now when the order of integration is changed, we sup- pose the lines which divide the area into strips taken parallel to the axis of X instead of that of y. Thus the integral breaks up into two parts — one corresponding to Change of Order of Integration. 3 15 the triangle OAB, the other to the remaining area : hence /(«> y) Ay^oi + f{p, y) a Jo dydx. As another example, let us interchange the variahles in the integral XI- ■a rlx- Vdxdy. II Jmz Here, let 00 and OD be the lines represented by y = & and y -mx; and let OA = a. Then the integral is extended to all points within the triangle OGB. Accordingly,changing the order, o' we get U= Vdydx + Vdy Jy dx. A X Examples. (. find the value of the double integral Jo J "^ •J [a - x)[x-y) Here, changing the order, the integral Tbeoomes f{y)dydx J J » V(a - X) But dx •^{a - x)(x - p) V(a -x){x- = IT ; hence U=it {/{«) -/(o) } . 2. Prove that rill ('° Niax-i' ra ra*\/it'-y' I f{x, y)dxdy=\ f{x, y) dyd: Jo ia-\la^-v* ^a-ya^-y* 316 Multiple Integrals. 3- Hence find the value of •'ft •'n Ans. ira" { u and W + m) Jo In like manner it is seen that if the multiple integral U= [[[f [ f-Y+ frY+ f-TI 0^-' y'^^ s"-' (?a^cf«/^2 be extended to all positive values, subject to the condition x\» (y\i (z\ we have U a'h'^o'' \p + V + n^lrl^ r^'* pqr \^ ^ ? ^ »•/ F{u) uf'i •■" du. (7) These results can be readily extended to any number of variables. Examples. I. Find the value of II x\-^ iir^ e^ dxch), extended to all positive values, subject \ax-Vy therefore -r- -^ -r- = r' sin ; dr dd djt and for the element of voliune dx dy dz we substitute »•' sin dr d6 d • \ ■> ■> «'2' y = >• sm A -v/ Jw'' + rt cos = sm d hw r + „ , ^ '^ ^\ cos^^ = tan ^ \/m^r^ cos"^ + «'s^ Hence jw^r^sin'^ -77 = seo'0 ,/m' »•" cos" A + w" s' 7= „ . (i ^^ ^ ymVoos^^ + w^'s' _ w'y' cos'^ + w'g' seo'^ _ r (ot' cos' + w' cos' 6) ^ •v/mV^o?^TwV cos v^ffi' + «' cos" 6 ' -—: = -r Bind cos d>. dti ^ and, finally, Hence for dx dy dz we substitute y' (ot" cos' j> + »' sin' (^) • sin 9 cos ^, w = »• sin 9 sin 1)/. The transformed expression is Ki Tic^ sin 9 cos edrdedijid^, where Vi is the new value of V. , „ «2 «3 M3 «1 «1 «2 2. It Xl = , «3 = ■, Xi = -, «1 Mj Mj prove that JJJ" Vdxi dx% dxa transforms into 4 JJJ Vi dui dU2 dut. 226. We shall next prove that 'fdu dv dw". , , , Tx^di'^lTz^^'"^^'^^^ U*0 lAitJ Uffi {u cos A + » cos /i + tc cos v) dS, where the integrations, respectively, extend over a closed surface 8, and through the volume contained by the surface : X, ju, V being the direction angles of the outward drawn normal at dS, and u, v, to being functions of x, y, z, which are supposed finite and continuous for all points within S. Here, since j8 is a closed surface, any intersecting right line meets it in an even number of points ; consequently ' du -7- dx du dz dx dydz'Si {Ui - Ml), where Mi and Mj represent the values u for two corresponding points of intersection with S, made by an indefinitely thin Greenes Theorem. 327 parallelepiped standing on dy d% ; and S denotes the summa- tion extended to all such points of intersection. Now, as in Art. 192, let dS^, dSi, dS^, &o., represent the corresponding elementary portions of the surface ; and Ai, Xj, A3, &o., the angles that the exterior normals make with the positive direc- tion of the axis oi x; we shall have dydz=- cos Ai dSi = cos A2 dSa = - cos A3 dS, = cos Ai dSi = &o. Accordingly, dx dx dy dz ■■ ucosXdS, under the same restrictions as to limits as before, follows immediately that (15) Hence it \\\ du do dw\ , , , ^- + -p + ^- dxdydz = dx dy dzj {uoos\-i-voosfi+woosv)dS. (16) This result obviously holds good when the triple integral is extended through any space which is bounded externally by one closed surface and internally by another, provided the double integral is extended over both the bounding sur- faces. Again, if for u we substitute u V, for v, v V, and for w, tv V, we get immediately IV + ■ m/ dv dv dv\_, ^ ^ W-d^^'di^"^}'^''^^'^' V{u eoa\ + V cos fj. + to cos v) dS - Hi dx dw dy de (17) under the same restrictions as above. 227. Oreen's Theorem. — We shall now give a brief notice of the very remarkable theorem given first by Green ("Essay on the application of Mathematics to Electricity and Magnetism," Nottingham, 1828, reprinted, 1871), as fol- lows : — If Z7and Fbe functions of x, y, z, the rectangular coordi- nates of a point ; then, provided U and V are finite and con- 328 Multiple Integrals. tinuousfor all points within a given closed surface S, we have fdUdV dUdV . dUdr\ dx dx dy ' d^U d^U d^U\_, ^ ^ + -jY ) ^^ ">!/ ^^ where the triple integrals are extended to all points within the surface S, and the doutle integrals to all points on S ; and dn is the element of the normal to the surface at dS, measured outwards. Hr ■ ^(n^I\=^JL^^ TJ^Z ' dx\dx)dx dx dx^ ' we have -;- ( U -r- 1 dx dy dz ■■ dx\ dx J I dUdV, _, _, — — r- »* "y "2 dx dx U-r-r dx dy d&, dxr the integrals heing extended to all points within 8. Again, by (15), we have dx U -;— ] dx dy dz = dx J U -r- ooa\ dS, dx under the same restrictions as to limits as before. Hence [dUdV_ dx dx dxdydz ■■ TJ-r- COS XdS dx -w TJ—rr dx dy dz, dx- along with corresponding equations for y and z. Green's Theorem. 329 Accordingly, -l\ [fdUdV dUdV dU dV\ , , , -; T- + —, r- + —■, 7- dx ay dz \dx dm dy dy ds dz j [„(dr . dV dV \,„ IJ\ -r- cos A + -r- cos u + -7- cos v do \djs dy dz J ^fd'V d'V d'V\^ , - Again, we obviously have dx dy ds cosA = -T-, cosu = -— , cos V = -7- ; dii dii dn ' ,, , dV . dV dV dV therefore -r- cos A + ^— cos u + -— cos 1/ = -7- . dx dy dz dn Hence nfdUdV dUdV dU dV\ ^ ^ ^ 1 1 — — r- + — + -r- -J— 1 dx dy dz \dx dx dy dy ds dz u'-^dS- dn v'^ds- dn U d'V d'V d^V dx' ^w''i^)'^"^y^')- ('S) ^^(d'U d'U d'U\^ ^ ^ The latter expression is obtained by the interchange of J/" and V in the preceding. If Cr=F, weget (V?J+(?J^(?''J^'^^'^' J-S--1J \ dx' dy" da dxdydz. (19) 330 Multiple Integrals. If, as in Biff. Calc, Art. 332, we denote dW d'V d'V . .^ dx" then equation (18) may be written in the following abridged form : — u'J:-vf\ds^ dn dn {Uv'V- Vv'U)dxdi/dz. (20) 228. Case where Z7 becomes Infinite. — We shall now determine the modification to be made when one of the func- tions, U for example, becomes infinite within S. Suppose this to take place at one point P only : moreover, infinitely near this point let V" be sensibly equal to - where r is the dis- tance from P. If we suppose an indefinitely small sphere, of radius a, described with its centre at P, it is clear that (18) is applicable to all points exterior to the sphere ; also, as 'd' d^ d^ dx' dy^ dz' U — dS, due to the sur- it is evident that the triple integrals may be supposed to ex- tend through the entire enclosed space, since the part arising from points within the sphere is a small quantity of the same order as a". Moreover, the part of face of the sphere, is indefinitely small of the order of the radius a. It only remains to consider the part of V — dS due to the spherical surface. Here, as V is supposed to vary conti- nuously, we may take for its value that { V) at the point P : also dJJ dn dU dr dr I T riTT consequently the value of V — d8, over the sphere, -4tF'. IS Integration through External Space. Thus (20) becomes in this case 331 w u'-Tds- j.^-d'V d'V d'V\, , , vfds- dn (21) ■where, as before, the integrals extend through the whole volume and over the whole exterior surface. The same method will evidently apply however great may be the number of points, such as P, at which either Z7 or V becomes infinite. 229. Integration through Kxternal Space. -^Let us next suppose a surface Si drawn inclosing another surface /Si, and let Green's theorem be applied to the space between these surfaces, we get dUdV dUdV dUdV\^^^ ^^ dx dx dy dy dz dz j U -7- dSi - u'^J-ds.- U^^Vdxdyds. Let us now suppose 82 to be a sphere of indefinitely great radius ; then, provided the double integral U ——dSi an become evanescent, Green's equation can be applied to in- tegration through the infinite space outside Si, as well as through the finite space within it. Moreover, since in this case w U -y- dSi = dn U ^r" sin 6 d9 dA, dn ^ we see that the double integral vanishes whenever r' U becomes evanescent when r is indefinitely increased; i. e. when- dV dn 332 Multiple Integrals. ever UV is of lower degree than - i in the coordinates ; a property which holds good in all physical applications of Green's theorem. 230. Application to Spherical Harmonics. — We shall conclude by establishing a few fundamental properties of spherical harmonic functions. In Green's theorem let U = Vi, V = Vm, where Vi is a solid harmonic (Diff'. Calc, Art. 333) of the degree /, and Vm another of the degree m. Now suppose a sphere of radius a, taken as the bounding surface 8, then equation (21) becomes Next, let Vi = r^ Yi, Vm = r'^T^, so that Yi and F,„ are surface harmonics {Diff'. Calc. Art. 334) ; then -'"-• Ym. dVi dVi jj.^^ ,dVm ~r- = -r- = Ir Yi, and — ; — = mr'' an dr dn Hence, since r = a over the surface 8, equation (22) be- comes ma' hm-\ or FiF„c?S=fe'+™-' Q-m) YmYid8, YiY,nd8 = o. Accordingly, so long as I and m are unequal, we have YiYmdS^o, (23) where the integration is extended to all points on the surface of the sphere. This may be written YiY,„dfid^ = o, (24) Jo J-1 adopting the usual notation {BiJ^. Calc., Art. 336). Application to Spherical Harmonics. 333 If we EulDstitute Pi and Pm for Ti and Ym, we get, so long as I and m are unequal, or j j PiP,„dfid^ = o, rPiP„d^^o, (25) since Pi and Pm are functions of fi only. Again, we have pTT r+i YiLmdnd^ = o, (26) where i™ is Laplace's coefiaoient of the »»'* order {Biff. Calc, Art. 337). 231. We can now find the value of TmLmdud^. For, let P be the point x, y, s, and P' the point x', y, z , then, since ^p> satisfies the equation vM -pw) = o, we have from (21), assuming ;8 to be a sphere of radius r, I d PP ?irm-Vml[-^)]ds = 4.rw (27) in which we suppose P' situated inside the sphere S. Again {Biff: Calc, Art. 337), I I "='=°Z„/" = - + S pjy «=i r ■n+i ' (28) hence dr\PF)~ r' "=i r"'' ' also, since Vm = »''"i^m, 'we have dV, dr = /wr*"-' F™ 334 Multiple Integrals. Sutstituting in (27), and observing from (26) that Ym Ln dfl ( except when w = m, we get = o {2m + 1) ■^m -^ m .„,^„-^clS=4Trr''"Y'^, or 21T- I'+l Lm T,„ dfi I J-i 47r 2OT + I Y' (29) where Y'm is the value that Ym assumes at the point P', i. e. when fi ^ fi, = = —^p', zm + I since all the other definite integrals vanish identically. Hence, since ao = -P'm {DiS Calc, p. 428), [PmYdix = Consequently, as Pm = zm + I I Id (35) (iu'-i)", we 2"* \m \diij Again, if we substitute for L^. in (33), we get in like manner a, (cos s^ TJ')y dfi dn', also D represents 336 Multiple Integrals. Here, by Leibnitz' Theorem {Biff. Calc, Art. 48), the general term in the development of D™^ [{x - a)'" {x - b)"'] is of the form im + n)im + n-i) . . .[m + n-r+ 1) ^ ^ , . ^ , „ V LS ' ^ L J)mH,-r^^ _ ^)m_ J)r^j. _ ^m. Moreover, as this is evanescent so long as r is less than n, we can assume r = n + p, and the preceding may be written \m+ n D*"-^ (a; - a)"" . I)'"'p{x - 5)*", \n + p \m —p \m+n \m \m or, ; — ~ f— -i !=^= (x-ay(x-b)'»-"-P. \n + p \m- p \p \m-n -p ^ ' ^ ' Accordingly, the expression {w-aY{x- J)» i)"^" ( {x - aY {x - b)"' } p=m-n \m+n \m \m = S i !=== \= I — !== {x-aY*P{x-b)'^. (39) ^"=0 \n +p \ni-p \p \m-n- p ' ^ ' * ' Again, the general term in D-^ {(»-«)•»(» -6)") may be written \m-n \p \m-n- p \m-n \in \m D^i-n-P (a, _ «)«• . 2)P(a; - J)» - (x-a)'^{x-b) m~p \p\m-n — pUi + p \m - p Comparing this with (39), the theorem in (37) follows im- mediately. Application to Spherical Sarmonics. 337 Again, if we substitute fi for x, s for n, and make J = i, « = - I, this result can he written in Rodrigues' form, viz., \m + s {n" - i)» D™" (ft" - if = ^^= Z)"-' (u' - i)". (40) Hence, since ■we get ^ T W = L^ + ^ (m'-i)''.Z)'»-V/x'-i)'» Iff* - s 2"\m^ Consequently, multiplying the two expressions, \m, + s 1 \m - s (2'"\my >-- ' y- / Therefore, Again, integrating by parts, and observing that the term outside the sign of integration vanishes for either limit, ■we get hence, by successive integration by parts, ■we get finally r B'^^'ifi' - iYB"'-'[f^ - lYdn " [22] 338 Multiple Integrals. Consequently \m -i- s Hence, &om (36), \m -^ s I ^ ' \ m~s J,/ ' '^ ^ ' \ m- s 2OT+I (41) 2 1 JW - s ffl, = (- i)' L COS s^' r„'W ; (42) L m + s and the complete expression for Lm can te immediately written down. (Compare Biff. Caic, p. 428.) 234. Expansion of a Function in Spherical Har- monics. — We next proceed to prove that any function f{fi, f), which is finite and continuous, can be expanded in a series of spherical harmonics, i. e. that /(;«, 0) = ro + Fi+ Fa + .. . + F„ + &c. (43) For if we assume this result, multiply both sides of the equation by Ln, and integrate, we get, from (26), r2ir r+l (•2ir r+1 .f{iH,^)Lndixdi^=\ Y„Lndfid^ Jo J-i Jo J-1 = Fn, from (27), 2«+ I ' ^ '" Again, writing ^', ^' for ;u, ^ in (43), we have /( /, 0') = F„' + F/ + F/ + . . . + F„' + &c. I «=oo r27r r+l ^ 4^ ^„=x ■^''' "^ 'Mo J -Z^"' *^ ■^" ^^^ ^'^' ^4^) Expansion of a Function in Spherical Harmonics. 339 We shall verify this result by proving that /(/*', ^') is the limit of the expression at the right hand side of (44) when « is increased indefinitely. Por, suppose h= — ; then since, by hypothesis, / is less than r, equation (28) may be written T = Lo + hLi + h^Li + ...+K'Ln + ..., (45) [i-zhX + K'f where A = cos PCF = fill +yi - fi' '/i- fi'^ cos { - )LndS {i-h')f(fi,^)d8 {i-2h\ + h^)i , (47) where the integrals are supposed to be extended over the surface of a unit sphere, of which dS is an element. Hence we infer that n=Qo rr is the limiting value- of '(^^^^i^^when;^=i. Again, when i - A is indefinitely small, the coefficient of [22 a] 340 Multiple Integrals. every element in the latter integral is indefinitely small except those for which (i - 2h\ + ¥)^, or PP", is indefinitely small, i. e. for — ^P which the point P is taken in- definitely near to the point A on the sphere. Consequently the integral has ultimately the same value as if it were only taken over a very small portion of the surface around the point A ; but throughout this portion we may assume/{/u, 0) =f{n', ^'), namely, its value at the point A. Hence the limiting value of Fig- S3- (i - 2h\ + h')^ /(i"',^') J(i- (i -h')d8 (i -2hX+h')i when A = I. Again, since X = cos ACP, we may write rfX di for dS, where ^i is the angle the plane ACP makes with a fixed plane drawn through CA, and we have }](i-2hX + h')i~]o J-(i -; "rrn 2k\ + h^)i d\ {i-2hX + h')^ Accordingly, for all values of h, (i - h') d8 4ir Hi .= 4T, (i -2h\ + h'')^ when taken over the surface of a unit sphere ; and we con- clude that 477 (2n + l) J J yO«, ^) Lndfid,l> =/(/, i,'), (48) thus verifying equation (44). This is the well-known general formula of Laplace ; from which we infer that every fimte continuous function of n and ^ Expansion of a Function in Spherical Harmonici. 341 can be expressed in a series of Spherical Harmonics. There is no difficulty in showing that the series is unique : i. e. that a given function can only be expanded in one way in a series of Spherical Harmonies. 235. It may be observed that the determination of the value in ^herical harmonics of a given function of 9 and ^ is usually best obtained by means of the corresponding solid harmonic functions. We shall illustrate this by an example. To transform «* = cos 6 sink's sin'^ cos ) wdw. These are deduced from (i) by differentiation under the sign of integration. 3. Show that the integral P' = 11/ (if + y) «'"^ y"'''^ ^x dy, supposed extended to all positive values suhjeot to the condition a; + y < i, can be reduced to a single definite integral, by the substitution x = im, y = «(i —■!'). Hence x + y = u, and dx dy becomes udv dv ; also the limits for u are o and h, and those for v are o and i ; hence T7= \ \ /(«) m'*™-! «)"{* - v)<^-^du dv Jo Jo _T{I)V{m)[^ — H^ /(«) u'*«'-^d!t. (Compare Art. 221). + »«) Jo r(l + m) Examples. 343 4. Show that the foregoing process can be extended to the integral V = ^lif{x + y + a) ic'-i y"--^ «"-' dxdy dz, ■when the Taiiahles are always positive and subject to the condition 1C + y + I -^ z»-' dudz T{l + m) Jo Jo This process is readily extended to any number of variables. 5. Find the value of the definite integral n ^i-i (i ™ i;)'"-'^dv Jo (4(1 ~v) + avy*^' By Art. 120 we have f°° r°° ,, 1 I , , r(0 r(m) Jo Jo ffl'J"* Transform by the substitution x= uv^y = u{i — «) ; then, since the limits for» are o and l, and those for u are o and oc , we get Jo Jo fl *•» oJo therefore 1 v-i(r -t.)w-i cosi|) in Surface Harmonics. Proceeding as in Art. 235, we easily get cos' S sin' S sin ^ aoi

y\ the sum of the products for these is half the whole sum ; hence M: ■■i\'Soy^'-^'^^''-''^^^y=h 2. A numher a is divided into three parts ; to find the mean value of one part Case of Two or More Independent Variables. 351 Let a;, 3^, a - it - 8» be the parts ; M I a ra Jo xdxdy ia na-x dxdy Jo I 3 This value might he deduced, without performing the integrations, by consider- ing that the expression is the abscissa of the centre of gravity of the triangle OAB ; OA, OB being lengths taken on two rectangular axes, each = a. Of course the result in this case requires no calculation ; as the sum of the mean values of the three parts must be = a ; and the three means must be equal. The mean square of a part is ^ a''. 6 3. A number a is divided at random into three parts : to find the mean value of the least of the three parts ; also of the greatest, and of the mean. Let X, y, a — X — y,he the greatest, mean, and least parts. The mean value of the greatest is M=~-^~ : the limits of both ^'' Ijdxdy integrations being given by x> y> a - x ~ y>o. If X, y be the coordinates of a point, referred to the axes OA, OB, taking OA = OB = a, the above limits resfiiot the point to the triangle A VS {AM being drawn to bisect OB) ; and the above value of M is the abscissa of the centre of gravity of this triangle ; i. e. - of the sum of the abscissas of its 3 angles ; hence Tig. 54- M = - [a + -a+-a] = —^a. 3 \ 2 3/18 The ordinate of the same centre of gravity, viz., {-a + -a\ = -^a, \2 3/18 is the mean value of the mean part ; hence the mean values of the three parts required are respectively II S I 18 ' 18 ' o 4. To find the mean square of the distance of a point within a given square (side = 2a), from the centre of the square. I ra ra ; Jf=— - {x' + y'')dxdy<=- ^a J,aJ~a ; 352 On Mean Value and ProlaUlity. It is obvious that the mean square of the distance of all points on any plan» area from any fixed point in the plane is the square of the radius of gyration of the area round that point. 5. To find the mean distance of a point on the circumference of a circle from aU points inside the circle. Taking the origin on the circumference, and the diameter for the axis, if dS he any element of the area, we have n M='^—^ = — ; Ur r^-dBdr=^--. Tta' Tta' J Jo 9ir 241. Many problems on Mean Values, as well as on Probability, may be solved by particular artifices, which, if attempted by direct calculation, lead to difficult multiple integrals which could hardly be dealt with. Examples, 1. To find the mean distance between two points within a given circle. If M be the required mean, the sum of the whole number of cases is repre- sented by {Trr'^fM. ITow let us consider what is the differential of this, that is, the sum of the new cases introduced by giving r the increment dr. If Ma be the mean distance of a point on the circumference from a point within the circle, the new cases intro- duced, by taking one of the two points A on the infinitesimal annulus 2itrdr, are TTc' Mit • Zirrdr : doubling this, for the cases where the point B is taken in the annulus, we get d . [ (ttj-")" Jf } = 47r' Moi^dr. Now Mo = — (Ex. S, Art. 240) ; therefore w'r^M = — ir I r*dr; 9 Jo „ 128 therefore M = — . 45'r 2. To find the mean square of the distance between two points taken on any plane area 0,. Let dS, dS' be any two elements of the area, A their mutual distance, and we have M=^jSSSA'>dSdS'. Case of Two or More Independent Variables. 353 No-w, fixing tie element iS, the integral of t^dS" is the moment of inertia of the area n round dS ; so that if iiThe the radius of gyration of the area round dS, I M=-[[K'idS: n let r = distance of dS from the centre of gravity Q of the area, h the radius of gyration round G ; then Xi' = »-2 + i2; therefore M = li^ \ - <{\ r^ AS = iTe^ \ a thus the mean square is twice the square of the radius of gyration of the area round its centre of gravity. 242. The mean distance of a point P witHn a given area from a fixed straight line (which does not meet the area) is CTidently the distance of the centre of gravity Q of the area from the line. Thus, if A, B are two fixed points on a line outside the area, the mean value of the area of the triangle APB is the triangle AGB. From this it will ioUow, that if X, T, Z are three points taken at random in three given spaces on a plane (such that they cannot all he cut by any one straight line), the mean value of the area of the triangle XYZ is the triangle GG'Q", determined by the three centres of gravity of the spaces. Example. I. A point F is taken at random within a triangle ABC, and joined with the three angles. To find the mean value of the greatest of the three triangles into which tbe whole is divided. Let G be the centre of gravity ; then if the greatest triangle stands on AB, P is restricted to the figure GB.GK, and the mtan value of AFB is the same as if i* were restricted to the triangle GGK; hence we have to find the area of the triangle , whose vertex is the centre of gravity of -^ ^ 6CK, and base AB ; Fig. 55. therefore M=l{ACB-\-AKB + AGB) = - li + - + -\aBG; hence the mean value is -3 of the whole triangle. The mean values of the least and mean triangles are respectively - and — 9 i8 of the whole. This question can readily be shown to be reducible to Question 3, Ait. 240. [23] 354 On Mean Value and Prohability. 243. If iff be the mean value of any quantity depending on the positions of two points (e.g. their distance) which are taken, one in a space A, the other in a space B (external to A) ; and if M' be the mean value when both points are taken indiscriminately in the whole space A + £ ; JP, Mj^, the mean values when both points are taken in A, or both in B, respectively; then {A + BfM' = 2ABM +A'M^ + B'Mj,. (4) If the space A = B, 4M' = 2M + M^ + M^; if, also, Mj^ = Mb, 2M' = M+M^: thus if M be the mean distance of a point within a semi- circle from a point in the opposite semicircle, Mi that of two points in one semicircle, we have (Art. 241), M+Mi = ^-^r. To determine M or Mi is rather difficult, though their sum is thus found. The value of M is — ^^ r. Examples. 1. Two points, X, Y are taien at random within a triangle. What is the mean area M of tiie triangle XYO, formed by joining them with one of the angles of the triangle ? Bisect the triangle by the line CD ; let Mi be the mean value when both points fall in the triangle ACD; Mi the value when one falls in ACB and the other in BCD ; then 2M= Ml + Mi. But Ml = -M; and M2 = GG'C, where G, G' are the centres of gravity of ACT), BCD, this being a case of the theorem in Art. 242 ; hence JIfz = - ABC, and M=— ABC. 9 27 2. To find the mean area of the triangle formed by joining an angle of a square with two points anywhere within it. Case of Two or More Independent Variables. By a similar method this is found to he 13 355 —= of the -whole square. 3. "What is the mean area of the triangle formed by joining the same two points with the centre of the square f We may take one of the points X always in the square OA ; take the whole square as unity ; then if ^ be the mean, the sum t. q of aU the cases is 4 ■-M: 42 42 4^ Ml, Mi, Ml being the mean areas when the second point Y is taken respectively in OA, OB, and 00. But Ml = Ml, for to any point Yin OG there cor- responds one Y' in OA, which gives the area OXS'=OXY; ,Y / / therefore But M = -Mx^^-Mi. 2 2 Fig. S6. Ml = — - . -, Jfu = —r ; hence M = — ^ of the whole square.* 108 4' 16 108 ^ 244, If two spaces A-v C,B-v G have a common part C?, and M. be any mean value relating to two points, one in ^ + C7, the other in 5 + C ; and if the whole space A-^ B -^G =W, and !!•„ be the mean value when both points are taken indis- criminately in W; M^ when taken in A, &c., then 2 {A'+ C) {B + G) M= WMy, + CM - A'M^ - B'M^, (5) as is easily seen by dividing the whole number W^ of cases into the different classes of cases which compose it. * In such questions as the above, relating to areas determined by points taken at random in a triangle or parallelogram, we may consider the triangle as «quilateral, and the parallelogram as a square. This wiU appear from orthogonal projection ; or by deforming the triangle into a second triangle on the same base and between the same parallels, when it is easy to see that to one or more random points in the former there correspond a like set in the latter, determining Ijhe same areas. This second triangle may be made to have a side equal, to a side of an equilateral triangle of the same area ; and then be deformed in like manner into the equilateral triangle itseH. Likewise a parallelogram may ba tleformed into a square. [2-3 a] 356 On Mean Value and Probability. EZAMPLD. Two segmenta, AS, CD, of a straight line have a common part CB ; to find the mean distance of two points taken, one in AB, the other in CD. 2AB.CJ).M=AD''.-AD+CB'.- CB-AC^.-AC-BD'.-BD, 3 3 3 3 since the mean distance of two points in any line is - of the line ; ,^ ^ „ AD' +CB'-AC'- DB^ therefore m = ■ 245. The consideration of probability may often be made to assist in determining mean values. Thus, if a given space 8 is included within a given space A, the chance of a point P, taken at random on A, falling on 8 is But if the space S be variable, and M {8) be its mean value, p.™. (6) For, if we suppose 8 to have n equally probable values 8„ 8„ 83 . . . ., the chance of any one 81 being taken, and of P falling on 81, is I 8, nA now the whole probability p = pi + Pi + p^ + . , . ; which leads at once to the above expression. The chance of two points falling on 8 is i' = -^- (7) In such a case, if the probability be known, the mean value follows, and vice versA. Thus, we might find the mean value of the distance of two points X, Y, taken at random in a line, Case of Two or More Independent Variables. 357 by tte consideration that if a third point Zbe taken at random in the line, the chance of it falling between X and Fis - ; as one of the three must be the middle one. Hence the mean 2ffl" distance is - of the whole line. 3 Again, the mean w'* power of the distance is , -— ,-, , [n + i){n + z) where a = whole line. For if p is the probability that n more points taken at random shall fall between X and Y, M{XYY = a"p. Now, the chance that out of the « + 2 points X shall be 2 one of the extreme points is ; and if it is so, the chance '^ n + 2 , ' that Y shall be the other extreme point is ■ + I Examples. I. From a point X, taken anywhere in a triangle, parallels are drawn to two of the sides. Find the mean value of the triangle UXV. If a second point X' be taken at random within ABC, the chance of its falling in XJTV is the same as the cliance of X falling in the correspond- ing triangle X' IT' V; that is, of X' falling in the parallelogram XC. Hence the mean value of UXV = mean value of XC. But the mean value ol {UXV ^ Fig. 57- + XC) is - ASC; as the whole triangle can he divided into three such parts by drawing through X a parallel to AS.' Thus M{UXV)=-tABC. The mean value of UV is-AB. For UV is the same fraction of AB that the 3 altitude of X is of that of C: see Art. 242. * The triangle may be considered equilateral : see note, Art. 243. 358 On Mean Value and Probability/. CoK. Hence, if ^ be the perpendicular from X on A3, h the altitude of the triangle ABC, we get If the area ^^C be taken as unity, we haye, since VXV:AXB = AXB:A3C, (AXBY = UXV. Thus the mean square of the triangle AXB is -. If two other points T. Z are taken at random in the triangle, the chance of both falling on AXB is thus the same as that of a single point falling on UXV; i.e. ^. Hence we may easily infer the following theorem : — If three points X, Y, Z are taken at random in a triangle, it is an even chance that 7, Z both fall on one of the triangles ti n AXB, AXO, BXG. ^ 2. In a parallelogram ABCB a point X is taken at random in die triangle ABO, and another, Y, in ADG. Find the chance that X is higher than Y. Draw XS. horizontal : the chance is mean area of ASK-^ ADO. Bvit ASK^XUF, and the mean area of ZZTris ~ ACB H I ° (Ex. I) ; hence the chance is -. 6 A 3. If be a point taken at random on a tiiangle, and lines be drawn through it from the angles, to find the mean value of the triangle DBF. (Mr. Millbk.) It will be sufficient to find the mean area of the triangle AEF, and subtract three times its value from ABO. If we put a, $, y for the triangles BOO, AGO, AOB, it is easy to prove that ai:f= Py (a + j8)(a + y) .ABO. If we put the whole area ABO = i, and if dS be the element of the area at 0, 0ydS -'^'^-\\w^ -y)' B he integration extending over the whole triangle. Fig. 59. But Up, q are the perpendiculars from on the sides b, c, it may be easily shown that the element of the area is dS = dpdq i BinA bcBiriA d$dy = 2dpdy. Case of Two or More Independent Variables. 359 Thus the mean value of AMF becomes JoJo (l-/3)(l-7) Jo' ° ' l-P Again, by Art. 95, the definite integral /3 log 3 '-T = therefore fi£h Jo I- jf=-x-z(i-^) = --3. Hence the mean value of the triangle SSF is 10 - tt", that of ABC being unity. It is remarkable that the same value, lo — ifi, has been found by Col. Clarke to be the mean area of a triangle formed by the jintersections of three lines, drawn from A, B, G to points taken at random in «, S, c respectively. 4. To find the average area of all triangles having a given perimeter (2s). By this is meant that the given perimeter is divided at random in every possible ■way into three parts, a, b, c, and only those cases are taken in which a, b, e can form a triangle ; then the mean value of A = is[s-a){s-b)[s-c) AX Y B Fig. 60. has to be found. Take AB = 2s, let X, Y be the two points of division, AX = x, AY= y : these are subject to the conditions x s, y-xow — = ^(s-x)iii-s){s-y + x); Va \ \ \/{s-x)[y-s){s-y + x) . dydx ••• -7.-3fM) = Tuf, Vi dydx J « jy-s Again, by Art. 132, we have Hence, Mean area = — - (25)2. 360 On Mean Value and Prohability. In the same case we should easily find Mean square of area = j-. 5. Three points are taken at random within a given triangle ; prove that the mean area of the triangle formed by them is — of the given triangle. Call the area of the given triangle A, the required mean M: we will first prove that if Mo he the mean area when one of the three points is restricted to a side of the given triangle, M = -Mo. 4 Let A receive an increment of area v vnll thus describe a curve starting from / ^ \\ N. I > ) ■century, and that A preceded B, and S preceded 0, let it he proposed to find how far this amount of knowledge alters the value of the chance for B. Let X he the numher of years from the heginning of the century to the event B ; then, for any assigned value of x, the numher of triple cases is a:(loo — a:) : hence thenumher of favourable cases divided by the whole number is !50+,i a;(loo - x) dx 50- n p = - _ n In ~ lOO \ 100 )■• (lUU a: (100 -x) dx [I 2. Two numbers, x, y, are chosen at random between o and a : find the ■chance that the product xy shall be less than — (its mean value). 4 „ r f dx dy Here p = •'•' „ ^ the integral being limited by a > a: > o, o > « > o, and xy < —. We have 4 a ■accordingly to integi'ate for y from a to o, when x is between o and - ; and from 4 a"^ ... " ■■ , — to o, when x is between - and a ; thus 4x 4 a fi Pa qZ fj^ fii adx +1 — dx= - -\ — log 4. Jn 4» 4 4 Hence I I , i' = - + - log 2. 4 2 3. Two points are taken at random in a given line a ; to find the chance that their distance asunder shall exceed a given value 0. It is easy to see that the distances of two such points from one end of the line are the coordinates of a point taken at random in a square whose side is a. Thus to every case •of partition of the line corresponds a point in the square — such points being uniformly distributed over its surface. Thus, if in the above question x, y stand for the distances of the two points, from one end of the line, y being greater than x, we have to find the chance of y - X exceeding e. The point P whose co- ordinates are x, y, in the square OB^ (side = a), may take all possible positions in the triangle OBI), if no condition is imposed on it. But ii y — x > c, then if we measure OS = c, the favourable cases ■E I I H / / V / / Fig. 63. Probalilities. 365- occur only when F is in the triangle BRI ; hence the probability required In fact this is only performing the integrations in the expression farn-e 1 dxdy p= . ■^ (■<■ rs I dxdy JoJo 4. Two points being taken at random in a line a, to find the chance that n» one of the three segments shall exceed a given -g Z K N D' length 0. The segments being as before, x, y — x, a — y, FS=x, PE=i be two eases :- PI=y -X. There -vrill H (I), li e>-a;t!siLeOV=Br=BZ=BN=e; then it is easy to see that the only favourable » cases are when P falls in the hexagon JJZNMJV; Pl-- 0BJD-3.USZ OBI) = -(•-?-•)" -OVI Fig. 64. (2). If o< -a; take OU=BV=c, as before; then the only favourable- ^ B K Z D cases are when P falls in the triangle It ST; EST OBD ^ 1 yn rx H since FST = - ET"^, and ET=VT+ ES- VM 2 U = 2e— (a — e). Such cases of discontinuity in the functions expressing probabilities frequently present them- selves. The functions are connected by very remarkable laws. Thus, in the present question, '' ■^ if Pi =/(«), Pi = F{c), we have Fig. 65. f(e)-f(fi-c) = F{a-c). S. A floor is ruled with equidistant parallel lines ; a rod, shorter than the- distance between each pair, being thrown at random on the floor, to find the chance of its falling on one of the lines (Buffori a problem). Let X be the distance of the centre of the rod from the nearest line, 6 the inclination of the rod to a perpendicular to the parallels, 2a the common distance- of the parallels, 20 the length of rod ; then as all values of x and 6 between their ■366 On Mean Value and Prohahility. extreme limits are equally probable, tlie whole number of cases will be repre- •sented by £1- '3 /*2 dxdS = Tta. Now, if the rod crosses one of the lines, we must have c > ; so that the cos fl iavourable cases will be measured by cnB(, dx = 2e. t2 rccnBfl Thus the probability required is = — . This question is remarkable as having been the first proposed on the subject now called Local Probability. It has been proposed, as a matter of curiosity, to determine the value of tt from this result, by making a large number of trials with a rod of length 2a : the difficulty, however, here consists in ensuring that the rod shall fall really at random. The circumstances under which it is thrown may be more favourable to certain positions of the rod than others. Though we may be unable to take account d priori of the causes of such a tendency, it will be found to reveal itself through the medium of repeated trials. 249. Sometimes a result depends upon a variable (or variables) all the values of which are not equally probable, but are such that the probability of a certain value for a variable ■depends, according to some law, on the magnitude of that value itself (and also, perhaps, on the values of other variables). Thus a point may be taken in a straight line so that all positions are not equally probable, but the probability of the distance from one end having the value x, being proportional to X itself. This would be in fact supposing the series of points in question as ranged along the line with a demity proportional to x ; as, e. g., if they were the projections, on the line, of points taken at random in the space between the line and another line drawn through one of its extremities. To give an example : — Two points are taken in a line a, with probabilities vary- ing as the distance from one end A ; to find the chance of their distance exceeding a length c. Let X, y, be the distances from A, and suppose y > x. Prohabilities. 367 Here the probability of a point falling between x and x + dx is not proportional to dx, but to xdx ; and the result will be ra I ydy ydy xdx . . xdx >j / \ The mean values of the three divisions of the line, in the same case, will be found to be 8 4 I — a, — a, -a. 15 15 5 The above value of p is also the value of the chance, that the difference of the altitudes of two points within a triangle shall exceed a given fraction — of the altitude of the triangle. a Examples. I. Two points being taken on the sides OA, OB, of a square a^, tie chance of their distance being less than a given value h is easily seen without calcula- tiou to be — J, provided J < a ; as it is tie cbanoe of a point taken at random in tbe square falling within a quadrant of a given circle. Suppose now that two points are taken on OA, and two on OB, and that we take X, Y, the two points furthest from on each side, to find the chance that their distance ZT is less than a given length J ; (b y . Hence p=\ \ — — =log2 — . ]\)yA y 2 2. An urn contains a large number of black and white balls, the proportion of each being unknown : if on drawing »» + « balls, m are found white and n. black, to find the probability that the ratio of the numbers of each colour lies between given limits. The question will not be altered if we suppose all the balls ranged in a line AB, the white ones on the left, the black on the right, the point X where they meet being unknown, and all posi- tions for it in AB being i priori equally A X~ .. _ probable ; then m + n points being taken -p. , at random in AB, m are found to fall on °' "' AX, n on X£. That is, aU we know of X is, that it is the (m + i)'* in order, [S4a] 372 On Mean Value and Probability. beginning from^, of m + » + i points falling at random oaAB. If AX = x, JLB = I, the number of cases for X between x and x + dxia measured by \m\n ^ ' Hence the probability that the ratio of the white balls in the um to the whole number lies between any two given limits o, /8 — that is, that the distance from .4 of the point X lies between a and 3 — is I »•»(!- x)« I a" (l - »)" Jo dx The curve of frequency for the point X will be one whose equation is y = a:*" (l — »)". The maximum ordinate KV occurs at a point K dividing AB in the ratio m : n. This is of course what we should expect : the ratio of the numbers of black and white balls is more likely to be that of the numbers drawn of each than any other. The value for p above is simply the area of the above curve between the values o, P, of x, dSvided by the whole area. Let us suppose, for instance, that 3 white and 2 black balls have been 2 4 drawn ; to find the chance that the proportion of white balls is between - and - of the whole — that is, that it differs by less than + - from -, its most natural value. Here II Jf ^ ' 2256 18 , x'-U-xfdx ^ ^ Jo The above results will apply to any event that must turn out in one of two ways which are mutually exclusive, this being the whole of our a priori knowledge with regard to it — the ratio of the black or white balls to the whole number, meaning the real probability of either event, as would be manifested by an infinite number of trials. We will give one more example of the same kind. 3. An event has happened m times and failed n times in »i + « trials. To find the probability that, onj) + j further trials, it shall happen i) times and fail ; times. * For a specified set of m points, out of the m + n, falling on AX, the \m + n number msc>"{i - xYdx; the number of such sets is , ; . Curve of Frequency. 873 That is, that p+qmoie points heing taken at random in AB, p shall fall in AX, and q in^X. The whole number of oases is as before : m + w I ml « {^AB]P*^ x">{i -xydx \m + n L^L». X"' (i — xYdx. When any particular set of p points, out of tliep + q additional trials, falls in AX, the number of f ayourable cases is \m + n \m\n X'i'*P{l -x)»*iclx. 1.2. 3 {p + 9) But the number of different sets of « points is I .2 . s . . .p . I .2 .3 . Hence the probability is, putting as before Ip for i . 2 . 3 . . . ^, pi = fpH-? J «""" (I -«:)"■" <^1' L^Li i: x'«{i-x)''dx By means of the known values of these definite integrals (p. 117), we find [p + q \m+p\n + q \m + n+ I i>i = \p\_g \m\n ' \m + n + p + q+l' For instance, the chance that in one further trial the event shall happen is . This is easily verified, as the line AB has been divided into m + n + 2 m + n + 2 sections by the m + n+ i points in it, including X Now, if one more trial is made, i. e. one more point taken at random, it is equally likely to fall in any section ; and m + l sections out of the entire number are favourable. 4. Trace the curve of frequency of the ratio - ; a and i being numbers taken at random within the limits + 1. If we measure the values of the ratio as abscissas along an axis OX, and make OA = 1 ; OA' = - 1, AB = A'B' = I ; „. then the line whose ordinates 5. are proportional to the fre- quency will be, for values of T comprised between the limits ± I, the straight line BB'; but, for values beyond these limits, wUl consist of the arcs BC, B'C of the curve x'y = 1. It is thus an even chance that the ratio - Kes itself between the limits + i : this would also appear by a construction such as that given in the next Article. 374 On Mean Value and Probability. 251. Errors of Observation. — One of the practically most important, as well as the most difficult, departments of the theory of ProbahUity is that which treats of Errors of Observation. We will give here an example of the simplest description. Two magnitudes A and B are measured ; each measure- ment being subject to an error, of excess or defect, which may amount to + a, all values between these limits being supposed equally probable.* To determine the probability that the error in the sum, A + S, of the two magnitudes, shall lie within given limits ; also its mean value. Thus the angular distance of two objects A, C is some- times found by measuring the angle between A and J3, an intermediate object ; and afterwards that between £ and C, and adding the two angles. If each measurement is liable to an error + 5', all values being equally probable, to iind the probability of the error of the result falling within assigned limits : its extreme limits being of course ± 10'. The question is more easily comprehended by moans of a geometrical construction > than by b' k integration. Take AB = 2a; then all the values of the first error are the distances from of points P taken at random in AB ; positive when in OB ; negative when in OA. Make also A'B' = 20 ; the values of the second error are given by points in A'B'. Take any values, OP = x for the first, OP' = x' for the second : these values taken as co-ordinates determine a point V corresponding to one case of the compound error x + x'; and such points V will be uniformly distributed over the square MK. The value of the compound error t corresponding to the point V is i = x+ x' = OS, if VS be drawn at 45° to the axes. Now all values of the • This supposition must not be taken to be practically correct. The Theory , of Errors shows that, the probability of an error of magnitude x is proportional to e <^'. p' T\. I H A Fig. 71. Errors of Observation, 375 errors x, of wliicli give x ■¥ af the same, give the same value for E ; hence all points on the line JI correspond to com- pound errors of amount OS. Take Ss = ds; the numher of compound errors between s and t + de ia the number of points between JI and a parallel to it through s. Now the area of this iafinitesimal strip is evidently (2a - e) d£. Hence the probability of the error being between e and E + fl^E is (2a - e) ds i' = -^— -1 — • 40 This holds for negative values of e, provided we only consider their arithmetical magnitude. Thus the frequency of an error of magnitude e = 08 is proportional to JI, the intercept of a line through 8 sloping at 45°. The probability of the error e falling between any two given limits OS, 08' is found by measiiring these lengths (with their proper signs) from 0, along AB, and dividing, by the area of the whole square, the area intercepted on the square by parallels through 8 and S', sloping at 45°. Thus the chance of the error falling between the limits ± a (those of the two component errors) is -. The mean value of the error, strictly speaking, is ; but it is evident that for this purpose we ought to consider negative errors as positive; and consequently take the mean of the arithmetical values of all the errors, which is the same as the mean of the positive errors only; hence the mean error required is Mh] = ±-a. 3 The most probable value, such that it is an even chance that the error exceeds it (since the triangle JKI must be— of the 4 whole square, for that value of 08), is ±0(2 - v/2) = + .5860. ^76 On Mean Value and Probability. Let it be now proposed to find the probability of a given error in the sum of A and B, assuming, according to the modern theory of errors, that the probability of an error between x and x+ dm ia either is v; e " dx; the coefficient — — being determined by the necessary con- CVTT dition that the differential, being integrated from oo to - oo, must give unity, as the error must lie between these limits.* Eef erring to the above construction, the number of values of the first error between x and x + dx being proportional to e ' dx, and the number of values of the second error between a^ and x'+ dx' being proportional to x' e'^dx', the corresponding number of values of the compound error is proportional to _ x' + x" e " dxdx. Hence the number of points, corresponding each to a case of the compound error, in any element dS of the plane at a distance r from the origin, is measured by e dS; which shows that the points have the same density along any * It ia of course absurd to consider infinite values for an error : but the curve y = e "' tends so rapidly to coincide with its asymptote, the axis of x, that the cases where x has any large values are so trifling in number, that it is indifferent whether we include them or not. Errors of Observation. 377 oircle whose centre is 0. Now the probability of this com- pound error being between e and a + rfa is proportional to the number of points between J'Zand the consecutive line ; making, as before, OS = e, 8s = ds. But this number is the same as when the strip JI is turned round through an angle of 45°, because the points lie in concentric circles of equal den- sity. Hence the number is proportional to e "" dx- '^^ -^d. — & ^^2 J-oo -v/z AS the perpendicular from on JI is — =. -•2 Thus the probability of a compound error between £ and t + c?£ is proportional to and as this, when integrated between the limits + oo , must give the probability i, the value of ^ is 1 -'- p = — -^e "'de. Cy^2TT It thus follows the same law as the two component errors, c^/a taking the place of c. 252. Various artifices have been employed for the solution of different interesting questions on Probability, which would be found extremely tedious, or impracticable, if attempted by direct integration. For example : Two points are taken at random within a sphere of radius r; to find the chance that their distance is less than a given value c. Let F = number of favourable cases, JF"= whole number; then ^•^ ^ Fig. 72. Let us consider the differential dF, or the additional favourable eases introduced by giving r the increment dr, c remaining unchanged. 378 On Mean Value and ProlabiUty. If one of the points A is taken anywhere (at P) in the infinitesimal shell between the two spheres, then drawing a sphere with centre P, radius c, all positions of the second point, B, in the lens EB common to the two spheres, are favourable ; let i = volume EB, then the number of favour- able cases when A is in the shell is ^irr^dr.L: doubling this, for the cases when B is in the same shell, dF=8Trr^Ldr. Now it may be easily proved, from the value for the volume of a segment of a sphere, that _ 27r , TTC* 3 4»- hence F= Sir'i- c^r^ --c^r^ + C \9 8 C being an unknown constant ; i.e. involving c, but not r ; P c^ g c" qC therefore p = — r =-t 7--r + --T- 1 6 , „ r' 1 6 r* 2 r" 9 Now the probability = i if >• = - c ; therefore i = 8 - 9 + - x 64 -; .-. -C= -^c'l 2 e" 2 64 , c' g c* 1 c' hence P = -i 7-4 + i- If the two points be taken within a circle, instead of a sphere, it may be proved by a similar process that c' 2 / c'\ . , c I c f e ffl = — + - I - -J sm-' .- 2 + - r tt\ r 2r a,Tr r\ r )J'-? Errors of Observation. 379' It is a remarkable fact, pointed out by Mr. S. Eoberts,, that if we draw the chord EI), the probability in the case of the circle is, 2 . segment EQD + segment EPD , area of circle EMI) and also, in the case of the sphere, 2 . volume EQD + volume EPD P volume of sphere EHD These results evidently suggest that there must be some manner of viewing the question which would conduct to them in a direct way. Examples. I. Three points 'being taken at random -within a sphere, to find the chance- that the triangle which they determine shall he aoute-angled. As the prohahility is independent of the radius of the sphere, it is easy to see that we may take the farthest from the centre of the three points as fixed on the surface of the sphere. For Up he the prohahility of an acute-angled triangle- in this case, p will also he the probability of an aoute-angled triangle for each position of the farthest point, as it travels over the whole volume of the sphere. Hence p will be the probability when no restriction is put on any of the points. Take then A, one of the points on the surface of the sphere ; two olhers, B, G, being taken at random within it, and let us find the chance of ABC being obtuse-angled : to do this, we will find separately the chance of the angles A, B, G being obtuse : the events being mutually exclusive, the probability req^uired will he the sum of these three. (l). To find the chance that^ is obtuse, let us fix B ; then, drawing the plane A V perpendicular to AB, the chance required is volume of segment ABV volume of sphere Let r he the radius of sphere, p =AB, fl = / OAB ; then the volume of th& segment jiifFis J x»^ (I - cos ef (2 + cos e) ; therefore when B is fixed the chance is J (I -cos 9)^(2 + cos 9). 330 On Mean Value and Prohdbilitij. Now let -B move over the whole volume of the sphere, and we have for Pa, the piobability that A is obtuse. Fa 6] n ~r» (2-3cos9 + oos3fl)p2sinfl(fflrfp. Hence Pa = — . 70 chance is I 4 (2). To find the chance, Pb, that B is obtuse. Fix B as before ; then the 4:hance that B is acute is segment J/iffiV ^here Now, voIume^B'i\r= ^m^ f- + l - cos9 J [ 2 + cos 9 -- ) ; so that the |2-3cose + cosS9 + 3^(i -cos2fl) + 3^cosfl -^|. Hence the whole probability (i — Pb) that B is acute ia IT 7 firarcosfl t 1? d'1 8^1 J p-3cose+cos'9 + 3^(i-cos»e) + 3!-jCosfl-y p2sin9(ffl(f;.. Performing the integrations, we find Ps = — . 7° The probability for C is, of course, the same as for B ; hence the whole pro- bability of an obtuse-angled triangle is P=PA + PB + Fa=^+^^+— = ^. 70 70 70 70 Hence, the chance of an acute-angled triangle is — . 70 For three points within a circle the chance of an acute-angled triangle is 7r2 8" 2. Two points, A, B, are taken at random in a triangle. If two other points, C, B, are also taken at random in the triangle, find the chance that they shall lie on opposite sides of the line AB. Errors of Observation. 381 The Bides of the triangle 2(5(7 produced divide the whole triangle into seven, spaces. Of these, the mean value of those marked (a) is the same, viz., the mean value of ABC; or, i^ of the ■whole triangle, as we have shown in Art. 245 ; the mean value of those marked (j8) heing f of the triangle. This is easily seen : for instance, if the whole area = i, the mean value of the space. PBQ gives the chance that if the foui'th point D be taken at random, B shall fall within the triangle ADC: now the mean value ot ABC gives the chance that 2) shall ■ fall within ABC ; but these two chances are equal. Fig. 74. Hence we see that if A, B, he taken at random, the mean value of that portion of the whole triangle which lies on the same side of AB as C does is ^ ot the whole ; that of the opposite portion is ■^. Hence the chance of C and D falling on opposite sides of AB is ■^. 253.. Random Straight Unes. — If an infinite number of straight lines be drawn at random in a plane, there will be as many parallel to any one given direction as to any other, all directions being equally probable ; also those having any given direction will be disposed with equal frequency all .over the plane. Hence, if a line be determined by the co- ordinates, p, b), the perpendicular on it from a fixed origin 0, and the inclination of that perpendicular to a fixed axis ; and if p, It) be made to vary by equal infinitesimal increments, the series of lines so given will represent the entire series of random straight lines. Thus the number of lines for which p falls between p and p + dp, and to between id and w + dw,. will be measured by dp duo, and the integral SSdpdio, between any limits, measures the number of lines within those limits. It is easy to show from this that the number of random lines which meet any closed convex contour of length L is measured by L. For, taking inside the contour, and integrating first for^, from o to p, the perpendicular on the tangent to the contour, we have ]pdw : taking this through four right angles- 582 On Mean Value and Probability. for b), we have by Legendre's theorem (p. 232), N being the measure of the number of lines, •>0 pdb) = L. Thus if a random line meet a given contour, of length L, the chance of its also meeting another convex contour, of length I, internal to the former, is I p^j-. If the given contour be not convex, or not closed, N will •evidently be the length of an endless string, drawn tight ^iTOund the contour. Examples. L. If a random line meet a closed convex contour, of length X, the chance of it meeting another such contour, external to the former, is X-T where X is the length of an endless hand ■enveloping both contours, and crossing between ^em, and Y that of a band also H -enveloping both, but not crossing. This may he shown by means of Legendre's integral above ; or as fol- lows : — Call, for shortness, lf[A)t}ie number of lines meeting an area A; N[A,A') the number which meet both A and A' ; then N{SJR.OQPB) + N{S'Q'OR'F'E') = N{SROQFE+ S'Q'OR'FE^ + N{SSOQI'S, S'QOKPST), since in the first member each line meeting both areas is counted twice. But the number of lines meeting the non-convex figure consisting of OQPSSS and OQ'S'S'JP'S' is measured by the hand T, and the nimiber meeting both these areas is identical with that of those meeting the given areas O, O'; hence x= T+N{a,a'). Thus the number meeting both the given axeas ia measured hy X— T. Hence the theorem follows. Random Straight Lines. 383 2. Two random chords cross a ^vkn convex boundary, of length J» and area n ; to find the chance that their intersection falls inside the boundary. Consider the first chord in any position ; let G be its length ; considering it as a closed area, the chance of the second chord meeting it is 20 L' and the whole chance of its co-ordinates falling in dp, da, and of the second chord meeting it in that position, is 20 dp da 2 _ , LUd^^'T^'^^^- But the whole chance is the sum of these chances for all its positions ; therefore prob. = — M Cdp da. Now, for a given value of a, the value oi\Cdp is evidently the area a ; then taking a from ir-tQ O, required probability = -=5- . The mean value of a chord drawn at random across the boundary is jl^^ SJOdpdu _':ra i Sjdp da Z ' 3. A straightband of breadth c being traced on a floor, and a circle of radius r thrown on it at random, to find the mean area of the band which is covered by the circle. (The cases are omitted where the circle falls outside the band.)* If S be the space covered, the chance of a random point on the circle falling on the band is _ M{S) This is the same as if the circle were fixed, and the band thrown on it at random. Now let .4 be a position of the ,— ■ — random point : the favourable cases are when y^^^ "^><^''^-- ^ KK, the Hseetor of the hand, meets a circle, /' /^ ^-"'^--X^i.^ centre A, radius Jo ; and the whole number /'' /,.^'^'\!--'"''l,^\^ are when SK meets a circle, centre 0, radius ^/-T^-" — ^.^^""'''^ \ 'i r + \e; hence (Art. 245) the probability is '^-r''' ',-^'''- I • _ 27r . !« _ c ^T \ i . ! / .' ~ 2ir{r+ Jc) ~ 2r + c \ ^v \....-/ / This is constant for all positions of A ; '^-.. -■'' hence, equating these two values of p, the Fig. 76. * Or the floor may be supposed painted with parallel bands, at a distance asunder equal to the diameter ; so that the circle must fall on one. 384 On Mean Value and ProbahUity. mean area required is jr(«) = 2r + c The mean value of the part of the circumference which falls on the band i» the same fraction of the whole circumference. 2r + o If any convex area n, of perimeter i, he thrown on the band, instead of » circle, the mean area covered is MiS) = - n. ^ ' £ + irc 254. Application to Evaluation of Definite Inte- grals. — ^The consideration of probability sometimes may h& applied to determine the values of Definite Integrals. For instance, if » + i points are taken at random in a Hne, I, and we consider the chance that one of them, X, shall be the last,, beginning from the end A of the line, the number of favour- able cases, vsrhen X is in the element dx, is, if AX = x^ measured by x''dx. Hence ■I Xfdll! / H+l but the chance must be : we thus have an independent n + I proof that as'^dx ■■ ;o «+ I when n is an integer. Again, ii m + n+ i points are taken, to find the chanefr that X shall be the (m + 1)** in order ; the number of favour- able cases when X falls in dx and a particular set of m point* fall to the left of X, is a;™ (1 - xYdx; taking 1= 1; hence the whole number of favourable cases is \m+nr^ (if{i - xYdx; Application to Evaluation of Definite Integrals. 385 tlu8 is the required probability, since ;»»+'»+i = i. But the value is , as every point is equaUy likely to fall in the (ot + i)"* place : we thus deduce the definite integral fi \m\n «'» (i - xY dx = , *— ^- . Jo I m + w + I when m, n are integers. (See Art. 92.) 255. To investigate the probability that the inclination of the line joining any two points in a given convex area Q, shall lie within given limits. We give here a method of reducing this question, to calculation, for the sake of an integral to which it leads, and which is not easy to deduce otherwise. First, let one of the points, A, be fixed ; draw through it a chord I'Q = C, Fig 77 at an inclination B to some fi±ed line ; put AP = r, AQ = r'; then the number of cases for which the direction of the line joining A and B lies between 8 and B + d9 ia measured by J {r' + r") dd. Now, let A range over the space between PQ and a parallel chord distant dp from it, the number of cases for which A lies in this space, and the direction of AB from 6 to 6 + dd, is (first considering A to lie in the element drdp) idpddl {f + r'^)dr = ^C^dpdB. Let p be the perpendicular on the chord G from a given origin 0, and let o) be the inclination of p (we may put dm for dB), then G will be a given function of ^, tu; and in- tegrating first for u) constant, the whole number of cases for which o) falls between given limits in', id", is 'A>\ C^dp; the integral \C^dp being taken for all positions of C between [26] 386 On Mean Value and Prohdbility. two tangents to the boundary parallel to PQ. The question is thus reduced to the evaluation of this integral; -which, of course, is generally difficult enough: we may, however, deduce from it a remarkable result ; for if the integral ^llC'dpdu, be extended to aU possible positions of C, it gives the whole number of pairs of positions of the points A, B which he inside the area ; but this number is Q' ; hence llC^dj[>du = 3Q', the integration extending to all possible positions of the chord C; its length being a given function of its coordinates Cor. Hence, if L, Q, be the perimeter and area of any closed convex contour, the mean value of the cube of a chord drawn across it at random is — =- . It follows that if a line cross such a contour at random, the chance that three other lines, also drawn at random, shall meet the first inside the contour is 24 — . JLd Some other cases of definite integrals deduced from the theory of Probability are given in a Paper in the Phib- sophical Transactions for 1868, pp. 1 81-199. See also Pro- ceedings, Lond. Math. 80c., vol. viii. Several Examples on Mean Values and Probability are annexed ; some of them, as also some of the questions which have been explained in this Chapter, are taken from the Papers on the subject in the Educational Times, by the Editor, Mr. Miller, as also by Professor Sylvester, Mr. Woodhouse, Col. Clarke, Messrs. Watson, Savage, and others. Some few are rather difficult; but want of space has prevented our giving the solutions in the text. We may refer to Todhunter's valuable History of Pro- hdbility for an account of the more profound and difficult questions treated by the great writers on the theory of Pro- bability. Examples. 387 Examples. 1. A oiord is drawn joining two points taken at random on a circle : find the mean area of the lesser of the two eegmente into which it divides the circle. Ans. . 4 T 2. Find the mean latitude of all places north of the Equator. Am. 32°.704. 3. Find the mean square of the velocity of a projectile in vaciio, taken at all instants of its flight tiU it regains the -velocity of projection. 'Ans. V^oow'a + iV sia^a, where K= initial velocity, and = angle of projection. 4. If X and y are two variables, each of which may take independently any value between two given limits (different for each), show that the mean value of the product xy is equal to the product of the mean values of sc and y. 5. If -X, F are points taken at random in a triangle ABC, what is the ■chance that the quadrilateral ASXYis convex ? . I AilS. - . i For, it is easy to see that of the three quadrilaterals ABXT, ACXY, JiGXY, ■one must be convex, and two re-entrant. 6. Find the mean area of the quadrilateral formed by four points taken at random on the circumference of a circle. 7 Ans. — (area of circle). TT 7. A class list at an examination is drawn up in alphabetical order ; the num- ber of names being n. If a name be selected at random, find the chance that the ■candidate shall not be more than m places from his place in the order of merit. . zm+i mlm + i) ,.- _ m,- • ^ p ^v i r^t Ans. ! — . (N.B. — Ihis is not, of course, the value of the n ■n' chance after the selection has been made : this may easily be found.) 8. A traveller starts from a point on a straight river, and travels a certain ■distance in a random direction. Having quite lost his way, he starts again at random the next morning, and travels the same distance as before. Find the ■chance of his reaching the river again in the second day's journey. Ans. - . 4 9. Two lengths, b, V, are laid down at random in a line a, greater than ■either :' find the chance that they shall not have a common part greater than c. [a-b-V + c)'' .a.ns. ; . [a-b)ifl-l>) [25 a] 388 Exampks, 10. A person in firing lo shots at a mark has hit J times, and missed 5 . Find the chance that in the next 10 shots he shall hit 5 times, and miss 5. 27 . 4 . 7 756 Am. ^ = -^—. If the first 10 shots had not been fired, so that nothing was known as to his skill, the chance would he — -if he- II had been found to hit the mark half the number of times out of a large number, the chance would be -~. 256 11. If a line I he divided at random into 4 parts, the mean square of one of the parts is — l^: but if the line be divided at random into 2 parts, and each part again divided into 2 parts, then the mean square of one of the 4 parts^ is -P. 9 12. Three points are taken at random in a line I. Find the mean distance of the intermediate point from the middle of the line. Alls. —pi. 16 13. A certain city is situated on a river. The probability that a specified inhabitant A lives on the right bank of the river is, of course, J, in the absence of any further information. But if we have found that an iniabitant B lives on the right bank, find the probability that A does so also. 2 Ans. -. (N.B. — It is here assumed that every possible pai'tition of the number of inhabitants into 2 parts, by the river, is equally probable a priori.) 14. If ^, B, C, D are four given points in directum, and 2 points are taken at random in AD, and one is taken in BC: find the chance that it shall fall between the former two. Ans. 2^j |i BG^ + BC{AB + CD) + 2A3 . CD j . 15. It x = X + i/, where x may have any value from o to a, and y any value from o to i : find the probability that z is less than an assigned value c (supposing Ans. (i) \i ec>l, p2 = ''~'^ . a (3) If Off, ^3=1-^ '- 2ab If we denote the functions expressing the probability in the three cases by /i(«, *, c), ft {a, b, c), fata, b, c), we shall find the rela- tion /i {«, *, c) +/a («, b, c) =/j (a, b, c) +/, (S, a, c) JExampks, 389 t6. In the cubic equation x' +px+ g = o, p and q may have any values between the limits + r. Find the chance that the three roots are real. Am. — V 3. 45 17. Two observations are taken of the same magnitude, and the mean of the results is taken as the true value. If the error of each observation is assumed to lie within the limits + o, and all its values be equally probable, show that it is an even chance that the error in the result lies between the limits + 0.293 a. 18. A point is taken at random in each of two given plane areas. Show that the mean square of the distance between the two points is F + k'^ + A» ; where A is the distance between the centres of gravity of the areas ; and A, k' are the radii of gyration of each area round its centre of gravity. 19. The mean square of the area of the triangle formed by joining any three 3 poiats taken in any given plane area is- /s^A* ; where A, k are the radii of gyra- tion of the area round the two principal axes of rotation in its plane. If one of the points is fixed at the centre of gravity, the value is fA^A*. {Mr.' 'Woot.HousE.) 20. A line is divided at random into 3 parts. Find the chance — (i) that they will form a triangle ; (2) an acute-angled triangle. Ans. (i)- P\ = 1- (2). i)2 = 3 log 2 - 2. 21. A line is divided into n parts. Find the chance that they cannot form a polygon. Ans. — -,• 22. If two stars are taken at random in the northern hemisphere, find the chance that their distance exceeds 90°. Ans. -, 23. The vertices of a spherical triangle are points taken at random on a sphere. Find the chance — (i) that aE its angles are acute ; (2) that all are obtuse. J T 31 24. Show that the mean value of -, where p is the distance of two points P taken at random within a circle, is — . 37rr 390 Examples. 25. Two equal lines of length a indude an angle 6 : find the chance that if two points P, Q are taken at random, one on each line, their distance PQ shall be less than a. Am. (I). When - > 9 > - ; ^1 = . , + 2 cos B. * ' 2 3 j 2 sin 9 (i). "When 9 > - ; Pi = ^^. ' 2 2sme Here the functions are connected by the relation J (9) + J(ir - 9) =/ (9) +/ (ir - fl.) 26. The density of a city population varies inversely as the distance from a central point. Find the chance that two inhabitants chosen at random within a radius r from the centre shall not live further than a distance r from each other. ir TT I 1 , 2 / Vix I t'^ede 3 f^ edB Ana. p= log3 + - |l-— | +— -r— r + — -:— ; ; 3 4'r\ 2/ 2irJoSin9 2irjjr8infl' a whence^ = 0.7771. This result is easily obtained by employing the values given in Question 25. 27. Four points are taken at random within an ellipse. Show that the 35 chance that they form a re-entrant quadrilateral is -. 36 28. Find the mean distance of two points within a sphere. Am. — r. 29. Three points A, B, C are taken within a circle, whose centre is 0. Find the chance that the quadrilateral ABCO is re-entrant. I 4 Ana. -+-^.. 4 3'" 30. Find the chance that the distance of two points within a square shall not exceed a side of the square. Ant. p = w---r. o 31. In the same case, find the chance that the distance shall not exceed an assigned value c ; a being the side of the square. 58 / 8 I \ Ans. (l). When c«;i, = 4^-5sm->--.-+l-^V.»-«»-2 --_, + -. 32. Three points are taken at random on a sphere : the chance that in the spherical triangle some one angle shall exceed the sum of the other two is -. Also the chance that its area shall exceed that of a great circle is -. 33. If a line be divided at random into 4 parts, show that it is an even «hance that one of the parts is greater than half the line. Examples. 391 34. Prove that the mean distance of a point within a triangle from the vertex C is lia + b {a-h){a'-V^) A^, a+i + c) where h is the altitude of the triangle. (See Ex. 6, Art. 242.) 35. The mean value of the distance hetween any two points in an equilateral triangle is ^=l.(i + ilog3). This question may be solved by proving that M= - Ma, where M^ is the mean distance of an angle of the triangle from any point within it. For, let Mti = /lAl, where /i is constant, and A = area of the triangle. Take now any element dS of the triangle ; draw from it parallels to the sides to meet the base ; let 8 be the area of the equilateral triangle so formed : the sum of the whole number of cases will be equal to 6 |[s./iS!.rf5=J!fA2, if dS is made to range over the whole triangle : if we call the whole triangle unity, and put dS = 2dad^, as in Ex. 3, Art. 245, B = o^, and the integral becomes — u. = M. The result then follows from 34. 10'^ 36. From a tower of height h particles are projected in all directions in space with a velocity due to a fall through h. Show that the mean value of the range is J(f = 2A I -^ 1 - x^. dx. (Pboi'. 'Wolstenholme.) 37. In » quantities a,h,c,d each of which takes independently a given series of values «i, 02, as, ... . ; h, h, h, ■ ■ -, &c., (the number of values is different for each), if we put 2a = a + b + c + d + .... + &c., and for shortness denote " the mean value of a; " by Mx; prove that if 2a = Jf« + Jfi + JKc +.... + &o. = SMa, M (2«)' = (iMaf - 2 {Ma)^ + 3M:{a^). 38. Two points are taken at random in a triangle. Find the mean area of the triangular portion which the line joining them cuts off from the whole 9 ( 392 ) CHAPTER XIII. ON FOURIER S THEOREM. 256. In many physical investigations it is of importance to express a function /(») in a series of sines and cosines of multiples of x. We propose to investigate the form of such expression, and the conditions under which it is possihle. Let us commence by assuming that f{x), between the limits + IT and - ir, is capable of being represented by a series of the required form : thus suppose fix) = tta + ai cos X + Ui 00s 2x + ... + On cosnx + ... + 61 sin X + b^ sin 2x + . . . + b„ ^ha. nx+ . . . (i) Here, since this relation is supposed to hold for all values of X between + tt, we get, on multiplying by cos nx and inte- grating, if if"' f[x) cos nxdx = - /(») cos nvdv. (2) T 1" J-ir «« = - Also and IT flo ■ /(») Binnvdv, Substituting in (i) it becomes if I ^-4""°° f" f(^r) = — f{v) dv + -'^ cos nx cos nv f{v) dv I _-4"=" . f + - V sin «a; sin nv/(v) dv I _. ^W = oo f"" fMdv+-S, cosn(v-x)/(v)dvj (3) 1 2ir Fourier's Theorem. 393 ^ It should be noticed that when /(a) is an even function of hK its development in general consists only of cosines ; if )• (7) Accordingly, lim. a=oo '*'-^V(* + ^)^.=;/(4 (8) Z 2 Likewise, " smas sins f{^)dz sina2 iz)dz, where 0(s)=^/(«) s sins' hence, provided A is less than v, we have H^^psina_s ^^^^ "-"Jo sms''^ ' 2-^ ■' If we change the sign of s, we have lim. a~oo f-* sin as " ^f ^ -^— /(-2) = "' fair 'r/(/3) = i f[e)dB + % /iB)cosn{d-l3)de. (17) Jo. n=i Jo This is usually called Fourier's theorem. Also, by aid of (15), TT . r^" -/;/3) = 2""°° sin MjSJ /(0) sin uO d9. 2 »=1 Jo 259. We shall next investigate the limit when a = co for the integral cos Ma; cos uidudi _ 1 - "2" Jo Jo '* sin a{x-t) X-t ^{t)dt + i '* sin a{x+t) x+t {t'j dt _ 1 - T Sin as (a + «) «fe + -^ 6 + a; smaz ^ (z - ») rfs. Now, by (6), the latter integral vanishes when a = 00 and x is positive ; and by (10), when x lies between a and b, the former • TT integral = -^(a;). Also, when x does not lie between a and J, the former in- tegral vanishes, and we have (^) cos ux cos «(^ <^M di = o. (18) 398 Fourier's Theorem. When X lies between a and b, ^ {t) cos ux ooautdudt = — ^ (»). (ig) » 2 Hence, if a; he positive, we have \ (»)<«'' + J ^_^^ J ^ "/>(") COS —-^-j ^ = c\ cos -7— (fe = — sin—-. Jo I ^^' Jo ^ 'w ' 400 Examples. 8. Find a function of 'x which is equal to kx when x lies between o and - and is k{l — x) when x lies between - and I. kl Ski 1 1 2itx I (nrx I \Qmx \ 2 , Sin mx = - sin mir I cos »i» = - sin iMir 1 I ^ (») cos -T— dv = \ kv cos -y- rfi) + L A (if - ») cos — dv. 1 ikl^ This = - ^-^—i, when n is of the form 4m + 2; and is zero for other values of n. "" 9. If (a;) = -, when x lies between o and o, and ^ (a;) = - when x lies between a and ir — o, and ^ {x) = , when x varies from ir - o to ir, prove that ,,i/. . I. . I. . \ ^ (a;) = - ( am a; Bin o + - Bin 3a; sin 30 H sin 5a! sin 5a + ... 1 . "■ \ 9 25 y 10. When X lies between + tt, prove the relations (sin X 2 sin 2a: 3 sin ^x \ I -»»« ~ 2»-m2 "*" 3» - »»■' ~ • • • j ' (I OT cos X m cos 2X m cos 3a; \ JOT i-OT» ~ 2''-«j' "^ 32 - m^' - •••J- 11. Hence prove the relation I 2U 3U cot M = - + -5 • + — + . . . 1 2. Find a function which shall be unity for all values of x between + I, and zero for all other values of x. F{x) = -\ rf/t 1 ^^(f) oosyttj cosAu;rf|= - dfi cos juj cos /ja;rfi IT Jo J. CO ^ -'0 J 2 f " cos /ta; sin u ^ irJo /« This result can be verified independently. 13. Find a function which shall be equal to cos x for all values of x between o and IT, and to - cos x for values between — ir and o. Here we easily find 1 f{x) cos nx dx =0, and we get 4(2. 4 . 6 . , ) cos « = - { — sm 2a; + — sin 4a; H sin 6a; + . . . > . t{i-3 3-S S.7 ) Miscellaneous Examples. 401 MrSCELLAlTEOTJS EXAMPLES. 1. Find the value of f^Jfl^. 2. Find the area of the inverse of a hyperbola, the centre being the pole of inversion ; and show that the area of the inverse of an ellipse, under the same circumstances, is an arithmetic mean between the areas of tiie circles described on its axes as diameters. 3. Find the integral of — ^^^Tb^ ■ /'' C Ana. tan-i \—. — 7; + -tan-i- \ ~ " . 4. Prove that f^ /W rf» = (J -«)/(!) log (^), where | lies between X and xo. _ 5. In a spiral of Archimedes, if P, Q and P', Q' be the points of section with any two branches of the curve made by a line passing through its pole ; prove that the area bounded by the right line and by the two branches is half the area of the ellipse whose semiaxes are TI" and F'Q. 6. If a be the sagitta of a circular segment whose base is i, prove that the area of the segment is, approximately, 2 , 8 «3 = -«} + — -- 3 IS * 7. If an ellipse roU upon a right line, show that the differential equation of the locus of its focus is (y" + *') ^ = \/(2ai/ + 2^i» + b') {2ay - y^ - i^). 8. A circle rolls from one end to the other of a curved line equal in length to the circumference of the circle, and then rolls back again on the other side of the curve ; prove that, if the curvature of the curve be throughout less than that of the circle, the area contained within the closed curve traced out by the point of the circle which was first in contact with the fixed curve is six times the area of the circle, (fiamb. Math. Tripos, 1871.) 9. In the same case show that the entire length of the path described is eight times the diameter of the circle. 10. Prove that the area of the locus formed by the points of intersection of normals to an ellipse, which cut at right angles, is 5r(a — i)". [S6] 402 Miscellaneous Examples. I r. Prove that the area hetween two focal radii of a parabola and the curve is half the area hetween the curve, the corresponding perpendiculars on the directrix, and the directrix. 12. Evaluate the following integrals : — JVtana ' 3 (I + k;^}{i + X*) x^ 4- ax 4- yM 13. If 5 = (a" + axf + bx, and u = log — , find the relation hetween the integrals I -^, f —^ • isiR J ViS . f xdx a t dx w Am. -— = _ 1 __ + J Vjj 3 J VJJ 14. If a curve be such that the area between any portion and a fixed right line is proportional to the corresponding length of the curve, show that it is a catenary. 15. Prove that the volume of a rectangular parallelepiped is to that of its circumsorihed ellipsoid as 2 : ir VJ. 16. Prove that — = , where sm g = « ain ». Jo VI - K^ sin'fl Jo Vk^ - siu^fl 17. If any number of triangles be inscribed in one ellipse and circumscribed to another ellipse, concentric and similar, prove that these triangles have all the same area. 18. Show that the value of the integral 1 may be exhibited by the JaVy^— I following geometrical construction. Let the curve whose equation is m — r- m r"'" cos (0=1 m +2 roll on the axis of x ; take the points {xi, yi) (a^, ya) on the roulette described by the pole, such that yi — a, y% — b; then -- Xi — x\. (Mr. Jellbtt.) 19. If « he the length of the arc of a spherical curve measured to any point T, and t be the intercept on the great circle touching at P, between the point of contact and the foot of the perpendicular from the pole, prove that s — < = J sin^tSa. The proof is similai' to that of the corresponding theorem in plana. See Art. 158. Miscellaneous Examples. 403 20. Prove that the volume of a, polyhedron, having for haaes any two polygons situated in parallel planes, and for lateral faces trapeziums, is ex- pressed hy the formula •where S is the distance tetween the parallel planes, S and JB' the areas of the polygonal bases, and S" the area of the section equidistant from the two bases. 21. If She the length of a loop of the curve »■» = «» cos«9, and A the area of a loop of the curve r^" = a^" cos 2n0, prove that ^ c. '^"^ Ax 8- — -. 2n 22. Find approximately the area, and also the length, of a loop of the curve »*S = at cos — . (See Diff. Calc, Art. 268.) Ans. area = a' x 0.56616 ; length —ax 2.72638. 23. Show from Art. 134 that if a parabola roll on a right line, the locus of its focus is a catenary. 24. If A be the area of any oval, B that of its pedal with respect to any internal origin 0, and C that of the locus of the point on the perpendicular whose distance from is equal to the distance of the point of contact from ; prove that A, B, are in arithmetical progression. 25. The arc of a curve is connected with the abscissa by the equation «'= Ja;; find the curve. 26. If the coordinates of a point on a curve given by the equations a: = c sin 28 (I + cos 29), y = c cos 29 (l - cos 20), ... 4 prove that the length of its arc, measured from its origin, is -c sin 38. 27. Show how to find the sum of every element of the periphery of an ellipse divided by any odd power {2r+ i) of the semi-diameter conjugate to that which passes through the element, and give the result in the case of the fifth power. — (W. EOBBKTS.) IT A fl (ai)2»^i Jo „, . . ir(a> + i'') , This gives ■ ., ,, — - when )■ = 2. a'o' 28. A sphere intersects a right cylinder ; prove that the entire surface of the cylinder included within the sphere is equal to the product of the diameter of the cylinder into the perimeter of an ellipse, whose axes are equal to the greatest and least intercepts made by the sphere on the edges of the cylinder. [26 a] 404 Miscellaneous Examples. 29. Show that the eqxiationa of the involute of a circle are of the form x — a cos ij> + a^ sin ^, y = a sin — ai^ ooa . 30. Show that the area of the cassinoid r* - la^r^ cos 20 + a* = J* is expressed by aid of an elliptic arc when b > a; and by a hyperbolic are when a>b. 31. A string AB, with its end A fixed, lies in contact with a plane convex curve ; the string is unwound, and B is made to move about A till the string is again wound on the curve, the final position of B being B'; prove that for variations of the position of A the' arc traced out by B will be a maximum or a minimum, when the tangents at B and B' are equaUy inclined to the tangent at A ; and will be the former or the latter, according as the curvature at A is greater or less than half the sum of the curvatures at B and B'. — ICamh. Math. Tripos, 187 1.) 32. Find the value of —7 e * . , Am. . /_ e Jo ^ ViS 33. Find the length, and also the area, of the pedal of a cissoid, the vertex being origin. Am. -llog(2 + VJ)- 4a; ^. 34. Prove that the length of an arc of the lemniscate r^ = fl^ cos 20 is repre- sented by the integral 4f- V2 J Vi-|sin2^ 35. Integrate the equation cos (cos 9 - sin a sin^) + A^cos'^, in the preceding, and it becomes ft Jo Jo 2 fa A' sin' (ft + Pcos'<)) - A' sin' 9 i/h^sm^ij, + /6'cos' V/fc'''- A'sin'fl n" VA'sin'A + A'oos'd. P (2 rz a Jo oi VA'sin'ift + A'cos*^ VA» sin' (j) + Fcos', =, or < a. „ „ . r Bin5a:sinaa: , tt ,,.,.,, , , . , 44. Prove that I ^ ax = - multipued by the leaser of th& numbers a and i. 45. If e be the eccentricity of an ellipse whose aemiaxis major is unity, and £ the length of its quadrant, prove that , = — -=^ . CW. ROBEKTS.) (i-e»)VA2-«2) 2\fr^^ 46. If S represent the length of a quadrant of the curve r™ = «»' cos mi, and S\ the quadrant of its first pedal, prove that 2m Here (Ex. 3, Art. 156), we have g « V 1- 2m r Also, since the first pedal {Dif. Calc.,Ait. 268) is derived by substituting -^ instead of »!, m+t l m+ i \ \ 2»» / Lrt. 268) is I _ (m+ l) gy/j V 2m / \ 2ml (m + I) ira' \2ml _ {m + l) vit' .■. SSi = -—r / =— r Miscellaneous Examples. 407 47- In general, if S„ he the quadrant of the ft"' pedal of the curve in the last, prove that , „ „ mn + i On-l On = Tra". Here it is readily seen that the »'* pedal is got hy suhstituting in- stead of m in the equation of the proposed; .-. &c. (W. Eobeets, Ziouville, 184s, p. 177.) 48. If an endless string, longer than the circumference of an ellipse, he passed round the ellipse and kept stretched, hy a moving pencil, prove that the pencil will trace out a confocal ellipse. 49. If two confocal ellipses he such that a polygon can he inscrihed in one and circumscribed to the other, prove that an indefinite number of such polygons can be described, and that they all have the same perimeter. (Chasles, Comp. Rend. 1843.) 50. To two arcs of a hyperbola, whose difference is reotifiahle, correspond equal arcs of the lemniscate which is the pedal of the hyperbola. (Ihid.) 51. Prove' that the tangents drawn at the extremities of two arcs of a conic, whose difference is reotifiahle, form a quadrilateral whose sides all touch the same circle. (Ibid.) 52. In the curve XS + 2/3 = (83, prove that any tangent divides the portion of the curve between two cusps into arcs which are to each other as the segments of the portion of the tangent intercepted by the axes. 53. If two tangents to a cycloid cut at a constant angle, prove that their sum bears a constant ratio to the arc of the curve between them. 54. li AB, ah, be quadrants of two concentric circles, their radii coinciding; show that if an arc ^4 of an involute of a circle be drawn to touch the circles at A, b, the arc ^S is an arithmetical mean between the arcs AS and ab. 55. If ds represent an infinitely small superficial element of area at a point outside any closed plane curve, and «, f the lengths of the tangents from the point to the curve, and fl the angle of intersection of these tangents ; prove that the sum of the elements represented by - — -r, taken for all points exterior to • the curve, is z-k^. (Prop. Cbofton, Phil. Trans., 1868.) 56. Show that, for all systems of rectangular axes drawn through a given point in a given plane area. { JJ(a;2 _ J.2) dx dyY + 4 { U^V ^» '^!'}^ taken over the whole of the area, is constant ; and that for a triangle, the point being its centre of gravity, this constant value is (-iie-A)2 (a* + J* + «* - S^c2 - C«a2 _ aH^ (Ma. J. J. ■Waikek.) 408 Miscellaneous Examples. 57. If ab = a'b', prove that Jo Jo Xff = log ^^,j log (±^{^{^)- ^(o)}, provided the limits ^(o) and 0(00 ) are both definite. (Mb. Elliott, Frooeeiimgs, Lond. Math. Soo., 1876.) ] 58. If 5' denote the surface, and F'the volume, of the cone standing on the focal ellipse of an ellipsoid, and having its vertex at an umbilio ; prove that S = wa (i» - c^)*, r = Jir« {b', - c% where a, b, e are the principal semiaxes of the ellipsoid. S9- Prove that, if ^ be positive and less than unity, \\xP + x^)los{_i+x)- = -^ 4 (I) Jo XV Bin «7r »' and (»P + x-f) log (I - a) — = - cot BIT- \, Jo ' = V ' a p p^ (2) where (i) may be deduced from (2) by putting x^ for x. (Prop. "Wolstenholmb.) 60. If n, V be the elliptic coordinates of a point in a plane, prove that the area of any portion of the plane is represented by w-^ {ji} — v") d/i dv I V(^2 - c«) (c* - i^) taken between proper limits. 6 1 . Prove that the differential equation, in elliptic coordinates, of any tan- gent to the ellipse /i = ^1, is dft. dv 62. Hence show that the preceding difierential equation in /t and v admits of an algebraic integral. 63 . Prove that the differential equation of the involute of the ellipse /i = ^1 is Miscellaneous Examples. 409 '64. Show that, for a homogeneous solid parallelepiped of any form and ■dimensions, the three principal axes at the centre of gravity coincide in direc- tion with those of the solid inscribed ellipsoid which touches at the six centres •of gravity of its' six faces ; and that, for each of the three coincident axes, and therefore for every axis passing through their common centre of gravity, the moment of inertia of the parallelepiped is to that of the eUipsoid in the same ■constant ratio, viz., that of lo to ir. (Townsestd.) 65. Show that the volumes of any tetrahedron, and of the inscribed ellipsoid which touches at the centres of gravity of its four faces, have the same principal axes at their common centre of gravity ; and that their moments of inertia for aU planes through that point have the same constant ratio (viz. :8 VT : ir). — {im.) 66. A quantity .Jf of matter is distributed over the surface of a sphere of radius a, so that the surface density varies inversely as the cube of the distance from a given internal point S, distant b from the centre ; prove that the sum of the principal moments of inertia of Jf at iS is equal to 2M (a' — b''). (Comb. Math. Tripos, 1876.) 67. If (I - 2ax + a«r* = 1 + aXi + a'^Xi . . . + a»i„ + . . . , prove that t+i /.+1 2 X„Xmdx = O, \ Xn^dx = . -1 J -1 2n+l 68. A closed central curve revolves round an arbitrary external axis in its plane. Prove that the moments of inertia / and /, with respect to the axis of revolution and to the perpendicular plane passing through the centre of inertia of the solid generated by the revolving area, are given respectively by the expressions J^m{a^ + SK'), r=»(/.»-g; where m represents the mass of the solid, a the distance of the centre of the generating area from the axis of revolution, h and k the radii of gyration of the area with respect to the parallel and perpendicular axes through its centre, and I the arm length of its product of inertia with respect to the same axes. (TowNSEND, Quarterly/ Joternal of Mathematics, 1879.) {x - «)»-! / (z) dz, find the value of -— . Ans. /(z). dsi^ 70. Prove that the superficial area of an ellipsoid is represented by , fi (I -eV2»2)(?a; 2tcc' + 2ir«S ' ^ I i(i-e^x'^)(l-e'^x^) where ««-*« = a^e'', b'^ - = r^Jf (CS - 3a'0+ 2a')dpda, 30" where C is a chord of the area, whose coordinates are p, a ; the integration ■extending to all values of p, a, which give a chord G> a. INDEX. AiLMAN, on properties of paraboloid, 268, 281. AmBler's plauimeter, 214. Annular solids, 261. Approximate methods of finding areas, 211. Arcliimedes, on solids, 254. spiral of, 194. area of, 194. rectification of, 227. Areas of plane curves, 176. Ball, on Amsler'a planimeter, 215. Bernoulli's series, by integration by parts, 128. Binet, on principal axes, 312. BufEon's problem, 365. Cardioid, area of, 192. rectification of, 227, 238. Cartesian oval, rectification of, 239. Catenary, equation to, 183. rectification of, 223. surface of revolution by, 260. Cauohy, on exceptional oases ia defi- nite integrals, 128. on principal and general values of a definite integral, 132. on singular definite integrals, 134. on hyperbolic paraboloid, 271. Chasles, on rectification of ellipse, 234, 248. on Legendre's formula, 405. Cone, right,. 256. Crofton, oa mean value and probabi- lity, 346-391, 407, 410. Cycloid, 189. Definite integrals, 30, 115. exceptional cases, 128. infinite limits, 131, 135. Definite integrals, principal and gene- ral values, 132. singular, 134. differentiatioa of, 143, 147. deduced by differentiation, 144. integration under the sign J, 148. double, 149, 313. Descartes, rectification of oval of, 239. DifEerentiation under the sign of inte- gration, 107. Dirichlet's theorem, 316. Elliott, extension of Holditch's theo- rem, 209. on FruUani's theorem, 157. Ellipse, arc of, 226. Ellipsoid, 266. quadrature of, 282. of gyration, 309, 312. momental, 309. central, 310. Elliptic integrals, 29, 173, 226, 232,, 235, 243, 279. coordinates, 249. Epitroohoid, rectification of, 237. Equimomental cone, 310. Errors of observation, 376. Euler, 102. theorem on parabolic sector, 198. Eulerian integrals, 117, 124, 159. definition of — r(«) and.B(m, «), 124, 160. B(m, ») = r(m) r(«) r (m + m) r(«)r(i-«)= ^^ 161. 162. value of r(i)r(^)...r(^ 164. table of log (r«), 169. 414 Index. Fagnani's theorem, 229. Folium of Descartes, 192, 218. Fom-ier's theorem, 392. Frequency, curve of, 368. FruUani, theorem of, 155, 408. Gamma functions, 124, 159. ■Gauss, on integration, over a closed surface, 287.