-JJ.. m MEMOREAM George Davidson 1825-1911 / A SYNOPSIS OF THE PRINCIPAL FORMULAE AND RESULTS OF purr |lilatftematic«i» CHARLES BROOKE, M. B. OF ST. JOHN S COLLEGE. CAMBRIDGE: Printed by J. Smith, Printer to the Univerait}-. SOLD BY J. ct J.J. DEIGHTOX, CAMBRIDGE; AND C. J. G. I'v: F. RIVINGTON, LONDON. 1829 PREFACE. The Author of the following pages, having experienced the want of a compendious collection of results, in common probably with most mathematical students, endeavoured, during his undergraduateship, to form such a collection for his o^^^^ use. This has since that time been arranged, and considerably enlarged, and is now laid before the Public, in the hope that it may in some measure facilitate the labour of the analyst, by enabling him to compare and apply the results of previous investigation. It was originally intended to restrict these pages to a mere collection of formulae, but it is hoped that in adding an outline of many useful methods of operation, sufficiently detailed to render them applicable to particular cases, the increased size of the work will not be deemed an objec- tion. The references that have been given will probably be found sufficiently minute to enable the reader to find without difficulty any demonstration he may require ; had tlie proof of each result been separately quoted, the size of the work must unavoidably have been much increased. Tlie object has been to refer to works of the greatest authority, and to those which can most readily be procured. The principles of Notation that have been adopted in the following pages are explained in the Appendix ; wliere will also be found an explanation of the very ingenious and ivi5133i0 F KEl' ACK. powerful symbols invented by Mr. Jarrett, of Catharine Hall ; to whom the Author is indebted for several of the theorems expressed by those symbols, that have been in- troduced in the notes. The great difficulty of printing mathematical works correctly, may be fairly estimated by the number of errors that occur in the works of the ablest writers on the subject. As it is probable that some few may still have escaped the Author's notice, he will feel much indebted to any of his readers, who will have the kindness to communicate to the publishers any errors they may happen to detect. The Author begs leave to take this opportunity of expressing his acknowledgements to the Syndics of the University Press, for the very liberal manner in which they have contributed to defray the expenses of the work. St John's College, May, 1829. TABLE OF CONTENTS. Algebra. Art. ' Page 1^ — 3. Definitions and rules 1 4 — 6. Powers and roots ib. 7. Surds 2 8, 9 Greatest common divisor 3 10 — 12. Fractions ib. 13 — 15. Equations, simple and quadratic 4 16, 17- Ratios and proportion 7 18, 19. Arithmetical progression 8 20. Harmonical progression 10 21. Figurate and polygonal numbers ib. 22. 23. Geometrical progression 11 24. Permutations and combinations 13 25. Binomial theorem 14 26. 27. Polynomial theorem 17 28. Indeterminate coefficients 22 29. Logarithms ib. 30. 31. Continued fractions 28 32. General properties of equations 36 S3. Transformation of equations 38 34. Elimination 40 35, 36". Depression of equations 43 37, 38. Symmetrical functions of the roots of equations . . 44 3Q. The equation of differences 47 40. Limits of the roots of equations 48 41. Commensurable roots .30 42. Incommensurable roots 51 43. Quadratic factors 53 44. Impossible roots ib. 45. Application oi" the ihcoi-}- of equations to surd? . . 54 VI CONTENTS. Art. Pack 46. Cubic equations 55 47. Biquadratic equations 56 48 — 50. The equation y» + 2) = 59 51. General solution of equations 62 52 — 56. Indeterminate equations of the first degree . ... 63 57. Forms of square numbers 73 58 — 64. Indeterminate equations of the second degree ... 77 65. Porms of cubes 86 66. Indeterminate equations of the third degree ... 87 67' Forms of biquadrates 88 68. Indeterminate equations of the fourth degree ... 89 69. Solution of homogeneous indeterminate equations. . 90 70. Solution of the equation x^' — b=:ai/ 9I 71. General properties of numbers 92 72. Properties of prime numbers 93 73. Quadratic forms of prime numbers 94 74. Resolution of numbers into squares 95 75. Quadratic divisors 96 76. Ternary divisors 97 77. Scales of notation 99 Trigonometry. I, 2. Divisions of the circle 100 3 — 6. Relations of the trigonometrical lines 101 7 — 9- Values of sin a, cos a, and tan a 102 10. Formulae relating to two arcs 104 II. Formulae relating to double arcs 105 12. Values of the sine, cosine, &c. of 30°, 45°, and 60°. IO6 13. Formula? relating to three arcs 107 14 — 18. Relations between the sides and angles of plane tri- angles ib. 19. Solution of right-angled plane triangles IO9 . 20. Solution of oblique-angled plane triangles . . . .110 21. General principles of spherical trigonometry . . .112 22. Relations between the sides and angles of spherical triangles 113 23. Formulae for the area of a spherical triangle . . . II6 24. Values of the radii of the inscribed and circumscribing circles .; II7 CONTENTS. Vll Art. 1'aoi: 25. Solution of riglit-angled spherical triangles . . .117 26. Solution of oblique-angled spherical triangles . . . 121 27. Series for the sine, cosine, &c. in terms of the arc 128 28. Series expressing the inverse circulai* functions . .129 2Q. Series for determining the value of tt 130 30. Formulae involving impossible quantities .... ib. 31 — 33. Formulae for the sums of arcs, and multiple arcs . 131 34. Powers of the sine and cosine of an arc . . . .136 35. Sums of trigonometrical series 137 36. Resolution of trigonometrical quantities into factors . 140 37. Approximate solution of triangles 141 38. Solution of triangles by series 142 39 — 41. Formula3 peculiar to geodetic operations . . . .143 42 — 48. Formulae for the construction of tables 144 49- Formulas for the verification of tables 150 50 — 53. Trigonometrical solution of equations 151 54. Properties of a quadrilateral inscribed in a circle . 153 55, 56. Properties of polygons 154 Analytical Geometry. 1 — 3 Definitions and principles 155 AnalytiCttl Geometry of two dimensions. 4, 5. The equation of a straight line, and its pi-operties . 155 6, 7- Transformation of co-ordinates 157 8—10. The circle 158 11 — 13. The parabola I60 •14 — 16. The ellipse Ifi3 17—20. The hyperbola I67 21, 22. Discussion of lines of the second order 172 23, 24 Summary of equations I74, Analytical Geometry of three dimensions. 25. The straight line I76 26, 27. The plane 177 28. The orthogonal projection of plane figures . . . .179 29« Oblique co-ordinates 180 30 — 33. Transformation of co-ordinates 181 34. The sphere . ^ 183 VIU CONTENTS. Art. Pack 35. The cylinder 184 S6, 37. The cone ib. 38 — 42. Surfaces of the second order 185 43 — iC). The intersection of a surface of the second order, and a plane ] 89 47. The tangent plane 19I Differential CALcirLtis. 1. Differentiation of algebraic functions igS 2. Differentiation of exponential functions 194 3 — 6. Successive differentiation ib. 7. Differentiation of functions of superior orders . . . I96 8 — 10. Development of functions 197 11. Implicit functions 199 12. Transformation of the independent variable .... ib. 13. Elimination of an arbiti-ary function 200 14. 15. Particular values of the variable ib. 16. Maxima and minima 203 Integral Calculus. 1. Fundamental formula^ 204 2. Inverse circular functions ib. 3. Logarithmic integrals 205 4. 5. Decomposition of rational fractions ib. 6. Formulae for rendering surds rational 210 7 — 15. Integration of irrational functions 211 16, 17- Integration of elliptic transcendents 218 18. Integration of exponential and logarithmic functions 222 19 — 22. Integration of circular functions 224 23. Approximate values of integrals 228 24, 25. Successive integration 230~ 26. Integration of functions of several variables . . . ib. Differential Equations. 27 — 32. Differential equations of the first order, and of one dimension in d^y 231 33 — 34,. The introduction of a factor which renders a differential equation integrable 233 35. Singular solutions 235 CONTENTS. IX Art. I'AGK 36, 37. Equations of more than one dimension in d^i/ . . . 236 38 — 42, Differential equations of the second order .... 239 43 — 45. Linear equations of the 71"' order . 240 4(> — 48. Simultaneous equations 24.0 49 — 52. Approximate integration of differential equations . . 248 53, 54. Comparison of elliptic transcendents 249 55. Total differential equations of several variables . . .251 56 — 61. Partial differential equations of the first order . . . 252 62 — 67. Partial differential equations of superior orders . . . 255 68. Integration of partial differential equations by series .261 6g. Singular solutions of partial differential equations . . ib. Application of the Differential and Integral Calculus to Geometry. 1 — 3. The contact of lines 263 4, 5. Asymptotes 265 6 — 8. Singular points 266 9 — 11. Curves referred to polar co-ordinates 267 12, 13. Circumscribing figures 268 14, 15. The contact of surfaces 269 16, 17. The contact of curves of double curvature . . . .271 18—23. Rectification 272 24—26. Quadrature 274 27. Cubature 275 28. Transformation of co-ordinates ib. 29. Conditions which render a curve quadrable .... 276 30. Conditions which render a curve rectifiable .... ib. 31. Trajectories ib. 32 — 38. Remarkable algebraical curves 277' 39_48. Remarkable transcendental curves 279 Calculus of Variations 288 Calculus of Finite Differences. Direct method of . differences: 1. Fundamental formula* 292 2, 3. Successive differences ib. 4. Series involving the differences and differential coeflicients 29 !• b X - CON TK NTS, AUT, I'AGt 5. Difrcronc'c'?: of functions of two or more variables . . 295 6 — f). Intcqjolation of series Of- 10. Differences of the trigonometrical lines 298 11 — 14. The variation of triangles 299 14 — 18. The construction of logarithmic and trigonometrical tables 305 Inverse method of differences. 19 — 20. Integration of algebraic functions ....... 309 21. The numbers of Bernoulli 310 22. Integration of exponential functions 311 23. 24. Successive integration^ 312 25 — 32. Equations of differences 314 33, 34. Equations of mixed differences 319 35 — 30. Summation of series 320 40 — 42. Theory of generating functions 326 Functional Equations. 1 — 3. Reduction of functional equations to equations of differences 330 4 — 9. General solution of functional equations obtained from a particular solution 332 10, 11. Differential functional equations 339 Appendix 341 The following symbols, although not original, may perhaps require explanation, as they have not yet been generally introduced. ' «;:}>& is read a is not greater than b, « .... < • -^98 14 after same insert time /^ rea(/ sin B v cos / ^19 '^ 131 13 for COS B sin • 14 cos sin -^ »/lS4 15 circumscribed inscribed •/181 9 ... --— -~ -' cosy ef . ^195 13 (sec x^) .... (secx)*'^ J 21 (1-.T)^ .... (1-x^)^ y/ 196 1, Note 7n in the denominator .... \ra v/ V 198 5, 6, 'Sote d^z^-Ai .... d;z:^:u ./ ;^,199 •^203 5 11 n/Vf^ ■ then does j»ser< , if u =iO,v/ not ^^ ^210 5 for a' + 6' ar" read a, + 6, a;" t/ *^230 14 functions constants ^ ^231 3 ^v .... r, y v^232 6/ro)n the bottom c. .... Calj V •^244 4 . ; drS:=^s .... dts — sihr*^ /267 last ... rp y ^A Gfrom the bottom (/9=j8 .... M/e=e^ y 286 4 /I ...../;{ ^ y 346 last r = ^{a + (ffi2_6)ij, ,r and y are rational only when a* — 6 is a complete square. (1^.258; Bour.W^ — 21.) Method of extracting the n*^' root of a binomial quadratic surd. ' (f^. 259.) Greatest common Divisor. (8.) The greatest common divisor of two quantities may be found by arranging both according to the powers of some one letter ; then dividing the greater by the less, and the preceding divisor always by the last remainder, until the remainder is equal to nothing ; the last divisor will be the greatest common divisor required. If any divisor contains a factor not contained in both the given quantities, that factor may be rejected. (5oi^= f | —p± (p" + 4-7)2}. The following are some of the principal forms of equations Avhich may be solved as quadratics : ~n^ a^ A- + 6j a; + c^ ' «j x- + b^cV + c^ a„ar^ + b„oo + cj\ a„x" + 6„ .r + c„ a cV- + bw + c\- + p .a x" + b.v + c = b, then a : c > b : c; c : a < c : h. a -\- .V : b + X > or < o : 6, according as a < or > b. (w. 157—70.) (17) Proportion. If - = - , then o ct a : b :: c : d; ad = bc ; b : a :: d : c; a : c :: b : d; a±b : b : c±d d a — c : b — d :: r d a : a — b :: c : c — d a + b : a — b :: c + d . c — d; and generally, ma + nb : pa + qb :: mc + nd : pc + qd. (ir. 171—182; E. 461—5.) If a : b :: c : d :: e : f :: &c. then a : b :: a + c + e -\- kc : b + d+f+ &c. c d ma : mb :: — '^ 9 n n b d ma : — :: mc • " • n n a" : h" :: c" : d" 8 AI-OKBRA. If fi : h :: r : d, and e : f :: g : fi ; tlien ae : bf :: eg : dh. OV. 184—8.) If two numbers be prime to each other they are the least in that proportion. ( ^^- 189.) Compound proportion. (E. 488 — 504.) Arithmetical Progression. (18.) Let «„, represent the w"' term of an arithmetical series, ft, the common difference, «, the sum of 7i terms ; «m = ^'i + (^*-l) ^• 71 «„_,„ + «„ + ,„ = 2 a„. To insert n arithmetical means between c, and e. The m'^ mean = c + (e — c). n + 1 Given the p^^ and q^^ terms, to find the 7'*'* term. (W. 212; E. 402—24; Bour. 193—8.) (19) Any three of the quantities a^, o^,, 6, n, s; being given, the others may be found from the following formulae : ra„ = ff 1 + w — 1 . h. ( .v= - |2r/, +71-1. h]. ARITHMETICAL PROGRESSION. W= 1 H ; («1» »«> *) i ■* = -T-'^ 26 6 — 2a, /2s 2ai — 6 \i ;a^ = a„ — n—l. b. s^-i^a^-n-l.b). n-l s /"'^ n 2 (6, w, s); 6. s n — 1. a,. — a, "6 = w-1 5 — a, C 6= 2 '- \ n{n— '^ / 2. ««= — -«r (a„ W, s): a„ + « i . g^ — «! r l) ■= — I 25-ai-a„ V 74 = . B 10 ALGEBRA. (^'„, '', *•); I 2 - ( " 2; } 26 - I 26 b J 2s o, = a. K' ^h s); n lb = 2 ''"/' ' . {G. p. 280.) Harmonical Progression. (20.) The reciprocals of quantities in arithmetical progression are in harmonical progression, and conversely. To insert n harmonical means between c, and e. The m^" mean = ^ -f . {9i — m-\-i)e + mc Given the p^^ and q^^ terms, to find the /^ term. «, = ^^-P'^^p-"^ . (jr. 381.) (r-p)aj,-(r-q)a^ FiGURATE, AND PoLYGONAL NuMBERS. (21) The n^^ term of the m*^^ order of figurate numbers = the sum of 71 terms of the (m — 1)*^ order n + m . n + m — 1 7i + 1 m .m-\-\ " The above proposition may be thus compendiously stated: n n, «„ \n + m SQL 01 1 _ '» + ! 12 m+l w + 1 For an explanation of this notation see the Appendix. GEOMETKICAL PROGRESSION. 11 The »i* term of the series of w-gonal numbers is the sum of n terms of the arithmetical progression 1, 1 + m - 2, l+2.m-2, &c. = - jw — 2. n^— m—'^.n\. 2 The sum of the reciprocals of n figurate numbers of the w* order 1.2.3. ..m I 1 1 I T" ( 1.2.3... (7/i-l) ~ (7i + l)(w + 2)...(w+w-l)j* TO — (G. A. 132, 3; L. C. 91, 2; £. 425—39; B. 27, 8.) Geometrical Progression. (22-) Let rt^ represent the m^^ term, r, the common ratio, *, the sum of n terms. — n^r'" - ^ r"-l s =a. r-1 1 — r" , if >• < 1 -"^ 1-r a. If n = CO , s = . 1 — r To insert ?i geometrical means between c and e. m The w''' mean = c ^-V ^ '. Given the /j'** and 9''' terms, to find the /'' term. "'="'€;) 1-p (ir. 214—22; £. 505—22; Bour. 199—208.) 12 ALGEBRA. The value of a circulating decimal is a fraction whose nu- merator is the period, and denominator a number consisting of as many nines as there are digits in the period. (W. 225; F. 50—2.) (23.) Any three of the five quantities «j, a^, r, w, s, being given, the other two may be found from the following formulae. ffl =a,r"~' («i, r, n); r= — a. <«!, «„) n); I n n M — 1 „ n — \ {a„, r, n ^*-(r_l)r'^-^- (r-l)s /■«, = . (r, w, «,); ^ TN »-i ^ J (r— i)sr" ^ v.a„ = :: :; • r«_l s s—a. r^ r H ^ = 0. U(*-««r"'-«i(*-«ir"=o. (««> Wj « fr" ^r-i+-^ =0. PERMUTATIONS AND COMBINATIONS. 13 ra,. — rt, r— 1 («„ a„, r); ^ AOff a„ — log a, W=— 2— /^ ^-'+1. « loff r '«.. = — — ■ («i5 »', «); 'w = ^"g («i+r-l.s)-logfti log ?* 5 — a, L= fe^ -_„ „_ log a, 7i = ^ ^ 2 1 \. 1. log («-«i)- log («-«„) (««» -^^ «); j ^ ^ log«,,-log(r«„-r-l.g) ^ ^ [^ log r (a p. 281.) Permutations and Combinations. (24.) The number of permutations of n things taken r at a time = w(w — l)(w — 2) (w — r + 1). ^5 The number of combinations = —^ ^^ / . 1.2.3 r (1^.226 — 31; £:. 352— 60; 5owr. 146— 50.) " The equations containing logarithms, are placed here on account of their connexion with the rest, although the properties of logarithms have not yet been stated. ^ The number of permutations = \n . r \n The number of combinations = JL_ • 14 ALGEBRA. Binomial Theorem. (25.) (a + 6)" = «"+-a"-'6+ Y^"^a" ^t* + &c. the w + 1*'' term being ^(n-l)(w-2)....(w-w + l) ^„ _ ^ ^^^ 1 . 2 ^ 3 w If w is a positive integer, then the number of terms 7i + 1 ; the coefficients of terms equally distant from both extremi- ties of the series are equal ; the coefficient of every term is an integer ; 11.2 n n{n—\) 1 1 . ji ^ 9i(n-l) w(n-l)(n-2)(w-3) 1 -J 1 i -j i — H &c. 1.2 1.2.3.4 71 n(n—l)(n — 2) = - H — :!^ h &c. 11.2 . 3 also the theorem may be put under the following forms : (a + b)" _ a" a"-' b 1.2.3...n ~~ 1.2.3...n "^ 1.2.3...(w-l).l n « (a + 6)"=S,„ — 2i::L_ «"-"' + * 6'" -1 m— 1 The product of n simple factors, on which the binomial theorem depends, may be thus expressed : •1 n+l m—\, n + BINOMIAL THEOREM. 15 1.2.3...(n-2).1.2 + &c. S 2n + ^^ ^ — l—L-i a« - 1 h" - ' (a- + h-) 1 . 2 ... n V -r ^ 1.2 ... w (a + 6)-»-i = («=«- ^ + 62»-i) 4- _!L a6(o2"-3 + /,2«-3^ (2w-l)(2w-2) ,,„^ „ 5 ,„„ , (2n-l)(2n-2)...(n + l) , ,, ,, _ , 1 . 2 ...(n — 1) P : = O ■/ (a + 6)"-"=S, \n — m + 1 . w — 1 l2w [2^ W— 1 I 71 \_2n-l (a + by - ' = S,„ , "'-' . (a by " ' . (a«« - sm + 1 ^ ;,c« - cm + 1^ hn^ — 1 16 ALGEBRA. (a + b)-" = a-" — ~ a-''-'b + _1 L r.-n-'ii" 1 . 2 " 12 3 ^ «-"-^6^+&c. « :: ::! m '^-i b m(m — n)'J}-2b {a + by = a" + -a" ■- + ^ ^ a" . -^ 1 7^ 1 . 2 7i + 7» (m — n) {in — 2n) 2i _ 3 b 1 . 2 w + &c. ^ ,,-i -i 1 i-i 6 n-1 1-2 6 (a + 0) = a + - a" . -— — a" '^ 1 n 1 .2 n (w-l)(2n-l) i-3 +1.2.3 "" &c. > -2 _i 1 _i_i 6 1(1+71) -i-2 6 (a + 6) " = a '• _ - « " . - + ■■ ^ , ^ « " . " ^ 1 w 1 . 2 w l(l+7i)(l+2/l)- -i_3 6 ^i ;:: 1:. « 1.2.3 n + &c. ([F. 232—8; G. Chap, xii; 5owr. 182—90.) jw— 1 r— 1 n y (a + 6)"=a"+ S„.(-l)' |»^--l I _ X TO— l,n i-m b [m »i 11 « (a+6)-"=a— "+ S,„(-l)'".-P^«-^-'".- (26.) Polynomial Theorem 17 values of r a. 1.2.3...W a„ = the sum of all possible (1.2.3...rO(1.2.3...n)(1.2.3...r3) (1.2.3...rJ subject to the condition that the values of r^^, r^, &c. being either 0, or a positive integer ; and the partial divisor for a value being 1. (27-) (a + Oi'*' + a^jv" + a^x^ + &c.)'' J. J. . ^ + -a" -Iff.. n{n-\){n-2) „__ + — ^^ — = — — — r — -a^ a 1.2.3 3^3 1 n{n — \) + t:-^ — : — - a a, a. 1 . 1 ^^3^ 5.^-l)(^--2)(.-3) ^„.,^, cr*+&c. 1. 2 . 3 + /^(/^-l)(/^-2) ,^_ 1. 2 a" 'ttj a^ 72(W— 1) + ^1 :i «" -a, a, 1.1 n(n—\) „ „ 1 . 2 ,?j-i -.1 (Lacr. Introd. Diff. Calc. 19 — 24; Arbogast, Calc. des Deriv. 33.) c 18 Continuation of Note. m = s. ALCEDUA. ,« - ;n + r — I ST' r—l ~»H — r+1 n — m + r — 1 m — r + l n — m \m a"- a n — ??i + 1 m — 2 a.. + a"- ■ n- -m + 2 + a"- -m + 3 \n — 7n + 3 + ri"- - m+i n- — m + 4) 1^ 74 + m — 4 j j-Oj 2 ^'l + — WJ + 4 (Im — 8' [4 Iw — 7"[2* " m — 6 ;n — 5 + Iw— o \ 2 / m — 5 ) n — m -\- 5 \ m ,ji — m + 5 lols a| a, r^ + _9[3" ' + a a + 71 — m+S n p (ttg . ^4 + a. . 0.5) + — ^ . «ct m — 7 m — o ^ J a,/" -12 a| «/"-" a^ + m-10 VI2|2 "^ 13* / + i^'"-- /a ^3+«,.a3.a,+ ^. at \ o,"-" /n| X a,'"-' 1 POLYNOMIAL THEOREM. Conlinualion of Note. „ii — jn + l / ri '« — !'* + \n — 'm-\- 7 (| m-14 '|7* |m-13 '|5' ' + m-lO w— 10 fim-9 ^ >rt-8 . \\m «2 + + 10 [w-m + 8 l[w-16'[8 |m-15 '|6' tti'""" /«! al «2 «2 \ a.'"-^- /at a^ a| pa| -, al \ + m ^•"*i «fl( + &c 20 ALGEBRA. Continuation of Note. .n - 1 ■gr-ff" •sr- . a' «' w-1 TTT + [7i |yi — 2 [2_ \n — \ a" 7^—3 • rr + 3 |3_ t + w — 1 a"- c 71 — 2 a«- 3 a„ a^ . a„ + 71-1 -. a, . -4 ■ [^ |_w-3 ■ [2 . a„ •ST . a t w a" a" -5' \5_ \n-4! |3_ . «o + n {ao.a^ + cti-a^] + — . a. n -nr^.a^' f/V a' r^ + [tj |n-6 ■ [6_ 1 71-5 [^ - «"-" (of ii ^ 1 + [n-4i^-L2+L3_-M + 1^32 t [2- +"'•"' + "■• "4 ""h^ a\ a" + liL h^-7 ' \7_ \n-6 ' [5 POLYNOMIAL THEOREM. 21 Continuation of Note. + + (d — S {"i • a, + o. • "s + "i ■ «65 + I T ■ "7 t • ^ + - 8 |_8_ |7^-7 ■ [6_ ^ ^ a"-^ (al a\ a\ a\ \ + In this theorem, n has been considered a positive integer ; if an expansion be required for any vaUie of w, we must expand in a similar manner •ZJT" a" = S ^[w . a" - '• 2ir"'-''.a, 22 ALGEBRA. Indeterminate Coefficients. (28.) If a + a^x + a„y + &c. = 6 + t^.T + h.^v- + &c. for every value of -r, then a ^ b, flj = ftj, &c. " If a + a^a; + n^oj"^ + &c. = 0, then ^ = = Gj = &c. (FT. 346; 5oMr. 187,8.) Logarithms. (29-) Theory of logarithms, and logarithmic tables. (E. 220 — 55 ; Bour. 209 — 24.) Let log X represent the logarithm of x to any base ; log^A', the logarithm of x to the base a. log X + log y = log xy. ®;/j - 1 ^^ "wi - 1 • This is only a particular case of the following more general theorems, which may be demonstrated in a similar manner : n tn — 1 n m — 1 whatever may be the form or value of m^, then n n If S„, a^ _ 1 I ■>' = S,„ 6„, _ 1 [^ , then m— 1, >■ m — 1, >• LOGARITHMS. 25 Either of these equations may be taken as the definition of a logarithm : the former is most usually adopted ; and the various properties are therefore given in the order in which they most naturally follow on that supposition. /og;a = l. log 1=0. « , log,, .V log„& log„6.1ogj,« = l. log cv, + log .r, + log x^ + &c. = log {a\ . ,v„ . .r>3 . &c.) v log X — log y = log - . li' m log a" = — lop; x. n ° , 1 2--^— 3 ^'- . log„a?=: a-1 (a-lf (a-iy — 2—-^ -3 ^'- ^ It will be hereafter seen that unity has an infinite number of logarithms, of which the above is the only possible one. V S„, log.r„, = logP,„,r,„. & S,„(-l)' / / m 24 ALGEBRA. log. '^■= -^ 2— + —3 &c. where e is such a number that 1 p-1 (.r-l)^ (..-If ,^^ l«g« '^^ r::f^ ]^^-r^ - ^^^"17^ + '"^^-^^ - ^^]- ' loffE all 2 «,'=! 4. _»i — + '^ ^; ^ ^ + ^ ''' J + &c. * 1.2 1.2.3 = 1 + ■: + -r^ + 1 1.2 1.2.3 + &C. . 11 1.9 6 = 1+ - + i + &C. ® ^ 1 1.2 1.2.3 = 2, 71828182845904523536028 &c. The number e is incommensurable. (Bour. Appendix.) « log,.x'=S,„(-iy ^(x-iy CO (^e_lV« 1 » /'^i i\> log. a m, g (log^a.A')'""^ s =s. I w — 1 m— 1 c ^ — m — 1 LOGARITHMS. = 0, 4342944819032518276511289. log. a. log,10 = 2, 3025850929940456840179914 &c. a& .1' X' 0! ,•? log.(l+.v)=- - 2 + 3 -^^- " log.(l-..)=-(j + 2 +3+&0.). 1 + .1' ^ (v x^ .t?' > H-i— =2|-+- + _ + &c.|. 7 , .r + 1 ,(1 1 1 „ ) log. .t =log. a + 2 ]— + - ( — — ) 1 /.v — «\* -) + -( ) + &c[. ' 5 \.r + a/ ) Mog,(i+,^)=s,„(-ir-^ - OS /«'» log.(l-,r)=-S^- *^'l_,r '"2w-l Mog"-±l = 2S ^ ' log„r = log.« + 2S,„^-^ ('^') iJw — 1 \,v + a/ Qm-l D 26 AI.CEBRA. log, ,. = log. (,. - 1) + 2 1^-- + 3^-i-- + &c.| . * 1 1 1 (a;"-l a>"-l)' (A'"-!)' > ((1-a^"") (l-.r""/ (l-a;'y ^ ) , log..r = 7i l'^ J ^ + ^^ ^-^ + 5^ ^-J- + &c.j . * (Lagrange, Calc. des Fo?icti(ms, Le^on V^.} (Bouvier, Ann. de Math. Tome 16.) + « 1 /c^— 1\ ^ log,a? = log.(^-l) + 2S V log, a> = w.S^(-l) (1 -..'") (2m-l).(2cr-l)-'"-^ « log.a? = w.S, w J CO ^"W 2 LOGARITHMS. 27 log.,fc' = ■-.V-' .v'-.v-' + 3 1 n 1 &c.| « Jog. (1 + .V) = (1 + .0 - "{^' - (§ - ^ 1 )'^' /l_,i 1 7^(^-1) IX /I ^ 1 ^ Vs 1 • 2 "^ 1 . 2 1/ Vi 13 -^17-2— 2" ! .2.3 '& ^^'i log, (v + a + b) = logXa — a + b)+ log,(^■ + «) + logX^' + &) — log, (.f — a) — log, (ci- - 6) ^j ab(a + b) 1/ «&(a + 6) x ) ^ " W-cr(a" + a6 + 6-) 3\,i?^-.)-".S„.r'"S.(-l)' "'2//? - 1 Va- - .V (a- + a 6 + 6') / h r — 1 . (/« — ;• -I- 1) -2S. (2?»-l)(2.r--4.r-l)-"'-i 28 ALGEBRA. log. (.f + 2) - 21og. {w + 1) + 21og. (.1- 1) -log, Or-2) 2 . / 2 log. (.1- + 5) - log. (.V + 4) - log. (.7; + 3) + 2 log. a; — log. (x — 3) — log. (.r — 4) + log. (^. 6 Be„,-i B„„, Acw-i-B2,„ — Ao„, -B^,,,-! = 1, A2,„ • Bo,,, + 1 — A2,„ + 1 • Bo„, = — 1. " -^ ill' < ^ < J-. . 6^B„, B,„.B,„^^ B„;-' 1 B„,(B„ + B,„ + ,) a A^, ^' A,„ 4. 1 ^ B„ i^ B„ + i 30 ALGEBRA. If p, 7 arc any minibcrs whatever < A,„, IJ„,, respectivelyj p a A„. a '^^ — > ■ - 'v. — <7 6 B., b If A,„, + ,,B,,,„ - A2„,B„,„ + , = n, then Ao,„ Ag„, + Ag,„+i Ae„, + 2 Ao,„ + i ^^_^ B, „. ' B, ,„ + B,, ,„ + , ' B„ „, + 2 B, ,„ + 1 ' Ag„>+(y^ — 1)B2;;, + i Ag„, + g . a form an increasing series, each term of which is < - . If A2„,_iB,„. + i - A,,„ + iBo,„_i = w, then B.,„-i ' B„„_i+ B,„, ' B,,„_i+ 2b,,„ ' Ag^-i + (y^ — 1) Ag,^ Ag„, + i Bg,„_i + (W-1) B,,/ B,.„ + / form a decreasing series, each term of which is > - No rational fraction whose denominator lies between the denominators of any two adjacent terms of these series can be inserted between those terms. A,„ 1 1 1 (-ir B^ Bi Bi-Bg Bg.B3 B,,.B,, + j To approximate to the value of a fraction, Avhose numerator and denominator are high numbers: obtain Cj, r*,,, Cg, &c. by actual division, and the series of converging fractions may be obtained from equation (1). CONTINUED FRACTIONS. 31 A continued fraction may fiv([ucntly be simplified by the introduction of negative quotients ; for a _| __^ = flr + 1 — 1 /)-l (G. Chap, xxxi; G. A. 12 — 20; E. Add. Art 1 ; Lagrange, Equ. Num. Chap, vi, Art. 1, 3.) To reduce — — — — - — - to a continued fraction : di- cr. + ffij + «,, + &c. viding the denominator by the numerator and taking one term of the quotient, we have ('».-2-''.) + («,-^"-^) + &< a + a^-ir &c. < 6+61+&C. "6 h + 6, + &c. a a Assume ffj — -.h^-=c, a„— -^ .h„ = c^^ &c. = &c. proceeding as before, the second quotient will be 6 + 61 + &C. _6 \' c ) \^ c ) C -f- Ci + &c. c C + '^l + &c. ' In the same manner, assume b , h foj + - . Cj = a, 60 + - Co = a,, &c. = &c. c ' c ~ and proceed as before : we at length obtain t±h±K±^ = L_ i_ 1. (G. A. n.) a + a, + «, + &c. a h c 7 + ■ - + ' •:i + &c. bed 32 AI.CltBUA. Periodic continued fractions. The value of every periodic fraction may be determined by the solution of a quadratic equation ; and the root of every quadratic equation may be expressed by a periodic fraction. a 1 If -- = - , « b c + c . 7 —1 = 0, and b li d = i i c,4-/c/ + c„.7 - -^=0. if?=^ Cj + ^c„ 4- ^3 a c^.Cg.Cg + Ci — Cg + Ca a ^ c„.c^ + \ _ h c^c^ + l ' b Cj.Cg + 1 ~ (G.J. 21, 2; 5.147—9.) To reduce the roots of a quadratic equation to continued fractions : let the equation be aar + bx + c = 0, (1) in which fi- — 4ac> 0. « The fnU point is placed over the first and last fraction of the period, as in circulating decimals, to denote the extent of the period. CONTINUED FRACTIONS. 33 from (1), a' = ~(-b+ 1? - iac\^) ; assume ,v = e -\ , cF beinff > e but < e -{-l, then OjA'f + 6jcj' + a = 0, (2) in which equation a^ = ae' + he + c, b,=2ae + h; from (2), a-,=: — (-6, + 6/-4a.«i|^), 2a, = e,+ — , and so on. , = 2flr, . e, + h, b.j = 2a„.e,, + b„ &c. and a: = e + b\-d 2a, 2 a -"'. bl-d 2a, 2 a bl-d " 2 a., &c. 2a \/d-k, 2 a., Vd-h, > e^ and < e, + 1 ; > e, and < e., + 1 ; > ^3 and < fj + 1- &c. 1 1 1 In the same manner we may find the value of the other root of (1), viz. la ' The values of e^, e.,, &c. are the same in this case, but they occur in an inverted order. (G. A. 23; Lagr. Erpt. Num. Cfi. 6. Arf. 2; Legendrc, Thenr. Nomb. 59 — 74.) 34 ALGEDRA. (31.) To find the value of ^/c: the equation (1) and the above formula? become respectively X- — c = 0, fe,=l.e-0 6., = Oj -6^ — 6^ 63 = a„ . e., — bo &c. and y^c = e + o, = — : — - 1 ~ > gj and < gj + 1 ; a„ = 1 c-bl &c. > ^2 and < 63 and < 63 + 1 . &c. + &c. (B. 143 — 8 ; Legendre, Theor. Nomh. 28 — 33.) The periods in the series Oj, CTo, ctg, &c. e^, e^, 63, &c. for all values of c from 1 to 100 will be found in the annexed table. 1 {a^ a„ Og &c. V \e^ e^ 63 &c. v/17{^ /QQf7471 V ''"^ 1 1 1 1 10 Vi8i!J /QQf838 1 V '^^112 110 V2U /1Q?352.531 V ^t2 13 128 /Q4f929 1 V '^*U4 110 V3!?J jmtl V351\%\, n/5U /Q1 554345 1 V tl 121 1 8 x/37{/. V65I1 /QOI632361 V '*'* 11242 18 -J^Kl. N/7i???l jmww 73911/, J^\\\ V2MSJ y^oii,". s/lOiJ V^eii v/«!ll,'2 ■ V"«J V27iL'„ -^*^\%k yi2i|J /QQp43 1 V '3 2 310 /4Q (763 92 93 6 7 1 V ^11315 131112 Vl3it??tJ V '^"121 1 210 /44.f85 7 4758 1 V ^Ul 121 1112 j-^^wiw x/305^/„ /AK(9454 9 1 V '1222 112 V15i?J /Ql 16532356 1 V "^^il 1353 1 1 10 /4()( 103 7 652 5 67 3 10 1 V ^^11311262113 1 12 CONTINUED FRACTIONS. 35 /4,7IU211 1 V*'l 1 5 1 12 /7J,rl0 7 7 10 1 V '*> 1 11 1 16 ^i^lK'l, /i7Ki 11 11 1 V ' ^ U 1 1 16 V50!/4 /76fl2 5 8 93 4 3 985 12 1 V ' ^ 1 2 1 1 5 4 5 1 1 2 1 16 N/31f?i /77fl3 47 4 13 1 V ' ' M 3 2 3 1 16 \/^'"Ul2L414 V78ri*IV,'o /KQf4 7 74 1 V "'"'^3 113 14 N/79iY?'»/e /kj,(59295 1 V ""^^2101214 x/80!',\'e ^S5\litl ys^ii V56Ui n/83!L'„ / ei7t87378 1 V ^7 ill 4 11 14 ^»Mlls / ejQ i 9 G 7 7 fi 9 1 V^^U 1111114 V "'^'4 114 18 / KQ no 5 2 5 10 1 V ^ ' 1 2 7 2 1 14 y86ii\ojvr,'f'?L'» x/601Vr.','4 y87,1,'s /(i1 n2 3 4 95 5 94 3 12 1 V ^1431221341 14 V°°i21 11218 y62i»|'/' v/89!lr3l/8 V63iY.'4 ySOiL's s/65!>. /91f 10 9 3 14 3 9 10 1 V ^^1 I 15 1 51 1 18 V66i|,'e /QO( 11 8 74 7 8 11 1 V ^'*^ 1 12421 1 18 /fi753C7 92 97 6 3 1 V ' ^5 2 11711 25 16 /Q3fl27 1143 4 117 12 1 V ^"^^ 1 1 1 464 1 1 1 18 V68U,'„ ^9i\TiiVn\TnTAV!}s /«Q( 5 411 3 11 45 1 V ^*33 1 4 1 33 16 V95iV»»,'8 /70'«9596 1 V ' '21 212 16 v^96!>/4.5,^ /7-I i7511 21157 1 V ' ^ t22 1 7 1 22 16 ^ /Q7'1«:* 11 8998 11 3 16 1 V I^ M 1 5 1 1 1 1 1 1 5 1 18 v/725«'. y98i'.'IV,'8 /7Q {983389 I V ' 1 1 1 5 5 1 1 16 v/99i'fi (£. Jdrf. 41.) root. The last quotieht in the period = 2e, e being the nearest (B. 149.) .3(i ALGEBRA. General Properties of EtiUAXioNs. (32.) Every equation of w dimensions in .r may be reduced to the form .r'' + o,,_j.r''-^ + a„_^.r"--+ +a^,r + a = 0; '■ which for convenience may be represented by (p (x) = 0. If (p (,v) be divided by .v — a, the remainder will be

. a \ aj 38 ALGKBKA. If a is positive, there is an even, and if negative, an odd, number of positive possible roots. If a + l//3 is a root of an equation of which the coefficients are rational, /3 not being a complete square, then a — 1//3 is also a root. (W. 280 — 97; Bonr. 275—80; L. 178 — 94.) TRANSFORMATlOlSr OF EQUATIONS. (33.) To transform the equation 0(.r) = into another whose roots are Oj — c, a„ — c, &c. a,^ — c: assume a? = ?/ + c, and by substituting this value of x in (p(x) ^ 0, we obtain " + nc y"-' 71 (n - + ~^— 1 . 2 r-' + «n-l + (n-l)a„_,c + n p«-l y+ c" + (n- -i)«„ .,c^-' + ««-iC"-' + {n- -2)«„ -2*- + «n-2C"-' + &C. + &c. = 0; which may for convenience be thus expressed : ^ 1.2. ..(n-1)^ + 1.2...(w-2) y^-' + kc. J~f+~y + cpic) = 0; or , =f/,, = «, ; />,. = ^,, — ^ -«,, = ^(2flr2 — 6r,.6i); ft, = 0, — ^ ^a„ + ^ ■'a,, = i(3a3 — 2a,.fc,, — a^.^i); &c. = &c. = &c. and let o = ^j . 6, + 2b., .f)^ + 3 £., . 6, + &c ; then is the final equation required; from which however it is necessary to exclude all products of the quantities jj, ^3, &c. exceeding m dimensions, and of Cj, a„, &c. exceeding n dimensions. (G. J. 44.) Third method. If a,v" = a^,v" " ^ + a^x" -" + a^x''-^+ &c. multiplying both sides by aw, and substituting for ov" its value, we obtain + (flj . ffg + ff • a4)(v) = has w — 1 of them: hence these two equa- tions will have a common divisor '^ (a — a)'"~^ which may be found, and the original equation thus reduced to another of (n — ni) dimensions. Also (b"(x)^=0 has m — 2of the equal roots, and therefore 0(cf ) = 0, 0'(a) = O, (p"(^) =0, will have a common divisor ^ (.v — ay"~^. If (p{'V) = has two roots ^^ +a, they may be found by changing the signs of all the roots, and finding the greatest common divisor of the original and resulting equation. (IF. 319, 20; Bour. 292 — 8; L. 205—8.) (36.) Recurring equations. Every recurring equation of 2w + 1 dimensions has one root = +1, according as the last term is negative or positive, and may be reduced to an equation of 2?i dimensions by division. (W. 295.) The roots of a recurring equation of 2n dimensions may be found by the solution of an equation of n dimensions, if ?^ > 1. (ir. 325 ; Bour. 299—300.) 44 ALGEBRA. Symmetrical Functions of the Roots. (37.) Sians of the powers of the roots. Let *S'i, *S'„, S^, &c. represent the sums of the 1st, 2nd, 3rd, &c. powers of the roots. Sn = — ff„ - 1 *S'i — 2ff„ _ 2 ; ^3 = ~ ^« - 1 *^2 — «« - 2 '^1 ~ 3rt^, _ 3 ; &c.= &c. ^n + TO = ~ ^« - 1 *^« + »J - 1 ~" ^'^« - 2 *^/! + OT - 2 ~ • • • ~ ^ ^m- Let aS.i, a^.j} &c. represent the sums of the negative powers of the roots; a. a a a «! On 3 a a a a a * a a &c. = &c. V V "' continually approximates to the greatest, and — =.^±1 m >i SYMMETRICAL FUNCTIONS OV THE ROOTS. 45 to the least root, as ?» increases, if all the roots are pos- sible. (W. 352, 3; Bour. 302—7; G. A. Chap, v; Gergonne, Ann. de Math. Tome. 3.) To express the sums of the powers of the roots in terms of the coefficients only, and conversely. So=- ^«„_e+ 2""-^' 3 3 3 4 4< 4 4< &c. = &c. {Waring^ Med. Alg. Cap. i; Arbogast, Calc. des Deriv. 68—78.) (38.) Let T (aj^i.a/i...a,/"') represent the sum of all possible transpositions of the products of the roots taken m at a time, the several roots in each transposition being raised to the r/**, r„*'^, &c, r^j*** powers respectively. Sr^.s,^ = s,^^,^+ r («/..«/'); Sr^ -Tia,'-^. a/^) = ^ («/' + ''3 . a/") + T (a/' . a/> + 'O + r(a/..a/..a3'-3); .S;.^. r(ai''.a/».a3''3) = T(a/' + '-4.a/».a3''3) + T(a,'-Ka/- + '-*.a/3) + J'(ai^'.a/».a3''3 + '-4) + r(a/'.a/^.a3'"3.a/4) ; &c. = &c ; n+l * If a^ is the r^^ root of the equation = S„,c„,_i.r"-"' + i, then S,a;" = wS,(-l)"'"''^' - 1 ^ m-r+l •ZeT .C 7/i — r + 1 ^d '- = S' u„-V + i ' f - = - ,-^ • S'°''"l jm 46 ALGEBUA. + T(a/-..«,'Va/3.. .«,„_;•.-. + '•".) + r(«,'-..a/^.a/3...a,/".). Hence T (a/> . aj\) = S,.^ . S,.^ - S\ + ,.^ . '1 a '3 'i^'a '3 'i^'3 'a '1 'a^ 3 'i^'a 3 If ri = ?•„ = r^, T (a^ . a. . 03 1**') = g *S'^J'^ — -|*S',._ . aS'2,., + 3 'S'3,.^ . Every symmetrical function of the roots of c^{x) = 0, and the coefficients of every equation whose roots are symmetrical functions of the roots of that equation, may be expressed in terms of a^, a^^ &c. the coefficients of <^{pG) = 0. (G. J. Chap, vii ; Bour. 303, 8.) The Equation of Differences. (39.) Let?/'' + K-^y"-' + K-of~' + ... + b^y + 6 = be the equation whose roots are the squares of the differences of the roots of d)(-v) = 0, and a^, cTo, &c. the sums of the 1st, 2nd, &c. powers of its roots. 2w ^ 2m{2m — l) THE EQUATION OF DirFERENCES. 47 2m (2m -1) (2m — 2) ^ ^ 1. 2 ... m 2^ '"'' 6,_,= -i(6,_.cri + ^-i — — , therefore the next greatest negative coefficient + 1, if > — —i is a sBperior limit. If the roots are all possible, and positive, each of the following quantities is a superior limit ; n(n-l) " a'„_i-2a„_, 71 1 . 2 n(n—l) (a\_2-2a„_,.a„_3 + 2a„_4) (W. 305.) The roots of the equation (pX1. [1 ] Lagrange's Method. Find a number 6j such that one of the roots is >6p buti + l, and for w substitute b^-\ — in the given equation. The resulting equation in .i\ must have at least one possiV)le root > 1 , let this root be > b. and < b. + 1 ; 52 ALGEBRA. for a?j substitute ft,, H , and repeat the same process with the resulting equation in a;^ : and so on, as far as may be thought necessary. Then 1 1 1 b^+/b,+^h^+ &c. [2] Newtoti's Method. Determine by experiment a number Cj such that one of the roots is ^ c^, but :|HCi + 0,1 ; for a? sub- stitute Cy+y, and from the resulting equation in y, obtain an approximate value of y, by neglecting y" and all superior powers: thus an approximate value of x is obtained. If greater accuracy is required, the same process may be repeated. (Bour. 341 — 9; Lagr. Equ. Num. Chap. 3.) Case 2nd. Suppose the difference between two possible roots to be < 1 . . . 1 Determine the inferior limit - of the positive roots of the equation of differences, and let k be the least integer > */l: transform the given equation into one whose roots are k times as great, in which equation if the series of natvu-al numbers be successively substituted for x, one root and only one will lie between any two numbers w, and m,-{-\, and therefore the k corresponding root of the given equation will lie between — , k and . A nearer approximation may be made to this root, w + 1 by either of the preceding methods. The equation must be cleared of equal roots, before this method can be applied. {Bour. 350—7; L 217—20.) Quadratic Factors. (43.) Every equation of n dimensions has at least one possible IMPOSSIBLE ROOTS. 53 quadratic factor, if every equation of ^w(w— 1) dimensions has one possible factor either of the first or second degree. Hence every equation of 2 m dimensions may be decom- posed into m quadratic factors. (L. C 38, 9) To find the quadratic factors of (p (7)=0, then f,(p, q) = 0, and /„ (p, q) = 0; from which equations q may be eliminated, and the correspond- ing possible values of p and q determined as above (Art. 34.) will give the quadratic factors required. {Bour. 338, 9-) Further investigation of the commensurable divisors of literal and numerical equations. (Clairauf, Elem. cCAlg. P. 3.) Impossible roots. (44.) If a + ^ ,y/— T is a root of an equation of which the coefficients are possible quantities, a and /3 being possible, a — (3/^ — 1 is also a root. Impossible quantities of all degrees may be reduced to the form A + BisJ — 1, A and B being possible quantities. To find the impossible roots : substitute y + x^ —1 for .r in the given equation, and another equation will be obtained b^,A + B V^ = 0. Determine all the corresponding values of y and z that satisfy the equations A=0, B = 0; let these be aj, a^, &c. ^j, /3,„ &c. then «i + /^i'v/ — Ij "2 + (^2\/ ~1' &c. are the required roots. 54 AL6EBKA. The scries of quantities, — 4/3f, — 4/3^, &c. will all be found amongst the negative roots of the equation of differences : to determine whether — c, a negative root of that equation, is one of these quantities, substitute ^^c for ^ in ^ and B, and if it be such, the resulting polynomials in y will have a com- mon divisor. If the equation of diiferences has other negative roots besides the above series, it must also have equal roots, which may be determined by preceding methods. If any series of quantities be successively substituted for a^, there can be only as many changes of sign in the results, as the equation has possible roots. The equation (p'(x) = 0, has at least as many possible roots as (,r) = 0, wanting one. (TF. 355—62; G. A. Ch.2; Bour. 385—92; F. 533 — 5.) , Newton's rule for discovering impossible roots. (W. 363.) Application of thk Theory of EauAxioxs to Surds. (45.) To extract the cube root of a + \^h : assume a + Vb = z{x+ V'yY, then or— y=. - . (a^ — h)z \ '■> z must be so assumed that (a* — b)z = c', c being an integer. Eliminating y we obtain 4i%x^ — 3czlV — a = ; which equation must have at least one possible root, in order that we may have a + Vh b}a z (cv + Vyy. To extract the n^^ root of a + \/b : assume CUBIC EQUATIONS. 55 then cr^ — y= - (a^ — b) z" " - " . z must be so assumed that (a- — b)z"~- = c", c being an integer. EHminating y between the equations y = ar — c, and n(n—l) „ „ , we obtain a final equation in ,v, which must have at least one possible root in order that we may have a+ \/b-4:iz(w+\/y)". {Bour. 403—5; LC 47—9; G. A. Chap. 15.) Cubic Equations. (46.) Cardan's Method. Let the equation be reduced to the form x^ + a^x + a = 0. Assume y + z= r, and 2/^ =— 5 ®i ( a a^ a\ \ r a a~ a\ ^l 2 ~ 4 "^ 27 3 27 \' i • The three values of x are y + ijr, a' a If — + r:; > 0, 1/ and r arc possible, and a result may be 4 27 56 ALGEBRA. obtained. In tliis case the equation lias one possible, and two impossible roots. " 4 + 27 ="' ""■ """' "'' (2) ' k) ' ""^-Hi) • If >" rJ? < 0» .V a"<^ ^ are impossible, and no result can be obtained. In this case all the roots are possible. Let the equation be ,v^ — a^x — «:=0, which has at least one possible root ; let this be a, the other two roots are 1 :ni - 2(a±4ai-3a-| ) from which their values may be obtained, if the value of a be determined by approximation. (W. 325—31 ; E. 734—49 ; Bour. 406—11 ; G. A. 58.) Solution of a cubic equation by the method of divisors. {E. 719—33.) Solution by the theory of symmetrical functions. {Bour. 415; G. A. 52.) Biquadratic Equations. (47-) [1] Descartes' s Method. Let the equation be reduced to the form w^ -\- a„x- -{- a^x + a-=0 :, the first side of this equation may be supposed to be the product of two quadratic factors x^ + ex +f, and x" — ex+ g. Multiplying these factors together, and equating the coefficients of = 0. 61 where a'^~' =a, '.' a"-! — 1 cj-,^. In this case the roots are evidently the same as before, !)ut arranged in a different order. Let /3, /3', /3', fi''~', &c. be the roots of the equation, ii^ ~ ^ — 1 = ; assume then €"-' = (a + /3a« + (i-a^" + ... + /S^-^a^^'V-S observing that all indices of a > a^~^, and of ^ > p— 1 may be respectively divided by those quantities, and the quotients neglected. Each of the quantities a, Ay, A^, &c. may be reduced to the form in which b and c are known quantities independent of a ; hence the values of a, A^, &c. are not altered by substituting succes- sively a", &c. in the place of a. Let Ci''"S cf~\ &c. c^_/~S be the values of c^'\ when /3-, /3^ &c. ft^-\ are substituted for /3, then p-1 p — 1 &c. = &c. 6'2 ALGEBRA. Application of this method to the solution of the equations .r-'-l=0; j'-l=0; .r^^-l=0; .x'^'-l=0. {Lagrajige, Equ. Num. Note 14 ; G. A. 77; Leg. 435-74.) General Solution of EauATiONs. (51.) The method of symmetrical functions is inapplicable to the solution of equations of more than four dimensions, because the reduced equation is of 1.2.3... (?i — 2) dimensions, if n is a prime number, and of — — ^^ ' ' ^ dimensions, if n is composed of two prime factors p, and q ; both of which quantities are > w, if m^ > 4. {Lagrange, Equ. Numer. Note 13.) Wronski has given what he asserts to be a general solution of equations of all degrees. {Resolution generate des Equations.) Torriani has published a Memoir, the object of which is to shew that Wronski's solution is incorrect. {Hist, da Acad, de Lisb. Tom. 6.) A general solution of equations of the fifth degree cannot be obtained. {Abel, Bulletin Univ. des Sciences; Ann. 1826.) A general solution of equations of any degree superior to the fourth cannot be obtained. {Rujffini, Theor. delle Equaz. Cap. xiii.) Indeterminate Analysis. EQUATIONS OF THE FIRST DECREE. (52.) Solution of one equation containing two unknown quantities. Let the equation be ax -\-by = e. (1) INDETERMINATE EaUATIONS OF THE FIRST DEGREE. 63 If a and h have a common factor c, this must also be a factor of e, otherwise the equation is impossible in integers ; a will therefore be considered prime to h. The above equation is always possible if c >ah — a — h. (B. 41.) First Method. Let the value of y obtained from (1) be e — a iV e, — a,x e — ax f—gx being the integral part of the fraction — - — . Assume e^ — a^x^^hu, . e,—bu „ e,, — a„u then x= =/i — jg'iWH =- ; e„ — a,w, = 72-^2^1 + e^ — a^u^=^a„u„y &c. = &c. 'yj + 1 ~ ^'n + 1 ^« - 1 = ^n ^«' Cn + x-anf^n ^H - 1 — I Jn + l~Sn + l ^y»» "» + ! from this last equation we obtain a value of ii^^ _ ^ which is an integer for all integral values of u^^ ; and thence an integral value of u^_„, and so on, until we obtain the values of x, and y, y=A.j(.„ + B„. (J4 ALGEBRA. If a and h have different signs, the number of solutions is infinite; in this case A^ and A,^ are both positive. If a t=^ 6, the number of solutions is finite ; in this case Ai and A^ will have different signs. Suppose A^ negative, then tlie number of possible solutions cannot exceed the number of values of u„< — ^ . A, (W. 367—9; E. Pt. ii. 1—23; Bour. 122—9.) Second Method, [l] Let the equation be aa) — by= +1. G A Convert the fraction - into a continued fraction, and let — ^ b B„ a be the converging fraction immediately preceding - , then (Art. 30.) aB^ — bA^=±l, + or — , according as 7 > or < - . If ax — by = 'l, and aB^^j — 6a„,= — 1, the given equation will be satisfied by assuming x = nb — B^, y = na — A^. If positive values only of a) and y are required, we must have n > — , and > — . b a If ax — by=^—\, and aB,„ — 6a,„ = 1, the values of x^ and y may be similarly obtained. If ax-by^aB^ — bA^, it will be sufficient to assume • b2>c„, then, Cg must be >C2ei, and < One^, in order that it may be possible to obtain a positive result. ' (B. 164 ; E. 24—30 ; Bow. 136—9.) (54.) Solution of one equation containing three unknown quantities. Let the equation be ax -{-hy + t'% = e^ in which at least two of the coefficients, as a, and b, are prime to each other. A . „ . « — ^ being, as before, the nearest convergmg fraction to - , the values of x and y are X = (e — c z)b,„ — nb, y = na—{e — c%) A,„ ; subject however to the following conditions, e e — cz e — cz %< -, n> A„,, n < — -— B,„. c a INDETERMINATE EQUATIONS OF THE FIRST DEGREE. 67 If any one of the coefficients is negative the number of so- lutions is infinite : suppose therefore that they are all three positive; in which case the number is finite. To determine the number of solutions : let k ; /^ fo, &c. /|., be respectively the integral parts of e e — c e — c e — 2c e— 2c ;; -T-B,„ — — A„, —7 — B;„ r— ^'«' ^^- CO a a e — kc e — kc o a then the total number of solutions =/j +/„ + &c. +fi.- If A; is a high number, the following method of determining the number of solutions will be more convenient : let i?j, R^^, &c. /?4, be the fractional parts of the terms e — c e — 2c e — kc -^B,„, — ^B,„, &c. — ^B,„, which fractional parts will recur in periods of b terms ; and let k g be the integral part of - , and s = Ri + Ro + Szc. + Ry Similarly, let r^, n, &c. r^, be the period of the fractional parts of e — c e — 2c . e — kc A,„, A„„ &C. A„, a a a k h the integral part - , and s = r^ + r^ + &c. + r^ ; then the total number of solutions = |y^ - — | \ke-\k{k + \)c.\ - {gS + 2?! + Ro + &C. + R^_^f;\ + {hs + r^ + r„-^ &c. + i\.na\- observing that h must be substituted for in the series ^j, 72,,, &c. 68 ALGEBRA. The period of /?,, if,,, &c. will frequently consist of a repe- tition of sliorter periods, in which case the summation of those quantities will be simplified. If no two of the quantities are prime to each other, suppose a = 7»flj, and b = mb^ then e — oz „ e,—c,z a,x + b^y= =f-gz+ — m The values of z which render gj — Cii?c-r^w may be found by the solution of the equation mu + Cji^ = e,. (B. 162, 3 ; Bour. 140.) (55.) Conditions of indeterminateness of equations of the first degree. If after elimination the values of the unknown quantities appear under the form g , the equations are indeterminate ; and conversely : this is always the case when the constants of one equation are equimultiples of the constants of another. (Missery, Ann. de Math. Tome 1.) (56.) Solution of n equations containing more than n un- known quantities. Let the equations be ^a^A\ + ^a„Xo + &c. + ^a^w^ = ^a, "a^.Ti + -a^ooo + &c. + -a^a?^ = ^a, 85c. &c. "a^x^ + "«2-^^2 + &c. + "a^x^ = "a. (^) Assume x^ = ^b + ^b-^u^ + ^b„u„ + ^637^3 + &c. x^ = "b + "b^u^ + "boU^ + "b^u.^ + &c. &c. " &C. cr^ = ''6 + '■fe^w^ + '62^2 + ''*3«'3 + &c. (B) s„'<'«"'',.='«.{;si- (A) ■^.=-6+s.'6„.7»,.,{::;i. (b) INDETERMINATE EQUATIONS OF THE FIRST DEGREE. 69 Substituting these values for .-Pp x„i &c. and putting the coefficients of u^, u„^ &c. separately = 0, we obtain ^«i . ^h + Vf, . -h + &c. + ^rt, . 7> = ^a , 'a^M)-\-"a„.'h -\r'i>i^- +'a^.''b = "a, &c. &c. "a,^b + "a., .-b + &c. + "a, . 'b = "a : 'a^ . \ + 'a„ . -61 + 'a., . \ + &c. + 'a^ .'b, = 0, ^a^ . \ + -«2 • '^1 + ""3 ■ \ + &c. + -a,. .% = 0, &c. &c. "a,.'b, + "a„.'b, + "o^.\ + &c. + X-''^ = ; (Q (1) \ . \ + iflj . \ + 'a^ .% + &c. + Iff, . '•6„ = 0, 8ec. &c. ^a^.^b^ + V^a.-feg + ^a^.^b^ + &c. + X-''^3 = 0, ^a^ . \ + -V/., .-63 + -rt.3 . -'63 + &c. + =tf, . % = 0, &c. &c. "a^ .\ + "a^ % + "a^ 'b., + &c. + "a^ % = 0. &c. &c. &c. (2) (3) The number of indeterminate quantities w^, Un, &c. will be r(r—l)(r-2)...(r-7i) 1. 2 . 3 ...(7i + l)' of which only — - — — ^^ enter mto the value of 1 . 2 ... w each unknown quantity. It appears from the equations (C) that ^b, ^b, &c. ''by may be any system of values that will satisfy the given equations (A). 70 ALGEBRA. [l] If we liave a single equation we need only consider the first equation of the several systems (1), (2), (3), &c. The equations of the same form which are essentially evanescent are .a^ — fl^-ffi + 03-0 +f/4.0 + &c. + cfy.O = 0, .a^ + a„.0 —a.^.a^ + a^.O +&c. + rty.O = 0, .0 -{■ a„.a^ — a^.a^-\- a^.O + &c. + cf,..0 = 0, .0 +ff„.«4 4-O3.0 — a4.a„ -t- &c. + a^.0 = 0, .0 +02-0 + a3.cr4 — a4.a3 + &c. + a^.0 = 0, &c. &c. We may therefore assume \ = 0, =&3 = ^64 =04, % = % = 0, % = , &c. \ = , &c. % = , &c. % = &c. 0, %= 0, %=-a^, &c. '64 = a4, %= 0, %=-a„, &c. '•65 = , %= a,, %=-a,, &c. '66 = &c. Particular cases of one equation. Cfj iTj + «„ "^2 ^ ^ ' a?j = ^6 + a^Wj, x„ = ^b — a^Uy x„ = ^b — a^Ui + a^ u^, ttj = ^6 — cf J u^ — a^Uy INDETERMINATE EQUATIONS OF THE FIRST DEGREE. 'Jl .r^ = ^b + Ooic^ -\- a^u„ + a^u.^^ x„=-'h — ayii^ + a^u^ + 04^5, a?3 = ^6 — a^Wg — ^2 ^3 + ^4^65 a?4 = "'ft — aj«^4 — Og^'s — a^Uf^. [2] If we have two equations, ^a^x^ + ^flo^'g + '031^3 + &c. + ^a^x^ = V, ^0^0?^ + 'a„x„ + "«3*'3 + &c. + ''a^A\ = "a ; it will be necessary to consider the two first equations of the systems (1), (2), (3), &c. The values of the quantities 6 that render these equations essentially evanescent are 'b,= ('a,.''a„-"a,M^),\= , &c. ^63= — {'ayra^—'a^.^ai), %,= ('ai-X — "ffi-'as)? &c. \= 5*^4= (^«3-'«4 — "«3-^«4)5 &c. &c. Particular cases of two equations. ^ttici^j + ^a^a^o + ^^3 "3^3 = *«5 ^aj.t?, + "0.2X2 + ^OsX-j = "a ; Xi = ^b + {^a.2."a^ — 'a2.^a^)u^y X2 = -6 — Cai . -a.^ — -Oi . ^ff 3) zSi 32 33 36 37 39 40 22 25 28 30 36 37 39 15 16 17 21 23 24 25 31 35 36 38 40 41 13 16 18 23 24 25 26 27 29 31 32 35 36 39 41 12 14 16 17 18 21 24 25 27 28 32 34 36 37 42 21 23 25 30 34 36 42 43 49 FORMS OF SaUARE NUMBERS. ']5 Let a be a prime number, and b prime to a ; tlien if 6^(26)^(36)^ &c. (^6)', be severally divided by «? the remainders will all be unequal. The product of a possible and impossible form of squares to the same modulus is always impossible. (B. 42 — 52.) Conditions of the possibility of the equation a" — ay^ = bx", in which a and b are positive integers, and a < h. Let a number c be found such that c > ^6, and c'^~rtc-r»6; and let the following system of quantities be constructed : c'^ —a = b b^ »/-, c^ = Wj />i + c l^^b^; Cf — a = 6^ 62 ^i'i C2 = w?2 *2 ± ^1 ^ 2 ^2 ; C„' — ff = 6263^3', &c. = &c. then if n, b, ftj, are such that any integral values of c^, C25 gj, e.^, will satisfy the conditions Pj' — a c-r^ 6, €{■ — b c-r^ a ; Cj* — tfc-r^ftj, e.^' — b^':^r^a\ tlie equation is always possible. The equation a x'^ + by^ = cz^, in which a, b, and c are prime to each other, is possible, if the conditions a ef + 6 c-tj c, ce^- — 6, and those of a, and c — h, when divided by c, tlie equation is possible ; if either of these three conditions fails, the equation is impossible. {B. 53, I78; Leg. 23 — 7-) Impossible pairs of quadratic equations : 2m2- \ or — y- = u^. \oc" —2y' = 2i ix^ + 2/" = 2«% ix" + ^y" = ^- (ct?^ — ?/- = 2 u^. \od"^ — '2y"-=u^ 1 2.372 _ 2/2 :== J^i (2 cr^ - r = 2 ^^^ rr m, then n n^ — nv 2mn INDETERMINATE KtiUATlONS OF THE SECOND UEGKEK. 77 m,v Second Method : assume ?/ = 1 -i , then 2m.n a?= — ;; mr — n~ Two square numbers whose sum, or whose difference, is also a square may be found from the equation To find the values of cc which satisfy the equation [l] Suppose c c|^ /'^ ; assume (a + hx -^f^'x^) =foD + — , then in" — n"a x = n"b~2mnf [2] Suppose rt 54^ /^ ; assume (f^ + bx + cx^Y =/H , then '2mnf—n-h x = n-c — m'^ 1 TYhX [3] Suppose a = 0; assume (6<2? + c (I. Assume a, the same process may be repeated, and we obtain •V -«!/„- = 6,, i?r/. In which equation b^<^b^: by pursuing this method we must at length arrive at an equation in which 6^„ is either 1, a square, or < a. In the latter case, by transposing, putting c for 6„,, and neglecting the subscript indices of ,v, y, and z^ we obtain af — cz- = ay-. Proceeding as before, this may be reduced to another equation ,v'^ — cz" = o^i^y', in which a^^< c. Then similarly putting e for a,^, we have .V' — ey'-=cz'. 80 ALGEBKA. By tliis process, the coefficient of cither y or % must be reduced to a square, or to unity : the indeterminate quantities in the equation last obtained may be expressed in terms of a-, y, and X, by successive substitution by means of the equations (1). The equation is thus reduced to oc" — y" =^ a x^ . The only systems of quantities to be computed are n- — a = bik- &c. = &c. T — ''m'^m - 1 n' a = f/j V- "1 = a" /'- &c. = &c. !-l a ^m ht - 1 -1 ar — ay" = hyx", af — ay-=^hr,x"^ &c. = &c. x^ — ay" =■ bj^z" = c%". x" — cx"=^a^y'^^ x' — cz" = ar,y", &c. = &c. x--cz- = a„^y- = eif. &c. &c. in which k", I", &c. are the greatest integral squares in the 71' — a n~ — c quotients — - — , , &c. b a (Lagr. Mem. Bert. 1767; B. 174-6 ; Leg. 15-22 ; E. Add. 52.) (61.) Solution of the equation x'^ — y- = az^. Let a=-ai.a„, acadi %-=Zi.z„, then y=^{a,.zi'-a,.z,'). (5.54; Leg. l^.) INDETERMINATE EQUATIONS OF THE SECOND DEGREE. 81 (62.) Solution of the equation cc- — ay"=- + 1. Let \^a be expressed by a continued fraction, (Art. 31.) and let — ^ be any converging fraction corresponding to the quotient 2e; the above equation will be satisfied by the values If the period consists of an even number of quotients, the equation txr — ay~ = 1 will be satisfied by every converging fraction corresponding to a complete period, and ,v" — ay-=.— 1 will be impossible. If the period consists of an odd number of quotients, the equations x" — ay" = — 1, and x" — ay" = 1, will be alternately satisfied by the converging fractions corres- ponding to complete periods. {B. 150; E. 96—111, Add. 37-) Having given one solution of the equation cV- — ay'= + 1, to determine the general values of x and y. [l] x" — ay" = l, and p" — aq- = l; x=^{ p + qVa I'" + p — q Va^] > [2] x" — ay^ = l, and p" — aq'= —1 ; ,v=L[p^q \/aJ"' + p — q\/a |""' } , y = 2y^ \p + fi v^«T"' - p-qVaT" \ . [3] x"^ — ay- = — 1 , and p- — aq'-= — 1 ; y=j^J'p~+WoT-'-P-gVor-']- (5.180.) 82 ALGEBRA. Table of tlie least Values of x and ^ in the Equation x^—mr all Values of « from 2, to 99. (£. v. 2; p. 89.) :1, for a •V y « A' y 2 3 2 53 66249 9100 3 2 1 54 485 66 5 9 4 55 89 12 6 5 2 56 15 2 7 8 3 57 151 20 8 3 1 58 19603 2574 10 19 6 59 530 69 11 10 3 60 31 4 12 7 2 61 1766319049 2261 53980 13 649 180 62 63 8 14 15 4 63 8 1 15 4 1 65 129 16 17 33 8 66 65 8 18 17 4 61 48842 5967 19 170 39 68 33 4 20 9 2 69 7775 936 21 55 12 70 251 30 22 197 42 71 3480 413 23 24 5 72 17 2 24 5 1 73 2281249 267000 26 51 10 74 3699 430 27 26 5 75 26 3 28 127 24 76 57799 6630 29 9801 1820 77 351 40 30 11 2 78 53 6 31 1520 273 79 80 9 32 17 3 80 9 1 33 23 4 82 163 18 34 35 6 83 82 9 35 6 1 84 . 55 6 37 73 12 85 285769 30996 38 37 6 86 10405 1122 39 25 4 87 28 3 40 19 3 88 197 21 41 2049 320 89 500001 53000 42 13 2 90 19 2 4-3 3482 531 91 1574 165 44 199 30 92 1151 120 45 161 24 93 12151 1260 46 24335 3588 94 2143295 221064 47 48 7 95 39 4 48 7 1 96 49 5 50 99 14 97 628O9633 6377352 51 50 7 98 99 10 52 649 90 99 10 1 TNDETERMINATK EQUATIONS OF THE SECOND DEGllEE. 83 (63.) Solution of the equation oc^ — ay" = +b, in which b < l/a. If this equation is possible, b will be found amongst the denominators of the complete quotients (a^, fin, a.,, &c. Art. 31.) of the converging fractions whicli express the value of ^/a. Given one solution of the above equation to determine the general values of x and y. Suppose that we have found, m, n, p, q, such that mr — an'=: +b, and p" — aq"= +1, then w=:mp + anq, y^np + mq, in which the general values of .v and y obtained in the preced- ing Art. must be substituted for p and q respectively. The equation p' — aq- = l, or p'^ — aq-= —I must be employed, according as the known solution and the given equation have the same or contrary signs. To determine the general values of x and y in the above equation, b being > v^a. [l] Suppose b to be composed of factors 6^ b^, &c. each a, and > c, the equation may be transformed into another in which b :j> a, or c. Suppose a Cj the same process must be repeated. In the successive transformed equations, the value of b" — 4ac is always the same. If 6" — 4rtc < 0, a and c are the least numbers contained in the transformed equation, in which b:lf> a, or c. (Leg. 53—8; B. 100—2.) INDETERMINATE EQUATIONS OF THE SECOND DEGREE. 85 [l] Suppose ft'^— 4«c <0, and=— /. Multiplying the equation by 4a, and assuming % = 2ax + hi/, we obtain z'^ -rfy^ = 4!ae. (Cl €\ * — ) which satisfies this equation, also renders by + %^^2a, a solution may be obtain- ed ; if both these conditions are not fulfilled, the equation is impossible in integers. [2] Suppose 6^ — 4«c = 0. In this case the first first side of the equation becomes a perfect square, and is therefore possible, if e is a square. [3] Suppose &- — 4ae>0, and =k'^. In this case the proposed equation may be decomposed into two simple equations a^x + h^y=-e^, and a^x + b„y = e^- [4] Suppose 6-— 4«c = 4/, /not being a square. First; let e be < l//: expand a root of the equation OiX' -\- bx + 6', in a continued fraction, {p. 32.) If amongst the quantities 2ai, 2a2, &c. the denominators of the complete quotients, we find the number e, the numerator and denominator of tlie cor- responding converging fraction, if substituted for x and y respectively, will satisfy the given equation. As often as e occurs, a different solution may be obtained ; if it does not occur at all, there is no integral solution of the proposed equation. Secondly; suppose e > \/f'- assume // = nx^ + e^/p BC ALGEBRA. an' + bn + c a,= , e 6j= 2na + b, c\ = ae; the equation is reduced to a^,t\" + h^Xyij^ + ^j2/i*= ± ^^ which must be solved for each value of n. The given equation is impossible in integers, unless n may be so assumed, that a^ may be an integer. {Leg. ^5 — 83.) Solution of the equation a.v- + bxy + cy" = + 1- Let it be transformed so as to fulfil the conditions 6 :^ a, or c, and a < c; multiply the equation by a, and assume z^ax + by, the equa- tion is reduced to z" + ey-= +a, ^vhich may be solved by preceding methods. (E. Add. 66 — 7^) FoEMs OF Cubes. (65.) Every cube bj:^ 4w, or4w + l; 5=1:5 7^, or7w + l; ^9n, or 9n±l. All cubes are of the same form to m.odulus a, as the cubes 0^ 1^ 2% &c. («-l)'. a^ h^^ a to modulus 6. The equations a?' + y^ = »^ , 0?' y^ !^ -3 + 5-3 = ^' w^ + y^ = 2z^, and generally mx^ + nay^ = %'^^ in which m is a number of an impossible form to modulus a, and n prime to o, are impossible in integers. No triangular number < 1 is a cube. (B. 60 — 70 ; Leg. 328—32.) Indeterminate Equations op the Third Degree. 87 {G6.) The equation a + o^.v + a„ or -\-a^x^ = 1/ admits of a direct solution only when a = b". First Method. Assume b" + a^x + a„a)- + a^x^={b -\- ca?)-, and a^ = 26c; c" — a„ a^ — ^b-tto then X = = — 7-77; ■ a^ ^b-a.^ Second Method. Assume 6^ + a^.v + a^x- + a^x^ = (6 + ex + ex")", and a^ = 2bc, a„ = c" + 2be; then ^ = ^iLl^. (^.26,7-) e- If a is not a square, one solution must be obtained by trial, let this be a + ff ^ A + a^ h" + cr 3 h^ = k", then by' assuming x = y + h, the equation is reducible to the above form. (B. 184, 5 ; E. Pf. 11, 112—27.) Solution of the equation a + a^x + a„x" + a^x'^ = y^. [1] Suppose a = 6': Assume 6^ + a^^x + a.^x~ + a.^x^ = (6 + exy, a. a„ — 36e" then e = — - , and x = -^ . 3b- e^ — Os [2] Suppose a.^ = c^: assume a + ayX + a^x" + e^x" = (e + cx^, a„ - a — e^ then e = — ^ , and x — 3c-' 3ee- — «! [3] Suppose a = b^, and 03 = 0"': assume 6' + «, v + a„ x" + c^x' = {b + c.r)\ ttj — 36-c then .V = ^Y~2 * 36c — «« 88 ALGEBRA. If neither of these conditions is fulfilled, one solution must be obtained ; let this be a + a J h + a„Ji- + a.Ji^ = k^, then by assuming a) = h-\-y, the equation may be reduced to the form [l]. {E. 147 — 61 ; B. 191—5.) Solution of the equation ax" + cy~ = z^. Assume tV = ap^ — 3cpq-, and y^3ap-q — cq^i then % = ap- + cq". {E.IS^.) Solution of the equation x" + axy -\-hy"-= %^. Assume x=t^ — 3btu' — abu^, y =3t"u-{-Satu" -\- {a" — h)u" ; then % =t' + atu + hir. (B. 196.) Forms of Biquadrates. (67.) Every even biquadrate % 2* . w . Every odd biquadrate c^ 2*. ?^ + 1 . All 4'^ powers are of the same form to modulus a as 0^, 1% 2^, &c. (—a)*, when a is even, 0*, l^ 2\ ^(a-l)p, when a is odd. Remainders of 4**^ powers from every modulus from 3 to 12. Mod. Rem. Mod. Remainders. Mod. Remainders. S 1 6 1 3 4 9 1 4 7 4 1 7 '124 10 1 5 6 5 1 8 1 11 12 1 3 4 .5 9 1 4 9 Impossible forms of equations of the fourth degree. 1 is a biquadrate. (Zeg-. 324— 7; B. 1^—6.) Indeterminate Equations of the Fourth Degree. (68.) Solution of the equation a + a^x + a„.v- + a.yV^ + ai.v'*=y"- [l] Suppose = 6^: assume b" + a^x + a^x" + a^cc^ + a^oc^ = {b-\-ex +/.2?")"» ffj a„ — e" ''^"^^ '=26' -^="26"' then x= —-, . f — «4 [2] Suppose a^ = c^: assume a + a^x + aox"^ + a^x^ + c"x'^ = (f+ ex + cx")-y 2c 2c f--a then x= ^ -— -. [3] Suppose a = 6", and a^ = c" : assume b" + a^x + a„x" + ag-^^ + c-o:'* = (6 + es = p- + eq". (^.198.) Solution of the equation .v^ + axy -\- by- = z'^. Assume .v = t^ — Gbfir — ^ahtu^ — (a^ — b)hu'^, y = 4tt^u + 6nftr + ^{a" — b)tu^ + {a"-2b)au^; then z = f + atu + bu". {B. 197) (69) Solution of the homogeneous equation ax" + ttj a?" ~ ^ 2/ + a^x"- ~ ~y" + &c. + a„ «/" = + e, or /C^»^)=±e; in which x and y, as well as y and 6, may be considered prime to each other. Assume x = cy -\-e%, c being such that ac" + «! c" -^ + a„c"-" + kc. + a„ ^^e; by substituting this value of x, the given equation is reduced to by" + b^y"-'z + b^y"-~z" + &c. 4- b,^ = ± 1. There cannot be more than n values of x, between the limits + ^ c and — ^ e, that render the integral polynomial (p{x)^e. (5.88.) THE EaUATION .v" — h=ay. 91 To determine p and q, the values of // and z, wliich render f(y, z) a minimum : let ttp a„, &c. a, + /3i v^ - 1, &c. be the roots of the equation /^r" + ftj .X'" - 1 + fe^o?" - = + &c. + fc„ = 0, w • • • then - is the converging fraction nearest to otie of the quantities a^, a«, &c. a,. Sec. which must be determined by experiment. {Leg. 120—8. E. Add. 28.) (70.) Solution of the equation x" — b = ay, a being a prime number, and b prime to a. If n and a — \ have a common factor c, there will be c solutions, or none. 0-1 If the given equation is possible, 6 " — 1 c-r^' a- If one solution, ci = e, has been obtained, the others may be found by multiplying e by the several roots of the equation ,v" — \=ay. The proposed equation is always possible, if n is prime to a — 1 . Solution of the equation r" — 1 = ay. If n is prime to a — 1, the only possible solution is d? = 1. Let « — 1 = ew, then x = %■", z being any number prime to a. If u is the remainder of ^^-f-«, all the values of w will be found amongst the remainders of the quantities M, u", u?, &c. r^^'^H-a, unless two or more of these remainders are equal. If r"' — 1 = at/ is impossible, r being a root of the equation ,i" — 1 = rty, and m a divisor of w, then r is a primitive root. If 7ij, w,^, &c. are the prime factors of n, the number of primitive roots of the equation x" — \=ay will be expressed by n. — 1 n., — 1 n . . — = . &c. n, Wo 92 ALGEBRA. If w is a prime number, every number prime to a is a primitive root. If n = ti/'' .iif'^ &c. and m^, m^, &c. are the remainders of n/'\ nf'\ &c. -;- a, and )\, r„, &c. the roots of the equations *'"'• — \=n //, x'"-^ — \=ay, &c. the values of x are the powers of rj, ?„, &c. Solution of the equation x"" + 1 = ay, a being a prime number, and a — \ ~ ■> — , &c, a a' a^ Prime Numbers. 93 (72.) Natiu-e of prime numbers. {E. 37 — 44.) All prime numbers > 2 c^ 4?^ + ^ ^ ^"^^ > 3 ^= 6 w + 1 . No algebraical formula can contain prime numbers only. The number of prime numbers is infinite. If a is any number whatever, and h^^ />„, 63, &c. the numbers prime to, and < 2 a, all prime numbers will be of one or other of tlie forms 4aw + 6j, 4aw + 62' 40^ + 63, Sic. The number of times that any prime factor a occurs in the product 1.2.3...W is the sum of the integral parts of the fractions There cannot be n prime numbers in arithmetical progres- sion, unless their common difference is divisible by the product of all prime numbers up to n ; except when n is the first term of the series, and then there cannot be more than n. The sum of any number of prime numbers is a composite number. If a and 6 are prime to each other, and each term of the series 6, 26, 3&, &c. (a — 1)6, be divided by a, the remainders will all be unequal. {B. 29 — 39 ; Leg. xvi— xxiii.) If n is a prime number, 1 . 2 . 3. . . (^i — 1) + 1 ^-^ w, and {x + 1) (c^? + 2) {x + 3) . . . (ct? + w - 1) - -i" " ^ + 1 ^.n. \i X is prime to w, ci'" 54^ ci', to modulus n ; a>" - ' — 1 c-Ti w ; ct?" ~ * i^ aw, or an + \:, _^i(n-i) 5^ ^^^ or aw + 1 . {Lagrange.) Mem. de PAcad. Berlin, Ann. 1771 ^ B. 86, 7) 94 ALGEBRA. If 2" — 1 is a prime number, 2"~'(2"— 1) is a j)erfect number. The only known perfect numbers are 2 (2--l) = 6, 2" (2^-1) = 28, 2' (2^-1) = 496, 2' (2^-1) = 8128, 2'2(2i'_l) =33550336, 2^6 (2i7_ 1) = 8589869056, 2^8 (2^9-1) = 137438691328, 230 (2'i_l) = 2305843008139952128. If 3.2"- 1, 6.2"- 1, and 18.2-«-l, arc prime numbers, 2» + ni8.2-"-l), and 2"+^ (3.2"- 1)(6.2"- 1), are amicable numbers. The only known amicable numbers are (w = 1) 220 and 284, (n = 3) 17296 .... 18416, (n = 6) 9363584 .... 9437056. (B. 25, 6 ; Schooten^ Eccercit. Math. § 9.) The number of prime numbers > a", is very nearly repre- ^"'^''y log..-lo8366 - (i^^- 389-96.) Quadratic Forms of Prime Numbers, (73.) Every prime number ■45 4}X + 1 is also i^^y- + z"i i^ 6.V + 1 c^jf + Sx'^; 4=5 8 A- + 1 =h^' + ^'5 andy-±2z'^'; ^^ 8a? + 3 ^:>,y^ + 2z'; bj= 8.V + 7 '^,y"-2z-; 54:5 12a' + 1 ^:i.V"^3%"-; 54:5l4a7 + l, +9, +11 y^.y-^^z-; ^:i20x±l,±9 c|3?/- — 5sr-; ^.20cT + l, +9 ^,y" + 5z-; cj:{24cf + l, +5 -^^y-^ez-. &c. &c. QUADUATTC FORMS OF PTIIME NUMBERS. 95 Every prime number '^y'' + az" is contained only once in that form. If c is a prime number ?4i?/- + «;j^*, and h is contained m times in the form f + ou", he is contained 2m times in that form, unless ftc-rsr. Every prime number h^,ay- + bs;-, or dpi ay" + 2byz + 2ez", is contained only once in that form. If «, ft, and r, are odd numbers, and unequal, no prime number can be contained more than once in the form ay" -r hyz + cz'^. If c is a prime number, any number a 1^ ^c is contained only once in the form t" + cu"^. (B. Ch. IX ; Leg. 142—7, 229—45.) Resolution of Numbers into Squares. (74.) (.r/- + yf )(*;-' + y^') . . . (.v„" + 2//) h^ ^- + y" ; and .V and y have 2" ~ ^ different values. (.vf + xj" + Ay)(yi" + yi) '^ z^" + z^" + z^" + z^: {x^" + x\j + a?3= + x^-){y^" + y„j) b|a z^" + z,- + z^" + ;jr/. (.rf + a?/ + ci'/ + .<-)(2/,- + y/ + 2/3^ + y^) N=i ^i" + ^2* + ^3* + ^/i and generally, (a'i^ — ao?/ — hx^" + abx^")(y^" - ay„j — by," + ahy^") ^:>z^" — az2" — bz^- + abz^'. (Lagr. Mem. Berl. 177^-) X,' - (a- + l)x,' b^ (a"^ + l)y,' - y^. The equation x^" + .Vj" + x^ + a?4" = ay, in which a is a prime number, is always possible. Every integer is a square, or the sum of 2, 3, or 4, squares. 96 ALGEBRA. Sa.v + (a + 2)(rt - 2)- h^(2aA\ - « + 2)- + {2a,v^ -a + 2y + &c. + (2«.r„ + „ - a + 2)'. (B. Ch. vii; Leg. 148—55.) Quadratic Divisors. (75.) If t is prime to w, every divisor of the formula t" + au", is also a divisor of w" + a. Every divisor of f" + au^ c^s a^y" + 2h^yz + c^z", in which a^c^ — b^- = a^ and &i >> -^ «i, or |- Cj ; or :|> ( - j . Every divisor of t" + 2?^- ^i^ c»" + 2y^. Every odd divisor of t" + 3u~ cj= oc" + Sy^, t" — 5u"^:iar — 5y^. (5.103 — 9; Le^. 136— 41.) Let R(a^^''~^^ -^ c) represent the remainder of a5(<^-i) _i. c, c being a prime number. R{2^<'-'^-^c)= +1, ifc-^,8n±l, i?(2^(--'>^c)= -1, ifo?4=8w + 3. Every prime divisor, p, of a" + 1 '^^29ix + 1 ; or at least is of the same form as the divisors of a*" + 1, r being the quo- tient of n divided by an odd number. Every prime divisor, p, of o'*— 1 «^ 27ix + 1, or is of the same form as the divisors of a' — 1, r being any submultiple of n. If m and n are prime numbers, then i?(w^('»-i'H-*w) = (~l)5<"^-^^"^-i'i2(m^<"-'>-H7i). The values of ,v in the equation i?(.^.^('-i)^e)= +1 are the remainders of any odd squares, < 2c— 1, -r-c. QUADBATIC DIVISORS. 97 If c is a prime number =j:; 4n + 1, and a an odd divisor of ar + cy-, then R(,a^^'~^^ -^ c) = + 1 ; + or — , according as a ?^ 4w + 1 : if c t^ 4w + 3, If c ^:^ 4/1 + 1, and a is a divisor ?4:j 8w + 1, or 8w + 3, and 6 a divisor ^:i 8w + 5, or 8w + 7} of the formula or + '2ci^ ; orif c?4^4w + 3, and « b^ 871 + 1, 6 54=8^ + 3, then i2(ai«^-') -^ c) = 1, and J?(6i<<'-'» -^ c) = - 1 . No prime divisor of t~ + oe*" can be expressed by more than one quadratic divisor of that form. {.Leg. 148 — 99, 232.) Properties of the quadratic divisors of 1r -\- au"y a being a prime number ?43 8w + l. {Leg. ^'^2 — 80.) Every quadratic divisor of t" •\- avr contains at least mie number prime to a, or to jO, and < a. {Leg. 410 — 5.) Further investigation of the linear and quadratic divisors of t" + cu". {Leg. 204 — 12.) Tables of linear and quadratic divisors explained. {Leg. 213 — 25.) Application of the above tables to determine [1] a prime number greater than a given number; [2] whether a given number is prime or not. {Leg, 246 — 59) Ternary Divisors. (76.) No number «4:^ 4w, or 8w + 7? can be of a ternary form. If 0=^471 + 1, or 47i + 2, the formula f + cu" has at least one ternary divisor. If c'^87i + 3, the formula f^ + cu- has at least one quad- ratic divisor a, such that either a, or 2 a is of a ternary form. {Leg. 260 — 6.) If a quadratic divisor of t^ + cu" can be decomposed into {a,y -{- b,%y- + {a^y -|- h.^zY + {a^y + h^z)\ then c=i{a^h^ — a,^h^)- + {a^h^ — a^h^"^ -\- {(lob^ — a^b^)-. N 98 ALUEBRA. To find the ternary divisor corresponding to Let F, G, H, be respectively resolved into fk^k^, g^i^si hk^k^^ fko and gk^, fk^ and hk^, gk^ and hk^, being respectively prime to each other. Assume a^ b„ — a„ b^ =/; the required divisor is k, {a^y + b^z)\' + k„{a„y ^b-z)\' + k^{a.^y ■\-b^z)\-. Since k^(a^y + bj^z) + k„(a„y + b^z) + k^(ajy -i- 63^) = 0, this divisor may be reduced by elimination to the form ay" + byz + ez^. If a = e, b = a or e, or e = 0, and at the same^he least of the quantities a and e > 2, this divisor will have two ternary forms, and no more : if these conditions are not fulfilled, there will be only one ternary form, corresponding to any given value of c. If either c, or 1 c, is a prime number, [1] two different ternary forms of c cannot correspond to the same ternary divisor of t" -i- au"; [2] the formula t" -{- au- will have as many ternary divisors as there are ternary forms of the number c ; [3] each ternary divisor of t" + au" has only one ternary form. If the number a is comprised in a ternary divisor f + cu", c will be comprised in a ternary divisor of f + au"; and the corresponding ternary values of a and c will be the same. If py" •{■2qyz + rz-, a quadratic divisor of t" + cu", has several ternary forms, and in these forms given numbers are substituted for a? and y, the results will all be different, if each of them<|-t'. SCALES OF \OTATIOV. 09 Suppose p prime to r; tlion if r is a divisor of t'- + pn'\ r will also be a divisor t'- + bii", h being any number con- tained in the formula py" + ^qy:i: + rz". Properties of reciprocal divisors. (Z,e^.278 — 313.) Scales of Notation. (77) Every number may be represented by a„r" + a^_^r'*-^ + a„ _ ^r" " = + &c. + a^r + a, the radix r being any number, and a„, &c. integers < r. To transform a number from one scale to another : divide the number in its own scale by the new radix, and repeat the same process with that and all succeeding quotients ; the re- mainders taken in order, beginning with the last, will be the digits which express the given number in the new scale. In any scale, radix r, if the number itself and the sum of its digits be respectively divided by r— 1, the remainders will be the same. If the sums of the odd and of the even digits be severally divided by r + 1, the difference of these remainders will equal the remainder of the number itself divided by r + 1 . Any number <^-r» 2", if the number represented by the n last digits '^^ 2". If the dimensions of any figure be expressed in feet, inches, and j^th parts, the area or solidity may be most readily found by transforming the number of feet into the duodenary scale, and multiplying together the results in that scale ; observing the same rules as in the multiplication of decimals. Every number < 2" "•'^ is the sum of some terms of the series- 1, 2, 2-, 2\ &c. 2". Every number < 3""*"^ may be expressed by the sums or differences of some terms of the series 1, 3, 3-, S\ &c. 3". (B. 121—9.) KK) TRIGONOMETRY. (1.) Divisions of the circle. The circle has usually been divided into four quadrants ; each quadrant into 90 degrees, (90°) ; each degree into 60 minutes, (60) ; each minute into 60 seconds, (60"). The more minute divisions are generally expressed in decimal jDarts of 1" ; they are however sometimes expressed by dividing each second into 60 thirds (60'"), and so on. The French have recently divided the quadrant into 100 grades Q.QO'); each grade into 100 minutes, (100') ; each minute into 100 seconds, (100*') ; and so on. In Astronomy the circle is sometimes divided into 12 signs, each of which contains 30°. (2.) To reduce a^ b' c" to the English scale. J° = integral part of (a^ b' c" - 0,1 x a*" 6' c"), B' = integral part of 0,6 {preceding remainder), C" = integral part of 0,6 {last remainder). F = 54'. i° = pirnv. r=32",4. I' = r85",185. V = 0",324<. 1" = 3",08641 &c. The arc equal in length to the radius = 57° 17 44" 48'" &c. = 57°,22577 &c. = 206265" nearly. = 63^66197 &c. {W. 19; L. 1—12; C. 631 ; Enc. Met. Vol. i, p. 672.) TRIGOKOMETRY. lOl (3.) To adapt any formula in which radius is unity to the general radius, r : multiply each term by such a power of r as shall render it of the same dimensions as the highest mentioned in the given formula. (jL. 24; JT. 18.) (4.) Mutual relations of the trigonometrical lines. (sin a)" + (cos a)" = 1 . cosec a . sin a = 1 . sin a 1 + (tan a)" = (sec of. cos a 1 + (cot a)" = (cosec a^. cos a verso = l —cos a. ""; ^^ cot a, sin a covers o = 1 — sin a. tan o.cota = 1. suvers a = 1 + cos a. sec a . cos a = 1. (L. 35.) (5.) The sine is positive in the Jirst and second quadrants, and negative in the third, and fourth ; the cosine is positive in the Jirst and fourth, and negative in the second and third quadrants : from these data the signs of all other trigonometrical lines may be determined. sin2w7r = 0, cos27i7r = l, tan2w7r = 0, cot27i7r=oo, sec2w7r = l, cosec2w7r= co • sin(27i4-^)'7r = l, cos(2w + -t)7r = 0, tan(2w + ^)7r=oo, cot (2w + i-)7r = 0, sec (2w + ^)7r= CO , cosec (2w + i)7r = l. sin(2w + l)7r = 0, cos (2n + l)7r= — 1, tan (27i + l)7r = 0, cot (27i + l)7r= — oo , sec (2»^ + l)7r= — 1, cosec (2n + l)7r= x . sin(2n + |)7r= -1, cos(2n + |)7r = 0, tan(2n + |)7r=-co, cot (2w + |)7r = 0, sec(2n + |)7r=co , cosec (2w + |)7r= — 1. 102 THICONOMKTRY. (6.) Different arm whose sine, or cosine is the same. sin(2w7r4:a) = sin (2w + 1 .ttT a), sin (2w7r + a) = — sin (2n + 1 .TT + a) ; = — sin (2mr — a), sin {2n + I.tt — a)= —sin ('2n + 1 .tt + a)- cos(2w7r + «) = cos(2w7r — a), cos(2w + l.TT — «)= cos(27^+ 1.7r + a), cos(2w7r4:a) = — cos (2n + 1.7r + a)j cos ( + a) = cos ( — o), sin ( + a) = — sin ( — a). sin(2w + j-7r + a)= cos(2w7r4:a), cos(2/i + j.7r + ff)= +sin(27i7r + a), sin {2?i + 1 . TT + a) = — cos (2w7r + a), (7-) cos (271 + 5 . TT + a) = + sin (2n7r + a). sin (^ TT + a) = cos a, cos (^TT + a) = ± sin a. Z. 36 — 41 ; C 64 — 81.) Values of sin a. cos a. tan a. cos a cot a 1 — (cos a) ^i 1 + (cot a)- tan a . 1 + (tan a)"] '• 2 sin i- a. cos -I. a. 2 2 i{l-(cos2«)^}f. 2tania l + (tani.a)- 2.cot-^o + tan^a] ^• I v/3 { sin (30° + a) - sin (30"- a) ] . 2.sin(45''+ 1«) -_1. l-2.sin(45*'-ia) -. l-tan(45°- j-«)|- 1 + tan (45° -i a)]'' ' tan (45° + }a)- tan (45° - f «) tan (45° + ^ a) + tan (45° -^a)' sin (60° + a) -sin (60° -r/). FORMUL.E RELATING TO TWO ARCS. 103 (8.) sin a tan a sin a . cot a. 1 — (sin a)" 1 + (tan af\ J. Values of cos a. l-(tanj-g)- 1 + (tan i- a)- ' cot \o, — tan ^ a cot i a + tan i- a 2 2 T J cot a . 1 + (cot a)- 1 (cosi-a)-— (sinXa)2. l-2(sinl-a)-. 2 (cos i «)--!. 1.(1+ cos 2a)f. 1 + tan a . tan 1 a tan (45° + ^ «) + cot (45° + ^ a) ' 2 cos (45° + |- a) cos (45° - J a) . cos (60° + a) + cos (60° - a). (9.) sma cos a 1 h cot a 1 1 (cos a)- sin a , 1 — (sii la)- 1 — (cos a)- 1 2 cos a 2 tan ^ a Values of tana.. 2 cot -i {cot Laf-\ cot i a — tan i- a 2 2 cot a — 2 cot 2 ff. 1 — cos 2 a sin 2 a sin 2 a 1 + cos 2 a 1 — cos 2 a 1 + cos 2a l-(tani-a)-' X {tan (45° + X a) - tan (45°- i a)}. (10.) Formulce relating to two Arcs. sin (a + 6) = sin a . cos 6 + cos a . sin 6. cos (a + /)) = cos a . cos 6 + sin n . sin />. 104 TRIGONOMETET. tan a + tan b tan (o + h) = , - cot (a + ft) = 1 + tan a . tan b cot a . cot ft + 1 cot ft + cot a sin(45" + ft)) 1 . , , • .s ^— >= -7— (cos 6 4- sin 6). 1 + tan . 6 cos (45 tan (45° + 6)= f^ tan (45° + i- 6) = 1 + tan . ft 1 + sin ft cos ft cos ft 1 + sin ft 1 4- sin ft tan (45° + ft) = -_ . , ^ — ^ 1 + sin ft sin (a + 6) tan a + tan ft cot ft + cot a sin {a — ft) tan a — tan ft cot ft — cot a cos (a + ft) cot 6 — tan a cot a — tan 6 cos (a — ft) cot ft + tan a cot a + tan 6 sin a + sin ft tan ^ (a + ft) sin o — sin ft tan X (a — ft) cos ft + cos a cot |- (a + ft) cos ft — cos a tan i- {a — ft) sin a + sin ft cos ft — cos a r = -. 7—, = tan l.{a + b) cos a + cos b sin o — sin o ^ sin a . cos ft = ^ sin (a + b) + ^ sin (« — ft) . cos a . sin ft = ^ sin (a + 6) — X sin (a — b). sin a. sin ft = -i cos (a — ft) — X cos (a + ft). cos a. cos 6 = X cos (a + ft) + X cos (a — ft). sin a + sin 6 = 2 sin i- (a + ft) . cos I. (a — 6) . cos a + cos 6 = 2 cos i (a + ft) . cos X (a — 6) . sina — sin6 = 2sin-I.(a — 6).cosi(a + ft). cos 6 — cos « = 2 sin i (a — ft) . sin i {a + ft) • FORMITL.E UELATIXG TO TWO ARCS. 105 tan a + tan h = sin (rt + h) cos a . cos h sin (a + 6) cot h -f- cot a = -: =—7- • ~ smrt.sino (sin a)- — (sin bY \ . ^ .... (cos b)-- (cos a)" S V -r y V J (cos a)- -(sin by I ( ,v' / • ..\=cos(a-^b).cos(a — b). (coso)-— (smf/)-) _ sin (a + b^ . sin (a — b) cos « . cos b tan ff — tan b ■ cot b — cot a = sin (a + b) . sin (a — b) sin a . sin 6 sin b = sin (a + b) . cos a — cos (a + 6) . sin a. cos 6 = sin (a + b) . sin a + cos (a + Z>) . cos a. sin (;i + 2) a + sin )ia = 2 sin (?i + 1) a . cos a. sin (;i + 2) « — sin wa = 2 cos (w + 1) a . sin a. cos no + cos (w + 2) « = 2 cos (n + 1) a. cos a. cosnn — cos (« + 2) a = 2 sin (n + 1) a. sin a. sin (n + 2)a. sin 7i a = sin (n + 1) a\' — (sin «)-. (L. 43 — 54 ; PF. Ch. ii.) (11.) Fovmulce relating to double arcs. sin 2a = 2 sin a . cos a . cos 2 a = (cos a)- — (sin a)" ; = 2 (cos a)- — 1 ; = 1 — 2 (sin a)'. 2 tan a tan 2a = 1 — (tan a)- ' 2cot o (cot af - 1 ' tan2o= - 2 cot 2a = cot a — tan a (cot a)- — 1 2 cot a = ^ (cot a — tan a). sec 2a = O (sec a)- 1 — (tan a)- ' 1 + (tan a)" 1 — (tan a)- ' 106 TRIGONOMETEY. sec 2a. cosec 2rt (cosec ff )' (cot a)- — 1 (cosec a)" 2 cot a ^ (cot a)= 4- 1 cosec 2a = ~^^ ; 2 cot a = i (tan a + cot a). sin^a = i.(l cos i. a = i- (1 + cos a)] . . sin a tan:i-a= :, ; ^ 1 + cos a 1 — cos a sni a 1 — cos a 1 + cos a tan a sec a + 1 sec a — 1 tana sec a — 1 sec a + 1 cosec a — cot a. cot -■ a — ? sin a sin a > 1 — cos a 1 + cos a 2 1 — cos a •> sec a + 1 tan a sec a + 1 i sec a — 1 ■> = cosec a + cot a. , 2 sec a 3 sec a + 1 » , 2 sec a ^ sec a — 1 1 ( L. 55 61.) (12.) FaZwe* o/ the sine, cosine, S^c. of 30°, 45°, and 60°. sin 30° = cos 60° = 1, cos 30° = sin 60° = 1/3, tan 30° = cot 60° = i ^3, cot 30° = tan 60° = \/3, sec 30° = cosec 60° = 1 1/3, cosec 30° = sec 60° = 2. sin 45° = cos 45° = ^ ^2, tan 45° = cot 45° = 1 , sec 45'' = cosec 45° = v'2. RELATIONS OF PLANE TRIANGLES. 107 (13.) FormulcB relating to three arcs. sin (a + 6 + c) = sin a . cos 6 . cos c + cos a . sin 6 , cos c + cos a . cos 6 . sin c — sin a . sin b . sin c. cos (a + 6 + c) = cos a . cos 6. cos c — cos a . sin 6 . sin c — sin a. cos & . sin c — sin a. sin 6 . cos c. . ^ tan rt 4- tan 6 + tan c — tan «. tan 6. tan c tan (a+6+c) = 1 — (tan a . tan b -\- tan a. tan c + tan b . tan c) (14.) Relations between the sides and angles of plane triangles. Let Af B, C, be the angles, a, 6, c, the sides subtending them ; s = ±(a + b + c). sin A sin 5 sin C cos A = b c b" + c" — a" 26 2 sin A = —■ . s (s - a)(s^ b)(s — c)Y be (c„si^)-liip^). 1^ . ._ (6-6)(^-~c) ^^ ~ s(s-a) {taniAy = sin 2 ^ + sin 2 5 + sin 2 C = sin A.sin B. sin C. cos2 A + cos 2 B + cos 2 C = 4> cos A . cos 5 . cos C— 1. tan ^ + tan jB + tan C = tan J . tan B . tan C (W. Ch. ii; L. 365.) The area = s{s — a){s — b) (s — c) ] ; 2a6c , ^ . „ , = ; cos ^ A . cos ^ 5 . cos jr C. a+b+c ^ - 2 108 TIUGONOMETKY. Let li^ be the radius of the inscribed, and if„ that of the circumscribing circle ; then = ^-.tanl- J.tani-^.taniC. -i Ji.^ = ^ abc .s (s — a) (s — b) {s — c) {Lege7idre, Geom. Note v ; Hind, Trig. 153 — 64.) (15.) Values of the side c. sin C ^ a — — -. * sin^ a cos B + b cos A. a" + b- - 2 ab cos cf. (a + 6)'--4fl6(cosiC)'-p. (a_/,y- + 4«6(sin|C)-f. b cos A±a. l-(sin5)-]'. ; .1 h cos J + a" — (6 sin A)" [. (16.) sin (A + B). c . - sni B. b cos 5 + sin 5 . cot C Values of sin C. V 2ab J \ sin J? csin B.a"-\-c~—2accosB\. (17) — cos {A + B). { a cos 5 + 6- - (a sin Bff ] . Values of cos C. a (sin B)" + cos B.b"— (a sin J5)'- 1_ -(sinBf a — c cos B 2s(s-c) ab -1. (a- + c" — 2 a c cos 5)2 (18.) Values of tan C. — tan (J + 5). c sin 5 . 6^ — (c sin 5)"]^. c sin B a — c cos B V«- + 6^ — cV acosB + b'^— (a sin jB)" a sin B—cotB. b"— (a sin 5)" |^ • Analogous formulae may be obtained by substituiiug^ J, a, for 5, i, and vice versa, in this, and several of the following formula?. Solution of Plane Triangles. 109 (19-) Right angled triangles : C, the right angle. fl] Given «, b: tan^= -•; log Tan A* = log r + log a — log 6. , I. c = (a- + b')^; or _ b cos A ' log c = log r + log b — • log Cos A. [2] Given a, c: b — {ir — r/")^ , log 6 = 1 log (c + «) + i log (c — a). cos ^ = - ; c log Cos A ■=■ log 6 — log c + log r. sin l> = cos^. [3] Given A, c: « = c sin J ; log a = log c + log Sin A — log r. IJ = 90°-J. 6 = c cos ^ ; log b = log (' + log Cos A — log r. or 6 may be determined as above. [4] Given A,b: B^W-A. a = b tan A ; log a = log b + log Tan J — log r. b c= -; cos yi log c = log ;• + log b — log Cos A ; (TF. O. v; L. 67—71; Z-e^. 48—52; C 529—58.) * Tlie sine, cosine, fkc. to the tabular radius r will, for the sake of distinction, be denoted by capital letters. See {Enc. Met. Art. Trigonom.) TRIHONOMETllY. (20.) Solution of oblique angled plane triangles. [l] Given A, B, a; sin B b = a- ; sni A log h = log a + log Sin B — log Sin A. c may be similarly determined. [2] Given B, a^ b„ a . sm ^ = - sin B ; b log Sin A = log a — log b + log Sin B. This result is ambiguous if o > b, and S is an acute angle. = 180" -(^ + 5). sin C c^b-:—-; sin B log c = log 6 + log Sin C — log Sin B : or c = a cos B + b"-^(a sin 5)^ r- [3] Given C, a, b; tan i- (J -5)= ^^ tan J ( J + 5) ; ^ ^ a + 6 2 V / log Tan § (^-5) =log (a-&) - log (« + 6) +log Tan ^ (J + 5), J + 5 = 180*'-C: or thus : suppose a > 6, and let tan = - ; log Tan 9 — log r + log a — log b : then tan ^{A-B)= tan i (J + B) . tan (0 - 45°), logTan K J-5) = log Tan I-(^ +5) +log Tan (0-45°) - logr. knowing A + B, and A — By A and B are determined. -- ?' •• SOLUTIOV OF PI-ANE TRIANGLES. Ill sin C sin A log c = log a + log Sin C — log Sin A. or c={a- + b' — 2ab cos C)^ : assume (tan 9)" = ; — . vers C ; {a-by log Tan 0=-^{log2 + logo+log&-flog Vers C + logr} —log {(i — b) ; . a — b then c = ; cos 6 log c = log (a — 6) + log r — log Cos 0. , . ..„ 2ab , or assume (sm t')" = -, -r-o • (1 + cos C), ^ ' {a + b)- ^ ' logSin0=^{log2+loga+log6 + log(r+CosC)+logr} — log(a+6); then c = (a + 6) cos Q ; log c = log (a 4- ft) + log Cos 6 — log r. [4] Given a, 6, c. 2 -,1 Firsi Method : sin A = —.s{s — a) {s — b){s — c) j^; log Sin J = log r + log 2 — log 6 - log c -r^ {logs + log (s-«) + log (s - 6) + log («- c)}. ^ (s-b){s-c) Second Method : (smi-J)-= ^ ; log Sin i- J = i- {log (« - 6) + log (s - c) - log 6 - log c} + log r. s Is — rt) Third Method : (cos i- A)-= — ; log Cos Xy/ = 1. {log s + log (.s- - cf) - log 6 - log c} + log r. 112 TRIGONOMETRY. Fourth Method : (tan i- Af = ^ ^ '^ ^ ' ; - ' .s (a — a) log Tan X y/ = i- {log {s-b) +log (s-c) -logs— log (s-fl) } + log r. If ^ iienrli/ = d0°, the first method is inapplicable. The second method is preferable if A < 90° ; the third if A > 90°. If A nearly = 180", the fourth method is inapplicable. (IF. CA. v; L. 72—84; Leg. 53—7; C 559—615.) Spherical Trigonometry. (21.) General Principles. Every section of a sphere made by a plane is a Circle. The distance between the poles of two great circles is equal to the inclination of their planes. ^he pole of a great circle is the pole of all parallel small circles. If the angle subtended by the arc a is invariable, then a oc sin dist. from pole. If the intersections of three great circles are the poles of three others, the intersections of the latter will be the poles of the former. The sum of the three sides is always < 360°. The sum of the three angles is always > 180°, and < 540°. The sum of any two sides is greater than the third side. The angles at the base of an isosceles spherical triangle are equal ; and conversely. The greater side subtends the greater angle ; and con- versely. {W. Ch. viii. prop. 1—12; L. 102—66; C 960—1010.) RELATIONS OF SPHERICAL TRIANGLES. 113 (22.) Relations hetiveeii the sijies and angles of spherical triangles. Let S = \{A + B + C), s = \{a -\- h -\- c) \ and N= -co^ S .co^{S - A).co^{S - B).co^{S - C)\^^ n = sin 6" . sin (.s — o) . sin {s — b) . sin {s — c) y- . cos a — cos h . cos c cos A = ; ; . Sin b . sin c cos yi + cos ^ . cos C fos a = -. ; . sin 5.sin C sin 6. sine. (sin 5^)' = sin (« — i).sin (.s — e). sin 6 . sin c . (cos | ^ )" = sin * . sin (.§ — a). sin ^ . sin 6 . sin c = 2n. sin (a + b) (sin i C)- = cos ~(A + B). cos ^ (^ — 5) . sin c. sin (a - 6) (cos h Cf = sin \{A + B). sin ^ (^ - 5) . sin c. sin i (^ + 5) = ^^^ cos i (rt - &). cos^c . 1 V Ti cosiC . , sinic -' ■ 1 y^ cosi(^ + fi)= ^ — i— cosUrt + />). COSjC COS |(J - -B) = ^-^^^— sin i(« + b). sin^c sin 5 . sin C . (sin \a)" = — cos S ■ cos {S — A). sin 5 . sin C . (cos j «)' = cos {S - B) . cos (.S' - C) . sin 5 . sin C . sin rt = 2 iV. sin {A + B) (cos \ry = cos j(r/ + b).cos];((i — b) . sin C P 114 TRir.OXOMETRY. sin (J — 5)(sin|c)' = sin7(« + />).sini(<7 — />).sin C. . 1 sin^c , _ sin ^(rt + h) = -,-i-cosl(J - /?). sin h (a - h) = ^r^ sin ], (A-B). 1 X , cos in , ^ cosH« + />) = ■ .' cosH^ + B). snijC " cos —C COS i (a - b) = —f- sin i (^ + i?). cos :7 c sin (a — 6) sin (a + 6) sin (A - 5) sin (A + B) sin i (« — 6) . cot jc sin j ( J — jB) . cot | C cosi(a + />) ~~ cos5(^ + J5) sin A . sin B -^ sin « . sin b = cos A . cos 5 . cos c + cos « . cos b . cos C. (sin C)^. sin (a + 6) . sin (a — &) = (sin c)". sin (A + B). sin (J — 5). cos c = cos (a — b) (cos | C)^ + cos (a + 6) (sin ~ C)". = (tan i C)^ tan i ( J + J5) . tan K^ - ^) ■ (cot5c)-.tani(« + 6).tani(a — &). (sin |c)^ = sin | (a — 6) . cos | C |" + sin | (a + 6) . sin | C]'-. .(cos|c)- = c (tan|0= = .(cos ic) ■ = COS |(a — 6 ) . COS i C I' + cosi(a + />).sinic]- sin (a — b) cos B + cos A sin (a + h) cos B — cos ^ cos C = — cos {A — B) (sin |c)- — cos {A + B) (cos \c)". (cos|C)'' = sin|(J — jB) .sin \cf + ^\xv\{A + 5) .cos \c\'. (sin \ Cy = cosi(^ — 5).sinic|' + cosi(^ + B).cofilc\'. , „ sin (J — B) cos /^ + cos a (cot jc) = -7-T-7— ^ • ; • sin {A + B) cos o — cos a sin(s-a)= —7 RELATIONS OF SI'HERICAL TRIANGLES. N 115 2siniJ.coslJ?.cos^C' 71 = 2 — coss^.sini^.siiisC; jy- 2 2 = (w.cot|J.tan~5.tanjC)^. 1 — cos A + cos B + cos C cos (.s — rt) = ■ ■ . 1 . TB wT • ' ' 4sin j^.cosjfi.cosjC 71 cos (6*— ^) = J 7-T-j — r-r ; N = 2 — sin7«-cosi6.cosic; sm(S-A) = = (iV.tan jrt.cot|6.cotic)5. 1 + cos a — cos b — cos c inhA-smlB.sm~C = sm 4 cos |a . sin |6. sin ^c sin (s — ff) . sin (s — 6) . sin (s — c) sin a . sin & . sin c n" co^~A.cos\B.co^hC= - tan^^.tan|5.tan^C = cos \a . cos j& . cos jc = sin s . sin a . sin 6 . sin c n . sin *• sin a . sin t . sin c 71 (sin s)" cos (^y - J) . cos {S - Jg) . cos {S - C) sin J! . sin jB . sin C N- sin^a.sin|6.sinio = cot ]; a . cot 7.6 . cot \-,c = cos aS*. sin ^.sin 5. sin C NcosS sin ^.sin 5. sin C (cos sy IIU TRlGOXO.ArETKY. n = j.(sin rt.sin ft. sin r)'-. sin ^.sin jS.sinC A'' = i .sina.sin ft. sin c.(sin^.sin i?.sinC)-l • n sin * = 2 — cos i^ . cos ijB . cos I C. N N n cos* = sin*S' = cos»S' = 2— sinia.sinift.sinlc. n J - - l-(si"M)'--(s i" \ Bf^ (s in-^C)- 2 sin i ^ , sin i 5 . sin I C 1 — cos A — cos B — cos C ^sinij.sin^^.sin jC 1 — (cos|r7)-— (cos I ft)-— (cosjc)- 2cosia.cos|ft.cos |c __ 1 + cos a + cos ft + cos c 4 cos i a . cos i-ft . cos I c (L. 181—201, tab. viii.; (23.) Formulce for the area (E) of a spherical triangle. E = 7r7'":(~-l\; = 2S—7r, the radius being the linear unit; = 57°,2957795(2^-180°), the value in square degrees. smlE=~ J —^ -. 2cos ja.cos jft.cos^c 1 „ cosja.cosift + sinifl.sinift.cosC cos'^E= — ^ ^ ^ i ; cos 2^0 _ cos a + cos ft 4- cos c + 1 4 cos|a.cosift.cosic _ (cosga)- + (cos g ft)- + (c os^c)- — 1 2 cos 5^f/ . cos jft . cos jC tan ji; = tanis.tani(s — r7).tani(« — ft).tani(5 — c)p. tan IE SOLUTION- OI" SPHERICAL TIUANGLES. H/ tan ja.tan jfc.sin C 1 +tan|a.tan|6.cosC sirig-a.sinlft.sin C "" 4cos|(0 + 0).cosi(0-0)' if cos = sin 5 a. sin J 6. (cos I C*)'^, and cos (f) = l cos j (« + />) • (W. Ch. viii. prop. 13; Leg. Geom. Note x; L. 177, 263 — 6.) (24.) Let /?p R^ be the circular radii of the inscribed, and circumscribing circles, _ jn N tan H^ — -: — -r \ - f- =r j— . sins zcos j^.cosjij.cos^C — coss Ssini'fl.sinife.sinic ., _,^- -. tan R„ = — — - = '- . (L. 270, 1.) (25.) Solution of right-angled spherical triangles. Let A, B, be the oblique angles, C, the right angle ; «, b, the sides, c, the hypothenuse. Naper''s Rules. The circular parts are 90°-^, 90° -5, a, 6, 90°- c; any one of which being called the middle part, m, the two adjacent to m on each side of it, ^^, ^^j and the two remaining or opposite parts, 0^,02; sin M = tan a^ . tan a„, = COS0i.COS Oo- [ 1 ] COS c = cot A . cot 5 ; [2] cos c = cos a. cos 6; [3] sin a = tan h . cot B ; [4] sin a = sin c . sin ^ ; [5] cos^ = tan h . cot ; [6] cos.// = cos«.sin J5. 118 TRIGONOMETRV. Tliese are all tlie forms essential/i/ different ; four more analagous to [3], [4], [5], [6], may be obtained by changing A, ff, into B, h, and vice versa. These rules may be applied to a quadrantal triangle, in which r- = 90% if A, B, -(90°-C), 90° -a, 90^ -fe, be taken as the circular parts. Any angle, and the side opposite are either both > 90°, or both < 90°. An oblique angle cannot be less, if acute, nor greater, if obtuse, than the opposite side. The sides are both > 90°, or both < 90°, if the hypothenuse < 90°; and one side > 90°, and the other < 90°, if the hypothenuse >90°. A side is > or < hypothenuse, according as it is > or < 90°. A + B and a + b are both > , or both < , 180°. A + B is always > 90°, and A'^B always < 90°. {W. Chap.x; L. 202—9.) [l] Given A, c : sin a = sin c. sin A ; log Sin a = log Sin c + log Sin A — log r. tan b = tan c . cos A ; log Tan 6 = log Tan c + log Cos A — log r. cot B = cos c . tan A ; log Cot B = log Cos c + log Tan A — log r. L2J Given a, c: sin ^ = -; — ; sm c log Sin A = log Sin a — log Sin c + log r. cos c cos 6 = cos a log Cos 6 = log Cos c — log Cos a + log r. cos B = tan a . cot u ; log Cos 5 = log Tan a + log Cot c ~ log r. SOLUTIONT OF SVHERICAL TRIANGLES. 119 [S] Given «, 6: cost^ = cos«.cos6; log Cos c = log Cos a + log Cos b — log r. tan a tan A = — — ; suio log Tan A = log Tan a — log Sin J> + log r. [4] Given A^ a: sin^= '— sin yi ' log Sin e = log Sin a — log Sin A + log r. sin h = tan a . cot A ; log Sin h = log Tan a + log Cot A — log r. . ^ cos A sm jB = ; cos a log Sin B = log Cos A — log Cos a + log r. [5J Given ^,0: tanc= -; cos^ log Tan c = log Tan 6 — log Cos A + log r. tan a = sin b . tan -i4 ; log Tan a = log Sin b + log Tan A — log r. cos B = sin ^ . cos 6 ; log Cos B = loo- Sin A + log Cos ft — los- r- t> o o o [6] Given vi, 5: cosc = cot ^.cot 5 ; log Cos c = log Cot A + log Cot 5 — log r. cos ^ . ^ cos a = , y logCosf/ = logCos^ — log>^5 + logr. ^<^/^. 120 TRIGOXOMETRY. The results obtained in [2], [4], arc doubtful; the am- bi«;uity may in some cases be removed by attending to the conditions stated in page 118. When a small angle is to be determined from its cosine, or an angle nearly = 90° from its sine, a small error in the sine or cosine gives a large one in the angle ; in such cases some of the following formulae will give more accurate results. -cos{A + B) ^"^^^= co.(A-B) i I tan(iJ + 5-45°) fin n 2 tan(|J-5-45°) tan i a = tan i (c + 6) . tan |(c — 6)]^ tanlA = sin (c — 6) sin (c -\- ft) tan (45° - ~a) = tan (45° — ^r) if tan ,%' = sin c . sin A. tan (45° + ^^) = tan J (c + a) tan h(c — a) /4Kn 1 \ tani'(^ + ffi) tan (45° + 7 c) = ^ — -—2 ^ - ' tani(J-«.) tan (45°+ 1 6) = sin (A + a) sin (A — a) tan (45°+ ii5) = f}- J ^ tan^(J — «) Formulce adapted to a table of natural shies. Sin a =iCos {c — A) — §'Cos(c + A). Cose =jCos (« + 6) + I Cos (a — ft). Cos J =|Sin (a + 5) - ^Sin (a- 5). (ir. Ch. x; L. ^. v; C. Ch. xvii.) ^\ SOLUTION OF SPHERICAL TRIANGLES. 121 (26.) Solution of oblique-angled spherical triangles. First Case: given a, b, c. 2 1 [l] sin^=-; — ; — ;; — .sin5.sin(.s — «).sin(.9 — ?>Vsin(.9 — r)] ; sin 6 . sm c ' ' ' ' log Sin ^ = log 2 + log r — log Sin b — log Sin r + II log Sin * + log Sin {s — a) + log Sin (.s — b) -f log Sin {s — c)}. foi /• 1 ^x" sin (* — 6) . sin (s — r) [2] (sin 1^ - = . / . ' ' ; sin o.sin c log Sin i J = J I log Sin {s — b) + log Sin {s — c) — log Sin b — log Sin c ] + log r. [3] (cosMf=^ill4ii^liii:f); sin b . sin c log Cos i^ = 1 1 log Sin s + log Sin {s — o) — log Sin 6 — log Sin c ] + log r. ATT/ 1 ,\o sin (5 — ft). sin (5 — c) [4] (tan id)- = ^ r-T—^ ^; ■- -^ ^ ' ' sin5.sin(s-a) A log Tan §• ji = |{ log Sin {s — b)+ log Sin {s — c) — log Sin s — log Sin (* — a) } + log r. [5] Assume cos 9 = cos b . cos r, — log Cos — log Cos 6 + log Cos c — log r ; then , 2sini(^ + ff).sinU0_a) cos A = . / . — — '- ; sin b . sm c log Cos ^ = log 2 + log Sin 1(0 + a) + log Sin i (0 - a) — log Sin 6 — log Sin c + log r. A — 90°, and — « have the same sign. r„T , , COS 6. cose [oj Assume tan = -, , ' sm a. log Tan = log Cos b + log Cos c — log Sin a ; then cos (rt + (h) cos ^ = • , • — ' cos . sin b . sin c log Cos ^ = log Cos {n + 0) + 3 log r — log Cos — log Sin b — log Sin c. Q 122 IKIGONOMETRY. Second Case: given A, B, C. 2 \ [1] sina= . ^ . ^ .-cos>S'.cos(.$'-^).cos(^-ig).cos(.S'-C)| • sin 5. sin C ^ 'I log Sin a = log 2 + log r — log Sin B — log sin C + 1 { 1^'g ( - Cos ^S*) + log Cos (5' - ^) + log Cos (S-B) + logCos(^-C)}. r^T / • 1 xo — cos6'.cos(aS'— J) [2] (smia)-= : ^ '; ^ ^ ^ ' ' sin 5. sin C log Sin I a = log r + H log ( - Cos S) + log Cos (^- ^) -log Sin B - log Sin C \ . [3] (cos|a)-=^"^^'^T^^"."^^^^"^^ ' smB. sm C log Cos|a = logr + 1 { log Cos {S-B) + log Cos(5' - C) - log Sin 5 - log Sin C } . r -. / 1 v^ — COS *S'.cos(). 1 / sin^(« — 6) , ^ tani(^ -B)= -T-j^ -^cotiC; '^ ' sini(« + />) log Tan i(J-5) =log Sin|(a-6) + log Cot jC-log Sin i(« + 6). A = \{A + B)+\{A-B), B = '^(A + B)--'^iA-B). [2] Assume tan a = tan « . cos C ; log Tan a = log Tan a + log Cos C — log ;• : and let l3 = h — a, then ^ sin a tan J = tan C -: — 77 ; smfi log Tan ^ = log Tan C + log Sin a — log Sin /3. To determine c. [1] 2 (sin |c)'- = vers (a — 6) + sin a. sin 6. vers C sin a. sin 6. vers C Assume (tan dy = ; ; vers (o — h) log Tan = 1 {log Sin a + log Sin b + log Vers C — log yers (a — ^>) } . then 2 (sin ^cY = vers (r/ — />) . (sec 6)- ; log Sin ic = log Sec 9 + ^{ log Vers (a — h) — log r — log 2 } . [2] (sin \c)- — (sin i . a + 6)- — sin « . sin b (cos g C)-. Assume (sin Qy = sin « . sin />. (cos |C)'* log Sin d = \\ log Sin a + log Sin b \ + log Cos \C — log r : 124 TRIGONOMETRY. then (sin ic)' = sin (h.a + h + 0). sin Q.a + h — 6); logSinic = niog Sin (i.a + h + 0) + log Sin (i.« -^.b-O)]. [s] Assume tan a = cos C . tan a ; . log Tan a = log Cos C + log Tan a — log r : and let /3 = /;'>^a, then cos/3 cos c = cos a. ; cos a log Cos c = log Cos a + log Cos /3 — log Cos a. If c is small, this formula is not sufficiently accurate. WsinC . sm c = ^ — ^ sm a ; log Sin c = log Sin C + log Sin a — log Sin A. Fourth Case: given A, B, c. To determine a and 6 : [1] tanKa + 6)=— f-^4?l|tanlc; COS |( J + 5) logTani(« + 6)=logCosi(^-5) + logTanic-logCosi(J-jB). J ^ , ^ sin i (^ — 5) , tan Ha — 6)= . i) , -^tan^c; logTan|(«-6)=logSini(J-5)-logSin^(^ + 5) + logTan|c. a = |(a + 6)+i(a-6), 6=|(a + 6)-i(a-6). [2] Assume cot a = tan A . cos c ; log Cot a = log Tan A + log Cos c — log r ; and let /3 = 5*^0, then cos a tan a = tan r -, ; cosp log Tan a = log Tan c + log Cos a — log Cos /3. SOLUTION 01" SPHERICAL TRIANGLES. 125 Required C : r -, . sin (r L 1 J sill C = sm A sm a log Sin C = log Sin A + log Sin c — log Sin a. [2] 2 (sin I Cy = 2(cos ~.A-\-By- + vers c . sin A . sin 5. / /iNo versc.sin J.sin jB Assume (tan dy = 2(cos|.^ + 5)' log Tan = h{ log Vers c + log Sin A + log Sin B + log r — log 2 J - log Cos i-( J + 5) : then sin I C = cos |(^ 4- -B) . sec ; log Sin i C = log Cos l(A + B) + log Sec 9 - log r. [s] Assume cot a = tan A . cos c ; log Cot a = log Tan A + log Cos c — log r : and let (i = B — a, then cos C = cos A — ; sma log Cos C = log Cos A + log Sin /3 — log Sin a. This is not sufficiently accurate, if C is small. Fifth Case: given a, 6, A. To determine 5: sin 6 sin 5 = sin ^ sma rvT~ ^.\ log Sin B = log Sin A + log Sin h — log Sin a. sin a . ^ 1 . '' If -: — < sin A, there is no solution. Y ' sin & ^ If -^— r- = sin A, B = 90°, when J and a are of the same sino species; when A and a are of different species, there is no solution. sin a . If -:—T >smA, and < s, there are two supplemental values sin o ^ ^ of B, when J and a are of the same species ; when of different species, there is no solution. 126 TUICONOMKTRV. If -: — 1 : if-7— r sm J, then C sin h sin h has two values, a + /3. [3] Assume tan = cos h . tan ^ ; log Tan 9 = log Cos h + log Tan A — log r : . ^ ^ tan h then sin (C + 0) = sin ; tan a log Sin (C + 0) = log Tan h + log Sin - log Tan a. To determine c r 1 . . sin C [IJ smc = sina-: ; sin^ log Sin c = log Sin a + log Sin C — log Sin A. T/^T 1 w , . cos ~(A-\- B) [2] tanlc = tani(« + 6)— f)-^-^; cos ^{A — B) logTanic=logTanl(a + ft)+logCosi(^ + i?)-logCosJ(^-.J5). SOLUTION OF SPHERICAL TRIANGLES. 127 [3] Assume tan a = cos A . tan h ; log Tan a = log Cos A + log Tan b — log r : ^, cos a. cos a and let cos B = ; cos 6 log Cos j3 = log Cos a + log Cos a — log Cos b : then c = a + l3, under the same conditions as above. Sixtfi Case: given A, B, a. To determine b: . /__ sin B sm o = sm a -; ; sm A log Sin b = log Sin a + log Sin B — log Sin A. _, , , sin « , , .. . sm A i he above remarks on -; — may here be applied to -: — - . sm b -^ ^^ sin B C, and c may be determined as above ; or thus, Assume cot a = tan B . cos a ; log Cot a = log Tan B + log Cos a — log r : , , . ^ sin a. cos J and let sm H = — ; cos B log Sin /3 = log Sin a + log Cos A — log Cos B : then C = a + /3, as above. Assume tan a = tan a . cos B, log Tan a = log Tan a + log Cos B — log r. . ^ . cos J sin « = sin a =; , COSjB log Sin /3 = log Sin a + log Cos A — log Cos B ; then c = a + /3, as above. (TF. Ch. xi ; C. Ch. xviii ; L. 221 — G2: Legendre, Trig. 84-91 ; Delambre^ Astron. 138 — 83.) 128 TRIGONOMF.TRY. (27.) Series for the sine^ cosine, S^c. in terms of the arc. .,7 ■7' K sin.r = .r— _ _ ,^ + 1.2.3 1.2.3.4.5 I....7 + &C. « COS ,r = 1 — T—z + .r sec 0? = 1 -f + 1.2 1.2.3.4 1....6 + &C. ^ 5x* 61 x"^ 5. 277 a" 50521 .r'^" 1.2 1.2.3.4 1....6 ' 1 8 ■ 1 10 5.540553.^1- + &C. y .12 1 a' Tw^ Slcv" 3.I27.V' cosec.'r= - + 1 + 1- — .r 1.2.3 3.1. ...5 3.1. ...7 5.1.. ..9 5.7.73^^^ 141 4477 .v^i + + 3.1. ...11 3. 5. 7.1. ...13 + &C. tan.r= ,r + + -- + _^^— - + -^—. + 1382a" 3.5 3-. 5. 7 3\5.7 3\5-.7.11 21 844. r^^ 929569 a" ■^ 3^5^7.11.13 ^ 3^5Vfn03 "^ ■ « sm,r' = S,„(-ir-^ -,2»l - 1 2m — 1 ' cos x = S, „(-!)""' 2m-2 (_ 12 m — 2 j m where 7er"*"^«"^ = S„ (-IV'-^ff-^.-ar"'- V//'-\ « cosec.T = ^„X-iy"-'^v~'"-\7!f"-'a-\ia,„_,= -^— — ^| . .an.=s,„(-ir-v-.;s„^;^,(„,„.,=^^l TRIGONOMETUIC SERIES. 129 1382,/" 4./^' 3617.i'' , „ 3^5^7^.11.13 3'\5-.7.11.13 3^. 5*. 7'. 11. 13. 17 (28.) Series e.vpressing the inverse circ7ilar functions. . , 1 a"' 1.3 -r^ 1.3.5 .r^ „ sin " \r = .v+ -. 1 . h -——z . — + &c. y 2 3 2.4 5 2.4.6 7 x^ x^ aP tan - 1.X' = ,r — - + - — — + &c. « 111 1.3 1 1.3.5 1 cosec ^a:= ~ + - . 1- . — r -1 ^ • z: — r + cCc. * X 2 3x' 2.4 5.i?^ ^ 2.4.6 7^-' vers .^ > (, 1 '^^ 1-3 r^ 1.3.5 x^ ^ \ COS ~ ^a? = - — sin x. cot c^' = - — tan \r. 2 2 -1 ^ -1 sec ^0? = - — cosec x. 2 » cota7 = S„,(-l)'"~''^^"' vers.= S,„(-ir-^ [^ 2m - 1 '" [m - 1 2™ - ^(2 m - 1 ) m-l, 2 eo ^."wi - 1 '"^ ; 2m- 1 cosec ~ ^r 7n-l (2m- 1)2'"-^ ' LL ^ {2m -I). [m \^J j R 130 IKIGONOMETIIY. (29) Series for determining the value of tt. -^ 1 1 1 1 TT 1 1 Eulcr's Mdhod. ~ = tan~^ - + tan~' - , 4' ii 3 TT 1 _ 1 Machin's Method. - = 4 tan ^ - — tan ^ , 4 5 239 _ 4 / 4 4^ 4^ \ ^ 5 V ~ STTOO "^ 5.100^ ~ 7.1003 + ^'C. j 1_ / _ 1 1 _ c. "i 239 ^^~ 3.57121 ■^5.571212 "'^V ' 7r = 3, 141 59 26.535 89793 23S46 26433 83279 50288 41971 ^9^99 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 0938446, &c. (Lacroix, Introd. Diff. Calc. 43 — 5.) (30.) Formulce involving impossible quantities. 1 . /- r- sm X — 7==^ (e'^-i — e--* '^^). 2V-I cos .1- = \ (e^^~i + 6-^^). tan cV 2 1 pS^vCTi _ ]|^ 6i-^^~ = cos cV + Ay/ — 1 sin ;f. cos ?/ ,r -f ^ — 1 sin Tic?? = (cos ^^ + \/ — 1 sin a')". wi y . m / — - . ^ — A- + ^ — 1 sm — A' = (cos tt- + >/ — 1 sin .37) . in , — ~ . m cos n ~ " n sin 7i.r = ; I (cos .r + ^/ — 1 sin cf)" — (cos x—^J — X sin a')" } cos nx = -^ { (cos X -\- fj — \ sin .r)" + (cos x — ^ — 1 sin .r)" } . MUl.TIPLK ARCS. 131 If (a + 6^ -1/+"^-' = Cos 6> + v^-1 Sin r, and - = tan ,r, then c = g.v + ^h. log (a" + Ir), a r=(a- + h-y^».e-'''. 1 1 + ^ — 1 tan c c = loc ■ • 2^/^l' ''l-.y^tanc (C. ch. viii ; Hifid, 247 ; ^- 357—62.) logj 1 = 2 (m — 1 ) TT ^Z — 1 ; m being any integer. Of these values only one is possible, which is obtained by putting m = l. {Lacr. Introd. SI.) (31.) Formihlce for the sums of arcs •> and multiple arcs. Let the sum of the continued products of each combination of the sines of m of the arcs, a^, a„, &c. a,,, into the cosines of the remaining n — m, arcs, be represented by C^_,^S„^ ; then M {a, + «2 + &c. + «„) = C, - C, _ , 5', + C„ .,S,- &c. - I'V ^ (a, + a, + &c. ^ «„) =^C„.,S,-C,.,S, + &c. ^^^^ Let T,„ be the sum of the continued products of each combination of m tangents, 7',-r, + r,-&c. then tan {a^ + «„ + ^c. ~ a„) = ^- ^ j, , j, _ ^^. • If Oj = cr,, = &c. = o„, then ^1 /"^ 1 ^ r^i 2^ sinwa = «(cosa)''-^sina— — ^— — (cosa)"-^(sin rOH&c. toi(n + 1) terms, if n is odd, and to^w terras, if n is even. 7i(n— 1) , N„ n / . .., cos na = (cos a)" - -^ — --^ (cos r/)" " - (sm a)' 1 . ^ «C«-l)(w-2)(n~3) ^^_ toi(n + 1) terms, if n is odd, and to ^ ^^ + 1 terms, if 7i is even. 132 THIGOXOMETRV. (32.) Sines and cosines of mufti pie Arcs, [l] Ascending series. sin 2,t = cos .r . 2 sin <» . sin 4.1- = cos 0? .{ 4 sin A- — 8 (sin x)'^ ] . sin 6(2wcosa). « sm wa? = sin X |(2 cos *)" " ^ - !-i-— (2 cos a)» " ^ + 1 Q (2cos.r)"-'^-&c.[ ^ If w is even, the number of terms is iw, and tlie last term, (_l)l«-i.7i cos X. If n is odd, the number of terms is i(?i + 1, and the last term, (-l)^'"-i>. 2 cos nx = {- 1)'«|(2 sin xy - n (2 sin .r)" " ^ + -—- — <- (2 sm xy - * ^ -^ ^ (2 sm *)" " ' + &c. / \n- m « 2cos7ia? = wS„,(— l)'"''-^^^^^^.— (2 cos cr)" -='" + ". I m — 1 r = Ln + 1 if w is even ; r = ^(w + 1) if 72 is odd. ^ I w — ?» ^ sin nx = sin a' . S,„ ( - 1)'" " ' ~^ (2 cos xy " ~"' + ^ Im— 1 r = ^7i if n is even ; r = ^(n + l) if n is odd. y If /I is even, • ,. \n — ')n 2 cos nx=:n S„, ( - 1)'" + ^^ " ' "'-'^ (2 sin xy " ^'" " " ; jm — 1 ,. \n — in sin nx - sin a?. S;„(-- !)'«+ 3» - ijn^^i (2 sin .2')« - ='" + >: jm— 1 iMULTIPLE ARCS. 135 sin nx={-l )i" . cos .r|(2 sin ,x')" " ' - ^^^ ■ (2 sin x)" " ^ (w — 3)(w — 4) ^ . , ) + ^ ^^ (2 sin .??)" - ' - &c. J 2 sin Wcv = (-1)'^*" - 1' |(2 sin a?)" — n{2 sin a?)" " * /i(w — 3) ^ . , 7i(n — 4)(w — 5) ^ . , r o ") + — -^^ -(2sin*')"-^ —-—-——— (2 sin.r)"-^ + &c.^ cos 7iA = (-l)^<"-^'.cos X I (2 sin aO"" ' -— (2 sinx)""" ^(»-3K«-t)(2,i„,„y.-_8,e.| ' n is even in the first and second of these series, and odd in the thii-d and fourth : the last term and number of terms may be determined as in the two first series. These series are true only when w is a positive integer. (L. 370-97; ir. Ch. iii; C 466-84; Lagr. Calc.desFonc.Le<^. 11.) (33.) Tangents of multiple arcs. 2 tan .V tan 2a? = tan3tr = tan 4'-^\^ W = -7^- ^12 3 » 71 2 1 ^ {Hind, Trig. 81.) (36.) Resolution of trigonometrical quantities into factors. . IT . Sir . Stt sin(2w — l)7r 1 sin — - . sm — . sin — . . . — ■■ — = -^ — , 2n 2n 2n 2n 2""^ 0) X CO -^ r» • "^ sin cV . sec - . sec — ■. sec — ...sec — =2"sm — . ' 2 2- 2^ 2" 2" " cos ma? , .^ . 1 . « S,-^^ = -log,(2sin4a.). n^.?,\nmx s 2/ = S.(-l)'"-^ W2 1 — w\"' sin 2 m .2' = / 1 — w\ = ''' + ^'»V~TT^J m. (2m-l)7r 1 * P,„ sin 2n " -J? .a? ' sina'.P^sec- =2"sin — APPROXIMATE SOLUTION OF TRIANGLES. 141 sm cos -/ \ (57r)V 7r_ 2.2.4.4.6.6.8.8. &c. 2~ 1.3.3.5.5.7.7.9. &e. ■ (£. 420—5 : Lacr. Diff. Calc. Tome 3, Ch. vi.) Approximate Solution of Triangles. (37.) Given A, B, c; A and B being small, a + b — c = ^ ABc^ nearly. Given a, 6, and the contained angle, tt — O; 9 being small, abO- c=a +b— 2(a + b) , nearly ; A=—-+ / ,,/ .Y. . nearly. (Leg. 97-9.) a + b (a + by b -^ v & « / « sind7 = A'.P„Jl-- ). "'V (niTryJ (mTrY 2^ cos ^■=H'-w;;r^Ay TT " y -. =P 2 '^ !w — l)7r (2my (2/»-l)(2m + l) 'A (2m-l)7r 7 _,^_.,,..,.(..:p,^.p,^(l_g_.-). 142 TEIGONOMKTRY. (38.) Sohition of triangles by series. [l] Given the sides a, 6, and the angle C of a plane triangle : h . h- . b^ B= -smC+ —-;sm2C+ — ^,sin3C+&:c. « a 2a- 3a^ Wc = log,a cosC— — — cos2C ^cosSC— &c. ^ a 2a- 3a^ [2] Given the sides a, b, and the angle C of a spherical tri- angle : tanlfe . _ /tanl/>\- sin2C ^ ~ cot la Vcotirt/ 2 i(J-5) = 90°~iC S-sinC-( f-) -— &c. « ^ ^ ^ tania Vtania/ 2 2^ ^"-""s tanift logj sin Ic = logj (sin ^a . cos 2^)~ f~ ^^^ ^ /tani6\-cos2C Vtan^a/ 2 logf cos ^f = log£ (cos "l^a . cos ^b) + tan la . tan Ifc . cos C , ,,„ cos2C . — (tan^«.tan-^o)-. — \- he. " ^ ^ . - . sin7??C -iO- - - g /6\'« cosmC log.e = log.a-S,„(-) .-^^. ■^ -^ \tan^a/ m s? /tanlfix'" cos//2<7 1 -1 1 r • I 17\ G /tan^ov" c log, sin if = log, (sni ^ a . cos ^ o) — p,„ ( ; f- ) • - vtan "o a/ ^ log4Cos^c=logj(cos-^a.cos-^6) + S,„(— 1)"' \tanla.tanl6) 2 1 7.,„<^osmC Wi FORMUL/E PECULIAR TO GkODETIC OPERATIONS. 143 (39.) Correction for a compound base. If the base be com- posed of two straight lines a and 6, inclined at an angle 180° — a", the correction is - ^^ 0,00000006001175. a + h (40.) Reduction of an oblique to a horizontal angle. Let a, 13, be the angles of elevation of two objects ; ' y, their angular distance ; C, the horizontal angle : then [1] C-7 = i{(a + /3)nani7-(a-/3)=cotl7}. cos a. COS p logSiniC = l{logSinl(« + a-/3)+logSinl(a-a-/3) — log Cos a — log Cos /3 } + log r. cos observed Z For a single object, cos reduced L = cos z of elevation (41.) Reduction of a spherical to a plane triangle. Let C be the spherical angle, C— X the corresponding plane angle ; then [1] .1'= — {(« + 6)-tanlC-(a-ft)'cotlC}. [2] Legendres Theorem: area of triangle "3^ a?= o^g =i(^ + ^ + C-180°). The value of a? in seconds is area of trianirle 3r- X 0,000004848 If the area be found in feet, and a degree on the Earth's surface = 365155 feet, then log,o d/;'i6ror = 9,803894. 144 TRIGONOMETRY. Reduction of a plane to a spherical triangle. Let 7 be the plane angle contained by the sides a, ft ; C the corresponding spherical angle ; then cos C = , , ^ T' • (l-laO(l-i/3^)|^ (^l, 293 — 313 ; C. ch. xx ; W. ch. xii ; Enc. Met. V. i. p. G98,9.) FORINIUL.E FOR THE CONSTRUCTION OF TaBLES. (42.) cos30° = l(l+i)f = Ci; cosi30° = i(l+q)f = c„; cosi30<' = i(l+Ce)f = ^3; &c. = &c. sinl5° = i(l +1)^-4(1-4)^=*,; sin4i5°=i(i + 0^-4a-*i)^=*c; sinil5° = 4(l+5,)^-4(l-«c)^ = 53; &c. = &c. 15° By continuing this process we obtain sin -^5 and thence sin 1' by taking it proportional to the arc. 1 Ko , y. 3 _ = ( 4" ) =52" 44'" 3"" 45". 21" V 4/ Another Method. 1 sin4a = 4-4{l-(sin«)"}*r' 1 &C. = &C. Sin 2 1 1 - - --a=i_i]l— ( sin - — - a\[ 2« 2 2 I y^ 2" - ' / j FORMUL.t FOR THE CONSTRUCTION OF TABLES. 145 Let 2 sin 6° = .7', then a,'^-5cr3 + 5ci'-l=0, which equation may be solved by approximation. sin (a + &) = sin a + { sin a — sin (a — 6) } — 4 sin a . (sin ^ h)" : Putting b = 1°, we obtain Delambre's formula, sin (a + 1°) = sin a +{ sin a — sin (o — 1°) } — 4 sin a . (sin 30')^. sin (60'' + a) = sin a + sin (60° - a). tan (45° + a) = 2 tan 2a + tan (45° — a). cosec a = cot a + tan ^ a. ( W. ch. iv ; C. ch. vi.) (43.) Sines of arcs expressed by surds. a/3 . sin6^=-i-(v/5 + l)+ ^^75-^/5. sin9°=^(/5 + l)-iV5-V5. l/3 1 sin 12°=- _-(^5-.l)+ ^JJTVS. 8 4\/2 sinl5°=2-1^^^3-^)- sinl8° = i(v/5-l). sin21°=^(v/5-.l)+ ^^5375. sin24°= ^(VS + I)--— ^5->/5. o ^V 'J sm2r-~^(/5-l) + iV5 + v^5. sin 30° = 4. V/3 + 1 , 1/3-1 ; '"'^^= "87^ ^^^-^^"^ -^r"^^^+^- ^'"^^272^^'-^''- 14C TRIGONOMETRY. t/3 sin 42"= - I (V5 - 1) + -^2 v/5 + v/5. sin 45"=^. 4/3 ;i sin 48" = — (1/5 - 1) + ;j^ ^5 + 1/5. sin5r= ^^(1/5 + 1) + ?^ VS=n75- sin 540 = ^(1/5 + 1). sin 57° = - '^ (VS - 1) + ^ ^STFS. sin 60° =i ^3. in 63°= J- (v/5-l)+i^5T75. sin 4V^2 V/3 sin66° = i-(/5 + l)+^^V5-v/5. sin 69° = ^ (1/5 + 1) + ^V 5=V^ ^^"^2° = 272^^"^^^- 4/3 sin 78°= i- (v/5 - 1) + ^-^ V5 + t/5. "« 81° = ^ (1/5 + 1) + i 753-75. sin 84° = — ( v/5 + 1) + ^2 ^5 - /5. sin 90° =1. (C. 304.) SERIES FOR THE CONSTRUCTION OF TABLES. 147 (44.) Lengths of arcs in terms of radius^ and their logarithms. V =0,017453292520 ; logio 1" + 10^ 8,2418773675. 1' =0,000290888209; log,o 1' +10 = 6,4637261171. 1" =0,000004848137; log,o 1"+10 = 4,6855748667- le =0,0157079632679; log,o 1» + I0 = 8,196ll98769- " log,o'n-= 0,497 1498726. {Enc. Met. V. i. p. 672.) (45.) Series for the construction of tables. 3 sin - . - = - X 1,570796326794827 - ^ x 0,645964097506246 71 2 n ir + -7 X 0,079692626246167 7 X 0,004681754135319 + -7r X 0,000160441184787 rr X 0,000003598843235 + -^ X 0,00000005692 1729 15 X 0,000000000668804 ,,,17 ?«!!> + -^ X 0,0000000000006067 TTT X 0,000000000000044. 2 4 cos- .- =1- ^-^ X 1,233700550136170 +— X 0,253669507901048 ?t 2 M^ « _ 'IL X 0,020863480763353 + -^ x 0,000919260274839 71" 71" X 0,000025202042373 + -r^- X 0,000000471087478 TX X 0,000000006386603 + -t« X 0,000000000065660 jg- X 0,000000000000529 + -w '^ 0,000000000000003. A few terms of these series will generally be sufficient, as it m will never be necessary to take — > ^ n The sines of the lesser divisions, as minutes, are usually found by the method of differences, which will be explained hereafter. Series for calculating the tangent and cotangent, inde- pendently of the sine and cosine. 148 TRIGONOMETRY. 9J2 TT $ 771 If tan^.- =-:— 2 X 0,6366197723675813 m ,3 + - X 0,297556782059734. + -^ x 0,018688650277330 n ?r + ^ X 0,001842475203510 + ^-^ x 0,000197580071520 + -g x 0,000021697737325 + -fi X 0,000002401 136991 711 m^^ + ^f^ X 0,000000266413303 + -r? X 0,000000029586468 -\ p X 0,000000003286788 + -r^ X 0,000000000365175 21 23 4- ^ X 0,000000000040754 + '^ X 0,000000000004508 + ^^ X 0,000000000000501 + -w X 0,000000000000056. cot-.- =- X 0,6366197723675813 n 2 ?« At*} n o X 0,3183098861837907 4 n'* — ?« X 0,205288889414508 ^ X 0,006551074788218 n ir 77?' tuv , X 0,00034529255397 7 X 0,000020279106052 if n' i7 X 0,000001236652718 rr X 0,000000076495882 rv' ir ' _ X 0,000000004759738 r? X 0,000000000296905 - ^ X0,000000000018541 -^—, X 0,000000000018541 — ^-^g X 0,000000000001 158 X 0,000000000000072 55 X 0,000000000000005. „2l ji^iJ (46.) Construction of logarithmic tables. logio sin - • J = logioW+logio (2 « - »«) 4- log,o (2 «+m) - 3 logio^ +9,594059885702190 _ X 0,070022826605902 T X 0,001 1 17266441662 71 fi 7)1 J. X 0,000039229146454 y X 0,000001729270798 jg-X 0,000000084362986 15X0,000000004348716 SERIES FOB THE CONSTRUCTION OF TABLES. 149 n X 0,000000000231931 j^v- X 0,000000000012659 i5i- X 0,000000000000703 3rr X 0,000000000000040. logio *^°5 ~ • 2 = ^°Sio (« — '«) -1- logiu (« + "0 - 2 logio » 5- X 0,101494859341893 - -r X 0,003187294065451 X 0,000209485800017 - -j- x 0,000016848348598 771 7/2 T- X 0,000001480193987 js X 0,000000136502272 TT X 0,000000012981715 r?? X 0,000000001261471 7n}^ 7r20 T« X 0,000000000124567 ott X 0,000000000012456 77i 7/2 90- X 0,000000000001258 ^ X 0,000000000000128. If m is small, log (1 -) may be expanded into the series -2m —; s+i(^7-^ -') +&c.[ I2n- — m- ^ \2n' — m-J J which will render the calculation independent of logarithmic tables. If we find the logarithmic cosines of arcs < 45°, and the logarithmic sines of arcs between 45°, and 90°, the rest may be found from the formula logio sin « = logic sin 2a - log,„ cos a + 9,698970004336019. {Enc. Met. Trig. §. 10.) (47-) Logarithmic series for the sine., cosine^ and tangent. log. sin ,r = log,..- {|^ + ^ + -^ + ^5^3^ .v'° 691 i'- 2,t" I" rCt I.O M 1 -■ ' ^ r»7 «.-» *T'i -1 1 TTi I 3'.5'.7.11 2.3'.5'.7-.11.13 3''.5-.7-.11.13 36l7.r^^ I ■^ 2'.3^5^7^11.13.17 "^ ^'"i 160 TRIGONOMETRY. log.cos,.= -|- + ~- + — +^5^3^^ + 3,-^ 691 x" 10922.1^^ 929569.1-^' "^ aTa'rs^iy.ii "^ 3^5-. 7^.11 .13 "^ 2*.3^5'.7-.ii.i3 "^ "" fx^ 7x' 62/ 127:r« log.tan^ = log.a.+ [3 + 2:3^5 + sTg:^ + 2^33. ¥7 146/° 1414477/' 32764/' 3\5-.ll 3l5\7-.11.13 3'.5-.7-.11.13 16931177^/- I (C. 405^-7.) 2.315^7.11.13.17 j ^ '^ (48.) To find the logarithmic sines and tangents of small arcs. log,„ Sin ^ = log^„^ + 4.G855749-^(10-log,„Coscr?); log,„ Tan X = log,„ n + 4.6855749 + 1 (10 - log,„ Cos 00) ; n being the value of os in seconds and decimal parts of seconds. {Taylor, Log. Introd. p. I7.) (49.) Formulce for the verification of tables. sin x + sin (36° - x) + sin (72° + a?) = sin (36° + x) + sin (72° - x). {Elder, Anal. Inf. V. i. p. 201.) sin (90° — x) = sin (54° + a?) + sin (54° — x) - sin (18° + x) - sin (18° - x). {Leg. 40.) cos (36° + x) + cos (36°-a^) = cos x + cos (72° + x) + cos (72°-a,') : this is only a particular case of the more general theorem COS X + cos (2 - + x) + COS (4 ~ + x) + &c. ^ n n I + COS (2 ^ — cv) + COS (4^ ~ x) + &c. j n n TRIGONOMETRICAL SOLUTION OF EQUATIONS. 151 ( COS ( \-^) + COS (3 - + a?) 4- &c. \ n n J TT TT (. + COS ( X) + COS (3 - — cc) + &c. n n in which each series is to be continued until the angle attains its greatest value next below 90°. {Enc. Met. V. i. p. 695.) Trigonometrical Solution of Equations. (50.) Quadratic equations. [l] ar — poi)-\-q=-0. 4g Assume (sin &)" = — ^^ ; then p~ a=p (sin ^ Oy, or p (cos \9y:> or cc = x^q . tan 1 9, or \/q . cot ^ 0. [2] a;^ + p.r-9 = 0. 4a Assume (tan ^)* = — | ; then .r = ijotan0.tani0, or — l^tan 0.cot J0; or cr = \/q tan J ^, or — \/q cot 1 0. The equations x" -'rpx -\-q, and ,r" — j3y — 1 . sin - TT ) (CD — COS - TT — ^/ — 1 .sin - tt ) > . . . w ^ w /J 4 1/ w— 1 /-^— - . w— 1 X J I .r — cos TT + -v/ — 1 . sm tt I (V w n / / n-1 y . 7^~l xj I a? — cos TT— \J — 1 . sin — — "" ) } > = (a'— 1) ( .^-— 2cos - 7r..x' + l ) ( A'"— 2cos- tt-cTH-I 1... f tV^ — 2 cos ir.ce -\-\\\ if w is odd : = (a?^ — 1) (a?^ — 2cos - 7r.a? + l ) f *■-— 2coS'-«7r.a;' + l 1 ... (n — 2 \ . a?" — 2 cos TT . a? + 1 I ; if w is even. cr" + l =(a? + 1) (a-— 2 cos- 7r..?;' + l) (a?^— 2cos- tt-.v + I )... (n — 2 \ • . a?" — 2 cos TT . a' + 1 ) ; if 7» is odd : = f a?^ — 2cos-H TT-.r + 1 j^.y^ — 2cos - 7^..^' + 1 )... (Vh •— 1 ^ a'^ — 2cos TT-'V + 1 1; if wis even. (G. ^.ch.xi.) PROPERTIES or AN INSCRIBED aUADRILATEBAL. 153 (53.) Solution of the equation a?-" - 2 cos 0. a" + 1 = 0. a?2» — 2 cos ^.a?" + 1 = (.1'^ — 2 cos - a? + 1) n (a'^-2cos ^ ■r + l).-.(^--2cos-^ ^ ^ + 1)- .v"^" + 2 cos . .r" + 1 = (x^ -.2 cos ^^^ aj + \) n Stt + O (2 n — 1)^ + (.r-- 2 cos —^^.1' + 1) ... (.r"- 2cos ^^ ^ a? + 1). ^ n n (Hind, 291,2; G. A. 71) Properties of a Quadrilateral inscribed in a Circlr (54.) Let Af B, C, D, be the angles ; a, b, c, rf, the sides AB, BC, CD, DA, respectively; a, /3, the diagonals ^C, BD; (p their angle of intersection ; s = ^(a + b + c + d): then a- + d" — 6- — c"^ cos ^ = — ;r-7 — ; r-r — • 2 (ad + be) 2 ^ ^1 sinJ= , . .{s — a)(s — b)(s — c)(s — d)\ . ■\ ad + be , 4^ = (^ -a){s- d) ad + be (^ -b){s- -) ad + be (s_a)(5-d)]^ tanlJ- ^^_,^^(^_^,>j a = (ac + />(/) «f/ + be ah + ed ^ = (nc + bd) ah + cd ad + 6c The area = (s — a){s — b)(s — c)(s — d)\ ; = Iaj3.sin0. 2 sin d) = , , ^ ae + bd U ..(s_a)(5_6)(*-c)(s-d)l* 154 TRKIONOMETRV. The radius of tlic circumscribed circle _j (ah + cd)(ac + hd)(ad + bc) = 4- ^s-a)(s-h)(.-cXc-d) (Hind, Trig. 166 — 74; Leg. Geom. Note \.) Properties of Polygons. (55.) Let A^, Jj,, &c. A„, be the exterior angles, ftp a,^, &c. a,,, the sides A^A,, A^A^, &c. A,^A^, respectively: 2 area = a^ { «„ . sin A^ + ffg • sin (Ar. + ^3) + &c. + a„. sin (A„ + A^ + &c. + A J ] + 0-2 { ttg . sin J3 + &c. + a„ . sin ( J3 + &c. + A^^) ] + &c. + a„_ 1 a,j-sin -4,j- " (Lhuilier, Polygonom. viii.) a J = f/„ . cos cf pffo + ^^3 • cos 0^,03 + &c. + a^j . cos «i,«„ • a^ = a„"-\-a^-'thc.-^aj^—'2 [a^a^.co^a^^a^ + a2a4. cos 02504 + &c.|. (Hamilton, Analytical Geometry, 41.) (56.) Let Oj = cr2 = &c. = a„, jch/yl*-'^/ Ri J the radius of the c k ' uuai u Liibu d circle, J?2 5 that of the circumscribed circle : then n — 2 ''T _^ „ . TT A = TT ; a = 2 i?, tan - = 2 ^„ sm - . n n ~ n n — 2 „ , n — 2 R z=^a tan — tt ; R„ = ^asec -— — ' -^ 2^* - 'J 2/i ?ja^ ?^ — 2 The area = — — tan — tt 4 2w TT- = ^7iaR^= \na (4!R„" — a-)^ i TT- n (Hind, Trig. 175—82.) 5= 71^1*. tan - =4?i/to-.sm 7i ■^ " n * 2 area =S,„«;„.S,«^ + ,. sin S,^, ANALYTICAL GEOMETRY. (1.) Method of representing algebraical quantities geometri- cally. {H. A. G. Introd. 6—10 ; G. G. A. 1—6 ; Biot, Ch. i.) (2.) Analytical solution of determinate geometrical problems. {H. A. G. 14 — 30 ; G.G.A.'J; Biot, Ch. ii.) (3.) Relation of indeterminate equations to Geometry ; and definitions. (H. A. G. 43 — 8 ; Biut, 29 — 40 ; L. A. G. §. 2.) Analytical Geometry of two Dimensions. (4.) The straight line. The equation to a straight line is y=:aoD -\-h. Let the line be represented by I, then if the co-ordinates sin / tC are oblique, a= —\ sin l,y if rectangular, a = tan l^w. The equation to a straight line may be put under the form •*' y a The equation in terms of p, the perpendicular from the origin is .v.cos p^v + y .cos p,y = p. {H. A. G. 49—51 ; G. G. A. Ch. ii; Bint, 41—7; L. A. G. §. 3.) (5.) The equation to a straight line passing through the point {A\,y,) is y-yi = fi{'V-a\). If the line passes through two points, (ip^i), (.p„,y..), the equation is y — y^=. (.v — d\). 156 ANALYTICAL GEOMETRY. The co-ordinates of the intersection of two lines 63 — fcj a.b„—a„h, are cc = , y = =: Oj — ag ff 1 — (In If a third line, y=^a^,x -\-h.^^ passes through the point of intersection, then Let the two given lines be represented by l^ , /., ; then tan/j,/2 = sinc'r,i/ Oj — O2 1 + (a^ + «„) cos c2^,?/ + «i «2 If the two lines are perpendicular to each other, 1 + (ttj + ffj) cos x,y + Oj tto = ^• If the co-ordinates are rectangular, then a, — a„ 1 + a^a^ a-, — a„ sin LJo= = 1 + Oj a„ cos ^j,^2 == (1 + 0(1 + «/) p If in this case the lines are perpendicular to each other, The distance between two points, (^'i,t/i), (^^j^c)' = {x, - .v.y + (y, -ynf\\ Let j9 be the perpendicular from a given point, (a?i,2/i), on the line y = ax-\-h, then y^-ax,-h p= H TT- ?rr-. - (1 + a-)^ If ("^u^i), {po^^y^, {^siys)^ ^^^ the angular points of a triangle, Area = 1 { (w^y. - a?„^i) - (cP.i/a - ^^s^/i) + (^^2 2/3 - -'^s^/e) 1 •■ (^. A. G. 52—9 ; -H". C. S. Ch. ii. ; Blot, 48—53; L. J. G. §. 4.) Teansfoemation of Co-oedinates. 157 (6.) Let X, y, be the co-ordinates of any line, d7j, y^, the co-ordinates of the same line in another system having the same origin, then 1 sm w^y (op^.sinw^.y + y^.smy^,y), 1 , . QIII tX aU If the axes in the original system are rectangular, and in the new system, oblique, the above formulae become x = .z'j cos .r^jj; + ^1 cos y^^a;, y = a.\ sin A'j,A' + 2/1 sin y^,ai. If the original axes are oblique, and the new ones rect- angular, then 1 , . X = -; {a\ sm x^,y + y^ cos .r?i,2/), sm x.fy y= — (.r sina\,a? + Vi cosa?,,^??). ^ sma?,y ^ ^ '^ If both systems are rectangular, then x = x^cosx^,x — yj^smx^,x, y = a?j sin a?j,a? -|- 1/^ cos x^^x. If it be required to change the origin, as well as the di- rection of the co-ordinates, then a, 6, the co-ordinates of the new origin, must be respectively added to the values of x and y in the above formulae. If the origin alone be changed, the direction of the co- ordinates remaining the same, then x = a + .i\, y = b + y,. (H. A. G. 71-6; L. A. G. §. vi; G. G. A. Ch. iii; Biot, 80-96.) 158 ANALYTICAL GEOMETRY. (7) Transformation of polar co-ordinates. Let rt, /->, be the co-ordinates of the new pole, w, the radius vector, and a the angle which the axis to which u is referred makes with the axis of x, then sin I x,y - (0 -f a) } • oo = a -\-u y = b + U sm a!,y sin (e + a) sni .v,y If the axes are rectangular, these formulae become .v = a + u cos (0 + a), y = b + usm(9 + a). (H. A. G. 77—80 ; Blot, 100.) The Circle. (8.) Let a, j3, be the co-ordinates of the centre, r the radius ; then the equation to the circle is (x — a)' + (y — (iy + 2(.x' — a) (2/ — /3) . cos x^y = r^. The equation between rectangular co-ordinates is ^v-af + {y-(iY = T\ (1) If the axis of i^, or y, passes through the centre, the equation becomes respectively, ^'^{y-(if = r\ . or (a? — a) + 2/' = r^. ^ ^ If the origin is in the circumference, (1), (2), become respectively, ,^,2 + 2/--2a.J?-2/32/ = 0; (3) y"^^rcG-x\ . ar = 2ry — y". ' If the origin is at the centre, then x^ + y~ = r'^. (5) . of these forms, (4) and (5) occur most frequently. THE CHICLE. 159 The general form when referred to rectangular co-ordinates is y" + x" + Ay + Ba!+C = 0. (9.) The polar equation. Let a, 6, be the co-ordinates of the pole, u the radius vector, and the angle w, a? = ; the polar equation is w- + 2 (a cos 4- 6 sin 9)tt + a" + h" — r"^ = :, + or — , acting as the pole is situated Avithin or without the circle. {H. A. G. Ch. v ; G. G. A. Ch. x ; Bloi 105 — 14.) (10.) The intersection of the circle <.v- + y- = r- and the straight line y = ax + b may be determined from the equation 26 b''-a-r- a + 1 a- -\-\ which having only two roots, it follows that a straight line cannot cut a circle in more than two points. The equation to a tangent at a given point {3L\,y^) in the circumference, is x^-\-y^co^w,y y — yi= (v — x,), if the axes are rectangular, y-yy= - -(*-j^i), or a.\v^+yy^ = r". To draw a tangent from a given point (A\,y„) without the circle: the co-ordinates a.\,y^, of the point of contact may be determined by the solution of the equations, ^i' + yi' = ^'• (H. A. G. Cli. vi; H. C. S. 37—43; Biot, 115—23.) IfiO analytical geometry. Conic Sections. THE PARABOLA. (11.) The parabola referred to its axis. Let ^S* be the focus, P any point in the curve, and PQ perpendicular to EQ the directrix., then PS = PQ. Q Z v^^ I i L / \ T E A K S N G Let PT be a tangent at the point P, and PG the normal. If the origin is at the vertex, and the axis coincides with the axis of <2?, the equation is or PN^=^AS.AN. /.QPT= aSPT. SP = AN+AS = ST = SG. The intersection of the straight line y =ax + b, and the parabola y~=:4!mx ; may be determined by the solution of the equation y — y + a = which has only two roots ; hence a straight line cannot cut a parabola in more than two points. The equation to a tangent at the point {x^^^y^) is 2 m y-yi= — (-v-^v,), o^* yyi = 2m{x + Xi). CONIC SECTIONS THE PARABOLA. 161 The subtatigetit, NT =2 AN. The equation to the normal is The subnormal, NG = 2 AS = 2m. To draw a tangent to the parabola from a given point (a,^>) without it : ci\,y^ the co-ordinates of the point of contact may be determined, from the equations 2m b= {a + a;^), y," = 4wij,-,. (,H. C. S. 46 — 62 ; Hust. Prop. 3 — 7.) (12.) The parabola referred to the focus i The polar equation, the focus being the pole, is 2 m m 1 1 _ 2 SP^ Sp~^ 1 + cos (sin i ey ' (See Fig. p. 160.) SP.Sp = AS.Pp. The tangent at any point, and the perpendicular on it from the focus intersect the axis ^ F in the same point. SP.SA = Sr: Let Z be the intersection of the tangent with the directrix, SZ is perpendicular to SP. (H. C. S. 63-71 ; Hust. Prop. 8.) X 162 ANALYTICAL GEOMETRY. (13.) The parabola referred to any diameter. A diameter and the tangent at its vertex being the axes, the equation is or QV^^^SP.PV. The parameter = 4 SP. 2m.qF=PF.FP'. PF" = 2m.PV. If 4w?j, 4^2 ^re the parameters corresponding to the ordinates Pp, Qq respectively, then 4, CP, respectively, intersect each other in 0, then QO.Oq : RO.Or :: CD" : CP\ If Oj, 6^, are any conjugate diameters, « J- + bj^ = a" + b". The area of all circumscribing parallelograms is constant, and = ^CD.PF=4>AC.BC. If «! = 6p then «i = i(a2 + 62)|^ ; sm Opfe^ = a- + 6- tan «,,«= - . a SP.HP = PT.Pt = CD"". (See Fig. p. 163.) AM.am = BC". Let P^ be the diameter drawn through any point P, whose co-ordinates are .r^, y^, and PQ, Qja the supplemental chords : the equation to PQ being 2/-yi = a('^-*'i)» the equation to Qp is «! a If the chords be drawn from the extremities of the axis major, their equations are y = a (v + a); b," «i a CONIC SECTIONS THE HYPERBOLA. 167 If two diameters be drawn parallel to any two supplemental chords, they are conjugate to each other. 2ab- tan AFa = -r-^ . {H. C. S. 127—64; misL Prop. 13—8; Biot, 161—90.) CD- Radius of curvature = PF ' Chord of curvature through C = 2CD- CP ' Chord through the focus = 2 CD- AC The Hyperbola. (17) The Hyperbola referred to its axes. (Hust. Prop. 19.) SP : PQ ■••• e : I; e being >1.* If the curve is referred to the axis, and a tangent at the vertex, the equation is AN.Na : NP" :: AC" : BC If the origin is at the centre, the equation is X a ^/'^Tc (•'■'""')' """"^'z ~r^ =^- * The lines in the above figure are analof^ous to those in the ellipse, on which the same letters are placed. 1C8 ANALYTICAL (iEOMETRY. The equation to the equilateral hyperbola is X' — y- =■ a'-. fr BC" SL=±- = -— . a AC The intersections of the hyperbola with the straight line, may be found by the solution of the equation a-a' — b- a'a —b- which has only two roots ; a straight line therefore cannot cut an hyperbola in more than two points. The equation to a tangent at the point (<*?i52/i). is or ;; -r = 1 . a- b- CN.CT=CA\ Cn.Ct =CB-. The subtangent, CT= ' ^' ~" . The equation to the normal is b~ BC^ The subnormal, NG = — w, = — — CN. a' AC' To draw a tangent from a point {x„^y^ without the hyper- bola : the point {'v^,y^ at which the tangent meets the curve must be determined from the equations a b fT _ y£ ^j {H. C, S. 172 — 94; Hust. Prop. 7, 8; Biot, 224 — 44.) CONIC SECTIONS THE HYPERBOLA. 169 (18.) The hyperbola referred to the focus. SP = ex — a ; and HP = e.v + a. HP^SP=2AC. The polar equation, the focus S being the pole, is e--l u = a . 1 + e cos d If H be the pole, then e--l 71= —a SP = 1—e cos G BC- AC-SC.co^PSN 11^ SP^ Sp'^ SL SP.Sp = ia(e--l){SP + Sp). If the centre is the pole, the equation is e--l 1 — (e CO?, oy L HPT= L SPT. If SY and HZ be drawn perpendicular to PT^ the locus of the points Y and Z is a circle described on A a. PI=Pi = AC. SY.HZ = B(f-. (H. C. S. 195—205 ; Biot, 259—63 ; Hust. Prop. 1—6.) (19.) The hyperbola referred to any system of conjugate diameters. The locus of the points of bisection of all parallel chords is a diameter. The co-ordinates of the points of intersection of any diameter y = ax with the curve are ab aba Y 170 ANALYTICAI- (;E0METUY. In order tliat the diameter may meet the curve, a must be < - , and > . a a If the diameter a^ is conjugate to b^, b^ is conjugate to a^ The equation to the diameter passing through {^\iyi) is the equation to the conjugate diameter is b" ci'j «■ ' ^1 * * The equation to an hyperbola referred to the , centre, and two conjugate diameters a^, b^, is, ^ _ i^ =1 ar br If P, the extremity of a^, is the origin, the equation is PV. Vp : QV" :: CP" : CD-. CV.CT=CP'; Cv.Ct=CD-. If from the several points of a straight line given in position, pairs of tangents be drawn to an hyperbola, the lines joining the corresponding points of contact will all pass through the same point. CONIC SECTIONS THE HYPERBOLA. 171 QO.Oq : RO.Or :: CD' : CP- a^' — 6j'' = «■ — b-, If tangents to the conjuorate hyperbolas be drawn at the extremities of the conjugate diameters, tlie area of the paral- lelogram is constant, and = 4 CD . PF ^^AC. BC- SP.HP=CD-. PT.Pt=CD-. If the equation to PM is y-yl = (^{^^-^^\)^ the equation to mp is af a If diameters be drawn parallel to any two supplemental chords, they are conjugate. (iy. C. S. 206 — 45 ; Biot, 245 — 58 ; Hust. Prop. 15—8.) (20.) The asymptotes of the hyperbola. If the origin is at the centre, and a, 6 are any system of conjugate diameters, the equation to the asymptotes is 6^ RQ, = rq. QI.Qi = BC-. QR.Qr = CD-. If the asymptotes be the axes, and the centre the origin, the equation to the hyperbola is 172 ANALYTICAL GEOMETRY. The equation to a tangent at the point (a'ijJ/i) is .vij^ + .v^y = ^(a" + b"). The area of a triangle contained between these axes and a tangent is constant, and = 1 («'■ + b") sin .i?,y. Let (I'pyi) be any point in the curve, and flj, ft^ the con- jugate diameters to that point, then «i = ± 2 Cj^i . 2/1)^ cos .v,y ; ^1 = ± 2 (r, . y,)^ sin .r,y. If C7^^^* is parallel to Ci, then UW= Wu. (H.C.S. 245—59; Biot, 250—8; Hust Prop. 11—5.) Discussion of Lines of the Second Order. (21.) The general form of an equation of the second degree containing two unknown quantities is ay" + ba/y + ex" + dy -\- ex +f= 0. From the solution of this equation y = _ ^±^ + L . (b'^-4!ae)x' + 2(bd-2ae)x + d^-4>aff; 2a - 2a by + e 1 + — . (6- — 4ac ' DISCUSSION OF LINES OF THE SECOND ORDER. 173 If b- — 4>ac<0, the curve is an ellipse; b' — 4ae>0, an hyperbola; 6" — 4ac = 0, a parabola. If 6 = 0, a and c must have the same sign in the ellipse, and different signs in the hyperbola. If the curve is an ellipse or hyperbola, and the axes parallel to any system of conjugate diameters, the form of the equation is ay- + cx^ + dy + ea!+f=0. If the origin is at the centre of the curve, the equation becomes ay" + bxy -\- car +f= : and if the curve is also referred to conjugate diameters, then ay- + ca?'^+/=0. If the axes are parallel to the asymptotes of an hyperbola, the form of the equation is bay + dy + ex +/= 0. If e->4!cf, the curve intersects the axis of a? in two points ; e-^^c/, touches the axis of .x"; e"<4c/, does not meet the axis of a ; The same conditions exist with regard to the axis of y, according as d" >, = , or < 4 a/. If B->AC, the curve intersects the diameter, y= '■ ; B^ = ACf touches that diameter; B" < AC, does not meet it. (22.) Determination of the species, and their varieties. [l] If ^ < 0, the curve is an ellipse. If the roots of the equation X-+ —a+ ~=0, (a) are equal, the ellipse is reduced to a point. If the roots are impossible, the ellipse becomes an imaginary line. If a = c, and ^ = 0, the ellipse becomes a cixcle. 1^4 ANALYTICAL OEOMETRY. [2] If A>0, the curve is an hyperbola. If tlic roots of the equation (a) are possible, the diameter, ba.' + d cuts the curve ; if impossible, that diameter does not meet it. , If the roots are equal, the hyperbola becomes two inter- secting straight lines. If a= —c, and 6 = 0, the hyperbola is equilateral. [3] If A = 0, the curve is a parabola. If B also = 0, and C > 0, the equation represents two parallel straight lines, C = 0, one straight line, C <0, an imaginary line. (H. A. G. 81-89, 184-213 ; G. G. A. Ch. iv ; Biot, 264-304.) Summary of Equations. (23.) Rectilinear equations. [1] Let the curve be referred to its centre, and principal diameter, then in the ellipse, 2/" = ~ («' — <'*''') ' .... hyperbola, y" = — (jxr — a!~) ; .... circle, y- = a^~ oc- ; .... equilateral hyperbola, y- = x" — a^. [2] Let the curve be referred to the principal diameter, ^nd a tangent at the vertex, then 6" in the ellipse, //- = — (2 a a? — ar) ; .... hyperbola, //== — (2acr + w) ; . . . .parabola, y-^^mx; . . . .circle, y- = '2ax — ar; .... equilateral hyperbola, y" = 2ax -\- x". The general form of the equation is y- = mx -\- nx". SUMMARY OF EQUATIONS. 175 The same equations subsist, if a and h are an}' system of conjugate diameters to Avhich the curve is referred. (24.) Polar equations. [l] Let the curve be referred to the radius vector, and angle contained between it, and the principal diameter, then 1 . The centre being the pole ; a-(l-e^) in the ellipse, ii-= .... hyperbola, u" = 1 — (e . cos d)i~ ' a{e'-l) (e . cos 6)" — 1 2. The focus being the pole ; aCl — e") in the elhpse, w= i ^, ^ 1+e.cosa .... hyperbola, 7i= , ^^ l+ecos^ 2m .... parabola, ii = . ^ 1 4- cos [2] Equations between the radius vector, and a perpendicular on the tangent from the pole. 1. The centre being the pole : n JO a'O' in the ellipse, P' = .... hyperbola, p" = a- + b- — u" a"h" r — (a- — 0-) .... equilateral h3'perbola, p = — : u 2. The focus being the pole ; in the ellipse, p- = , 2a — w .... hyperbola, p- = 2n + u' . parabola, p" = mu. (H. A. G. 183.) 176 analytical geometry. Analytical Geometry of three Dime-nsions. (25.) Tlie straight line. The equations to a straight line referred to three rectangular co-ordinates, are which are the equations to its projections on the planes of o?^, yx respectively- The equations to a line passing through the point (■^p2/i»^i)' are x-.v^ = a{z-z^), y-y^ = b(z-z,). The equations to a line passing through two points (a?j,yi,»j), (0^^,2/2,^2), are ^1 — ^1 ^2 ~~ •^i In order that the two lines \y = b,z + (i,; \y = b,z + (i„, may intersect each other, it is necessary that «1 — «2 ^ ^1 — f^ '2 ^1 "" ^2 ^1 — ^2 The co-ordinates of the point of intersection are a^ — Oo '^i — ^2 ^1 ~ ^2 The distance (dj of a point (■i\^y^,Zj) from the origin; (h = C^'i' + y' + -1')^ = - (1 + «' + ^')^ ' if x = oz, y = bz are the equations to the line passing through the given point, and the origin. The distance (Z>) between two points {xi,yy,Zi), {xc.^yny%n) ; D = (c^i - .^2)"^ + (^1 - 2/2)' + (^1 - ^2)" = rff + C?2- — 2 (cTi072 +yiy2 + ^1 %) I' » do being the distance of (<^?2,2/2)«o) from the origin. THE PLANE. 177 The inclination of a straight line / to each of the axes : cos Lz = 7:; -, — T^Ti . (cos l,ivy + (cos l,yy- + (cos l,^)'- = 1 . If lyvy, &c. denote the angles which / makes with the planes a-y, Sec. then (sin I,.vyy- + (sin lyvz)'- + (sin Z,^;^)" = 1. The mutual inclination of two lines, l^, I, : cosli,lo = cosli,x.cosloya) + cos li^y. cos l^fy + cos li,z. cos l^,z. 1 + aia„ + byb^ ~- (l + «f + 6r)a + «/ + &./) I'' The equations to a line in terms of the angles it makes with the axes are cos l,.v cos 1,1/ „ a;= —z + u, y= j^z + fi. cos l,.^ cos l^Z (G. G. A. Ch. xvi; H. A. G. 229—51 ; Biot, 54—69.) (26.) The plane. The most general form of the equation to a plane is Aa+By+Cz + D = 0, or >i: = AiX + B^y + Cy The equations to the traces of the given plane on the planes of xy, xz, yz, respectively are Ax+ By + D = 0, Aa + Cz + D^O, By+Cz + D = 0. If a, b, c, are the distances from the origin at which the plane cuts the axes of a', y, z, respectively, the equation becomes X y z ^ - + f + - =1. a b c If p is the perpendicular on the plane from the origin, then p = a: cofi p,x + y cos p,y + zco^ p,z. Z 178 ANALYTICAL GEOMETRY. The equation to a plane passing through the point {x^^y^yZ^), and parallel to the plane An + By + Cz -i- Z) = 0, is A{,v-x,) + B{y-y,) + C{z-z,) = 0. The co-ordinates of the intersection of a straight line x = az -\- a, y = bz + fi and the plane Aof + By + Cz + Z> = 0, are a(Aa + Bb + C)-a(Aa + B(i + D) 0? = Aa + Bh + C _ fi{Aa + Bb + C)-b{Aa + B(i + D) y~ Aa + Bb+C _ Aa + B(i + D ^~~ Aa + Bb + C' When the line and plane are parallel, then Aa + Bb + C = 0. The equations to a perpendicular p from the point (a7j,^i,^J on the plane, are - _ Ax, + By, + Czi + D ^~- (A' + B'^+Cy ' The equation to a plane drawn through a point (.ri,2/i,^i) perpendicular to a given line, is a {.V — cTi) + b{y — yi)+z — Zi = 0. The angle contained between the line /, and the plane P ; Aa + Bb+C sin LP = — — — - ^ — ^ T . (1 + a- + b"-){A- + B" + C"-)\^ (27.) The inclination of a plane P to the co-ordinate planes: cosP,.2/=^^,^±,^^,^,, + B cos P,xz = (A' + B"+C-)h ORTHOGONAL PROJECTION. ^V^ (cos Pyvy)- + (cos P^xz)- + (cos P^yx)- = 1- The mutual inclination of two planes P^, Pj ' cos Pi,P2 = cos Pi,.r2/ . cos PoyVy + cos P^^OCZ . cos Pjj-J^^ + cos Pi,y% • COS P^^y- + (yi - VoJ' + (-1 - ^^Y + 2 { (.rj — oe^{yy — ?/„) cos x,y + {x^ — x„){z^ — z^) cos x,% + (y 1 - ^2) (-1 - ^J cos y,z]. The inclination of two lines sin^,,y^ , sin/„,cr« sinZ„,.^^?/ cos ^,,/<,= -: — = cos LyV + -; — cosL,y + - — = cos Z^,;?, smx,yz fiiny,xz sm z^xy sm l.^yz ^ sin l^yvz sin l^,xy = • cos Z„, — » ^^^ plane cuts both sheets of the surface, and the curve is an hyperbola. The equation to the intersection of the oblique cone y- (x" + if) + (a" — r") z- — 2ayw% + 2ar-x + y'r" = 0, and a plane situated as above is {(^cos^-asin^)-— (rsin^)-} 2/^ + 7- a?/ + 2ar^ cos ;^.yi—7-r-=0. The curve is an ellipse, a parabola, or an hyperbola, ac- 7 cording; as tan v <, = , or > . 2a7 If the section is a circle, tan ^ = a~ — y — r- {H. A. G. 335 — 46 ; G. G. A. Ch. xv ; Biot, Ch. vi.) Surfaces of the Secoxd Order. (38.) The general form of the equation to a surface of the second order is Ax- + By" + Cz"-\-2{A,yz-\-B,.vz-\-C^xy + A„x+B,y + C„z)-\-D=0. If the axes are parallel to any system of conjugate diametral planes, the equation l)econies A.v- + Bif + Cz- + 2 (^,,.r + B.y + C^z) + Z) = 0. (1) A straight line cannot intersect a surface of the second order in more than two points. Aa 186 A\AI,YTICAI. GEOMETKY. The Ofjuation to a diametral plane, which is the locus of the points of bisection of all chords parallel to the line ;v:=mx, y=^n%, is {4m + C,?i + B^) X + (CiW -\-Bn + A^)y+ (B^m + A^n + C) z + A„m + B^ji + C = 0. The number of systems of conjugate diametral planes is unlimited ; of these, however, only one system can be rectangular. The co-ordinates of the centre are _ A„ (A,- - BC) + B, (CC, - A, B,) + C {BB, - A, Cy) '^ ~ ABC + 2A,B,C,- A a;' - BB;' - CCy' _ B, (B,- - AC) + q (A A, - B, C,) + Ao (CC, - A, B,) ^~ ABC + 2A,B,C,-AAf-BB,--CC;' _ C. (Ci" - AB) + J, {BB, - A, CQ - B„ (A A, - B, C,) '^~ ABC + 2A,B,C,-AA,--BBr-CCf If the surface is finite, then A,--BC<0, B,--AC<0, C,--AB<0, A A,' + BB;- + CC," - ABC - 2A^B,C, < 0. If the surface has not a centre, then AA," + BB," + CC;~ = ABC + 2A,B,C,. (39.) Surfaces of revolution. If the axis of revolution coin- cides with one of the co-ordinate axes, as = 0. If the axes coincide with any system of conjugate diameters, the equation is reduced to Ax''-\-By--\-C%" + D=^0, 2 2" X V Z' or - + ^ + - = 1 ; a- b- c a, b, c being the semi-axes ; in which case, the co-ordinates arc rectangular. Let ffj, 61, Cj be any system of conjugate diameters, then, «i" + b^^ + c^ — a" + b" + c". If the surface be referred to the diameters Wj, ftj, Cj, then (oj 6j . sin x,y)" + (a, Cj . sin x,z)- + (6^ Cj . sin y,zy=a-b' + a"c' + b"c'. and a'^ b^ (? = a^ b^c^ {1 — 2 (cos x,y . cos x,z . cos y,«) — (cos x^y)- — (cos x,zY — (cos y,zy\. (41.) The species of the class of surfaces defined by the equation x" y" z~ -+^ + -=1, a* b c- depend on the signs of a*, 6", and c" ; the equation to X" tf ^" the ellipsoid is — + =-;+ — =1; a'- b- c" o o n the hyperboloid of one sheet, 1 — — = 1 ; a- b' a'- x" y' z' the hyperboloid of two sheets, — =:1. a- b' c" 188 AXALYTICAL GEOMETRY. In the ellipsoid, the three traces, or principal sections are ellipses, their equations are y' ^' — 4- 2 ~^' ^'^^ trace on y%; or %- — a + — =1, . . . . . . .00%', a- c- a If two of the quantities a", 6^, c" are equal, the surface is a spheroid, if they are all equal, a sphere. In the hyperboloid of one sheet, the principal sections are an ellipse, — -\ — ^ =1, a- b- an hyperbola, — :r =!> a' c~ ^ — — =1 6^ c" The conical surface the equation to which is a,^ y" z" a^ b^ c- is an asymptote to the hyperboloid. In the hyperboloid of two sheets the principal sections are an hyperbola, a" — b' = 1. ,ir z" a" — c" 1, 2/' z"^ an imaginary curve, ■— H — ^ = — 1 . The plane yz does not meet the surface. The equation to the asymptotic coiae ia a" b- c {H. A G. 359—81 ; G. G. A. 200—24 ; Biot, 322—30.) INTERSECTION OF A SURFACE AND A PLANE. 189 (42.) Surfaces ivhich have not a centre. If the origin is at the vertex, and one of the co-ordinate axes, as a.*, coincides with the axis of the surface, the general form of the equation is My- + Nz- + Px = 0. This class consists of two species, the elliptic paraboloid, ny- + mz" = 4!mn.v, the hyperbolic paraboloid, ny" — mz" =^ ^mnx. , The principal sections of the elliptic paraboloid are a parabola, y-=^^mx, 5r- = 4w = ; 190 ANALYTICAL GKOMETRY. the equation to a curve of the second order, which will l)c an ellipse, a ])arabola, or an hyperbola, according as Ali (cos -x)'- + C (sin -x)' I B (sin 0)'- + A (cos 0)- } > , = , or < 0. In the ellipsoid, every section is an ellipse, or one of its varieties. In the hyperboloids, the section may be either an ellipse, a parabola, or an hyperbola. (44.) If the surface have not a centre, the equation is The equation to the line of intersection is [M (cos (p. cos -^y + iV(sin^)-} y" + N (sin (pYar — 2Msm (p .cos (p .cos -y^. X y + P cos (p. cos -^.y + Pcos 0..r = O; the equation to a curve of the second order, which will be an ellipse, a parabola, or an hyperbola according as MN (sin

, = , or < 0. In the elliptic paraboloid the section is an ellipse, and in the hyperbolic paraboloid an hyperbola, except when = 0, or Y = 0, in which case the section is a parabola. (45.) If the surface has a centre, the section will be a circle, when A (sin (py + B (cos (py == C (sin ■^)- + (cos y)-{ A (sm(py + B (cos (p)"]. ^(A — B) sin (p . cos (p . cos % = 0. These equations will be fulfilled if the plane is perpendicular to yss, and tan^= + | \ , /A-B\i 'TX, = + I I , of which three quantities only one is possible. If the quantities a-, b", c", be arranged in the order of magnitude, the section intersects the mean axis 26 in the ellipsoid, and hyperboloid of two sheets, and the greatest axis 2 a in the hyperboloid of one sheet. THE TANGENT I'LANK. 191 The inclination of the section to the plane xy is c /a- — b"\^ in the ellipsoid, tan + - ( jz rr ] > c /a" — 6'\2 .... hyperboloid of one sheet, tan + r ( — :, ) » two sheets, tan H — ( — ^ I • ~ a \ir — c- / (46.) If the surface has not a centre, the section will be a circle, when M (cos p^. cos (p)- + N { (sin y)" — (sin <^)- { = 0, 2 M sin (p . cos cp • cos }( = 0. These conditions will be fulfilled if the plane is perpendicular to ^\?, and sin p^= + ^ — J , ,vy, and sin = + / — j . (H. A. G. 396—401 ; Biot, 332—4.) The Tangent Plane. (47.) If the equation to the surface is A.v- + By^ + C;^- + 2 {A„cc + B^y -t- C„z) + D = ; the equation a plane touching the surface at the point (yi?^i) is {Ax,+ A^)x + {By^^ B^)y-^{Cx, + C„)z + A„x^ + Byy, + C.z^+ D^O. If a; — a\ = a{z — %^).) y — 2/i = ^(*~^i) are the equations to a line passing through two points (^,y,5f), (A\,yi,~i)' ^'^^ point {v„,y„,z^ in which the plane passing through these points touches the surface may be determined from the equations ^< + 5s// + Cz.j + 2 ( J„.r, + B„y., + C.z^) + /) = 0, {Ax„ + A„^ .x\ + {By. + 5,) y, + (C^, + C);?-, + A„x„ + i?,t/. + C„^,, + Z> = 0, ^a.i'. + 5 />//,, + 6'^ , = 0. 192 ANALYTICAL GEOMETRY. If tlie equation to the surface is A.V- + By" + Cz- + D = 0, the equation to the tangent plane is A AW, + Byy, + Cxz, + D = 0; and the equations to the normal are If the equation to the surface is My- + Nz- + Px = 0, the equation to the tangent plane is Myy^ + Nzz, + ^P(a) + ,v,) = 0. If three planes perpendicular to each other are tangents to the surface iv" v" z" - + 7. + -=l, a- b- c- the locus of their intersection is the sphere or -\- y" -\- z^ =■ a^ -\- h" -^ c". If a conical surface circumscribes a surface of the second order, the line of contact is a plane curve. If any number of planes passing through a given point intersect a surface of the second order, and at the lines of intersection conical surfaces be circumscribed, the locus of their vertices is a plane. {H. A. G. 364, 85 ; G. G. A. Ch. xx ; Biot, 335—8.) DIFFERENTIAL CALCULUS. (1.) Differentiation of algebraic functions. d^Uy + u^ + &c. + u„ + const.) = d^u^ + d^u., + &c. d^u„. « d^ au = ad^n, if a is independent of w. d^.v"'=via)"'-\ rf u Uo = u, d^Uo + t(>n djii, = u, Uc, ( 1 I . d,u,u,...ti,=u,u,...ut--—+ — -+&C- + -— )• ^ ^ u/d^u_d^\ ■ 7* vd ji — tidj) u /d„u d^v a,— = ;r- V v~ d^ -^^ ^" = -!-? "' ^ -^-i + ^^- + &e. + X m U. dx^i d^Vo ^ d^v — — czc " d^u expresses the same quantity wliicli has visually been denoted by — . (See L. C D. Vol. ii. p. .527.) ^.r(S„,?^„ + const,) = S„,(/^t/ m X -■ in n n n fj ^^ X -^ III 111 m m d ^^T = ?i^ (s ^- s ^^4 Bi? 194 DIFFERENTIAL CALCULUS. (2.) Differentiation of exponential functions. d^\og,x= —^. d ^a" = \og^a . a' . "j-'"0'' X d,e' = e X d^ sin X = cos X. d^ tan X = (sec o?)'^. d^ sec X = tan X . sec x. d^ vers x = sin ir. c/^sin-' X a a (a" — cV-)5 ' d^tan-' ^ .r- a" a a' + X- d, sec " ' X a" a x(x~ — a-yi' d_vers ~ . 1 *' ^ d^ cos .f = — sin x. dj. cot x=^ — (cosec x)". d^ cosec X = — cot + ' df-'u.d^v + &c. «(«-l)...(„- r+0 ^^^ _^ 1 . 2 ... r « d;:Mu = S,«cZ/-"' + ^:w-d;"-':t;. SUCCESSIVE DIFFERENTIATION. 195 (4.) rf;«' = (log,a)"a'. d. d^ sin x = cos x, d^sva.a;=i —cos a?, &c. = &c. d^^' " ^ sin a; = ( — 1 )" - ^ cos x, d^ tan .r? = 1 + (tan T— 1 V" ~ " sin x " 2m — 9.11 ^ 1 " '^ " '^" 2 Wi — 2 7i : tan 07 « (_!)'« -"sin .17 „„ „ , "-jK-l) =s 2 ?» — 2 ;i 2^^«-^«-l^-S„ 2 m — 211—1 w-"-\a-\ (— l)''sin.r] j _ (— l)'"'cos.r __(— 1/sm.r^ 196 DIFFERENTIAL CALCULUS. d; tan - ' .<• = — 'J .r (l + x") ' ". d^ tan - * d! =—2(1+ a.") " * + 2^ a?" (l + .r") " ^ &c. = &c. dj. sec ~ \r = .i- ~ ^ (.v- — l) ~ i. d^ sec " ^07 = — a^ - - (a- _ j ) - 2 _ (a>2 _!)-§. - - - •■' _ * d^sec-'^oo = 2w-^{o(r—i) - + w''^ {,v"—\) ''' -{-3x{.x-—\) ~\ The coefficients of cos ~ \r, cot ~ \r, cosec ~ '^x may be found from these as above. {L. D. C. 34. — 4-8.) (6-) d^(piu) = d^,(p{^(')-d^u. « (L. C.D.U,L. D. C. l6.) Differentiation of Functions of Superior Orders. (7.) Letrf,0(AO = 0'O^), then f/_^. sin"cV = cos X . cos sin cV . cos sin-a? . . . cos sin" ~'^x. >* M — ??J „>« ^ " C:0(^O = S,„C0OO y^. » I «.- i=f/;:?*}- *S'ee Appendix?, A few applications of this important theorem may here be added : d;:w"' = T3-" . a'% I a, _ ^ = d/ " ^iw | . d^:sin ic 2m— \ \2m d^:cos u r ,—31 — 2m + 1 fJim - 1 n—r __m — 2;n -,0»i = S,„(- 1)'« sin u ^ -^ + S„, (- 1)'" cosz. ^ -^ , 2 m — 1 2 m { ^w - 1 = ^J'-^ ' r = i w, if ?^ is even, r = :| (7^ — 1), if n is odd. } n V cL sin"c^ = P„, cos sin'" ~ ^ .x'. DEVELOPMENT OV FUNCTIONS. 197 d^cos'\r^ (— l)"sin.i'.sincos,x'.sincos".r...sincos'*~^i'. * rf,.log/^^• = .l7-^(log.cr)-^(log;^^•)-^..(log,«-^^)-^ • Development or Functions. (8.) dj(,v + y) = djia; + y). Taylor''s Theorem. Let E^{it) represent what ti becomes when ,1' + Dx is substituted for x, then E^{u) = ti + d^u?^ +d^u^-^^ +d^u^^^ + &c. « 1 1.2 1.2.3 If the 71 first terms of this series be taken, the limits of (Dxy error are the greatest and least values of d" u ^ 1.2. 3.. .W provided there be no value of x between x, and x + Dx that renders u or any of its differential coefficients infinite. (L. C. D. 169.) Maclaurhi's Theorem. Let (?Ox=o> ^»=o^* represent the values of ii, d"u, respectively, when x = 0, then 2 3 * d^ cos" A' = ( - 1 )" P,„ sin cos'" " ^ x. Let Ej.tc — Ti = Dj.ti, then (/_,,?« " "^ ' '" \n-m+^ Lr;;; (r/^?<) where a^ _ ^ = (//+': ?^ . 198 DIFFEUENTIAI- CALCULUS. (9.) Taylor's theorem applied to two variables. + ^ [ d;u (Da-r + 2 dj^u .D.V . Dy + dpc (Dyf } + -~{dXD.vy + 3d!d^u.{Da^yDy + 3dJ;uDx.{DyY + d^u^Dyf] + &c. « c/™ rf> = d; d^u. (L. C. D. 25 — 30.) (10). Laplace's Theorem. Let y = ^[x -\- x.(p{tj)]., in which % is independent of x and y, then .v Lagrange'' s Theorem. J^et y = z + x.c^{y), then This theorem is a particular case of the former, in which y^{z + x.(l){y)] =% -\- X .(f>i;y). CO m « ^,.» = S,„S„(2^^)'"-«.(Z>2/)«-^rf/'--rf/-^:w. If a third variable z be supposed to receive an increment, then E,,y.M = S,„ S„ S,. {Dwy-'^ . (Z>2/)"- . (Z)^)'-i . d:^--:d--^:dr\u. Maclaurin's theorem applied to a function of two variables. if ' am / {Jarrett, Camb. Phil. Trans. Vol. iii, Pt. 1.) CO iV"' ' Ay)=m^) + ^,nT^d'r{cp^l.{z)\'\dJ^{z)\. CO rpm ' f(y) =.m + K T- • d:-' \ 0(^"'- rf./(^) } • TRANSFORMATION OF THE INDEPENDENT VARIABLE. 199 ^^ f(y)=y^ then {L. C. D. 107—9 ; Tr. L. Note E ; L. D. C 49— 6'1.) Implicit Functions. (11.) Let d^u — M, dyU = N, then ^ /^^:( ^ =^ CP, M-^Nd^y = o, ■" from which equation the value of d^y may be found. By successive differentiation, the following equations may be ob- tained, in which w^, u„, &c. are functions of x and y : u^ + ic„ d^y + u^ {d^yf + Nd'^y = 0, u, + u,d^y + ii^{d^y)- + it^id^yf + {u^ + u.jd,y)d;y + Nd^y=0, &c. &c. from which the values of d^y, d^y, &c. may be obtained. (L. C. D. 41 8 ; L. D. C. 100 3.) Transformation of the Independent Variable. (12.) ci,= ^y dy " {d^yf d;,= '(d'yy-^^^ , &c. = &c. If o! and y are considered as functions of a third variable t, then _ d,x.dfy-d^x.d,y '^'^~ '(d;wy ' {d,xf .dly - 3d,x .d^v .dfy + 3 {dfr^.d^y- d,x.dfx.d,y d.V= r-; — 77^ > (c/,a) &c. = &c. (L. C. D. 51 — 69.) y=^'^~^^n}--dr'W)\''- 200 DIFFERENTIAL CALCULUS. (13.) Elimination of an arbitrary function. Let z = 0/(<'',y) = 0(0 suppose, tlien d^z = dfCp{t).dJ, and d^z = df(l)(f).d^f, and hence d„^ dyf{x,y) ' In the same manner, by repeating the differentiation of any given equation, two or more arbitrary functions may be eliminated. (L. C D. 77 — 9-) Particular Values of the Variable. (14.) Let ?^ = 0, when x=^a, then u must be of the form v{x — a)"\ V being a function of w not containing the factor x—a. d^u = v^ {x — a)"* ~ ^. &c. = &c. n being the greatest integer 1^ m. If m = n, then d"u = v^^, which is the first differential co- efficient that does not vanish, when x = a. If m is not an integer, then d^^^u and all succeeding co- efficients become infinite, when x = a. Let u= - when x = a, then V, (x — a)'" u = v„(x-ay Let 71, r be the greatest integers :|> in, p respectively, then when x = a, if 7i>r, u = 0; n p, u = 0\ "1 jn = p, u= — ; mcr?)'"' + a^C-D.r)'"' + &c. 7< = If m^ > Wj, ?* = a, (D.v)'"^ - "' + ao(Dw)"'^ " "■ + &c. If m^()2 DIFFEltENTIAL CALCULUS. (15.) Let /(,?',;/) = 0, be an equation cleared of irrational quan- tities in wliich y is an implicit function of a-, then (Art. 11.) M-^Nd^y = 0, (1) u, + «, d^^j + u, (d^y)- + Nd^y = 0, (2) and by eliminating d^y we obtain an equation of the form Mn + Nd^y = 0, and generally an equation may be obtained of the form M, + Nd:y = 0. If the values of a; and y which satisfy the equations f(x,y) = 0, M„ = 0, do not satisfy iV=0, then d^y = 0. If the values of x and y which satisfy the equations /Cr,2/) = 0, iV=0, do not satisfy il/^^ = 0, then d"y= oo . If f(x,y) = 0, M=0, iV=0 be satisfied by the same values of 07 and y, then d^y= , the value of which may be obtained from (2) since i\^=0, unless ti^^, ti„^ u.. vanish for the same values of x and ?/, in which case we must differentiate again, and obtain an equation of the form ^1 + «2 d^y + ^3 {d^yf + «4 (d^yf = o ; and so on until we arrive at an equation the coefficients of which do not vanish for the proposed values of x and y. If d"y= - , its values may be similarly found. If for any system of values of the variables, any differential coefficient has more than one value, then that coefficient obtained from the differential equation of the corresponding order must appear under the form - . (Z. C. D. 135 — 7 ; L. D. C 109 — 13.) Maxima and Minima. 203 (16.) Let the values of m corresponding to a + D.i\ a — Dx be 7«i, u _ J respectively ; then if u — u^ and ii — u_^ have the same sign for all values of x from a to a + Dx^ the value of u when that does^anish with any given value of x is of an even order. /\ Let the value of x which makes d"'u = 0, make dj^"^ hi= : u — ti^ and u — u_^ may be thus developed in ascending powers of Dx ., u — u_^ = a^(-Dx)"'^ + a„(-Dx)"'^ + &c. u — w^ = a^(Dx)'"^ + a„{Dxy"i + &c. then if any of the indices m^, m„, &c. have an even denominator, or if 771^ has an odd numerator, there is no maximum or mini- mum. If TOj has an even numerator, ti will be a maximum or minimum, according as a^ is positive or negative. {L.C.D. 154 — 6i.) (I7.) Let u=f(x,y): the values of x and y which make d^u=.0, d^u = 0, give a maximum or minimum value of u, provided d^n, d^d^u, and d^u do not vanish for the same values of x and y, and that d^u.d^u>(d^d^uy. dju, and d^u must evidently have the same sign : and generally there will be no maximum or minimum, unless the first series of differential coefficients which do not vanish for any given values of x and y be of an even order ; and unless the series of vanishing; differential coefficients might be the coefficients of an equation of the same dimensions, which has no possible root. {Gamier, Diff. Calc. Ch. xxv.) INTEGRAL CALCULUS. Fundamental Formulae. (1 . ) f^d^tL=-u-\- const. J'^aio = af^u^ if a is independent of w. / .v'»= + const. '' m + 1 (2.) Inverse circular functions. 2 « .00 - = COS ~ — I- const. w const. y-» a " {a^ — ar) y-> —a ^ {a" — ct?-) y-» a" .2? -^- — 2 = tan - ^ - + / -5 -„ = cot " - + const. .J T a~ -\- 0D~ a /a" OG —7—c ^1 = sec ~ ^ - + const. CO {x~ — w-Y « y—a^ .X —7-5 --t = cosec ~ ^ — h const. ' a;{x^—a-)^ a pa X J rTc ^ overs'' - ^C(mst. p —a CO I Tv; ;^TT = covers ~ ^ — H cows/. DECOMPOSITION OF RATIONAL FRACTIONS. 205 (3.) Logarithmic integrals. r. 1 / _ z=\o^^x -\- const. / {."-±2aa>) l = ^"^^ ^ " ± ^ + ("' ± '"^^'^ ^ + '''^''' yl 1 , /a + cfc'X yl 1 /,!' — a\ — r, = — loojj I I + const. 1 00- — a- 2 a xx + a/ I -7—„ 7;rr = — IoAe ( „~ ', I + const. Decomposition of Rational Fractions. (4.) Every rational fraction may be reduced to the form ^,„cg"' + ^„,-i.3?"^-^ + &c. ^A^x-^A tlJ\ S,,cj?» + 5„_i.r"-^ + &c. ^B^x^-B" \ v) in which m may be considered less than n. Let a^, a„, &c. a„ be the roots of the equation F=0; and [l] let these quantities be possible, and unequal. If ( t^j^, d^^a ^ respectively represent the values of U, d^ V, when x=:a, then Or thus: let F=(.r-— aJQ, and assume U _ K, P y ~ .v-a, "^ Q ' then ^.= (^') ,and/'=i^^£i«. The other partial fractions may be similarly found : U— K^Q being always divisible by .v — a^. 206 INTEGRAL CALCULUS. [2] I.ct the equation V=0 have r equal possible roots. UK, K„ ^ K, P Assume — = -. + 7 '-—Ti + ^^- "I ^ 7T ' 1 ,u 1.2...(r— 1) Q ^ 1 U-K,Q ,, , ,^ /t^\ Or thus: assume = c/j, then a„= ( — - ) ....£.=^ = ,;,„..^3=(§) .... &c. =&c. . . . K^- (-—) • [3] Let the equation F=0 have unequal impossible roots, and let Q not containing the factor or + c^x + e,. U A\ + L,(x + ic,) P Assume — = — ■ + — , y cV -r c-^w + e^ Q and let M + N^^/^l be the value o{2i—{K, + L, (a? + Ici) } Q when oi + (ii-^ — 1 is substituted for x; from the equations M = 0, N=0, K, and L, may be obtamed, and P= ^ V ^ ^ ^^ ^^ . ^ •' oc' + c^x + e, the other partial fractions, which will be of the same form, may be similarly obtained. The values of K, and Li may sometimes be more conve- niently obtained by substituting — — ^=r-.c?^Ffor Q in the 2pi\/ — 1 quantity U— { Ki + Li{x + Jc,) } Q, and proceeding as above. DECOMPOSITION OF RATIONAL FRACTIONS. 207 [4] Let the equation V=0 have r equal impossible roots. Assume — = -^, ^^^ %r^ + -r^ ^-^^ rfzTY + &C. , jr, + L> + ic) , p A- + C.V + e " or -\- ex + e &c. = &c. the constants K^^ Z^, ^"2, Z/^' ^^- ^^^7 ^^ obtained from the quantities U, U^, &c. as in the preceding case. The resolution may in all these cases be effected by the method of indeterminate coefficients, by clearing the equation of fractions, and equating like powers of x ; this method is however usually mucli more laborious. (L. C. D. 380 — 2 ; Hirsch, Int. Tab.) or (5.) Decomposition of the fraction .t?" + 1 rir (r+l)7r rw {r-\-l)'ir cos cos X cos 3 — —cos 3 X oT 2 \ n n n n + — +... -'-\: x"-\-l n [ „ TT ^ „7r I .a?- — 2 cos — A' + 1 0?- — 2 cos 3 — ^ n n TT TT cos(27^^— l)r ^ cos (27?i— 1) (r 4- 1) — .a- n n ... H ■ — — — + &^c- o / ^ ■"■ a?- — 2 cos (2W — 1 ) — ] by assuming v" = ?/, if ^ is a positive integer. It Dn 210 INTEGRAL CALCULUS. [ 1 0] .v""-^ R { .r", (a + 6'a?-")4, [bx" + (a + i-cr-")i] ] ; Assume y = baf + (a + b"x"")i. .v" -^R{ .v""', (a + b,v"y , (rt + b.v"y , &c. | ; Assume (« + fecr") =2/'*" " ^^^/. <:^.uj / a + bx" \ Assume ( ; ) =«'*•• (L. Z>. C. 215 — 21.) (6.) The preceding are particular cases of the following more general forms, in which Ry, R„, &c. denote any rational functions, and 5i~S ^2~S ^c* the corresponding inverse functions. [11] R{w,R-\oo)]; Assume Ri^{ao) = v. [12] R {w, R-\w), R,-'Rr\v), &e. R:'R,:l,...Rr\^v)} ; Assume /2„-'i2„-i, ... 2?r'(*0 =^- [13] R { .V, R,-\-v), R,-'(.v)f, i?rV)|% &c. } ; Assume TJ-'Gr) =v"'"-- [14] d,i2„(,37).{i22(A0, i2f^/22('^)l ; Assume R„(,R,„^v\; this formula may be rendered rational under the same condition as the preceding. [17] R{.v,RC\,v),(l),Rr\a^),(p,Rr\aj),kc.} may be reduced to R }t', (jyM, ^sC^)' ^^] by assuming Rr%T)^v. {Bromhead, Phil, Trans. 1816.) INTEGRATION OF IRRATIONAL FUNCTIONS. 211 INTEGRATION OF x"' " \a + bx")". (7.) Let a + h.v" = X. If- is a positive integer, for .r substitute its vali^ in terms of X obtained from the above equation. If — + jj is a negative integer, assume ax'" + b=Zy and n substitute for x its value in terms of Z. If neither of these conditions is fulfilled, the indices of 0; and X may be reduced by some of the following formulae : from which however no result can be obtained if any of the coeffi- cients become infinite. A result may in these cases be obtamed by substituting in the given formulae for «, h, m, &c. the values which render the coefficients infinite. (m + np) b (m -f 71 p) b'^'' m + np m + np'^ — -^''"^^^' _ (in + n.p+ 1)6 f ,r"' + " - '^ X'' ma ma (p + l)na (p + l)na ^^ yl 1 m + n(p — l) r 1 ,1F^^^ ^ (p-l)nav"'X''-' "^ {p-l)na J .^''^^'X"''' 1 {ni + 7i(p-l)]b p \ 212 INTEGRAL CALCULUS. f^af^ - ^^P = { A^ x"' -"-Ao .t''" - •" + A^a>"' " ^" - &c. + (- iy-'Ai,v"'-'"}XP + ' + (- i)'J,.(m-i7i)«y;a?"'-'"-'^''. 1 , (m — n)a ^ , (m — 27i)a ^ {m + np)b' - {m+7i(p-i)\b ^' ^~ {m+n(p—2)\b "' {7n + n{p — i+l)\b '"^' + Ai{p - i + \)naf^x'" - ^X" " ' : A = ' • J = ^^^" J • J - J^ini)!!^ J . m + np ' m + n (p—l) •* m + w (p — 2) ' (p — i + 2)na m+ (p — i+ l)n + (- iyAi(m + n.p + i) b/^.v"' + '" " 'X' J _ ' . J _ (m + n.p + l)b . . _ {m^n.p+2 )b ■"1 — "~ » •"" — ~7 ; ' — -"1 ' -"■? — 7 ^^ -"2 » wa ' {m + n)a {rn-\-2n)a _ {m + w(p + i— 1)}6 {m + {i—l)n]a = - J J.xY^+i + ^2-^^"^" + A.,X^^^ + + AiX^^'\x''' + J,- (wj + 71 . pTi)/X' ~ '^^ ^ ' : 1 m + ?i ( p + 1) . . . _ ^w + ^^(j> + 2) ^ (p + l)wa ^ (p + 2)wa {p + 3)7ia 7n + 7i{p + i—l) Ai= , .. Ai_^. (p + i)7ia (8.) ax'' + bx'' + '' = X. A;+ 1 m H-jo/c - •^'^' m-\-pk+l Tn+pk + l*^ " Lco'^X^^ — ■ = z — ~ I x"'-"X'\ INTEGRATION OF IRRATIONAL FUNCTIONS. 213 J^x"'~ {m-pk-l)ax"'^''-'^ (jn-pk-\)a ^ ' cc'"-''' ^»« ^,;„_A:-„ + i m — pk — n+\ ,y»<-fc-» J ^ir^^~ {p-\)nhX^-' ^ {p-\)nh J' X"-^ y X ^ = (p-l)naX''-^ (p- l)na ^ ' X"-' ' (9.) a + b.v=X. y-» iv a; a p 'll~mb~ hJ - X f x'" _ x"' ma p w'"-^ J - X~ ~ (m-l)bX ~ {m-l)hJ' T^ ■ y-. 1 _ 1 h p 1 ' x"'X ~ ~ {m—\)ax"'-^ ~ "aJ ' x"'-''X' yi 1 1 7)ih p 1 ' x"'X- ~ ~ {m—\)ax"'-^X~ {m—l)a*y'x"'-^X-' (10.) a + 6,v- = X y^ 1 _ X 2p — 3p\ ' X^ ^ (p-l)2aX^-' "^ {p-l)2aJ - Xp'^ ' ^ -Y ~ (m-l)b ~ hJ - X ^£»_ a-'"-^ {m-\)a px"^-"^ J ' X- ~ {m-3)bX ~ {m-'3)bJ ^ ~F" ■ y» 1 _ 1 h p \ 'afY~ ~ {m — l)ax'"-' ~ a^ - x'"--X' p 1 1 (m + l)b p 1 J^ .t.'»X= (tn—l) ax"*-^X ~~ {vi—\)a'J * x'^-^X^' 214 INTEGUAL CALCULUS. t/* m + 1 m,+ l^y r --X' 1)6 (w + 71+1)6-^^ ^ " m + n+l m + n+ 1^^ ' (11.) ax + bx"=X. ' 2 7n. + ^i + 2 am + ^i + St^'^ ,f"'-' JS'"' (2m + «)a '^'^ (m + w+l)6 (m^-?^^- 1)26«-^^ ^ X^ X^ na p X^^ -1 f^"__ aJT'^ (m-w-2)26 ^ ^' J X af^ ~ (2m — n—2)acv"' . (2m — tj — 2)a«>^^ a?'"~^ y» w"^ _ 2a^'" 2(m — 7iH-2) ^ w'"-^ X^^ (n-2)aX^-' (*^-2)« ♦^'^i-^ INTEGRATION OF IRRATIONAL FUNCTIONS. 215 (12.) a->rhx" + cx^"=:.X. y, A'"'X^ pnb r _.,,„. 2«7ic /> „,,2„_i ,^_i (m + 2pn)c {m + 2pn)c^ ^ {m + n(p-l)]b r ^,n-n-ij^P. (m + 2pti)c ^' = + — i- / x'"-^ x^~ "^-^ / ^"0 ^A'^ ': m-^'ipn 7n + 2pji*^ ^ m+2pn*^ " ^ w-X^^' _ {^-^ 72 (/) + !)} 6 ^ ^,„^„_, ^, */i « m.n. J X _ {m + 292(j> + l)}c ^ ^,„. + .„_i^p (13.) « + 6.2?= + cx^ = X- y» 1 bcx^ + (b" — 2ac)ic (4'p — 5)bc /•_f!_ 'X^^(i3-l)(6--4ac)2aA:^-' "^ (p-l)(6--4ffc)2a»>''A'^~' 2(p — 1)(6- — 4ffc) + 2ac — 6- y. 1 2(p— 1)(6- — 4ac)« "^^ X^~^' V. a?"* 6ca^'"+' + (6"-2ac).t?'" + ^ (4!p — m — 5)bc f f^^ Jx~]p>~ 2a{p-l)iJr-A>ac)X''~^'^ 2(p-l)(6=-4ac)a«/* ^^'^ 2(p— 1)(6= — 4ac) + (w + l)(2ac — 6") y. a?"* "^ 2a(;j-l)(/r-4ac) ~ -^^ ^T^ ' ic^ a- (w — 4p + l)c^^-* (w — 4p + l)c»/' ^^ (m — 2p—l)b p.y^-^ (w — 4p + 1 ) 1 ^ u /• *' ^ 1 _ 1 (m + 2jo — .3)6 y. I Jx g^m-^p — ~ (w_ 1)0.1.'" -i;fP-i (m— i)a «/ ' .r"*-- 4/) — ;>)(■ ^ 1 (m-l)ff J'.T'^-^X'' 216 INTEGRAL CALCULUS. (14.) a + hx + ca)- = X. r J__ 2CcV + b (2p — 3)2c p 1 '^ ' X {m—\)c hJ ' X ~ c^' X ^"^ X^~ (m-3)cX (m-3)c^^ x" ~ {m--3)cJ ^'W ' y^ 1 1 b p 1 e p 1 * a!"'X ~ (m—l)a{v'"-^ ~~ a^^ x'" - ^X ~ a^* af^ " "X ' y"* 1 _ 1 mb p 1 ' .r'"X- ~ ~ 0/i-l)aa?'"-^Y ~ (m-l)aJ^ x'"-''X^ (m+ l)c p 1 (7W— l)a«/^.r''"--X'' •^ * m + 1 W + W* w + 1*'* f'^— 0?'"-^ (m-l)a pd?"-- «^* JT^ ~ (w — 2p + l)cX^-' ~ {m — '" ~ ^ ~ (?» — 2j3 + l)c«/^ JT^ f—-^ -^^ jJ& /• ^y^"' 2pc p X"-^ J X .^'» ~ (ttj — 2j5 — 1) .1'"' - ^ ~ m — 9.p—\J=' x'" pb p x^-^ p \ 1 (m+p — 2)6 >^ 1 y* ai^Xv ~ " (w— 1 )«,«?'" -^X^-i (w— l)a •/' a^'»-ix^ (m + 2p — 3)c p 1 (w-l)a */='w"'--X^' (Hirsch, Int. Tab. ; Z,. C Z>. 391 — 404.) INTEGRATION OF IRRATIONAL FUNCTIONS. 217 (15.) Development of f^ a?'" ~ ^ (a + hx'^y by series. f^a;"'-\a + bx"Y = \m a m + n l.2.a^' m + 2n + ^' p(p-i)(p-2).v"^ ^ ^^ I Im+np b m+n(p—i) 1.2.6" m + n(p—2) j ^«^P + i f_Jll! (m-7i)ax--" {(m + np)b (m+np){m+7i.p-l)b- (m — ii) (m — 2 71) a" x ~^" „ ) + 5^ ^^ ^ , - &c. [ (m + 7ip)(m + 7i.p — l)(m + 7i.p — 2)b > \ma 7n(m + 7i)a- + (ni + 7i.p + l)(m + 7i.p + 2)b- ^o„ _ ^^} m (m + n) (m + 2 7i) a^ j _^^m^P + if__i__ ^ m + n(p + l) ,^ U? +!)**« (7>+ l)(j^ + 2)?i-a- + (^ + M-p + l)(w + n.p + 2) ^ ^ ^^\ (p+l)(p + 2)(p + 3)w3a^ i (m + wp Cm + yiwVm + w.©— 1) ^i> (m + 7ip)(m + »i.p— 1) «(«— l)w-a- „ ■> + . \ / =-^-"- + &c.[ (W + W/>) . . . (W + 7Z . p — 2) J pwfe r" p{p—\)7i-b- (m m(w + w) A' m(w + w)(w + 2w) VjiT/ j , , ( 1 m — 7i X I(p + l)w6 (p + l)(p + 2)7i-6- * a?'* (w — w)(m — 2w) /-^Y ;5-y"-^ (/j + l)(p + 2)(p + 3)n^6^ (Tr. L. 175 — 80; Hirsch, Int. Tab.) Ee 218 INTEGRAL CALCULUS. (16.) Reduction of 5— , • ^ ' {a-^ a^x -\- a„x- -\-a.yV^ -\-a^arp h -\- cv Assume ^r = . then the denominator, which may be 1 +ij represented by X^ becomes (e + e^y + e„y^ + e.,y' + e^y^)^ in which e, e^, &c. are functions of a, Oj, &c. 6, and c. Let the values of h and c be determined from the equations ^1 = 0, 63 = ; then / — — - becomes / -— ~ ^, , or / — — : this again may be separated into Jy Y ^y Y rV'Rniv")^ r R(%) Let y" = z, then / - — —^ becomes / — ^^ — i , which may be integrated by preceding methods. pRiiy") If R^ is an integral function, each term of / — — — may be integrated, or reduced by preceding methods to the form If Ri(y') is a fraction, it may be decomposed, and each separate fraction may be integrated, or reduced to the form / a + a X + a„ar R(x) b + b^x + b„x^ y (2/2 + a) Y may be reduced to the form — — - by multiplying the numerator and denominator of the fraction by the denominator. INTEGRATION OF IRRATIONAL FUNCTIONS. 219 r ^(^) / ;; 5 g-7 may be separated into two parts /. R(x) ^ J X ~~X~ ^^ separating R{oc) into R^{pc-)^ and x.R„{x-), and putting x- = y. / , -„ -r-r may be separated into two parts ^^ ' {a ^- a^x- -\- a„x^)^ ^ r r «/*(« + a^x" + a^'^^)^ ' »^-^ (a + ai*'- + agO;'*)* ' to reduce the first of these, assume (« + a^x" + On A'*)4 = tX'y ; to reduce the second, assume (a + ffjO?'^ + Og"^*)* == * ' then by substituting for x its value, we obtain respectively /^ jR {x) . a + a^.T- 4- 050?* | an d y^ i2 (cv) . a + a^x-+ anX*\ "' may be transformed in the same manner as the preceding formula. To reduce y^, R(x).a + a^x + a„x- + 030?^ j"* assume « + a,a? + aoa?^ + agOJ^f = a* + x%, or = a^^x + %, by either of which assumptions, the term a^x^ will disappear from the radical. To reduce / ; —, TT ,6 _^ ^,«Ca ' separate i2(.r) into R,{x") and a-.iJjOr-), and put y = x-: the part corresponding to R^ix") may be further reduced by assuming 1 y+ - =X. y {LCD. 406 — 11.) 220 INTEGEAL CALCULUS. Let a + a^x'^ + a„o(!^ = (61 + c^ar) (b„ + c^x"), = 61.6.(1 +ei^.r2)(l +e/.a?«). [ij Let the quantities Cj, €„ be very unequal, and e^ > e„; assume e^".v^ = y", then /•, ^|f> becomes ^^Z ^W ^ and since ( l H — ^2/" ) may beexpanded by thebinomial theorem g in a series which converges rapidly, if — is small, the required integral will depend on / ,y . '/y(l-ay-).TTfY [2] If e^ and e„ are nearly equal, put the given function under the form 1 R(.v) "•^M(?-^')(z-=--)F 1 1 . . and assume —^=r^ — k, and — ^ = r" + fc, then by substitutmg , . /. ' R(a;) these values we obtam / — — ; r-^ — -rn •^ * { (V + lV^Y — A;- j 2 /• f ^ ^ A;" 1.3 A;"* | a series which converges rapidly if r^ is positive, and k small : the required integral is thus made to depend on in ' Laffranee's Method. Assume y= —, -i L_^ , then 6^ 62 + CgcV- y- \-^ 4-3 becomes / R(y")-\- f J' „ ==ttj INTEGEATION OF IRRATIONAL FUNCTIONS. 221 where J9"y^ and (fy" are either both positive or both negative, and y=(-{-c,6,)i-{+(c,6,-6,c,)}i, or 7={±(Ci^-^C2)}i-(H:Ci&2); + or — according as c^ and h^ have the same or different signs : and the second value of q must be taken, if c^fcg ^^'^ ^1^2 have different signs. The same transformation may be repeated by assuming yQ i±9 y y'~ = 1 ■ ' %, 2 ' P2 = Pi + iVv" - fi,-)K qi=Px- {Px - qi')^ ; ^ IE 7i y 1 &c. &c. where Q,„ represents { (l ± p^.y^^) (l + q^.y^) }i : or the values of p^,7i, &c. may be thus calculated ; let p -Vq =r, p —q =s, Pi + 9i='>\^ Pi-gi=s,, &c. &c. then 7\ =?• + «, and s^ = 2 (rs)i. &c. &c. by this process we obtain a series of the form /ij(r)+/_i?(yr) + &c. in which, since p '^ q continually increases, Pn'^q„ may be made as large as we please. Similarly by assuming p=p-x + (p^i-q-x)K q=p-,-(pli-qI,)K &EC. &C. or p-i=i(p + q)> q-i = (pq)^* he. &c. 2S3 INTKGRAt CALCULUS, we may obtain a series of the form in which P-n^9-n ™^y ^^ made as small as we please: in either case, the value of the last term may be expressed by a converging series, by the methods already given. (L. C. D. 423—6.) Integration of Transcendental Functions. (18.) Exponential and logarithmic functions. y- = logj {V + const. r .V const. ya" a'=-, + * logj a /a^'.u a^.d,.u a'^.dhi of.u= ^ + '-~ - &c. log,, a (log, a)- (log^a)^ a!-a^=.- Iv-- + -A^ 1- &c.[ * log, a t log, a Uogs ^) ■' - =\og,cc + ^^ + ^: ^^—L + ^^ ^LJ^ + &c. * 00 *= 1.1 1.2.2 1.2.3.3 ' (\m + i (w + 2)- (ot + 3)^ / nx.Xog^x / 1 nx (nxy \ 1 \m + 2 (m + 3)- (m + 4)^ '/ (wcr-.logjcT)^ /I ?Zci? (/ic^)^ \ 1.2 \w + 3 (m + 4)- (m + 5)^ / 4- &c.| « ^^■^ -^ -^'^ r-l •'^H^ + r + s-l)*" INTEGRATION OF TRANSCENDENTAL FUNCTIONS. 223 d,.v / .dog. t) = log. t. ./..-/ -i- / ... f u (loge wY = (logj a?)" r u — n (log. ,r)" " ^ /^ - J^ u yt ti uoc d^u.v ^- (log,.r)« "" ~ (w-l)Gog,.r)"-' ~ (w-l)(n-2)(log,a>)"-- — &c. (w — 1) (it — 2) (?^ — 3) (log. .t')" - ^ a;"'(log,cr)"= { (log, aO" Hog, xY'^ n(7i—l)^ ^ „ ") a ycT" ^, f 1 m + 1 __^,'n+ij i : -(log,A)« ((n-l)(log,a)"-l ' 0i-l)(yi-2)(log,^)"-- + (!!i±l)^ +&cl V (^ _ i)(n _ 2)(w - 3)(log, A')" - 3 j -s+l (m+ 1) •^ * J^ (log. a)« ' ' \n-\ . Gog, a)« - ' {m+ 1) "^ "pr^rr/^ (iog,.r)"-' 224 INTEGRAL CALCULUS. =logr.r + -^=^ + \-^^l. + L_5i_2_ + &c. « 'log, cV ^ 1 1.2.2 1.2.3.3 (L. C. D. 427 — 37 ; Tr. L. 181 — 91 ; Hirsch, Int. Tab.) (19.) Integration of circular functions. J'^ cos x = sin )" " ''"' + ^ + iii£ a?, (^j even), mj— 2 t".2 I ?Z — 1 i(n+l) L = sincr.S,, '"-''-^ (cosj7)"--"'+S (w odd). »;2,— 2 /" — ^^ = - cos 07 . S„, "'^1^=^, (sin x) - <» - -"' + ^>, (w even), ^ ' (sin ci')" bi — 1 i ("-1) 1.^-2 Li = - cos x . S,„ '"-''-' (sin A') - <" - "'" + 1' + ^'"-^''-log, tan i^r, (w odd). */ j; (COSA^)' »H,-2 4 (n-l),2 Ui — 2 in = sin X . S^ "'-1'-- cos a? - <" - g"' + 1), (n even). In— 1 i(«-i. L^iz_2 [3 ^ sin .r . S,„ f^i^ cos ^-("-2"'+i) + Ah;J)£ log, cot (45°-ict7), (w odd) [n-l [^ 2 "1.-2 i(n-l),2 INTEGRATION OF CIRCULAR FUNCTIONS. 227 e" (cos a) " ~ ' (a COS x + n sin .r) /, e"' (cos x) " = — 5^ n r^ ' Jle"' .sinbx = a- + ?i- H ^ TT- I e (cosct)""-. a--\-n- ^^ e'"' (ff sin hx — b cos 6 a?) «^ + b- ' - 6 (a cos bx + b sm bx) /,. e"* . cos bx = s rs ^^ • From these formvilae y^e"^. (sinfe.t?)". (cosca?)" may be obtained, if (sinfev)" and (cos c.x")" be expanded in series of multiple sines and cosines. (22.) f ~ ^ ■* « + cos X 1 1 (b + «)(1 + cos x)T + (b-a)(l-cosx)Y v'^-^^' (b + a)(l + cos x) 1^ - (6 - a)(l -cos o?)]^ 1 6 + acosif + sin.r(/r — a^)2 ; log£ h const, n a b. (a- — b")^ a + b cos .v sin X 1 y^ SXU X 1 ; = — - lo- (;a — l)(f^ff^ — ft/>j) + (/^ — 'i){(ib^ — a^ b) cos x {n—\){a-—b")^' (rt + ftcoscv)""* 228 INTEGRAL CALCULUS. /*,.(! + e.cos.r)'" = Aa; + A^ sin ,v + ^A„ sinSa? + -^^3 sin S.r + &c. m(m—\) 1 „ m...(m — 3) 1.3 . „ 1. 2 2 1... 4 2.4 [m 1 w(m— l)(/« — 2) \.3 „ fm 1 mim — l^ = 2e\-. +~ ^ (12 1.2 3 2.4 + ^^ . e^ + &c. \ 1... 5 2.4.6 J _2me^ — 2J1 _ (m— l)e^i — 2-^g ''~ (m + 2)e ' '"■ (m + 3)e (m-peA-2^A,^ &c. = &c. (m4-4)e If y^ (1 + e.cos .r) -"* = A,v + A^ sin a' + -J ^2 sin 2cX- + &c. and f^ (1 — e . cos a?) " '" " ^ = 5.r + B^ sin a? + i^j sin 2a? + &c., _ 2mA — (m—l)eA, . e , . then B = ^ ^ ' =A+ - d^A, 2A-B , e , ^ B,= =A,+ -d^A,, e w 2(A,-B,)-2eB , e ^ ^ e ~ m ' &c. = &c. = &c. 2k y^ log, (1 + e . cos x) = — X log, — + 2\k. cosa? — ^A;".cos2cr + ^k^- cos3 „ , r ,1 + 1 >± i a?« - sy^ ux" - &c. + (- 1 )" - ^J^ ux'' - ^ J « (25.) Series independent of the integral sign. yi2 X- (1 X ,V- ,„ „ ) u= — {~2i . dAi H . a, w — &c. ( + c, X + c„. 1 (2 1.3 1.2.4 J 1.2 (3 1.4 ' 1.2.5 J 1 - 3 &c. = &C. y->n A'" ri cl' X~ ,, ) \.9....{n—\)\.n l(ri + l) ^ 1.2(«+2) j + c,ct?" - ^ + C2 .r" - - + &c. + c,, ; '^ i/Zt.Aj Cj, ^2 &c. being arbitrary £ifj<[*ftiofis. (L. C Z>. 483 — 5.) Integration of Functions of Several Variables. (26.) If z* is a function of x and «/, then an arbitrary function of y^ which would have disappeared by differentiation, being added instead of an arbitrary constant. \n—\ « /'«^=_i_.S,,(-i)--if:i .i?«-- r ux"^-\ '' r'w= - — .s,„(-i)'"-^— ^: — d:-\iL^'^,o,x-\ J =' n—\ «^^ ■> n + m—l'' DIFFERENTIAL EaUATIOXS. 231 li d^dyU = v, then d^u=J'j^v +f(x), and Similarly if J^f/^c?.7* = wT then ^^ ^ =fJJ> + 0iC^') + eing the integrals obtained from the equations (l) respectively, u = u^-\-f^{P—d.u^, = '^h +f. {M- d.'ii,). (L. D. C. 282 — G.) (29.) Separation of the variables. All equations of the form X+Yd^y = 0, in which JC and Y are functions of cV and y respectively, are immediately integrable ; the integral is The equations Y+ Xd^y = 0, and XiY+XY^d^y = may be reduced to the above form by dividing them by JlY. (30.) Homogeneous eqtiations. Any homogeneous equation between two variables, each term of which is of m dimensions may be rendered integrable by dividing every term by <;r'", and assuming y = zA\ then and the given equation may be reduced to Zi + Z„(z + £od^z) = 0, in which the variables may readily be separated. The equation «j + b^.v + c^y + (^a^ + bn^v + c^y)dj.y = may be rendered homogeneous by assuming a^+b^x + c^y=^u, a„ + bo.v+jlj^ = v, whence d.y = — ; . Ci'd^v—c„ (31.) The linear equation, d^y + JC^y + X„ = 0. This equation may be rendered integrable by multiplying every term by e-^ ^^ : the required integral is y = €~^''^^{ f^X^e-^""^' + const. ] DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 233 (32.) RiccaWs equation ; d^y + ay^ + fea?"* = 0. If m = 0, the variables are immediately separable : otherwise assume y= — H , then x ax , _d^z 2« I ' or x^ ax" substituting these values for y and d^y respectively, we obtain x-d^z -\- az" + 6,j?'"+* = 0. This equation is homogeneous, if w= — 2 ; and if m-= —4, the variables are immediately separable : otherwise assume 1 Vi i?»» + 3^.rj, and by substituting these values, the equation will be reduced to its original form, and the same process may be repeated. — 4?^ The given equation is integrable if m c^ , the value of n being 0, or any positive integer. THE INTRODUCTION OF A FACTOR WHICH RENDERS A DIFFERENTIAL EQUATION INTEGRABLE. (33.) Let M + Ndj.y=0 be an equation which does not fulfil the condition d^M=d,N, and let V^=c be the primitive equation, and d V d V u u then the given equation may be rendered integrable by intro- ducing the factor u . (w) : in the simplest case (p{u) =■ 1 . To determine u we have u[d^M-d^N\=Nd^u-Md^u, this however usually presents greater difficulties than the given equation. Gg 234 INTEGRAL CALCDLUS. If M is a function of a? only, then rfj,t< = 0, and dii 1 hence w = e-^x-^. If u is a function of y only, it may be determined in a similar manner. If M and iV^ are homogeneous functions, then 1 Mx + Ny (34.) The required factor may sometimes be conveniently found by transforming the equation M+Nd^.y = into ir+ Ldfti = 0, in which the relations between x, y, #, and u, are known, and K L finding a factor V such that — may be a function of t, and — of u : then if the values of t and ii, in terms of x and y, be substituted in F, the quantity obtained will render the given equation integrable. The equation P + QdtU = may be rendered integrable by *^ ^"""^ PT-QU '^ V being any function of t and ti, and T and C7, of ^ and w, respectively. The equation ^i2/ + (y + Xz)^^^ = ^ ^^Y ^^ rendered inteffrable by the factor — -^ . ^ ^ y^ + Xy-+{X-a + b{2a-Xy-]ay The equation X^y + (X„y + X^)d^y = may be rendered integrable by the factor if ^ ^ (1 + XyY SINGULAR SOLUTIONS. 235 X^ = hX'\d,X+{n-\)aX"-''.d,X, X^ = {m — n-\-2) aX'\ ^3 = 6X'" + ^ + {m^\)aX"-\ (L. D. C. 316 — 23 ; L. C D. 567 — 80.) Singular Solutions. (35.) Let f(,v,y,c) = 0, c being an arbitrary constant, and let (p(.v,y,y',c) = 0, in which y' represents d^y, be derived from the former by differentiation ; from these equations, another of the form "^(x^y^y) = may be obtained by the elimination of c. If u is such a function of x and y, that (f>(-^',y,y\u) = 0, may be derived by differentiation from f(x,y,u) = 0, then \l/(w,y,y') = will result from the elimination of Uy and f{x,y,n) = may be considered as the primitive equation. Such values of f(x,y,u) = as are not included in the general form fi^v^yiC) = are singular solutions. The equation d^f(.v,y,u) = must be satisfied by the values of u which give singular solutions. All singular solutions will be found amongst the equations obtained by substituting in f(x,y,c) = those variable values of c which satisfy the equation dj{x,y,c) = 0. The values of c which give singular solutions satisfy the equations d^.c= oc , f/^c= =c ; they also render d^y= - . If/(.r,y,c)=0, the general solution of a differential equation, represents a class of curves, each of which is determined by assigning a particular value to the arbitrary constant r, then the singular solution, which is independent of c, represents the bounding or circumscribing curve. 236 INTEGRAL CALCULUS. If a given differential equation f(^,y,y) = can be solved for y\ then a singular solution may be obtained by substituting for y a value which satisfies the equations {L. D. C. 324 — 41.) For singular solutions of differential equations of higher orders, see L. C. D. 638 — 58. (36.) Equations of more than one dimension in y'. If a constant of n dimensions be eliminated between any equation fipc,y) = and its derived equation f\x,y) =■ 0, an equation may be obtained of the form y" + ««- 12/'"~' + &c. + a,y' + = 0: let the roots of this be p^, p„, &c. and let the equations y' — p^ = 0, y' — p„ = 0, &C. be integrated separately ; then the primitive equation will be the product of the integrals thus obtained. If we have an equation f(,x,y,y') = 0, homogeneous with regard to x and y, then by assuming y = ux, we obtain a; = € ^ ", y = U€ *"", whence we may frequently obtain an integrable equation be- tween y and u. (37-) Clairaulfs formula : y = y'^+f(y')- By differentiating the given equation we obtain l^ + d,Ay')}d!y = 0, whence d^y = 0, and .-. y' = c, by substituting which we have y = ca^+f(c). This is the general solution : a singular solution may be obtained by eliminating y' between the original equation, and o^ + d,J(y) = 0. (L. D. C. 342 — 7 ; L. C D. 582 — (^.) DiFFEBENTIAL EQUATIONS OF THE SECOND OrDER. 237 (38.) Let the equation be f(d^y^x) = 0. If this equation can be solved for d^y, another may be obtained of the form d^y = X ; whence d^y =f^ X + c,, and y =f^ X + c^x + c„. Let the equation be f(d^y,y} = 0. If this can be solved for d^y, we obtain d!y=V; whence, since d^y = ydyy', (d^yy = 2fyY+c, and x=Jl(2fyV+c)-^ + c,. Let the equation be /(d^y^d^y) = : this being solved for d^y gives d^y = (p(y), or d^y' = + (d,«)^}; &c. by substituting these values of u and its differential coefficients, and dividing by e", we obtain d:u + B,d/-hi + kc. + B^_,d^u + A„ = 0; (2) where B„_, = (d^vy-' + A,{d^uy-- + kc. + J„_,. This equation is seldom integrable ; if however A^^^ An, &c. the coefficients of (l) are constant, (2) will be satisfied by a constant value of d^ti; in which case since d^ii=^0, d^u = 0, &c. (d^uy + A,(d,uy-' + A,(d^uy-"- + ...A,^_,d^u + A,=0. let the roots of this equation be a^ a„ &c., then from dj. u = a^ we obtain y = Cj^ €"> ', .... d^u = a„ y = C2€°''i', &c. &c. and y = C,€"''+a6«-' + &c. + C„€"-' is the complete integral required. DIFFERENTIAL EQUATIONS OF THE n ORDER. 241 If two roots, as a,, aj, ^ "T + 6^y — 1, then the correspond- ing part of tlie vab^e of y \i ay=-a„=hc. =^aj,i the corresponding part of the value oiy fcj=j 6«.'{ir, + ir.o? + ... + iT^.T?"-^}. If there are two equal impossible roots, the corresponding part of the value of y b}=>6«''[(/'i + /'„-r)cos6a? + {G^ + G,.x)smhx]. (44.) Solution of the equation d:y + Jirf;-'y + ... + A,^_,d^y + A„y + X=0. (l) [l] Elder's solution. Let e"*' be the factor which will render this equation integrable, and let and f,e-'''\d:y + A, d^-'y + ... + A,^_^d^y + A„y\ = e-''^{dr'y + B,dr"-y+...+B„_,y}; by differentiating which, and equating coefficients, B^, &c. may be determined in terms of A^, &c. and k will have n values determined by the equation k'^ + A,k"-' + ... + A„_,k + A,^ = o, the equation (l) will therefore have n immediate integrals of the form dr'y + B,dr'y+.:+B„_,y = e''{X, + K], the value of X^^ depending on that of k : from these n equations the n—l differential coefficients may be eliminated, and the result will be the primitive equation. [2] Lagroiige'^s solution. Let the equation y = Cje"'' +C26*«' + &c. = (2) be the integral of d:y + A,d:-'y + ... + A„_,d,y + A„y = Oi Hh 242 INTEGRAL CALCULUS. it is required to deteniiine into what functions of x, Cj, c„, &c. must be changed, that (2) may he the complete integral of (l) : in order to render the investigation more intelligible, it may be applied to an equation of the third order, d^y + J,d^y + J,d^y + J^y + X=0. (3) Let y = X,y^ + X„y,, + X^y^ be the required integral, in which ^/u 2/2' Vz^ represent t"", e'^', e^ai' respectively. By differentiating this equation we obtain d,y = Xx-d^yi+X„.d^y., + X^.d^y^^y^.d^X^-Vy.,.d^X„ + y.,.d^X^; assume y^.d,X^+yo-d^X2 + y^.d^X^ = 0, (1) then by differentiating again, we obtain d;y = X,.d!y,-\-X._.d^y„_ + X^.d^y, + d^y^.d.X, + d,y„.d^Xo + djj,.d,X^ ; assume d^^/i • ^z ^1 + ^^2/2 • 4 X + d^y^ . d^ Zg = 0, (5) and by repeating the differentiation, d!y = X\ . rf iVi + X, . rff tj., + X, . d,'y, + d^y, .d,X, + d, 2/n . d, X. + d^y, . rf, X, ; substituting these values of d^y, d^y, d^y in (l), we have d!y,.d^X,+d!y,.d,.X, + d^y,.d,X, + X = 0, (6) since either of the three values of y will satisfy the equation from (4), (5), (6), Xi, Xj, and JT, may be determined by elimination, and thus the primitive equation may be obtained. If ?/3 is unknown then (5) is inadmissible; X^, Xn, X^ may however be determined l^y the solution of a differential equation of the first order: if y„, and y^ are both unknown, (4) is likewise inadmissible ; the values of X^, Xn, X3 may however be found ^ by means of a differential equation of the second order. The same method of investigation is applicable to the equation of the 01^^ order : and if only p particvilar values of y can be found, the integration of a differential equation, of the {n — jiy^ order must be effected in order to determine the quantities Xj, Xj, &c. DIFFERENTIAL EQUATIONS OF THE 7i"* ORDER. 243 (45.) The complete integral of the equation (l) is («i — «...)(«i — «3)---(«i — ««) (a. — ai)(ao-a3)...(a2-a„) + &C. + - \ n Jx J («« — «i)(a« — "c) . . . (a« — -a„ - i) If two of the quantities ap Oo, &c. 14s a + 6 ^ — 1, the two corresponding terms in the values of y become Sc^'^C^ cos ba! + B sin bx){Ci — cos bx/^ Xe ~ "") A- + B- 26"' (^ sin b.v — B cos b.v)(c2 — sin fcj^y^ Xe " "•'") + A -\- B ^ — I, and A — B >^ — \ being the denominators of the fractions. If two of the above quantities are equal, the corresponding terms become ~~^ ^T""^ ~ "' ^. ^"' Vx^^-"'^; where £0 = («i — «3)(ai — "J • • • («i — «„)• If Oi = aj = 03, then the corresponding value of y becomes ■^3 where £3 = (a^ - aJCaj - 05) . . . (a^ - a„). (£. Z). C. 370 — 8 ; /.. C. D. 603 — H.) 244 integral calculus. Simultaneous Equations. (46.) Let there be three equations, flj u + a^o) + a^y + a^d,u + a^ d^x + a^ d^y = 0, b^u + h^x + h^y + b^d^u + 65 d^x + b^ d^y = 0, (l) c^u + c^x + c^y + c^dfU + Cj d^x + Cg dfy = 0, in which the coefficients a^, a„,...b^, b„, &c. are constants. Assume x = mu, y=:^nu, then in order that the equations (l) may subsist, a^-'t a^ni-\- a^n b^+b^m + b^n Cj^ + c„m-i-c^n O4 + a^m + a^n b^ + b^m + b^n c^ + c^m + c^n Let a,-^-a,m + a,n^^^ then an equation of three dimensions may be obtained for determining k; let the roots of this equation be k^, k„, k^, and let W7p m„, 7W3, and n^, n„^ n^, be the corresponding values of Qiiy and n, and C^, C„, C^ arbitrary constants ; then u = C^e-^'' + C.e-"^' + C,e-\\ y^C.n^e-''^' + C^n^e-^' + C^n^e'"*'. The equations a^u + a„x + a^y + (04 + a-J^ia^dtU + a^d^x + a^dfy) = 0, fe^w + 62-2? + ^zV + (&4 + b.t)(bQdfti + ftyrfjO? + &8^<2/) = 0> Cj2^ + c^a? + c^y + (C4 + c^t)(cgdfU + Cyd^x + c^dfy) = 0, may be reduced to the same form as (l) by assuming S^tSd^^ a + bt = bs, and^^fe^ If any of the quantities k^, k„, k^ are equal, or impossible, the changes which take place in the values of u, x, and y, are analogous to those in the preceding article. SIMULTANEOUS EQUATIONS. 245 (47) The equations of the second order a^x + ar.y + a^d^x + a^d^y + a^df.v + ffgrff ?/ = 0, b^x + 622/ + ^3^^^ + ^4^ty + h^t^ + ^6^/2/ = 0> may be reduced to the equations of the first order 6^0? + h.,y -T b.^p + h^q + hr^d^p + fegd,*/ = 0, by assuming dfX = p, dfy = q. (3) Let y = lx^ p = mx, q = nx, then in order that the equations (2) and (3) may subsist we must have a,+a^l + a^m-ra,n b. + b„l + h^m + b.n n — z i 1 = _i z 1 1_ = 7W = — . a^m -\- a^n b-m + b^n I from these equations we may obtain y = C^y^ + Coy^ + C^y^ + 04^4, a?j, a?2, 2/i5 2/2? Sec. being the particular values of x and y, corresponding to the roots of an equation of four dimensions. The solution of the equations OjW + cLnX + a^y + a^dfU + a^d^x + a^d^y + T^ = 0, b^u + 60.^' + 632/ + b^dtU + ^sf/t-^ + b^d^y + T^ = 0, CjW + Co-a; + c^y + 04^^?^ 4- Cgrf^.i? + c^d^y + T'g = ; a^.t? + a^y + 03^,0? + a^d^y + Ojrff cT + fle^^f 2/ + "^^i = 0> fcjcr + 622/ + ^A«' + ^4<^(y + ^5<^'^^ + K^h + r, = ; may be determined from the solution of the same equations wanting the last term by a method analogous to that of Lagrange (Art. 44..) : the same methods of solution may be applied to a greater number of equations involving a greater number of independent variables. (/.. C. D. 615—9.) 246 INTKGRAL CALCULUS. (48.) D'Alemberfs Method. r,d,.v + T„d,y + T^x + T,y + T, = 0, (l) t, d,x + t„d,y + t.yv + t^y ^t, = 0, (2) are equations of the first degree between three variables in their most general form, in which t is the independent variable, and Tp T„...#i, ^o, &c. functions of/. INIultiplying (2) by an arbitrary function of t, we have + Tr, + %e = 0: (3) this equation is linear with respect to in which case from (4) we obtain from which equations 6 may be eliminated, and the resulting equation between T^, t^, &c. Avill determine the condition on which the integrability of (l), (2) depends. The equations (l), (2) may be thus integrated : by alternately eliminating d^x and d^y, we obtain equations of the form d.o! + U^x + U„y + 17^ = 0, (a) d,y+V,x+V,y+V, = 0; (b) putting u for x + yO in (a) + (6) x 0, we have d,u + u(U, + V,e)-y{d,9 + {U, + ¥,6)0 -{U, + V„6) ] + (73+ ^30 = 0. Let be determined by the equation SIMULTANEOUS EtiUATIONS. 247 and let the value thus found be substituted in d,u + u{U,+ V,e)^U^+ r,0 = 0, (c) which may be integrated by preceding methods. If f/j, Fj, are constant, then c/, = 0, and in this case (c) becomes d^u + Au + B = 0; by integrating which we obtain the primitive equations Op &p «„, &2> being the values of A and B when Oj, a.^ are substituted for 0. If we have two equations each containing three variables, we may by elimination obtain three equations of the form dfX + M^x + N^y -V P^% -^ Q^ = 0, (o) d,y + M,^x + N„y + PnZ -\- Q„ = 0, (b) d,z + M. a; + N,y + P^z+Q, = 0, (c) in (a) -f- (6) X 6*^ + (c) X Oo = put zt = .v + 6iy + 9.z, then by making the coefficients of y and z respectively equal to nothing we obtain d,e, + (M, + M„.e, + M,.e.:)9, - (n, + N,.e, + N,.e,)=o, d,e, + (M, + M„.e, + M.,.e.;)e.-{P^ + Po.e, + p,.e,)=o, whence 9^ and 9o may be determined ; substituting their values in d.zi + (J/^ + M„.9, + J/3.0,)w + Q, + Q„.9, + Q.,.9. = o, u, and thence the primitive equations, may be obtained. If the coefficients J/j, J/.^, &c. are constant, the primitive equations may be determined as in the preceding instance: the same method of investigation may be applied to a greater number of equations, and to an equation of any order, in which none of the differential coefficients exceed the first degree. {L. D. C. 387 — ,01 ; L. C. D. (V22 — t ; Tr. L. 286, 7.) 248 INTEGRAL CALCULUS. Appboximate Integration of Differential EauATioNs. (49.) Assume the primitive equation to be yz=a^,v"i + a„x"-> + a^x''i + &c. the exponents tIj, Wj, &c. being an ascending order, then d^y = 7ij a^x"^ ~ ' + n„a„x"^~^ + &c. d^y =. n^ {n^ — 1) «^a"i ~~ + n^, {7i„ — 1) a^a:"'^ ~^ + &c. &c. = &c. By substituting these values of y, d^y, &c. in the proposed differential equation, and equating corresponding indices and coefficients, we may obtain as many terms of the series as we please : the law both of the indices and coefficients is frequently evident from two or three terms. {L. D. C. 392, 3 : L.C. D. 660—7.) (50.) Approximation by a continued fraction. Assume the primitive equation to be a^x"^ a^x", a^x"3 ^ ~ 1 +/ 1+/ 1 + &C. ' and substitute a-^x''^^ w^a^.r"'"^, &c. for y,d^y, &c. respectively in the given differential equation ; then by a comparison of corresponding terms, a^ and ji^ may be determined. Then assume y= , ^ 1 + a2x"^ and for y, d^y, &c. substitute their values obtained from this equation, by which means o.j and ^Zg maybe determined; the same process may be continued as far as we please. (L. D. C. 394, 5 ; L.C. D. 668 — 71.) (51.) If we have an equation of the form d:'y=f(^v, y, d^y, &c. dr'y), then x—a (x—a)- , (x—a)"'^ y = {y).=a + d.^ay- -7- + d^^y 5—-^ + &c. +d;-^ ^ 1.2 '-" 1.2...(7Z— 1) ,„-!,. (^'-«r + f(a, (2/)_, d^^y, &c. dr» ^V^ + ^^^- This series may always be made covergent by taking a value of x sufficiently near to a. (L. C D. 659.) COMPARISON OF ELLIPTIC TRANSCENDENTS. 249 (52.) Lagrange s Method. Let y = (li{xy c^, c^, &c. c„), in which c^, c.^, &c. arc arbitrary constants, be the complete integral of d'^y + m = o : in order to extend this result to the equation in which u and v are functions of .r, y, d^y, &c. d"~^y, the quantities c^ c^, &c. c,,, must be considered variable ; ii equa- tions may in this case be obtained, of the form «^c, y ' d^c^ + d.j.d^c, + &c. + d^j.d^c„ = 0, dc, d^y-d^'^i + dc, d^y.d^c.., + &c. + d^J^y.d^c„ = 0, &c. &c. dc, dr'y.d^c, + d,^d:-'y.d^c, + &'c. + d,^ d^'y.d^c,, = av. From these equations the values of rfj.c^, d^Cr., &c. d^c„ may be obtained cfr; c/_^c = aT.i'', (l) Tj, 7^0, &c. being functions of cV, c^, Co, &c. which being constant when a = 0, may be expanded in the form c^ = Ci + a^^i + orz{ + &c. &c. = &c. these values being substituted in the equations (l), an ap- proximation may be obtained by neglecting all powers of a superior to the first. (L. C. i). Gji, 5; Lagrange, Berlin Mem. an. 1781.) Comparison of Elliptic Transcendents. (53.) Let the given equation be {a + a^x + a^d'-\-ayV^ + a^a.^)~^ + (a + ffi^ + a^y- + a^y^ + a^y^) -id^y = 0. Suppose lV and y to be functions of a new variable ty such that dfX = -]- (a + a^x + a„x" + a^oj"^ + a^x*)^, d,y= -(a + a,y + a^y" + ay + ay)i; Ii 250 INTEGRAL CALCULUS. differentiating, and adding the results, we obtain d'tP = ^1 + «cP + I «.!(/>' + 7') + ^",P (p- + 3q"), (1) 2 Cp + 7) ^"•^^ 2 (P ~ 7) ^^^"g substituted for a; and y respectively ; also d,p.d,q = fi^q + a,,pq + ia^qiSp" + q") + ^a^pqip" + q') : (2) (1) X q— (2) gives by integration a + a^ ,r + a„.v^ + a.^x'^ + a^.v^ \ — a + a^y -\- a„y- + a^y^ + a^y^ \ ■ — ■ 1 J = {^' -y).c- + a.,{.v + y) + a, (x + yy\. in which c is an arbitrary constant. Another integral may be obtained in a similar manner, by substituting in (2) the value of d^p, whence d,q = «i + a„P + 1^3 (3p- + q") + \a^p (p" + q") (c- + a^p + a^p-)^ If the quantity a + a^x + a^/v- + a^x^ + a^x* is a perfect square, no result will be obtained ; but by assuming ^1 .i a + a^w + a„oe" + ^30?^ + a^x'^ = (6 + ca? + ex'Y ■\-k\ , and in the development, neglecting the powers of k superior to the first, the above integral may be reduced to that of the equation ^ '^— — -„ = 0. a-\-hx -\- ex" a + by + cy" (54.) Let the given equation be - J ±^ = 0. { 1 — (e sin lt)" } 3 {1 — (esin^)-}3 Assume rf^a? = { 1 — (e sin x)" } ^ d^y = { 1^— (e sin y)" ] 2, then by a method analogous to the preceding we obtain 1 — (e sin x)" p — 1 — (e sin y)" y = c^ sin {x — y), or 1 — (e sin xf |^ + l — (e sin yfY = c^ sin (a? + y) ; either of which may be the required integral ; or if ^ represents the value of x when y = 0, the integral assumes the form cos X . cos y — sinx. sin y . 1 — (esin^)-|'' = cos %. (L. CD. 690— 711.) TOTAL DIFFERENTIAL EQUATIONS OF SEVERAL VARIABLES. 251 (55.) The most general form of a total differential equation containing; three variables is M+Nd^y + Pd^z = 0: (i) if this equation answers the criterion of integrability (Art. 28.) its integral may be obtained under the form (p{w,y,z) = c. If (l) is not immediately integrable, but may be rendered so by the introduction of a factor Q, then P{dyM- d^N) - N{d.M- d^P) + M{d,N- d^P) = 0. If this equation is satisfied, the given equation may be thus integrated: let (7=0 be the integral of Q3f + QNd^y = 0, then [J+Z = will be the complete integral. Z may be found by differentiating and comparing the result with the given equation. If in the proposed equation the quantities d^y, d^x exceed the first degree, these methods are applicable only when it may be resolved into factors of the form (l). {L. D. C. 414 — 7 ; L. C. D. lis — 21.) If the equation (l) is not integrable by the above method, a solution may be thus obtained : let Q be the factor which renders the equation J/ + A^rf^y integrable, and let the integral be U=c, ;^ being considered constant. By differentiating this last equation considering cV,y,z, and c, all variable, and com- paring the result with (l) x Q, we obtain the equation dM-QP = d,c: putting c = 0(;»), and d,c = 0'(^)> we have d.^U-QP = (p'(z) which system of equations will satisfy the proposed equation (l) whatever be the form of the function (p. If the equation contain four variables, a solution may be obtained by an analogous method. (L. D. C. 418 21 ; /.. C. D. 808 13.) 252 INTEGRAL CALCULUS. Partial Differential Equations of the First Order. (56.) The simplest class of partial differential equations con- sists of those which contain only one differential coefficient of the first order : let the proposed equation be then z =y; f(.v,y) +(p(y). Let the equation be d^x=f(,v,y,!s)y and let zi^=c be the integral of this equation, y being considered constant ; then the required integral is u = (p(y). If any other variables are contained iny, they must likewise be included in the function (p. (57) The most general equation containing two partial dif- ferential coefficients of the first order is Pd^z + QdyZ + R = 0. Since d^ (z) * = d^x + d^z. d^y, (l ) by eliminating dj.z, we obtain (Pd^.y-Q)d^z = Pd,(z) + R. If P and Q contain .v and y only, and P and i?, a* and z only, then the equations Pd^y—Q = 0, Pd^z + R = 0, may respectively be rendered integrable by the factors M, N; let the integrals obtained from these equations be ^7=0, r=o, Nd z then if we assume — -j- =(p'(U), the required integral will be V=cp(U). If the two equations U=a, F=&, may be obtained by integrating conjointly any two of the equations * For an explanation of this notation, see Tr. L. Art. 12(j. PARTIAL DIFFERENTIAL EftUATIONS OF THE FIRST ORDER. 253 Pd^y-Q = 0, Pd^Z+R = 0, Qd^Z + R = 0, (2) containing .r, y, and x, then V=(p{U) will be tlie required integral. When P, Q, and R are functions of all three variables, the integration of (2) cannot generally be effected ; an integral may however sometimes be obtained by means of the following formulae derived from (l) ; z = xd^z +/^ {d^z - xdJyZ), (58.) The integration of partial differential equations of the first order may frequently be effected by the introduction of an indeterminate quantity : let the proposed equation be reduced to the form Md^%,^) =f.(d^z,y) ; assume f, (d^z,x) = ti =f„d^z,y), whence dj.z=(p^{uyv), dyZ = cl)oiu,y)' let X^i ("'^^0 = ^' ancl/y 02 0^>!/) = Q' from these equations we may obtain ^ + >/,(m) = P+Q, d„v/,(w) = 4P + rf„Q; from which equations the integral of the proposed equation may be obtained : a similar method of introducing an arbitrary quan- tity may be applied to some other equations. (59.) If in the equation Pd^z + Qd^z + i? = 0, P, Q, and R are homogeneous functions of .r, y, and z, and of n dimensions, assume ^v = uz, y=.vz, and let P = P,z", Q = Q^z", R = R,z'\ then from (2) we obtain (P, + uR,)d„v-iQ, + vR,)=0, which may be integrated, since Pp Q^ and R^ contain only 264 INTEGRAL CALCULUS. u and V : eliminating u or v from tlie equation thus found, and one of the equations (Q, + R,v)d^z + R,z = 0, the resulting equation may be integrated, and the values of u and V being substituted, the required integral will be obtained. (60.) If we have an equation of the form by substituting this value of d^z in the equation we obtain d^ (z) =f(d^ %,x,y,z) + d^z.d^y. Let u be such a function of d^z as, being considered constant, will render this equation integrable, and let the integral be (p{u,x,y,z) 4-x/,(zf) = o, the function x|r being such that du(piu,a!,y,z) + d,^\l/(u) = : the primitive equation may be obtained by eliminating u from these equations. (L. D. C. 422 — S5 ; L. C D. 728 — 34 ; Tr. L. 311 — 5.) (61.) A method analogous to that of (Art. 57.) may be applied to equations containing more than three variables : let the proposed equation be Nd^z + Pd^z + QdyZ + i? = 0, then since d^ {z) = d^^z + d^z .d^x + d^z . d^y, by eliminating d^^z we obtain Nd,{z) + R = d^z(Nd,.v-P) + d^z(Nd^y-Q). If the equations Nd^z + R = 0, Nd,^x-P = 0, Nd^y-Q = o, contain only u and z^ u and x, u and y respectively, they may PARTIAL DIFFEKENTIAI^ EQUATIONS OF SUPERIOR ORDERS. 255 be rendered integrable by the factors M^, M„t M.^: let the integrals obtained from these be F=0, T = 0, U=0, then if the required integral will be V=cp(T) + yp(U). {L. CD. 735.) For the investigation of partial differential equations in which the diiferential coefficients exceed the first degree, see (Z. C. D. 740—9.) Partial Differential Equations of Superior Orders. (62.) Differential equations of the form f(a;,y, d^z, dj^z, &c. d;'d;^) = may, by assuming d"z = u, be reduced to f{cc,y,u, d^u, &c. d^'u) = 0. All equations of this form, in which the differential co- efficients involve only two variables may be considered as equations containing only two variables, and integrated by preceding methods: arbitrary functions of all other variables contained in the function / must however be substituted for arbitrary constants in the integrals thus obtained. The following are some of the simplest forms belonging to this class of equations. [1] d^z+f{.v,y) = 0; ^ +f'f(^^y) + '^' • (p(y) + ^(y) = o- [2] d!z + d,z.f(.v.y) = 0; ^ ^jy.Ar.y)+^i^) + v|,(j/) =0. [3] d!z+fiw,y)d^z+f,ix,y)=0; z+f^e-y^^^-^'{fy^^-''^Ma',y) + (p(y)]+f{y) = 0. [4] dj,,z +fia:,y) d^z +f,{^v,y) = ; 266 INTEGRAL CALCULUS. (63.) The most general partial differential equation of the second order is Rd^z + SdJ,^x + Td^z + r= ; (l) in wliich li, S, &c. are functions of d^z, d^z, x, y, and z. let d^z, dyZ, d'^z, d^dj,^} d'^z; be represented by p, 7, r, *, f, respectively ; then (1) becomes Rr + Ss + Tt + V = 0; also d^(p) = r + sd^y, eliminating r and t between these three equations, and putting d^y = m, we obtain Rmd^(p) + Td,(q) + Vm = s {Rjn- Sm+ T] which must be satisfied independently of s; let M=a, N=b be two equations which satisfy Rmd^p+Td^q+ Vm = 0, (2) Ri7i"-Sm+T = 0; (3) then N=(p{M) is the required integral. If R, S, and T, are constant, the values of M are constant, let them be a^, a„ ; from (2), (3) we obtain Ra^p +Tq + a,f^ V=f(y - a,.v), Ra,p +Tq + a,f^ V=f(y - a„.v) ; eliminating p between either of these equations and and integrating the result, we obtain an equation of the form Rz=/J'V+(j)(y-a,.v) + yl.{y-a,w). In determining y^- F, a^^x + b^ must be substituted for y in F, and y — a^x for b^ after the first integration; then for y substitute a„x-\-b„, and y — a„x for b^ after the second inte- gration. The equation to vibrating chords is r — c"t = 0, in this case a^ = c, a„= —c, -ff = 1, and f^ V= ; and z = (p(y — c.v) + \|/(y + c.v). PARTIAL DIFFEREKTIAL EQUATIONS OF SUPERIOR ORDERS. 257 (64.) The introduction of an arbitrary function is in some cases applicable to equations of the second order ; suppose we have the general equation to developable surfaces, rt-s' = 0; (L. CD. 339.) . r s let - =u= -, then r = su, and s = tu, s t whence r + sd^y = u{s + td^y), or d^ip) = U'd^(q) '■> the inte- gral of this equation is p = (p{q)i and therefore by the integration of which we olitain z = a,(p{q)-\- qy+f(q), also = x(p\q) + y +f{q) : from which equations the value of z may be obtained. (L.D. C. 436 — 46; Tr. L. 3l6 — 20.) (65.) In the equation Rr + Ss+ Tt+Pp + Qq + N% = M let the coefficients R, S, &c. be functions of .r and y only. Putting d^y = m, as in Art. (63.) we obtain d^y-m = 0, Rmd^(p) + Td^iq) + Pp + Qq + Nx-m = : the value of y in terms of a? obtained from the first of these equations being substituted in the second will reduce it to an equation containing four variables, x, z, p, and 4- Ed'^z= V, in which A, B, &c. are functions of oc, y, x, and the differential coefficients of the first and second order. By eliminating rf^ ^j Sec between this equation and dA'>')=d!^ + d!d^%.d,y, ^xi^) = d^dy^ + d^d^^.d^y, and putting d^y=zm, we obtain d^y-'m = 0, Ad^{r) + {B-Am)dXs) + (C-Bm + Am'^)d^(t) - V=0, in which the value of m depends on the equation Am^ — Bm'^ + Cm — E = 0: If from these we can obtain two primitive equations M=a, N=b, then N=(p{M) will be one of the required first integrals. When A, B, C, and E, are constant, and V a function of a? and y only, then the three first integrals of the proposed equation will be Ar + {B-Am,)s + (C - Bm,+Amf)t-J\.V, = (p{y-m,.v), Ar + (B-Am2)s + {C- Bm, + Am',)t-/^ ^2 = x(2/-^"2^^)» Ar + {B-A m,) s-\-iC-Bm, + Ami) t -f^ V, = f(y- m^ x) ; If F=0, the primitive equation will be % = (j){y- m^w) + x(y- 'rn^x) + y\f{y-m^x). If two of the roots, as m^ and m^ are equal, the correspond- ing part of the value of % will be (p{y — m^w) + xy^{y — m^a;). PARTIAL DIFFERENTIAL EQUATIONS OF SUPERIOR ORDERS. 259 The same method of investigation may be extended to equations of any order, in which the partial differential co- efficients appear only in the first degree. (L. C. Z>. 758— 6'1.) (67) Reduction of partial differential equations of superior orders to the first order. Let 00 and y be supposed to be functions of some new variables u and y, then d_,{%)=d^z.d,.u + d„z.d^v ; hence p = d,^z.d^.u + d^z.'di.v, q = d^z.d,jU + d^ss.dj^v: by differentiating these equations we obtain r = d^z.(d^uy + 2d^d„!i!.d^u.d^v + d^%.(d^vY + d^z.d^u + d^z.d;v, s^d^z.d^u.dyU + d^^d^z.d^u.dyV + d^d^z.d^u.d^v + d^z.d^v.dyV + d^z.d^dyU + d^z.d^d^v, t = d,fz.(dy2l)- + 2dJ„Z.dyU.dyV + rf>. {dyVf + d^z.dyU + d^z.dyV. By substituting these values in the equation Rr + Ss+Tt + Pp + Qq + Nz = M, we obtain {R(d^uy.+ Sd^u.d^u + T{d^uy\d^z + {2Rd^u.d^v + S(djt.dyV + d^u.d^v) + 2 T d^u .dyV\d^d„z + {R{d^vY -^ Sd,v.d,j^ + T{dyVf]dlz + {Rd^.u + Sd^d,jU + Td^yU + Pd^u + Qd^u\d^z + {Rdlv+Sd^dyV + Td;v + Pd^v + Qd^v]d„z + Nz = M. (1) The relations between 21, v, .r, and y may be determined by assuming R {d^ti)" + Sd^u .d^u-r T{dyUy = 0, R (rf, y)- + Sd, v.d^v+T (d^ y)- = ; 260 INTEGRAL CALCULUS. which equations, hy putting d^u = md^u, d^v = ndyV, become Rm- + Sm+ T=0, R7i- + Sn+ T = o. The values of u and v thus obtained being substituted in (l), the resulting equation is d^d„z + P,d^z+Q^d^ls + N,x = M„ (2) which may be put under the form du{d„z + P,z) + Q,{d„z+^iN,-d,P,)z}=M,; and this may be reduced to an equation of the first order if d,P, + P,Q,-N, = 0; (3) for by putting d^z + P^z = z.^, (2) becomes d,,z, + Q,z, = M,. The complete integral of (2) obtained from the integrals of these equations is If the equation (3) is not satisfied, assume N,-P,Q,-d,P, = a, then putting, as before, d^z -{• P^^z — z-^^ (2) becomes duZy + Qi^i + az = M^: the values of z and d^z determined from this being substituted in the preceding equation, we have, for the determination of z-^y the equation dud.z^ + P^^^i + Q-A^i + N^%x = M„ ; in which P„ = P^ .d^ a, N^ = a-^d,a + d^,Q, + P,Q,, M„ = d„M, - ^ d„a + P^My a The equation (2) is thus reduced to another of the same form, with which the same process may be repeated. (L. C. D. 764—7.) Integration of Partial Diff. Equations by Series. 261 (G8.) If we have an equation containing x, y, and x^ then by Maclaurin^s theorem of" ^ = (-),=o + '^ • 4=0^ + ^ • Co^ + &c. « If the equation is d^z=^{d;'-'d,^z,hc.w,y), then («),=o5 ^i=o^j &c. d^~^%, are arbitrary functions of y. If any coefficient becomes infinite, when x = 0, then x '\- a may be substituted for x in the given function, and a series obtained by developing according to the powers of y) =0, M + Nd^y = 0, then the values of x and y which satisfy the equations (P(.v,y)=0, M = 0, but do not satisfy iV!=0, give a maximum or minimum ordinate, according as d^y is negative or positive : and those values which satisfy 0(cr,?/)=O, N=0, but not Jf =0, give a maximum or minimum abscissa. If d^y = 0, {'V,y) is a point of infiexion, or contrary ^exure, at which point the tangent cuts the curve : if however d^y = is satisfied by the same values of x and y, {v,y) is a point of undulation, at which point the tangent merely meets the curve. Generally, if several successive differential coefficients, as far as the n*'^, vanish for the same values of x and y, the curve has a contact of the n^^ order with its tangent, which intersects the curve or meets it, according as n is even or odd. (7-) If the same values of x and y satisfy the equations (p(x,y)=0, M=0, N=0, then d^y = - , in which case the values of d^y may be found by preceding methods. (Diff'. Calc. 15.) If the equation d^.y= - has two or more unequal possible roots, then (x,y) is a multiple point, through which as many branches of the curve pass as the above equation has possible roots. If all the roots of d^y= - are impossible, then (x,y) is an insulated or conjugate point. If the same values of x and y render d^y = - , the roots of this equation may be determined as above : if it has two or more unequal possible roots, (x,y) is a point of osculation ; SINGULAR POINTS. 267 the branches which pass through this point have a common tangent : if all the roots are impossible, there is, as before, a conjugate point. (8.) Singular points at which y or any of its differential coefficients become infinite. Let a? + Do? be substituted for a?, and let the corresponding value of y be expanded in a series of ascending powers of Dx^ then E^y = y + a, {D.v)'"^ + a„ (Da-)'"- + &c. and suppose that neither of the indices is a fraction having an even denominator : then if the numerator of m^ is odd, there is a point of contrary flexure, the tangent at that point being parallel to the axis of y, or .r, according as w^ > or < 1. If the numerator of m^ is even, there is a maximum or minimum ordinate, if m^>l ; and which is also a poi7it of regression, or citsp, of the first species, if Wj 1 : if jWg alone has an even denominator, there is a cusp of the first species, but no maximum or minimum ordinate : if i7i^ alone, or any succeeding index, has an even denominator, there is a cusp of the second species. If some terms are rendered impossible by + Dx, and some by — Z>) = 0, also since \|/(a,6) is a maximum, or minimum, >//'(«,&) = 0, from which equations a and 6 may be determined. ' (13.) Method of determining the curve tohich circumscribes any number of curves described according to a given law. Let the equation (p{x,y,a) = represent a system of curves depending on the value of the parameter a ; then by eliminating a between the equations (p('V,y,a) = 0, d^ (b{x,y,a) = 0, we obtain an equation between w and y whicli is the equation of the curve required. The Contact of Surfaces. (14.) Let X, y, ^, be the co-ordinates of a given surface, ■^1' Vii ^1' those of the tangent plane ; «> i3, y, the angles at wliich the tangent plane is inclined to the planes of yz, xz, wy, respectively. The equation of the tangent plane is %^-z = d^z{x^ -x)+ dyZ (y, - y). (cos a) - - = 1 + (rfy x)" + {d^ x)-, (cos^)-"-=l + {d_^yy + {d.^yr, (cos 7)-- = 1 + (d^zf + {d^zf. If ^j, ^j, z^^ are the co-ordinates of the normal, its equa- tions are w^ — x = dj.z(Zi — z) , If a, j3, y, are the angles which the normal makes with the axes of x, ?/, z, respectively, their values are the same as above. The length of the normal intercepted between tlic surface, and the plane of xy=fz\l + (d^z)" + (d^^z)- j i. 2^0 APPLICATION OF DIFF. AND INT. CALC. TO GEOMETRY. (15.) Let x = (p(r,y) be the equation of a given surface, and ;^j =\^(.r'j,i/,) the equation of a surface touching the former, and let dj.z, d^z, d^z, d^d^z, d^z be, respectively represented by Pi 7, r, s, f, in the first surface, and by p^, g^, r^, «j, f^, in the second. If these surfaces have a contact of the first order, then -i = ^» P=Pi^ '7 = ?i» to satisfy which, the equation z^ = ^{^\,yi) must contain three arbitrary constants. For a contact of the second order, we must also have rj = r, 5j = s, /, = f, to satisfy which conditions in addition to the former, the equation %i=-^(A\,'y^) must contain six arbitrary constants: hence an oscii- lating sphere cannot be found generally, as a circle of curvature to a plane curve. If however we suppose a section of the surface to be made by a plane passing through the normal, \ the tangent of the angle which the projection of the section on the plane xy makes with the axis of .r, and p the radius of the sphere oscu- lating this section, then The planes of greatest and least curvature are perpendi- cular to each other. Let R^ and R^ be the radii of greatest and least curvature, and let the co-ordinates be so transformed, that the tangent plane, and the planes of greatest and least curvature may become the planes of xy, yz, xz respectively ; then if is the angular distance of any section from the plane xzy and p the radius of the sphere osculating that section, R, . R. ^~ R, (cos ey + Ro (sin 6)"' 1 1 , R.= , and R.= , hence r ' t - - =r(smey + t(cosey. p (L. D. C. 168 — 78 ; L. C. D. 313 — 24.) Contact of Curves of Double Curvatube. 27I (16.) Let y =d)j(,i), z =0, (1), be the equations of a given curve, and ^i = ^/'i('i'i)? ^1 = ^:('*'i)? those of the touching line; then in order that there may be a contact of the first order, we must have for a contact of the second order we must also have and so on for the superior orders. (17) The equations of the tangent are yi-y = d.yi^i - «') > z,-^=d^x {x\ - x) . Let a, /3, 7, be the angles which the tangent makes with the axes of a, y, %, respectively, then (cos a) --= 1 + {d^yf, + {d^%f, (cos /3) -- = 1 + {d^xf + (rf^^)% (cos 7) - 2 = 1 + {d.jv)- + {d.yf. The length of the tangent intercepted between the curve and the plane ooy = z\l + {d^z)" + (c?,y)'} ^• The equation of the normal plane is G^i - ^) + dxV ivi - y) + ^x- (^1 - ^) = 0. If a, /3, 7, are the inclinations of the normal plane to the planes yz, xz, .vy, respectively, their values are the same as above. The distances from the origin at which the plane cuts the axes of a?, y, z, respectively are ^v + y.d^y + z.d^z, y + x.d^.v + z.d^z, z + x.d,x + y.d^y. The equation of that plane passing through the tangent in which the curvature takes place is \a\ — .1" / \x\ — X / 272 APPLICATION OF DIFK. AND INT. CALC. TO GEOMETRY. P Let p be the radius of curvature, then - = {L. D. C. 179 — 86 ; /.. C. D. 351 — 60 ; L. Mec. Cel. Liv. l. Ch. 2.) Rectification. (18.) Let s be the length of the arc of the plane curve, yz=(p{x). then d^s=\+{d^yf\\ and s=f^ I + \d^^w)Y\\ The value of s between the Hmits <» = «, and a? = 6, is (19.) Let?/ = 0(0) be the equation of the curve referred to polar co-ordinates, then de5 = w- + (rf9w)1% ands=^{0(0)}'-+{c/9(^(0)}-^|'. (20.) Lett/ = 0(ci'), z=^y\/{x), be the equations of a curve of double curvature, then d^s = \+{d^yy + {d^%f\\ and s =Xl + \d^(l>{x)Y+{d,^{cc)\-\\ (21.) Rectification of the ellipse. •/^ l 1 — a?- J ./x 1(1 _ a;'){\ — e'o)-) \^ comparinff this with / — „ „^, , — „ „.,. ■? (/wf. Calc. 17-) ^ ^ •^M(i±pV)(i + rr)l^ we have ^^ = 1, q = e, y, = ^'\^ tt^] , (1 — e'x^} p-, = l + (l-e"')K q^ = l-(l-e")i. Assume e, = K P.y.=^v„ and .,= / . ^^ _ ^^^^ff .^^^^^ , then 2(1 + ei)s = 2eia7i — (1 — ef)(5i — ej.rf^ *j) -|-25j. (l) * This notation will be explained in the Appendix. RECTIFICATION OF THE ELLIPSE. 273 By a similar transformation, we may obtain an equation of the same form between s^ and s„ ; from these equations we have e,.ri + «i — (I +«,)•? l+e. e„.r., + «. — (! +e,.)«i ^ >. __ — . . (Z) i—e, 2 l—eo if l-(l-ej)i i-(i-ej)i € = we should have a similar equation between (i_„, ^'-p and s; and generally we may obtain a series of equations ^ l—e.: 2 l — e 2 from which, any two consecutive values of s being known, any others in the series may be determined. By this process the rectification of any proposed ellipse may be made to depend on that of another ellipse of either much greater or much less eccentricity : in either case an approxima- tion may be made by a series, as in (Art. 17-) (22.) The arc of an hyperbola may be thus referred to elliptic arcs : let 1 be the eccentricity, e the semi-axis major, and S the arc ; then the equation of the hyperbola is K ., ... e Let ,v = e[l + (1 — e-)(tan (p)" \ ^ , then t/ = ( 1 — e') tan (j) ; I oil/* (cos 0)' and S = ta.n (h\ I — (esm (by y^ — e- I — r-— — rrr; ^ ^ ^' ^ ^^ { 1 — (esin0)-|i = tan { 1 — (e sin 0)'}* — ^ + (l — 0(« - e.d^s). By changing e, 0, s, S^ into Cp 0i, «i, -Sj, and substituting for s^ — e^.dc Si its value obtained from (l), we obtain 5', = tan0j{l — (eisin0L)-}* + 2eisin0i -|-«, — 2(1 -ve^s. ^23.") f • ; z TTT which depends on the Mm 274 APPLICATION OF DIFF. AND INT. CALC. TO GEOMETRY. where Q= {(l +jj''w^)(l +Q^u-)}^, may be reduced to the form y

in which Xy is an algebraic function of .v, and f^ X^^ a function involving transcendants, which however vanishes when oe=^a, .v = b, &c. and let in which i? = a? + X^ (.v — a) (a) — 6) &c. X^ not containing any factors of the first degree, then if y = d^Xy + (pix) — d^z. (p(x), the curve is quradrable the limits .v = a, .v = b, &c. (30.) Conditions which render a curve rectijiable. Let dj.y^p, then y=p.v-/j,.v, and s = .v(l+p"')i-^-^^^^r, (l) assume J at=P, and /- ^rT=Q» then p=- ^ , ^^„,i : -^P Jp(l+p"')h ^ {l + (c/^Q)=}5 by eliminating a; between the equations (l), the ordinate of a rectifiable curve will be obtained from the first, and the expression for the arc of that curve, from the second. (L. C. D. 735—7.) Trajectories. (31.) Let 0(a7,2/,a) = O be the equation of a system of curves, of which a is the variable parameter, and let the derived equation be M + Nd^y =^ ; (l ) then by eliminating a between these equations, we obtain x//(rf,y,a;,2/) = 0. REMARKABLK ALGEURAIC CURVES. 5277 Let u be the value of d^y obtained from this equation, and Q the angle at which the trajectory is required to intersect the given curves, then the differential equation of the trajectory is (1 + M tan Q) d^y — u^- tan = 0. (2) If V is the value of rf^y obtained from (l), then the equation of the trajectory may be found by eliminating a between the equations ^C'^'j^/jo) = 0, (1 +vtan0)rf^y — v + tan0 = O. (3) If the orthogonal trajectory is required, (2) becomes If the elimination of a presents any difficulties, and one of the variables may be readily separated in the proposed equation, then a being considered variable, we may obtain from (3) the equation (l -Hf") tan^.rf^.x'— (l — utan0)c?„i/ = O, (4) from which the relation between a and x may be determined, by means of which the required trajectory may be described. If only d^y = v, the differential equation of the proposed system of curves is given, then (4) becomes (1 + v") tan e . d^.v -(l-v tan e)f,d„v = 0, which is an equation of the first order, if d^v is integrable with respect to ,v. (L. D. C. 402 — 5 ; L. C D. 681 — 3.) Remarkable Algebraic Curves. (32.) The conchoid of Nicomedes. Let the pole be in the axis of J.', at a distance h from the origin, then the equation between rectangular co-ordinates is artf- = (6 + cf) - . (a" — x"). The equation of the tangent is (.f' + a b") .T^ — 2a Ir.v — hrx"' 4- o x^ The points of inflexion may be determined from the equation x^ -\-3hx- — 2 n'^ 6 = 0, 278 APPLICATION or DIFF. AND INT. CALC. TO GEOMETRY. ^, , . h The polar equation is w = h a. cosO If a < b, there are four points of inflexion, and the pole is a conjugate point: if a = 6, there is a cusp, and if «>6, a node, at the pole, and the curve has only two points of inflexion. The axis of y is an asymptote to the curve. (33.) The cissoid of Diodes. The equation between rectangular co-ordinates is y'= — • The equation of the tangent is I Ihe subtangent= — -. 3a — 2a; The curve has a cusp at the origin, and two hyperbolic branches, to which the ordinate corresponding to ,v = a, is an asymptote. rri. . . 2 a (sin 0)- Ine polar equation is m= — —. cos 6 The area contained by the curve and the asymptote = I TT a-. ( Wood, Jig. 496, 7.) (34.) The Witch : y=- (ax - w~)^. The equation of the tangent is a mi 1 2 (ax — or) T he subtangent = — . The axis of y is an asymptote to the curve. (Agnesiy Analyt. Inst. V. i. Art. 242.) (35). The Lemniscata. The equation between rectangular co-ordinates is (x" + y")" = a" (x" — ^^). The polar equation is u = a. cos 20. The area of the curve = ar. REMARKABLE TRANSCENDENTAL CURVES. 279 (36.) The semicubical parabola. The equation is ay-=zx^. This curve is the evolute of the common parabola : the length of the arc measured from the origin is 8a (■/ 9^\* 1 (37) The equation of the evolute of the ellipse or hyperbola a- o* IS (^^ (!;)'=- , . , a- + b" a"-\-b" m which . a, = , and 6, = . 'a ^ b (38.) The Trisectrix. The equation is Qc^ — 2ax + y-y = a- (x" -\- y") . The polar equation is u = a (2 cos 6 + l). {Peacock, Ex. pp. l60, 70, 7.3.) Remarkable Transcendental Curves. (39-) The quadratrix of Dinostratus. y = {\—.i)ta.xi^irx. The polar equation is w = ; — ;. . Trsmd The quadratrix of Tschirnhausen. y = sin i irx. The polar equation is u sin 9 = cos (^ttw cos 0). (40.) The logarithmic curve. y = a'. The subtangent = log, a The axis of — y is an asymptote to the curve. 280 APPLICATIOK OF DIFF. AXD IN'T. CALC. TO GEOMETRY. (41.) The cycloid, and cycloidal curves. Let the vertex of the curve be the origin, and its diameter the axis of .r, then the general form of the equations of a cycloidal curve is .r = a.vers0, y = a{nmiO -rvnO). The equations of a trochoidal curve are A' = a vers ^, y = ff(sin + 7n0). The equations of the common cycloid are X = a vers 0, y = « (sin -\-d). The length of the arc is four times the diameter of the generating circle. The area is three times the area of the generating circle. The radius of curvature = 4 «. cos ^0. The evolute of each semi-cycloid is an equal semi-cycloid on a parallel base, the vertex of which coincides with the cusp of the former. The maximum ordinate of the curtate cycloid corresponds*" to a; = a{l +m). The 'prolate cycloid has a point of inflexion corresponding a to )(l — cos— ) a 2h 46 = — (a + 6) vers 10; a 46 hypocycloid = — (a — b) vers 10. The radius of curvature of the epicycloid = -^^ sin 10 : the evolute is an epicycloid similar to the original curve, the radii of the base, and generating circle being respectively ar ah a-\-2h^ a + 2b' The equations between p and ?/, the centre of the base being the pole; in the epicycloid, jt> = e(-^ -) , where e = a + 26, a'^ — u-'Y —^ ;; I, ....e = a — 2o. The area contained by the axis, the radius vector, and the curve = - (a + b){a + 2b){0 — - sin 0), in the epicycloid, = - (a — h){a — 2b)(0 sin0), .... hypocycloid. If a : ft is a finite ratio, the curve may be represented by a finite algebraical equation. The equation of the cardioide, in which a = b, is y" + A' (c^? — a)]" = a"{y- + {x- a)-} . If the diameter of the curve is the axis of ,v, and the cusp the origin, the polar equation is ;t = r/(] — cos\//). REMARKABLE TRANSCENDENTAL CURVES. 283 The equation of the epitrochoid in which a = h, the origin being at the first point of contact, is cr'* + 2 6^ ci'^ — {a^ — 6f ) x" + 2 (a" + 6 J xy" — a"y" + i/'' = 0. The equation of the epicycloid with two cusps, in which 6 = ^a, is a^y" = (a" + y" — a") '. The equations of the hypotrochoid, in which 6 = ^a, are x={^a + 6J cos6>, y = {^a — b^) sine, the equations of an ellipse, of which the semi-axes are ^a + b^, and ^a — by (Peacock, Ex. p. 192.) (43.) Spirals. The equation of the spiral of Archimedes, is u = a9; U' the equation between p and w ; p = -— ;; ^—^ . ^ /' /- (a- + M-)2 a~ The equation of the Lituus is u-= . V The first radius produced is an asymptote to the curve, and there is a point of contrary flexure. (44.) Coles's Spirals. r T ® [IJ The hyperbolic spiral; u= -. _, , , - au The equation between p and u; » = —-^ —j . The subtangent = a. The area = ^au. n The length of the arc =alos:, — —; H (a^-\-u-)h — a. ^ ^ {a"+u-)^ + a ^^ ^ ^ r T bu ^ b, c+(c' — 7r)i [2] p= 7-. ;;rT, or 0= -log. ^^ ^, {a- — tc-p c u where c = {a- — 6") . 284 APPLICATION OF DIFF. AND INT. CALC. TO GEOMETEY. [3 J p= -—^ — . , and a = T-T ^71 5 ^"d a>b; then (a" + ?<-)2 c ^ «^ = c . sec - 0, 6 where c=(b" — «")^. [5] The logarithmic spiral: p= -u^ and {a—b-)^ a The angle contained between the radius vector and the curve IS constant, and = tan '■ -—z — ^ • (a- — b'Y The evolute is a similar spiral. (45.) The polar equation of the involute of the circle, is = + sec {u- — a-)i _ ^ _ J ?^ (46.) The tractory : The differential equation is {a"-y")l {a+ ifi- — y-)h\ whence ^ x = a\.ogA r — («" — 2/')^ • The evolute of this curve is the catenary. The equation of the syntractory is h^-(b"-y")h ,v = a\og, —y-^ -ib'~-f)-' THE ELASTIC CURVE. 285 If 6< a, this curve has a point of inflexion, corresponding to _ h . al ^~ (2a + 6)i* If 6>«, there is no point of inflexion, but when y= (ah)', the tangent is perpendicular to the axis. If 6 < 0, there is a point of inflexion, when -b.a^ (/= --T. (Peacock, E.v. p. 171-.) •^ {2a + h)i ' r / (47) The catenary. Let the curve be referred to its di- ameter, and a tangent at the vertex, and let * be the length of the arc ; the differential equation is (c- + *-)2 d^s= , whence d' + c = (s" + c")^, or s = (,v" + 2 cx)K' The equations between y and s are y s + (s- + c'y^ f the equations between w and y are 'V + C = ^C(€^ +€ ^), - =log. '- . c c (Whewell, Meek. Ill — 3.) (48.) The elastic curve. Let the extremities of the elastic line be situated in the axis of y, at equal distances from the origin, and let a be the angle at which the curve passes through the origin; 6'=— a- sin a, and c' = a- — b', then the differential equations of the curve are , a- — c- + x" d^y = - , (c"-.x-)(2a--c^ + a?2)P d^s = . 28G APPLICATION OF DIFF. AND INT. CALC. TO GEOMETRY. Species [l]. Let - be very small: then, neglecting c and .v", we have a . y\/2 d^y= ■ : ■ , „ o. 1 . , whence .r = c . sin , •'-^ |2(r-— A'-)}a' a c~ c- . 21/2.V s = y -i r.y^ 7 — sin .. 2cr^ 41/2.0 a p [2] - < 1. In this case the values of y and s can only be a obtained by series. [3] - = 1. In this case rf, v= 7-3 iri > and a {a —x)^ Let I be the length of the curve, and h the distance of the extremity of the curve from the origin, then 1/2 i 2-2 2-. 4- 4 J 7r«r 1- 3 1 1-.3- 5 1 ^ ) V/2I 2-12 2-. 4- 3 4 j //i = 7ra-. [4] ->1, andA>0. In this case there is a maximum a ordinate when x" = «-.sin a. [5] - > 1, and h = 0. In this case the curve returns into a itself. [6] - >1, and h<0. The branches of the curve cross a each other. [7] - =V'2. The axis of y is an asymptote, and the a equation of the curve is THE ELASTIC CURVE. 287 c + (c- — w")\ y= —{c- — x-)i + Ic log, [8 J ~>\/2. Let c' = 2a-+g-, then '^"-i^'-i^ g is the minimum, and c the maximum, value of x : beyond these limits x is impossible. y is a maximum, when x = ^{cr + g'')\- [9] - = 00 . In this case the curve is a circle. a (Whewell, Mech. Appendix.) CALCULUS OF VARIATIONS. '7L Jr OR a history of the origin and principles of the calculus of variations, see {Woodhouse's Isoper. Prob. ; L. C. D. 825—43.) (1.) Sd^u=^Ju; S/^ti=fJu. U:u = d:U, Wu=fJ^lu. If u:=(j)(.v, y, y\ y% he), then ^u = d^uhx 4- dyii^y + dy,uhy + &c. ^y = d^ Oy -y^x)+ y". Ix, &c. = &c. U:y = d:{ly-y'lcc) ^.d^^^yXv; from which ^w may be obtained in terms oi^y and ^x. (2.) ^fM^^y^^^y"^^^-)=U^v =.V^x+{d^.V-dJy„V^hc.]{^y-y'hv) + {d,yV-hc.]d,{^y-y'lx) + &c. +fAdJ-dAj'V+d',d^„V-hc.\m-y%v) (2) In order that V may be integrable with respect to x, we must have d V -dd.V + d^d^nV— he. =0. From this last equation it follows that the quantities f.{fAfAV-d,,v) + d,„vu &c. may always be determined. (L. D. C. 456 — 67; L. C. D. 844 — 53.) CALCULUS OF VARIATIOX^. 28d (3.) If r= {.i',y,2/',?/", &c./ >//Cr,y,i/', &c.) } let ^y - y'^x = a,, / F, = 7^ and / d„ V = U, then V* ^= f'^-^ + { d,- V- d,dy, V + d'^d,y„ V- &C. } CU + &c. {d,F,-dJ.,.V, + d;d^,„V,-kc.]co {d,.J\-dJ^.,J\ + kc.]d^co I + &c. ' + X { ^. '^. - dA.^ l\ + f//^,. l\ - &c } a> - { Udy r, - rf, f7(/^„ V, + kc}w - { Udy, l\ - d^ Udy„ r, + &c. } d^ a. ^{Udy„V^-^hc.\d;io - &c. - /. 1 ^'^. ^1 - ^x Ud^, V, + rf/ t7d,„ r, - &c. } a,. (/v. C. i). 854.) (4.) Let F=<^{.r,j/,a?, yj, g', r, «, ^...&c.}, In which ;?, q, r, Sy f, u, v, iv, o, represent rf^ij^, d^^r, rfjsf, rf^rf^^Jf, fZ/s-, ^>, d^d^z, c/.d^^r, r/^^^v^ respectively, and let w = ^z — p^.v — qSy, then -d,rf.„r + &c.l + d^w{d,V-kc.} +kc. Oo IftH) CAT.CUI.US or VAHTATIOMS - d,d,V+d,d^d,, V-kc.i + djd,V-kc.} +J>{drV-dAV + d'^d„V-&cc. -d^d,V+d^d^d„V-kc. -Vd'^d^V-hc^ ■^/J,u>{d,V-d^d,V+kc. -d^d,J+kc.. + &C.. -d/,.V+ke. +J\d^w\d„V-kc.\+hc. +Jld!w\d„V-kc.]+kc. ^fJM^J- "^rdf V + did, V- &c. -cl,^d^V+d,d^d,V-kc. + d^d, V- &c. (L. C. D. 862.) Maxima and Minima of Indeterminate Integrals. (5.) Since the part (l) of the value of Sj^^ V has been obtained by integration, it must be taken between limits : let the values of V, x, y, y ; Sic. corresponding to the first limit be ^i» ^i.» 2/i» V\ '" &c. and to the second v^^ of^, y^, y'^ ; &c. then + \d.j, V, - d,^d^,^ V, + &c. I {hh - y'.^-^.) - {d^,^ V, - rf,. c/,<, V, + &c. } m, - y\ ^.r,) (3) MAXIMA AND MINIMA OF INDETERMINATE INTliC;RALS. 291 - { d^,^ F, - &c. } . rf^, (^y, - ij\ ^a-,) + &c. — &c. +/Ad,f'-dA'^+d!^v'V-^'^] (h-y^^r (2) (6.) If a maximum or minimum value of J]. V be required, then ^Jl V=0, and hence the quantities (2), (3) must be separately equal to nothing. Eliminating therefore between (3) and any given conditional equations as many as possible of the quantities ^a?i, ^.Vo, ^r/p ^^2' &^-> ^^^ making the coefficients of the rest severally =0, we may obtain the value required. The required result may frequently be more easily obtained from the condition dj- d,d^, V + d'^d^., V- &c. = 0. (7-) If in addition to the condition Uy^=maximum or minimum^ we have also U^ = 0, then hU„ = Q. Both these conditions may be expressed by the equation a being a constant to be determined by the nature of the ques- tion. From this equation ^0?^, hxc^ &,c. must be eliminated, as in the preceding case. Similarly if the condition f/, = be also given, then and so on for any number of independent conditional equations. {Airy, Math. Tracts.) CALCULUS OF FINITE DIFFERENCES. DiiiECT Method of Diffeuences. (1.) /\^Cn^-{-'hi^+kc.+"u^+const.) = A^ht^+kc. + A;'u^. " A^ff .?<,=« Aji^j, if a is either independent of + /i ^*j • ■^ar + 1 • • • ^'.r + w + 1 (L. D. C. 497, 8 ; Tr. L. App. 342, 3.) (2.) Successive differences. ^x («»'^'" + ^''u - 1 *'" " ' + &c. + «! cv + a) = w a,,.^'" " ^ + 6cV" " - + &c. A,V« -'*''" +««-r^"~^ + Sic. + a^.v + a) = n{n— l)^,^^^?"-^ + &c. &c. = &c. A;(a,X + «,.- i'^""' + &C' + «i<^ + «) = 1 -S-S^.n-a,, ; from which it appears that the n^^ difference of an algebraic function of n dimensions is constant, and therefore that the differences of all orders superior to the n*^ are equal to nothing. « A,S„/"%. + const. = S„, A,,"V/.^. DIRECT MKTHOD OF DIFFERENCES, 293 &c. = &c. n n(n — 1) . „ „ o = (l + A,)«zv 1 1 • ^ » w(w— l) , A'^O"' = »'" (n— 1)'" H ^^ (n — 2)"' + &c. * 1 ^ ^ 1.2 w 7i(w— l) , n" (n—lY+ — -(n — 2)" — kc. = l.2.3...n. 1 ^ ^ 1.2 » + i • a;m,= S,.(-i)"'"'. m-l ?« > + « — )« + i' I m— 1 m— 1 « + 1 I ^* y A^a"' = S,(- 1)'' - i-Lj — (a- + w - r + 1)" [r — 1 " (-ly-i (n-r + l)'" or A":0"' = S/ I r — 1 [» — r + 1 SOi CALCULUS OF FINITE DIFFEEKNCES. (3) A,'w,X---'X = {(i+'A)(i+=A)...(i +"'A)-l j.^t/.^.X-X in which 'A, &c. ai-e merely used to denote that ^u^^ &c. are to be respectively annexed to them after the expansion. " A;'m,.-« ..'X={ (1 +'AJ(1 +^A,)...(l +'"A,)- 1 \\'u,.-u,..."'u,. (Tr. L. App. 344 — 5G ; L. D. C. 499 — 507 ; L. C D. 882 — 7-) (4.) Series involving the differences and differential coefficients of a function. ^xU^ — - dx^ir + — f^Sx + d^Ux + ^^- ^ 1 ' 1,2 ' 1.2.3 = (e'^'-l)2f,- n , ti^ . ««x + n = u, + - d, u^ + — d>, + &c. * d,?*, = A^?*, - 1 A; ?/., + i A>., - &c. * = log,(i + A,)?v <«*,= lloge(i + AJ}X- " {Tr, L. App. 357 — 6l ; Z,. C D. 929 — 38.) m m—r,r « A,P,X=S. C,,(X.A,X): this notation will be explained in the Appendix. =° A'" 11 w » f j\W -1 «i '" a; tf, = s^ A" 0" + '« - ^ rf;+'"-i:!/.. direct method of differencks. 295 Functions of Two or more Variables. (5.) A,^w,„ = A^?^,.^ + \u,^y + A, A^?/„^ . = {0+AJ(l + A,)-i}w„^. Ky'^^,v^{ (1 + A.)(i + A,) - I {"w..,. {L. CD. 919, 20.) 1*2 »H 1*2 m = I (1 + A,. )(l + A,^)...(l + A J - 1 JX.^v—-- ^>^^.,y = W,+„,j,+„ - W . ?^,+„_i,y+„_i + &C. 71 n(n—\). (Tr. L. App. 362 — 6 ; L. C. D. 919, 20; 33, 4.) Interpolation of Series. (6.) Let M, Mj, W2> Sec. ?/,,, and i', v^, Vj, &c, r„, be corres- ponding known values of u and v, in the equation tc = (p(v), in which the form of the function (p is unknown, and let it be required to determine the value of ic^, corresponding to any proposed value t\, which is either between the limits v, v„. ee m+1 « A^ ,, _ C C //"•-"+'•//"-'•,/ 296 » CALCULUS OF FINITE DIFFERENCES. or very near to one of thcin. Let v^ be the value of v preceding v^, then if the intervals between the values of v are small, we may assume u^. = a + a, (v, - v J + a„ (v^ - v^)"" + &c. + a,^ (v^ - 1),)", which equation, when arranged according to powers of v^, becomes u^. = b + b^v^ + b^v^ + 8ec. + b^v", in which 6, tj, &c. remain to be determined. We have the 71 + 1 equations 71 =b + h^v + b„y + &c. + b„v", u^=h + h^v^ + h„v^ + &c. + 6„<, &c. &c. Let!^l^=J, ^^^^^=^, &c. ^^^iZl^^^^,..,; = 5, ::^iZL^=5„ &c. ^-^~^-^=Jg,.-,; = C, ?^^^=C„ &c. V, — V ^4 — I'] then h = u — A.v + B.vv^— Cvv^Vo + &c. b^ = A — B(v + vj + C(Wi + «'W2 + 'Vi'y2) — &c. b, = B— C(v + i^^ + v„) + &c. and 7* ^. = ?< + .4 (t*^ — 'w ) + 5 (u^, — v)(v^. — vj + C(v^-v)(v^.-v,){v^-v,) + &c. (1) The value of ti^ may likewise be put under the form , (^■r-^00'.r-^'l)---K-O ,, . c , + 7 . Un + &C. ^1 — 'i A- ■A ^2" • -?? B,- -5 INTERPOLATION OF SERIES. (7-) Let Vi — v = v„ — r\ = kc. = Dv, then v.^ = v + vDt, Uv Vv 1.2 Z>v" '1.2 Dv- Cj = &c. 297 1.2.3 Dv^ by substituting these values in (l) we obtain or u.=^u+^-D^u+— -D^u + kc. 1 " 1.2 (8.) If V, = 0?, and n = 2m, let the values of «* be «*_^, w_,„_i, 8ec. w_i, w, Wj, &c. u,„.i, u^i then w,. = w+ ^-.i(^xW+^-^-i)+^-7:^7^'2(^>-i+^>-2) a?(.r--l2)(.i'--2") , , , n X o + 1.2.34.5 4(^>-e + A>-3) + &c. 1,2 1.2.3.4 ' '^ + —5^ ^^^^^ — -^ A>_3 + &c. 1.2.3.4.5.6 ^ If n = 2TO— 1, let the values of u be M_2;;» + l, W_2„, + 3» &C. W_i, Wi, &C. W2m-3> «^2m-n then w, = i(Wi + M-i)+ ^-^^.i(^xW-i + ^>-3) "^ 2.4.6.8 2 ' -1 ' 2.4.6 2.4.6.8.10 ' -5 -TV* (Tr. L. App. 401 — 7 ; J^. C. D. 898 — 908.) Pp 298 CALCULUS OF IINITE DIFFERENCES. (9) If u^ ,j is a function of two variables, then ''" 1 ' 1.2 ' 1 . 2 . .S + - A,w+ -^A,A,z^+ -5^ ^A.-A,,^ \ ■' 1.1' 1.2.1' 1.2 ^ 1.1.2 ' " 1 . 2 . .9 (L. C. Z). 914, 5.) Application of the Calculus of Differences TO Trigonometry. (10.) Differences of the trigonometrical lines. D^ sin .V = 2 sin i Doo . cos {x + ^Dx). D^ cos 00 = — 2 sin i Z).r . sin (a? + \ Dx), sin Dx D, tan ti? : cos 07. cos {x + Z)c^?) - sin Dx D,coix= — sin a?. sin {x + Dx) D^ (sin xY = sin Z) a' . sin (2 a? + Z) a'). X)^ (cos xY = — sin Z) a? . sin (2 a + Z><»). sin Z)a .sin (2a? + Z)a) D, (tan a?)-= (cos a)-, {cos {x + Z)a?) }- „ sin Z) A . sin (2 x + Dx) D, (cot a)-= - (gina?f.{sin(<2; + Z>cr)}2 * (Z. C. D. 892 ; Ca^^. Trig. 212 — 33.) n m+l j /y I ^ ^^.,. = ^^ + S., S. ,b:;T • !:rr • A;'-^^: A;-^i*. VAllIATION OF PLANE TIUA-NGLES. 299 (11.) The variation of triangles. Let X, F, be respectively functions of -r, y, any parts of a triangle, and let X=mY\ then from a given error in one, to determine the corresponding error in the other, we have ' 1.2 1.2.S = m{i>2/.rf,F+^".c/;F+&c. « All terms except the first may in most cases be neglected, then Dx.d,X=mDy.d,jY. If great accuracy be requisite, the relation between Doc and Dy may be determined by the solution of the quadratic equation Dx.d,X-\-^{D.vY.dlX=m{Dy.d^Y+\(,Dyy-.diY]. (Woodh. Trig. Ch. xiii.) (12.) Corresponding variatiotis of plane triangles. [l] Let A, c be invariable; then DB= —DC, and Dh : sinZ>B :: a : sin(C4-Z>C), :: a + Da : smC; d.B= . a \Da : tSLXi^DB :: a^^Da : tan (C + ^DC), . 1 ^ ^ sin (C + DC) , „ tan C ^Da : ain IDB :: a : j^ — i-^; d^B= . 2 ^ cos{C + ^DC) " a Da : Db :: cos(C + ^DC) : cos iZ>C; df,a = cosC. [2] Let A, ft, be invariable; then DB= —DC, and I2)c :sm 1Z)C ::^: ,,,(e + iZ>C) ^ '^^^ = "T- COS C7 i>6 : -£>c :: cos(B + ^DB) : cos (C+^2)C) ; f/ /> = . '^ '' cosB « S,XD.v)"'-'.dr'-^=m{SADyy-'.d:;-':Y\ 300 CALCULUS OF FINITE DIFFERENCES. [3] Let b, c, be invariable ; tjien tan^DB : tan^DC :: tan (B + ^DB) : tan(C + 4Z)C); ^ tan 5 -^Da : tan ^DB :: a + ^Da : cot (C + ^DC); cotC d^B=- —- . a -siniZ)J : siniZ>5 :: « + iZ)a : b . cos (C + iDC); d.B= — -cosC. ^ a sinlDA : ^Da :: cos ^DB : b.smCC -\- ^DC); dA= ^ . ^ . (Ca^w. rri^. 632 — 67.) (13.) Corresponding variations of spherical triangles. [1] Let A, c, be invariable; then sin 2)6 : s\n DB :: sin(a4-X)ff) : sin C :: sin a : sS.n{C-\-DC), :: sin a . sin (a + Da) : sin c . sin A, :: sine, sin J : sin C. sin (C + Z)C) ; d^B= . sm a If J, or c = 90°, then sinDb : sin DB :: sin 6. cos (6 + Z> 6) : sin 5.cos(5 + Z)5) ; sin 2 6 tan^Dfe : — sin iZ>C :: tan(a + ^Da) : sin (C + ^DC); sin C d,C = tana sin^Da : tan^DB :: sin (a + ^Da) : tan (C + ^DC); tanC d^B = sin a VARIATION OF SPHERICAL TRIANGLES. 301 If ^ = 90°, then sin^Db : -IsinDC :: _,_ .f" i r.rx - sin (C + DC), cos C ; 2 2 . 2—- •• cos (b + ^Dh) 2 tan 6 1 . .1 cos B ism Da : sm ^DB :: sin (a + D a) cos a : - — -— ^ ^^, ; ^ 2 V >» iim{B + ^DB) 2coti? d.B = sin 2 a If c = 90% then si„ii,6 : i sin DC :: ^^^^^^ : sin (C + BC) cos C ; '^ 2 cot 6 . . • sin 5 — i sin Z)a : sin^Z^^ :: sin (a + Z) a) cos a : rm^^ "^ 'i ^ ' cos {B + 2 -^^) 2 tan 5 rf„5= - -, . sin 2 a taniZ>a : — tan^DC :: tan (a + li>a) : tan (C + iZ>C) ; tan a cot 5 tan 5 If ^ = 90°, thenrf„C= -. ; if c = 90°, a :: cos \DC : cos (C + ^DC) ; rf^a = cos C . tanl2>5 : —iaxi \DC :: cos^DJ : cos {a -\- \ Da) ■, dgC= — COS «. If v4 = 90% then COS b cos « sini2>6 : sin*Z>a :: -: — -. : -r—. — — tTT^ » cot a rfj,a=— — . cot o 302 CALCULUS OF FINITE DIFFERENCES. sin B cos C ^\\\hDB : -sinlDC :: cos(5 + ii>5) ■ sin (C + 4Z>C)' If c = 90°, then 2 cotC tan jB siniZ)6 : — sini/>a sin 6 cos a ^ • ="'2^" • cos(6 + iZ>6) • sin(a + 4Z)«)' cot « d^a= - - — -. tano cos B cos C .m\DB : sin^DC : ^^^s + ^DB) ' sin (C + ^i^C)' cLC = 2 cotC tan B [2] Let -4, «, be invariable ; then — tanAZ>6 : iaxi ^Dc :: cos (5 +12)5) cos (C + ^Z)C) 2 • ^""2^- •• cos^DB ' cos^DC cos C ^^0= — -taniZ>5 : tan^DC cosB cos (b + ^Db) cos (c + ^Dc) ^ ' ^ " ■ cos ^Dh ' cos ^Dc cos c d„C=- siniJDc : sin IDC :: cos 6 sin c sin C 2^c . Miig^L. .. ^^^(^^1^^^ • cos (C + IDC)' tan C d,C = ±B=- tanc cos 6. tan C sine vauiation of spherical triangles. 303 If ^ = 90°, then . , • 1 r» COS 6 cos (c + X>c) — sin4Z)6 : sin:^Z>c :: -: — z : ^^ r -; ^ 2 sm (b + ^Db) sin (c + iZ)c) cote d^c= — cot 6 cose — sinZ>5 : sin DC ;: cos 6 : cos (c + Dc); (LC— — cos ft isini>e : — sin^D^ :: sine.cos (c + X>c) : -; - ., _^ ; 2 cot 5 d,B= ^ . sin 2 c If = 90°, then cos C — sin Db : sin Dc :: cosjE? : cos (C + DC); df^c = cosB « ^ sm(B + lDB) sin (C + iX»C) ^ cotB cos e sin^De : -isini>5 :: ^- ^— -— : sin 5 . cos (5 + Z)5) ; ^ ^ sin (c + i2>c) '^ . „ sin 2 5 d^B= . 2 cot c [3] Let 6, e, be invariable ; then tanjZ>5 : tan^ZJC :: ta.n(B + ^DB) : tan(C + ^DC); * tan5 - sin iZ)a : tan ^DB :: sin(a + iZ)a) : cot (C + ^DC); d.B^-^. Sin a sin^Pi^ : sin ^2>a :: sin (a+^D a) : sin6.sinc.sin(J+^i>J); sine, sin 5 304 CALCULUS OF FINITE DIFFERENCES. 1 ^r. { sin ( a -^l Da)}- sm{A+^DA).smC ^ ■^ sma.sm6 sin^.tan(C + ix>C) d.^ = 2- — sin a sin h . cos C Ifc = 90% then COS C i sin Da : sin 4Z)C :: sin (a + Do). cos a : -:— ttttTTTTT: ' ■< '' sin(c+^i>'C) Scot C d^C = sin 2 a cos a cos a siniD-4 : siniDa :: — — 7- — rrm • "^""7 — TTTwi' ^ ^ sin (J + i-D^) sm (a + ^Da) 2 cot -4 cot a siniD^ : isinDJ5 :: ,T, . r, ^x '■ sm B. cos {B + DB); 2 tan J[ dgA= . ^ ■_, . " sin 25 cbsS — sinD-4 : sin DC :: sin (a + Da) : cos 5; d.A= — ^ ^ " sin a [4] Let B, C, be invariable, then tan c tanlD6 : tan ^Dc : tan(6+|D6) : tan (c+-|Dc); c^6C= — - sin A sin^ DA : tan-^Dft : sin(J + iD^) : cot (c+^Dc); dt,A= -^^ 1 X. X sin (J + IDA) sin^DA : sin ^Da : sin (a + :|Da) : — -. — p f „ ; 2 -^ V .4 ^ sm S . sm C d„-4 = sin5.sinc. { sin (^ + ID J)}- sin (a + iDa). sine sin iDa : taniD6 : 2 ' sin J. sin 5 ' sina.tan(c + ^Dc) , _ sin J3. cose d„h= : — -j — sm A CONSTRUCTION OF LOGARITHMIC TABLES. 306 If C = 90°, then cos 6 ^sinDA : sin^Dc :: sm(A + DA)cosA : r r, 1 ^^tt' '^ -i ^ ' sin(o + ^i>0) sin 2^ ^ 2 cot 6 ,„ 1 sm DA : sin 1 Da :: — cos A cos a 2^.x . ,,...2 .. gij^(^_^X2)^) • sin(a + lZ>a)' cot J cot a . 1 1 sin « . , ^, ^, sin ^Drt : isinD6 :: z , ^ , : sui 6. cos (6 + />6); ^ ^ cos(a + ^jDa) sin 26 2 tan a sin Da : sinDc :: sin (a + -Da), cos a : sin (c + i)c) cos c ; sin 2 c d„c = sm 2 a (Cagnoliy Trig. 121.9 — 1420.) Construction of Logarithmic and Trigonometrical Tables, (14.) Logarithmic series. Dd , /Da!\- , /D.v "Z),log,.r=S„.-^-.(-) . 2'"— 1 /Dx\'"-^^ 306 CALCULUS OF FINITE DIFFERENCES^ D^\og,a=2{—\ -&c. If Dv is small, these terms will be sufficient ; the following series are however still more convergent : 2>,log,a;=2 \-\{ \ +i(- -I +&c.>. « Two terms of the first series, and the first term of the second, will in most cases be sufficient: if a? =10000, and Dx = 1, the common logarithms will be exact to fifteen decimal places. ' ^ ^ I 1 1.2 1.2.3 j (Cagn. Trig. 372 — 91 ; L. C D. 889, 90.) (15.) The sines of arcs for intervals of a degree having been found by preceding methods, {Trig. 42 — 5.) the sines for minutes may be more easily found by differences : any two differences having been found, the others may be determined by the equation Z); sin a? = - (2 sin \Dxf. \D^-^ sin x + D^-^ smx\. (16.) The series of differences for any given intervals having been found, the differences for smaller intervals may be found by the following general method : let x be the larger interval, y the smaller, and let x=.py, then ^1 o 1 r Dx \ « D, log, a; = 2 8,;, .( =r-) ^ "'2m— 1 \2x + Dx/ ^ D!log^x=-2S„A(-^T' ' ^ "'2711 \x + DxJ y D,\ogr'x=^iogr'x.^ Dx — '"-^ ) Dx ^2/« {Dx)' CONSTEUCTION OF TRIGONOMETRICAL TABLES. 307 p 2p- 6jP' _ (;>-i)(2;?-i)(3;>-i) ^, 24>p* (p-l)(op-l)(3p-l)(i.p-l) ^, ^^ ]20p^ "~ p~ P' ' 12/ (p_l)(2p_l)(5;j-3) 12/>^ D' + &c. these differences, which are given as far as the 5'** order, are quite sufficient for the calculation of tables. A"o'* (17.) d:u= dyi.{D.vy ^ ^ 1.2...W 1.2...(7i+l) ' ^ ^ In this series, the values of d'Ju, &c. and Dx must be determined in each particular case : for intervals of l'. Da? = 0,000290888208665721596, &c. A" 0'" The values of — — will be the same in every case ; these 1.2...W ^ for all values of m and w, from 1 to 12 inclusive, are given in the annexed table : V z);w=s \n + m- 1 ^ '^ 308 CALCULUS OF FINITE DIFFERENCES. A"o"' - , . Table of the values of , m and n beinor i> 12 t> t> t> > l> cc c-. t> t> t> \> © 1 © © 1 © © © © c © © i-i ►-10 « 1 © © 1 © © © © © © - bil- ;:i^-- © © 1 © © © © © © - - CI -> 00 © © © © © © © © - u,\ w -h Si- i^ *» © © © © © © © - IC ■fc"! C/T ^I- ll - 01 9. 1 j © © © - u|i-. ^1 w >u|of il" "■'I M -- b~. 0-. © © © © - w ^ «| a. 11 fe ©1 "" © : ^ © © © - ul-1 *'|3 *|S ccl oc il^ c ^ 4.> © OD ^ © © © - i(^ Elh H^. £l K _ 1- ; © © - 14|0 «|g c^l^ ^ 1 © 25 Kl^ © w JC g M 0: M or 1 - © ^ 0, ujt* — » m is OS cr to © ~ 0; 1 M I cc 2 - • 9. - u| = •^ %\ cc p II = 1 © © © ^1 Si ©: g c © •A © c © i - IS INVERSE METHOD OF DIFFEEENCES. 309 (18.) If the log. sines be calculated for large intervals, as for every 10°, the difrerences for every degree may be thus found : ^ - . ("sin Lv + Z>.J') — sin ,. sin.r , / Z)_sin.r \'' „ ) (2 sm 0? + 2>,. sin x -^ \2 sin x + D^ sm x/ ) T^ 01 • r (sinZ).*")- D' log. sm a? = — 2 ^ ^ ^„ -. rr— - "^ * ' \ (cos Dx)---^ cos (2 cv + Z> cr) (sin Dx) / (smD-r)- y^^^^) ^ \ (cos Z>cr)- + cos (2 X + Dx)/ j this series converges very rapidly. (Efic. Met. Trig. 194 — 209 ; L. C. D. 893 — 6.) Inverse Method of Differences. (19.) Integration of algebraical functions. ^^A^u^ — u^ + const. ^^a.u^. = a'2_^u^; if a is either independent of , + , - &c. + (-i)"-4a;-x-2;x+„.,-2,(a>,.2:x+„)}; ^ = w,2,t^, - A,«,(E,'y, + 2,«J + A>, (2>, + 2 2>,. + 2,1^ J - &c. 2f («,.0 = t^..2^,-2A,7...2^', + , + 3 A>,.2>, + ,- &c. -(-i)"-M2;(A>„.2>^„) + n2.(A;w,.2;^^wJl- &c. = &c. 2;K.i;.)=7^,.2>,- ^ A,«,.Sr^t^^,+ 'y^^ A>..2;+^r.^, -ecc.+^-ij .^ ^ ^^^ (r— 1) ^ ' "^ ^^ + ,«-i « 2,(a^i.J = S,„(-i)--^^3^.Ar-^^,+ ^5 2,K.zv) = S^(-l)"'-^-Ar-^^^..2;'tv+.-i + (-i)".X(A;t^,.2;t;, + J. INVERSE METHOD OF DIFFERENCES. 313 (Tr. L. App. 376, 7 ; /.. C. D. 9^9 — 62.) (24.) 2,?*,=/:w,-47/,+ -^d,%+ ^J..-^—^ +&C. V / . T ./x r 2 ^ 2 4 1.2. a = (6"— 1)-'W,. ^ 2,(6\wJ = 6^(6.e^'.-l)-^f/,. « 2:;(6^^^,)=ft'(6.e''^-l)-".^^.• {Tr. L. App. 378 ; L. C. D. 9^3 — 80.) .„ \m -\-r — '■2 yr-i , \ n + s — 2 m - 1 „ - Rr 314 calculus of finite differences. Equations of Differences. (25.) Tlic complete integral of an equation of differences of the 7i"' order O = 0(.r, U^, M^^j, &c. w^.^„) must contain 7i independent arbitrary constants. (26.) The general equation of the first order and degree is Let u, = V, .a^.a„... a^ _ , = v, . P,„ (a J , * then {v., + ,-v^)V^(aJ = b^,, or A,u^.P,„(«J = &.; .-. u^ = P„,(« J . ] 2, , " + const. \ . (Tr. L. App. 379 — 81 ; L. C. D. 1038.) (27) The general equation of the n*'' order and first degree is the integration of which is reducible to that of the equation which latter equation is always integrable if w — 1 particular integrals can be obtained ; and if m particular integrals can be obtained, it maybe reduced to an equation of the (71 — m)"' order. The general method may, to avoid complexity, be - illustrated by its application to the equation Let this equation be obtained by elimination from the equations * See^ppeudix. • EQUATIONS OF DIFFERENCES. 315 then \. = 'v^ + „+ \^. + 1 + ^v^ , From (3), ^.. = P,„(- '"J | 'C + E, ^-^' \ , x-l , 2^^ -J ^t.,=p,„(>x.)rc+2,i — - — , (5) -■- OTV m/ in which 'C, "C, and ^C are arbitrary constants. From these equations we may obtain by elimination u^='c.'u,-^'C."-u, + 'c:'u,+ r,, in which 'f7,, = P,„(- '^J, x-\ P (—-v \ P„X-^^J r-1 »^*=Pm(- O -2, j - — -^A- A 1 j j 'P.(-\«) ^p,«(-x,) p,„(-y„)«i If b^. = 0, then F, = 0, and the equation is the complete integral of 1^^ + ,j + '''.• ^^x + c: + '«x •", + ! + v.,. 7A, = 0. 316 CALCULUS OF FINITE DIFFERENCES. If this equation is known, then 'r,, -'u^, &c. may be deter- mined, for from (6) &c. = &c. these quantities being substituted in V^, and the result added to the complete integral of (2), the complete integral of (l) will be obtained. If n— 1 of the quantities f/^. are known, the n"* may be obtained from the equations (3). (28.) If the coefficients of (2) are constant, that equation will be satisfied by assuming w^. = e* ; then, dividing by e% we have e" + ^a.e"-^ + *a.e"-- + &c. + "-'«.e + "a = 0. Let gj, e„, &c. e,,, be the roots of this equation, then w^ = 'C.e^ + -C. el + &c. + "C. < is the complete integral required. If any of the quantities Cj, e,,, &c. are equal, or impossible, the changes which take place in the value of u^ are analogous to those in {Int. Calc. 43.) (29) If we wish to extend the integral of (2) to (l), then U = ^C.e' + &C. + "C.e^ + -, ;; -— . ^ " 1+ ^a + -« + &c. + "a {Tr. L. App. 382 — 5; L. C. D. 1036 — 52.)' (30.) An equation of the second order and first degree ^*.r + C + '«x-Wx + i+X-W'x = 0, (1) may be thus solved : assume z*^ = i;.,..P,„(-'aJ, and — 'a. ^.+2 = ^«.-y,- EaUATIONS OF DIFFERENCES. 317 then ^x + 2-i'r+i-«r + c-^'a: = 0' (2) "^X ^ ^T . Cj. Cj. _ I and = 1 + —^ = 1 + -" , •^"' = &c. Vx-i Vj:-, 1 +/rl_o = 1 + 1 +^1 +&C..../1 +c,' -^^^ being constant, and assumed equal to unity. Let this continued fraction be represented by F(c^, then will X P^{F(c,„)} be a particular integral of Vj. in (2), and ^C and "C being arbitrary constants: this value of v^ multiplied x-2 by P,„ ( — 'ff,„) will give the required value of u^. From this the value of u^. in the equation may be obtained : the result is «x=P,„!-'''„.-ffe,+,)!=< in which a constant must be added after each integration. (31.) The equation of the second degree, u^j,y.u^. — au^ + ^+hu^ + c = 0, (1) in which the coefficients are constant, may be reduced to z\ + o + (a + h) r, ^^ + (ah + c) v, = 0, (2) by assuming u^ = — — ^ + a ; let the complete integral of (2) be 318 CALCULUS OF FINITE DIFFERENCES. and let — =k, then the integral of (l) is e' + Ar.e/ If /> = «, and c=l, then ?^^ = tan { (a; + const.) . tan " ^ (— «) ~ * | . The equation w^ 4 j • tt-, + «a • W;r + 1 + 6^. w^ + c^ = 0, may be reduced to ^x + 2 + (^-«r + i)^. + i + (c.-«x-&x)«. = 0, (1) V by assuming «^^= — a^'- let ^i>^, ^y^, be the two particular V integrals of (l), then X + 1 + ^-X + i (32.) The equation of the third degree may, by assuming u^ = t3inv^.s^a, be reduced to the complete integral of which is v^ = Cj^. cos "I TT . ct? + Co . sin "I TT. cr, then u^ = v'a . tan (c^ . cos I- tt . *' + c^ . sin |- tt . a) . (Tr. L. App. 386; Mem. Analyt. Soc. 1813. pp. 84 — 95.) Equations of Mixed Differences. (33.) Equations of mixed differences, in which ?f^, ?<^ + i, &c. and their differential coefficients do not exceed the first degree, and in which the coefficients are constant, may be rendered integrable by assuming ii^,^=-v^ + k: the equation EQUATIONS OF MIXED DIFFERENCES. 319 f may by assuming u^=-v^ '- — be reduced to which is satisfied by putting ^^ = 6*"', and Cp Cg, &c. being the roots of the equation 1 +ae^ + hk-\- cke^ = 0, / u^ = 'c . ef + ^c.e^ + &c. — 1 + a The equation may be thus reduced to an integrable form: let m^ = ^,.u,, in which %j. is determined by the differential equation from which we obtain %^ = ^ , then the proposed equation becomes v,. + 6 r_j + ^ — rfjr_j + c = 0. (34.) The equation may, by assuming tc^ + i + «^-Wj =Uj,5 be reduced to d^v, + b^.v, = e,+ {d,a^ + a^.b,- cj w„ which becomes d^v, + b^.v, = e„ if d,a^ + a^.b^ - c^ = : if this latter condition is not satisfied, assume d^a^ + a^.b^-c^ = g,y d^v^-\b.v^—e. then u^= -^-^ — ?, and w^ + 1 = g. x + l and if — — — = a^, gs + i= c.^ ox Ox CALC'TLUS OF FINITE DIFFERENCES. the equation 7l^_^_^ + a^.u^ = ^)^ becomes d.% + 1 + 'ffx-c?^iV + ^x + i-^. + i + ^c,.v, = \, which is of the same form as the original equation, and with which the same transformation may be repeated, until we arrive at an equation in which when this condition is satisfied, a value of v^ may be determined from the equation d^t\ + h^ . v^ = e,, and from that, by successive substitutions, the value of tc^. If we begin by assuming the above transformations may be effected in an inverse order : the steps of the operation are analogous to the preceding. The values of .v obtained by the first method are i^. = A + ^.-c + C^. .p^, if ^, = 0, Sec. &c. and those obtained by the second are u^ = A, + B,.c + C,f,Q,-p„ if ^. = 0, u, = 'A:, + 'B,.c + 'Cj,'Q;,.p, + \/,'R^.p,, if V. = 0; &c. &c. in which p is any periodic function. (Tr. L. .S87, 8 ; L. C. D. 1256 — ^^.) Summation of Series. (35.) Let Wj + Uo + &c. + u^ be represented by S„,w„,, then X S„,?*„, = 2.w,^i + const. = 2,7^^+1 — S^=o^*a- + r SUMMATION OF SERIES. 391 (36.) s„(-ir-^n- ^ ^ l^ 1.2 1.2.3.4 ^ ^ am + b a 2(aa; + b) 2 (o^r + 6)" 4(atJ? + ft) + . ..4 — &c. + const. ^ ' 1 1 1 m^o r'" (rn—\)x'"-^ 2.r"* 2.1?'"+^ m(w,+ l)(w + 2)^4 ^ ^ ^ + — ^^ V,.., — 8ec. + const, y S,„log,m = llog,27r + (^ + 4)log.a.-.^+-j^-^, +&C. « (L. C. Z>.'990 — 1008.) + const. * 1 1 1 " /f w"'" ~ ^ ^'"^^;r+6~a ^^' "^ 2(aa? + 6) "^ '"^ ^ 2w(rta^ + 6)''" + const. y c _!_ I L J. lS (—\Y \m -" + const. X * S„ log, m = 4 log, 2 TT + Cx' + 4) log, oc - cc tgm 2?«(2 7rt — 1)^?"*""' Ss 322 CALCULUS OF FINITE DIFFERENCES. (37) The two following series will be found very useful in renderin»- the general term of a series integrable. 1 (t,""'" ni »j-m+l n-m-«+l,n = A^P,.K) H- S,„-7^.P.K+,.-i)-S,(-ir^A-o'"-^C,(a,). SUMMATION' OF SERIES. 323 (38.) Recurring series. The general form of a recurring series is by the integration of whicli the general term may be obtained, and thence the sum of the series. By tlie following method the sum of a series may be ob- tained from its equation, without knowing the general term : for .r write tr+l, and substitute then (l) becomes = aA,w, + „ + (a + a,) A,m, + „_i + &c. + (a + «i + &cc. + a,^_^)A^u^ + i + (a + a, + &c. + nju^ + ^, .'. "^^u^j^j^^ const. _ a^r + „ + (a + ai)u^ + „-i + ^^- + (a + fh + &C. + «„-i)m, + i a + «! + &c. + o„ (Tr. L. App. 390 — 5.) (39.) Application of the integral calculus to series. The sum either of an infinite or a limited number of terms of many series may be represented by a definite integral : the following methods are applicable to numerous classes of series, consistino- of ascending powers of some quantity, the coefficients of which are composed of arithmetical factorials. Let the series be 5 = at'" + (a + 6) t'" + " + (a + 2 6) t'" + "" + &c. + {a + (.r-l)6}r + *'-'*'': n \n-m-\ multiplymg by -^ , we obtam n f -H- III -\ t — t n p ji l—t" by difierentiating which, the value of 5 may be obtained. 324 CALCULUS OF FINITE DIFFEEENCES. If » = (a + b)(c + e)r + {2a + h){2c + e)r + " + &c. -\-(ax + b)(c.v + e) r + <' - ^> '* , . , . , W n(- + i)-m-l , . . J . then multiplying by - ^ '^ , and integrating, and again n „(5._l)_i . multiplying the result by -^ " " , and integrating, we nave ■ ft ^" ' I St ^* =# " from which 5 may be obtained, after two differentiations. The same method may be applied to the series of which the 0?"* term is {a^x + h^{a^x + 6„)...(a,^ + 6J r +<*-^K ^ fm + ji 4m. + (^ — 1)« LetS= r+ :+&C. + -, a + 6 2a + ax-\-b Cl { — hl)« — Wl then multiplying by — # " , and differentiating, we obtain a series from which may be deduced S= - t ft a ^* \ — V' t f Let S = -r- : H r— + &C. + (a + 6)(c+e) (2a + 6X2c+e) {ax-\-h){cx+ey then by a similar process we obtain If the 0?"* term is -— — — , then [ax + h){cx + e){fx + g) acff" ^ ^ If the tX""* term is ;: — , then {ax + by SUMMATION OF SERIES. 325 f s = l.2...(m— l)a'".<" l{(log.O"'-^/^"-^ -~(\og.tr-'ft'^o^.t.'^'' t b * ^" *^ %r —( f" n \ ^" 1" — 2" 3" — ' \c/t'=log|t c/w=-»/ 1 _|_ e" ' where 'U:=logj#. a + 6 2a + 6„ aa? + 6 Let 5= 1-\- — 2—t'' + hc.-\- 1f\ c + e 2c + e c.v + e e multiplying this by ct', and differentiating, and then multiplying 1 !l — ' the result by -#" ", and integrating, we obtain Let s = {a + b)t+{a-[-h){2a + h)f^-hc. + (a + 6)(2a + 6)...(a.r + 6)^% 1 --1 then multiplying by - ^" , we obtain the equation afd,s+\{a-\-h)t-\}s = {a + h)...{x-itl.a + h)f^^-{a + h)t. If the a?'" term is : , ^^— ^ — \ ; f, then mul- (c + e)(2 c + e) . . . (c.v + e) 1 --1 tiplying by - ^" , and integrating, and then multiplying the 326 CALCULUS OF FINITE DIFFERENCES. £ _ 6 quantity thus obtained by ct" ", and differentiating, the result is af ^ ^' ^ (c+e)...(c.x^+e) {Tr. L. App. 412 — 5; L. C. D. 1140 — 8.) Theory of Generating Functions. (40.) Let the indefinite series &c. + u_J-"' + hc. + ii_J-^^u-{- u^t + &c. be represented by 0(0 : this is the generating function of u^.. The generating function of ?«^ ^. „ is t~".d){t), A,r«*x - Q -l)0(O» a:u^... Q -iy, -(^-i) ^0(0; from the development of t~"(p(^t), when put under the form |l+ ( A] (f)(t), and of ( IJ cp{t), the values of u^ + ni and A"w^ in Art. 2 may be obtained. (41.) Let y w_^ be used to represent the series the generating function of which is («, n^ a\ , ^ -55, CO « 0(o = S,?*.r (5 V?''.r=S„,«„,-i?^. ■Vm-V THEORY OF GENERATING FUNCTIONS. 327 and let '^^u^ = a\7u^ + a^\7ti^ + ^-\- kc. + a„Vw, + „, &c. = &c. : then the generating function of V"'w7. is and that of A''v"'w^+p is If we assume z = t( l) , and develope — in a series of powers of z, we may obtain the formulae in Ai't. 8. (42.) A very general formula for interpolation may be obtained in the following manner. Let ar = a + — + — - + &c. + — ; y t t- t'" it is required to obtain an expression for t~", arranged accord- ing to powers of sr, and containing no higher negative powers of t than t~"\ Multiplying the numerator and denominator of the fraction ; by (a — %) v'" + «! ■u'" - ^ + Sec a,„ - 1^ + cf„„ and substituting for ^ its value in the numerator, we obtain + ^ («3«'"'' + &c. + «,„_iv + aj + &c. + a,„ J Vl — 1 '« /»J - 1 a u'" + a^v'" - '■ + &c. + a„, _ i-y + a„, — «*" m+l Q. m+l r- 1 328 CALCULUS OF FINITE DIFFERENCES. let ««"• + «!«'""* + &c. -i-ff,„_iU + «„= r, « and let Z,_, ,. represent the coefficient of v' in the expansion of P, then the value of — which is the coefficient of v" in the development of the above fraction is ^ = «1 ^O.n - m + 1 + «2 ^0,« - m + 2 + &C- + ^m ^O.n + sr(a,Zi,„_„,„ + i + a2Zi.„_2,„ + 2 + &c.+a;„Zi,„_J + %^{a^ Z2,„ _ 3m + 1 + «2 ^2.» - 3m + 2 + &c. + a,„ Z2,„ _ „,„) + &c. !«2'^0,«-m + l + «3^0,«-m + 2 + &C. + a,„Zo,„_i + Z(a2Zi,„_2„, + i + a3^i,,,-2m + 2 + &c. + a„,Zi,„_„,_i) + &C. r a3^o.n-m + i + &c. + a^Zo,„_2 + Lj +Z(a3Zi.„_2;;, + i + &c. +a,„Z,,„_^_2) I + &c. + &C. multiplying by 0(0 j and passing from the generating functions to the quantities which they represent, we have + V ^x{«l^I,7.-2m + l + «2^1.«-2m + 2 + &C. + a„,Zi,„_,„} + VX { «1 ^2,« - 3m + 1 + »2 ^2.« - 3m + 2 + &C- + «m ^2.n - 2m } + &C. TO+1 1 "t 1 CO m— p+1 THEORY OF OEXKKATINO FUNCTIONS. 329 + w^ + , {a. Zo.„ _ „,+, + a, Zn.„ _ ,„ + c + &c- + «m ^o,» - . } + V "a- + 1 1 ^2 ^l.n-2,n+l + «3 ^l.n - 2m + 2 + ^^^ + "m ^l.n-tn-l I + &C. + W^, + „ _ 1 a„ Zo.„ _ „ + 1 + V «^.r + ,« - 1 ^'m ■^1 ,« - 2,« + 1 + &^- ' If a value can be assigned to q that will render y'?*^ = 0, then the above value of 21^. + ,^ will consist of a finite number of terms. If yw^ = 0, the above series will afford a complete solution of a linear equation of differences of the n"' order : in that case the quantities u, u^. See. m„_i, represent the arbitrary con- stants. (Laplace, T/i^or. Analyt. des Proh. Pt. 1 ; L. CD. II09 — 33; Tr. L. App. .346 — 9) m 05 m—p+l Tt FUNCTIONAL EQUATIONS. Reduction of Functional Equations to Equations OF DlFFEUENCES. (1.) This reduction can in very few cases be effected, so that - the result may be an integrable equation of differences : in the . equation = F\.v,^f.{a,(.v)],^l.{a,(^)]\, (l) in which F, a^^ and a.^, are known functions* assume in which x is a function of ,v. Eliminating a: from these equations we obtain from which the function

M^(^i,...^«-l)i J an equation of partial differences from the solution of which we may obtain i^\(p(^i,--.x„)\ =v|,jai(.r)5 =F,{z„...z,,, Zp Z, &c.) in which Zp Z„, &c. represent the arbitrary functions p^,, he. in (4). If the values of ij, Zo, &c. *„, obtained by elimination from (4) be substituted in this equation, F, {z^ &c.) will become a function of ai(.i), and the function \// may be determined. (3.) The general equation containing two variables 332 FUNCTIONAL EQUATIONS, may be treated in a similar manner by assuming two systems of equations, which may be solved in the same manner as the equations (2). If a(/,(a?) = /3(a>).x/,{a(a.)J, (l) and /, of the equation ''l^(v) = ^{a(x)], then the general solution of (1) will be If /„ is a particular solution of the equation then the general solution of (l) will be Let f{x) be a particular solution of the equation then by assuming \//(ci') =f(x) + (pi^), this may be reduced to the preceding case. By a similar assumption, the equation = >/,(a?)+/3i(a?).v/.Ja,(a?)j + &c. + /3>0-^{«„(''P)i + 7^^) may be reduced to = x/,(.7') + ^\(.v) . V, ) a, (.r) j. + &c. + (i,X^v) . x/, \ a,X^) \ . (5.) Given \|,(.r) = >|r)a,O^OS = &^c- = "^1««0^)S' °^ ^^ *^^ same system of equations may be more conveniently expressed, to determine the function \^: let/p/g, &c./„ be so assumed as successively to satisfy the system of equations 112 m then ^^(.^•) = 0|/J..•)/l(•^')^•.i, n n-1 1 in which (p is arbitrary, will be the general solution. If it be required to determine \|/ from the conditions 384 lUNCTIONAL KtiUATlOXS. let/,, y!,' ^^■- ^^^ ^" assumed as to satisfy the equations I 1 m m 1 1 HI m I 1 m-1 ni m m n n n then ^('V) = (l>\fn-- \A \a,i...u,X^)...\ If a general solution of the equation be required, having given \^(,r)=/(.r, a^, a, -,,... a^), a particular solution containing r arbitrary constants ; let (p be determined such that then the general solution will be in which 0i, &c. are particular values of (p. (6.) Functional equations of the second and higher orders. The equation ^//'(<^?) =-3^ will be satisfied by any value of v// obtained from the equation (l)\oi;,ylf{x)\ =0: let /be a particular value of \^ in this equation, tlien the general solution will be (h being arbitrary ; it is however necessary to observe that those values only of (pT^ should be taken, which satisfy the equation Iff is a particular value of \/y in the equation then the general solution will be (iENERAT. SOI.UTIOKS. 335 Let the proposcnl equation be \//'(.i) = a(^>) ; assume then r-{|/"(x') = u{v) may be reduced to /" { (p(.v) } = { a(.i') } . Given ^Z' ) «{ "^j^C^O } \ = '^' ■ assume x/,(*) = (j)'' \ f{ 0(a) } j , then a \ /,"-{ a,(.v) } , . . . ^y' { «„ _ ,(.r) } \ may be reduced to an equation of the first order = F\ cp-\y), -' {f(y) ],...|a(.r)|, &c. \//^a"(a){, may be eliminated, and the resulting equation will be 0=:/5.r, a{x), a%v),,..a'\a>), v//(a7)} , from which the function \// may be found. This method is inapplicable if the proposed equation either does not contain x ; or is homogeneous with respect to ^^^fijc), >// |a(.r) \ , &c. ; or can be put under the form unless fix) =f { x, a(v), a.-(x), . . . a"(x) \ . (Babbage, Phil Trans. I8I6, pp. 229 — 34.) (8.) Functional equations containing two unknown quantities. Let the proposed equation be and let/p/o, be functions which satisfy the equations ./■l(^)=/^laG^)^ fM^f.my)], then the general solution will be If/ is a particular value of \^, the general solution may be put under the form The general solution of the equation y\^{x,y) = x// ja(c^■,y), ft{x,y) \ (1) is -^i^^^y) = ^ \fii-v,y),Ma^,y) \ , in which/, and/„ are particular values of v|/, and ^ arbitrary. FUNCTIONAL EQUATIONS CONTAINING TWO VARIABLES. 337 The general solution of the equation is i^('V,y) =f(a',y) . x/, {f,(i,y),M^v,y) \ , in which / is a particular value of \^ in the proposed equation, and /,, /„, particular solutions of (l). The equation x/,(.r,2/) = 7, G^,2/) • f \ai^v,y), ^{w,y) ] + y^{x,y) may be reduced to (2) by assuming in which f is a particular value of yj/. (9.) Functional equations of the second and superior orders^ containing two variables. The equations y\f^'\ce,y) = x, i^^'\v,y) = a{,v,y), may be treated in the same manner as the equations x//2(a?) = .r, \//%r) = a(.r), arbitrary functions of y being substituted for the arbitrary constants contained in the general solutions of the latter equa- tions. A general solution of the equation \^"(.r,?/)* = a is in which ^, , and Y being as above, and ^ determined from the equation The same solution applies to the equation ^ being in this case determined by the equation r(^)=7(^)- The equations a: .^^'^(x,y) = y .\j/^'\x,y), yl^^'\a;,y) . yl^'%v,y) = ocy, y{x,y^^'\a!,y)] =y\-^'''\x,y), y\ , may be solved by assuming ^{x,y) = {y)]\, f being determined by the equations f'\a>,y) = a,, f'%x,y)^y- The equations w.\\/'^(a!,y') = a.y\/'^{x,y), y{y\r{x,y),x\ =y{a,ylf'''\x,y)], may be solved by the same assumption ; / being in the case determined by the equations nx,y) = cp{a), f'\x,y) = x. (Babbagey Phil. Trans. 1816, pp. 184 — 222.) Differential Functional EauATioNs. 339 (10.) Given the equation >//|a(//(*'), and its differential coefficients may be eliminated. (11.) Given >//(// {a7,a(i/) ] , in which a-(y) = y, then >//(a?,y) = 6^0l I y, «(?/)! + e-'{a(y)--y\ .(p^{y, a(y)\, is the general solution, in which <^, and 0„ are arbitrary functions. If \l/(>oiati'm. 342 APPKNDIX. *' To employ as many symbols of operation, and as few of quantity as possible is a precept which is now found to ensure elegance and brevity."* In addition to these maxims it nmy be stated that those symbols will be found most convenient, which contain the greatest number of indications, and which, provided such in- dications are clearly and distinctly expressed, are calculated to afford the most effectual assistance to the memory. The various branches of analysis have been discovered at different periods, and enriched by the labours of different individuals, and it may not therefore be unreasonably as- sumed, that the modes of notation which they have respectively introduced in their writings, although perhaps the best adapted to the then existing state of the science, should yet be suscep- tible of considerable improvement, in proportion as the mutual relations of distinct and hitherto unconnected branches of analysis are more clearly and accurately investigated. The different systems of notation which have been adopted in the differential calculus appear to involve, to a certain extent, the different theories on which that calculus has been founded. Among these, the establishment of the mutual relations of certain functions by Lagrange is perhaps the simplest, and at the same time the most disencumbered of considerations foreign to the subject : although on the other hand the symbols by which those relations were originally expressed are liable to almost as many objections as any that have been devised. The method of Lagrange may be thus stated : let 0(a?) represent any function of .r, and let cV be supposed to receive an increment Dx, then the difference between (p(.v) and (p(a! + Dx) may always be developed in a series consisting of ascending powers of the quantity Dx multiplied by some functions of oc and of any other quantities contained in (p(x). Pielacc to the Memoirs of the AnaUjlical Sociely, Cambridge, 181.?. y APPENDIX. 343 The coefficient of Dx in this development has been called the first differential coefficient of 0(a?), and the coefficients of ~, &c. -^: — , the second, &c. m'*, differential coefficients 1.2 1.2.. .m respectively : for these the symbols proposed by Lacroix have been adopted in the preceding pages. On whatever hypothesis the differential calculus may have been established, the correctness of the results obtained by the application of the inverse process, or integral calculus, must necessarily depend on this one axiom ; that two essentially different developments of the above-mentioned form cannot be obtained from the same function, or in other words, two different functions cannot have the same differential coefficient ; or the same differential, or Jluaiion, according to the theory adopted. The object of the various artifices of the integral calculus may be stated to be the separation of any proposed quantity into a sum of other quantities having the form of previously ascertained differential coefficients, from which the corresponding integrals are known by an inductive process. Considering the intimate connexion that exists between the differential calculus, and the calculus of finite differences, it is obviously desirable to render the analogy between their symbols as complete as possible : it is with this view that the previously adopted symbols ~^ or A~^ ; they may however be retained for the convenience of writing, more particularly as they involve no breach of analogy. With regard to partial differential equations, the notation that has been adopted in the preceding pages, though it appears more in accordance with the general system, and with other parts of the calculus, than that ordinarily used, must never- theless be acknowledged to be far fi'om satisfactory. The nature of these equations, and their connexion with the general theory of functions is probably not yet sufficiently understood, to render them subservient to any general system of notation. In denoting the powers of the trigonometrical functions, the notation of Arbogast, as (sin a)"', has been adopted. The index has been placed in the same situation by Gauss, Delambre, and others, but the parentheses have been omitted by these writers, without perhaps sufficient reason ; for although the trigonometrical functions of the powers of any quantity have not hitherto been the subject of investigation, there does not appear any reason why they should not hereafter form the basis of important analytical relations, in which case the introduction of parentheses would be indispensable : and it appears to be a desirable object in the construction of symbols to render them applicable not only to existing circumstances, but to provide as far as possible for those cases which we have it in oui* power to contemplate. As the sine, cosine, &c. may be considered merely as functions, independently of all geometrical considera- APPFXDix. 345 tions, and as sin"'.!?, &c. have been used to represent the corresponding inverse functions, tlie more usual notation of sin"'if cannot longer be retained without an entire disregard of that uniformity of system, which it appears so desirable to establish. The convenience of denoting the value which any function , these functions d'" usually appear under the form , in these cases the •^ ^^ 1.2. ..m adoption of c?"*: instead of — will afford an additional facility, thus ^/ 1.2...W. ' " 1.2. ..m. 1.2. ..n APPENDIX. 347 " The letter S may be taken as an abbreviation of the word »» snm, thus S„, ^/,„ will represent the sum of n terms of a series of which the w'" is «,„. It follows from this that the symbol S,„ S„a,„ „ will correctly represent the sum of a series a consisting of r terms, of which the m"* term is the series S„a,„,„; and the same notation may be extended to any number of symbols of summation." A method analogous to the preceding has been devised by Mr. Jarrett for expressing by means of brackets the relation that exists between the different parts of a formula, when they are not connected either by addition or multiplication. An index is placed over the first bracket to denote the number of parts of which an expression consists, and a second bracket is so placed as to exhibit the connexion between two consecutive parts : thus r r+l r r+1 n+1 = \a, + b,{a„ + b^\...{c\...\, 1 2 3 n+l n+1 1 = {a,+ \b,M„_+{h,.b„.a,+ \...+ {b,.b,..\_;.a,^+{b,.b,..\.c\...], 12 3 4 n »»+l n+l ' =a^ + b,.a„ + bi.b„_.a.^+ ... -^b^\...b^_^.a,^ + b^.b^^...b„.c, n ni— 1 m :^S««,„.P..(6.) + c.P,(6,). The brackets may be rejected after the expansion, when- ever they are found not to possess any analytical significa- tion : as in the preceding example. In many investigations, particularly those connected with the theory of equations, it will be found convenient to denote by a symbol the sum of all the combinations of n quantities taken m at a time. Let therefore the letter C be taken as an abbre- 348 APPENDIX. m,n viation of the word combination, then C,(a^) may be used to denote tlie sum of all possible combinations of n quantities of which the r"' is ff,, and of which m are to be taken at a time. In page 294- will be found the application of another symbol which lias been devised by Mr. Jarrett to express what may be termed the symmetrical combinations of quantities, and which may be thus explained. Let there be two series flj, a„, &c. a^^, fej, ig, &c. 6„, each consisting of n terms, and let every possible combination be formed out of the first series taken w at a time ; tlien if each combination be mvdtiplied by w — m terms of the second series, such that those indices subscript which did not occur amongst the terms of the first series, may be found amongst those of the second, the sum of all the combinations m,n—m thus formed may be denoted by C^,^(a^.6,). This will probably be best understood by an example : ^r,ti^r-h) = «i«2^3^4^5 + «i«2^4^3^5 + a^a„a^b^b^ + a^a^aj)„b^ + a^a^a^b^b^ + a^a^a^bri^b^ + a^a^a^b^b^-^ a„a^a^b^b^ + a„a^aj)^b,. -\- a^a^aj)ji„. Some general theorems relating to these symbols will now be given, by means of which various analytical operations may be considerably curtailed. The demonstrations of some of these theorems will be found in the Paper by Mr. Jarrett above referred to : the others may be demonstrated in a similar manner. n » [1] (1) PM(0 = P.»(«n-n.+l); by this theorem the order of factors in a given factorial may be inverted. n r n-r (2) P^ (a„,) = P,„ {a J : P^ («,+„J ; by which any number of factors may be separated from the rest. (3) If -^^ = c^, m being any positive integer, then a f»« = «vPr(0- APPENDIX. 349 (4) l{^=c,„ then m m-1 m-1 «2m-2 = « • Pr (C2r-2) ' and aj^,, = »!.?, (c^^.i) . (5) If 6 is independent of m, then (6) P,„ (a,„) = 1 , whatever may be the form of a,„. The following formulae very frequently occur ; they may be considered as particular cases of the preceding : (7) \a=\a + (m-l)r ^ ^ Lli i n ni,r m. (11) il« = i; \m \n — m (8) \a = \a.[ a + rs ; (12) [« = 1, and [o= 1 ; m,r s,r m—s,r (9) [«=la6.^; (is) \a_ ' m,—r I- m [2] (1) To invert a given series; n n (2) To divide a given series into two parts ; n f n—r (3) To separate the even and odd terms of a series ; 2n n n 2n— J n n— 1 S„«,„ = S;„«2m-i + S^«2m» and S,„C,„ = S;;,a2m-1 + S^Og^. (4) If b is independent of w, then n » (5) (S,.0(SA) = S^«.in*n. 360 APl'KNUIX. (0) If r is independent of n, and s of m, then r X it r CO a 00 m (7) S,„ S„«„,,„ = S,„ S„a„,_„^.j^„. CO '«! «'*-! = S„,j S,„^. . . S„,(p(m, - m„ + 1 , . . . w,_, -m, + l, m,). (9) If ««+i-«,„ = ^„, then n «„+! = «1 + S,„6,„. [10) If «^+i + «,„ = &,„, then m n m m— «+l 11) S„ S^ a„^r = S« S^ ««+,-!,»•• TO TO— n+1 12) =S„ S^a„+y_i,„. 1 3) S„ S, ff'„,,. = b,j b^ (tt2n-r.r + ^2n-)+l,»') ' 2m— 1 n m— 1 n 2»i— 1 14) S„ SrCtn,y = S,j S,. {Cl2n-r,r + ^2«-r+l,y) + Sr«2m-1.,- • [3] [*] CO TO CO oo 15) 8^8,1 a„,,n-6„ = S,j&„. S;;ja„,+,i_i,„. 05 » CO m 16) (S,a^_i..r'"-^)(S„6„_,.^'"-^) = S,„ct.-^S„a_,.6„_, l) If 6 is independent of r, then m,n m,n m,n n n-m,n / J \ 2) aK)=p.K).a(-) n— m+l,TO n—m,m n—m+l,m—l n—m,m » »">*• /fe \ 2) C,,(«,..feJ = P.K).C,(-^). APPENDIX. 351 Of the particular theorems which will here be considered, the first, and probably the most important, is the binomial theorem ; the true development of which as first given by Mr. Swinburne and the Rev. T. Tylecote,* and as more compendiously demonstrated by Mr. Jarrett,t will afford a solution of all those contradictory results which have insinuated themselves into modern analysis, in consequence of neglecting the remainders in the development of diverging series : to obviate which, some writers have proposed a distinction between the mathematical signification of the term eqiiality, and that which is implied in the ordinary acceptation of the term. Cr-iy ,,*-! » Lf f L n being a positive integer, (a— 1) .'F-"=1 + SJ-W. + (l-a;y-\S ^n + t-s .y 1 n r+i - [m = 0, j;=^_i j ; and n'+^ . I - = [w - t+T l^ r+l r+l n \ s-l L_ L__ \„ ; r+l m r i^-|5-[7.-S.-[7^-bhi(!:::f±i)-(--('--+0) * The true development uf the Binomial Theorem, Cambridge, 1827. t Otmbridge Philosophical Ttiinnnclioiis, Vol. III. 352 Al'l'E>iDIX. The following theorem, which is nearly allied to the binomial, may easily be demonstrated by means of the theorems already given : la + 6 \a . \b L- — : ti+i 1— I — n,r CJ n—m+ l,r m—\ ,r — t3, \7i '" Iw — m + 1 . Im— 1 If r = 0, this series is reduced to (/3), p. 15. (L. C. D. 987.) The polynomial theorem has been investigated by Arbogast, but with the assistance of a very cumbrous and obscure notation : the symbol devised by Mr. Jarrett, which has been adopted in the preceding pages may be thus explained. Let the character •nr be taken as an abbreviation of the terra polynomial coefficient; then the coefficient of x"* in the polynomial function may be represented by •z3-„"'0(a). The index subscript of -nr is the letter according to the indices subscript of which the different powers of x ascend, and the quantity following the functional symbol is the term independent of x in the series If the index subscript of -zar is omitted, that letter is under- stood which immediately follows it, and if the function is a power of the polynomial, the parentheses including the first term of the polynomial may also be omitted : thus CO 'sr^ + a-.-ar'-a^, = S^-; ; '- . -; , Since w"'al=0, •^ \n—m+T-\ I TO — r + 1 [2](i), =S.- The development of the first eight polvnomial coefficients has already been given, (pp. 17 — 21.) The value of the polynomial coefficients, in the particular case in which ?i = — 1 , may be obtained from the formula given in Note y, p. 128 ; they may however be more readily deduced in the following manner : let tjr"'a~' = 6„,, then l=(S.«„_,..r"-')(S^6_...r'"-'), m from which we obtain 6,„= — ^/~'.S^cr„_^^.,.6^_i. The actual values of some of the coefficients 6, 6p b„, &c. may here be given, as the several terms of many series may be expressed by means of these quantities. Yy 354 APPENDIX. fe=a~'; b^=—a~^.a^; i^ = -f- a~'.a,' — a~'.a, ; 63 = — a~' .03 + 2 o~^ .a^.a,^ — a~* a^ ; 6g = + a-^ of —5 a-*. a * . Og + a"® (60^.0^ + 4 af . ag) — o-*(6a,.ag.a3 + 3af.a4 + a2^)+a-^(2ai.a5 + 2o2.a4 + a3^) — a~'a"; + 4 a"° (3 af . ff 2 • «3 + «i • «2^ + «f . a^) — 3 a"* (a^ . a^ + Og^ . a. The two following theorems may be placed here on account of their connexion with the preceding : < CI m— !,)• V -^ Q n—y,r \ Q m—\,r , (^'" [m-1 W^« \n-l ) ~ "• |m--l ' l^"' \m-\ S ~^"' \m-l ' the first of these is Stainville's theorem. Values of the logarithm of a factorial : log, [a = w (log, a-l) + (n-^+ -^ -^ log, (^+~A> r" = 7ilog,a-S„,(-ir-^-— .s. s" ma The three following theorems frequently occur in the theory of equations : n ^ n+1 n—m+l,n n t—l s,n n t,n APPENDIX. 365 nt,n nH-1 I i— l,n \n — m » n+l n— m+l,m— 1 from this very useful theorem we may obtain the following : see Trig. (31.) n CO n— 2m+2,2m— 2 COS S//, = S,„ ( - 1 )"'"' . Q, (cos ov • sin a,) ; n ea n— 2m+l,2m— 1 sin S,a, = S,„ (- I )""' . C,,,(cos a,. sin a,) ; n » 2ni— l,n « sinS.a. S„,(-l)"'-^C,(tana ,.) ,..,... tan b,a,= „ — = ^^,^zo;;i , by [4] (2). cosS.rr, S,„(-ir-'.C,(tanaJ The series (a, p. 294-) may likewise be readily deduced from this theorem. The theorem (Note a, p. 196) may be thus demonstrated : d":0(w) is the coefficient of (Dxy in Ej^(p(u), and E^(p(u) = (p{u + D^u), e, p. 197, = s„rf:-':0(w)- (A ?')"-'. ■=kdr'-(p(u)\L(D^^y"dr.u\""* [2] (15), = S„(D.r)"-'.S„//r'":<^(«)-2r'"-^«"-" ; = S,„rfr"'^':0{f/).7ir'"-^a'-'"^' + 0(M).'nr''a\ 3|§ APPKNDIX. = S„. d;-'"-"': 0(7/) . ©■'"-'«"-"*+', since tir" a° = 0, The following* theorems may be proved in a similar manner : {a) . ,;--;;—- • n The first of these may be applied to the following examples : eo =0 „ m-1 n-»n+l m—\ ., n— m+1 n—m+l ' The following theorems will be found very useful in integration : ./• {u-v) = S..(- 0'"" . dr' " . /p" r + (- l)'/(c/^. ./^ .). If a,, = «^^ _ . .6 -;- a^, _o-c, n being a positive integer, then ^ I /?. — /;i — 1 ^ bi — pi — o ■ I m — 1 I wi — 1 If a^^=^h,^ + c,^-(in+f> where t is any quantity whatever, then •■ m-l >• APPENDIX. 357 The following may be added to the theorems already given in finite differences. ct(a)-§ \v 5^5!i^) ^(^')=s.«^^zn-^(^ + ^)^"'" ,« — m + 1 )L — ni 4- 1 X);" Lr = i±w. (2),a7)"'. [a' + mDx _ +n,Dj m +n—m,Dx '' D^ \w = I ±n . ( - Z).r)'" . [^ mDai ±n,—Dx m +n—m,—IXe For the theorems which have here been given, as instances of the great power of this system of notation, the Author is indebted to the unpublished researches of Mr. Jarrett : many of these are entirely original, and the others appear under such an altered form, that a reference to the authors from whose works they have been taken would not at all assist the reader in deducing them. In conckuling this subject, a slight modification of the symbols P and C may be mentioned, which Mr. Jarrett has found to afford considerable facility in many important ti;t m,n;t investigations : the symbols P^ (a^) and C, (ffy) express the n m,n values which P^O ^"^ Q(^'r) respectively assume, when a^ is omitted, wherever it is found to occur. This will be best understood by an example : given to find 7,. {Cauchy^ cours cC Analyse, Pt. 1, p. 71.) 868 APPENDIX. Multiply by C,( — «J, then n—r,nit n n—r,n;t fcr-i • Q (-0 = S„^^„. . <-' . C, (-a.), H n~r,n;t n n n— r,n;< = S„.A'^.P,(o,„-0- But Pr(a,„ — a^) = O5 f°r every value from m = 1, to w = n, except when m = t\ n n~r,n\t S,6._,.C,(-«.)=<^.-Pr(«.-«r). n n~r,n;t , S,&,_i.C,(-a*) and «,= —^^ . If 6^_, = 6'-», then UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. 25Sep51LU 19: U.A •d:ii LIBRARY USE OCT 11 1954 OCT 1 1 1954 lU REC'D Lp MAY 1195fl LD 21-95»i-ll,'50(2877slC)476 REC'D LD DEC ^^ JM 2 1 1^ MAY 1 7 1390 APR le mQ CiRQULAtiori