HfTTH VJb4 CORNELL UNIVERSITY LIBRARIES Mathematics Library White Hall CORNELL UNIVERSITY LIBRARY 924 059 55 62 DATE DUE 1 CAVLOffO PRIHTeOINU.t A. Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059551162 Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39. 48-1984. The production of this volume was supported in part by the commission on Preservation and Access and the Xerox Corporation. 1990. A TRACT ON THE ADDITION Or ^' ELLIPTIC AND HYPER-ELLIPTIC INTEGRALS. BY MICHAEL EGBERTS, M.A., ERASMUS smith's PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF DUBLIN. DUBLIN: HODGES, FOSTER AND CO., GRAFTON STREET. PUBLISHERS TO THE UNIVERSITY. 1871. S 4 ." ^ CAMBRIDGE : PRINTED BY W. METCALFE AND SONS, GEEEN STREET. PREFACE. This little work was written with the view of presenting in a simple and elementary way the fundamental properties of those transcendents which, in the ascending order of classification, immediately succeed elliptic functions. I have adopted as the basis of my investigations the following memoirs of Jacobi, " Considerationes generales de transcendentibus Abelianis" {Crelle, vol. 9, p. 394), "De functiouibus quadrupliciter periodicis, quibus theoria transcendentium Abe- lianarum innititur" (Crelle, vol. 13, p. 55), " De- monstratio nova theorematis Abeliani" (Crelle, vol. 24, p. 28), and of the greater part of these memoirs much of what appears in the following pages may be regarded as a free translation. By the employment of the method given in the last of the above-mentioned memoirs, I have con- PUEFACE. structed a trigonometry of the functions in ques- tion, bj which the reader is introduced to the study of them in the same way as he has already become acquainted with the properties of circular functions by the formulae of ordinary trigonometry. Since the appearance of Jacobi's papers, the subject has received large develop- ments at the hands of Riemann, Weierstrass, and other distinguished mathematicians, and I venture to hope that what I have put together may invite the attention of mathematical students to the higher theories and more abstract prin- ciples of the Grerman analysts. The perfect uniformity of treatment of elliptic and hyper-elliptic functions, by which their fun- damental properties are derived from the em- ployment of Jacobi's method, is shewn in the second, third, fourth, and fifth chapters; the sixth chapter is devoted to an accurate exami- nation of those cases where the first class of hyper-elliptic integrals depends on elliptic func- tions, a question which appears, ever since the subject was started, to have been invested with interest, and the analogues of Fagnano's theorem, which I have given in the seventh chapter, bear PREFACE. out the well-known analogy between conic sec- tions and the lines of curvature of an ellipsoid. I have to acknowledge, in the first place, my obligations to Mr. Cathcart, Fellow of Trinity College, Dublin, for the readiness with which he undertook the revision of the proof sheets, by which greater freedom from typographical errors has been attained than could have been secured without such aid, and also for sugges- tions of which I gladly availed myself. I have also to thank the Board of Trinity College for their liberality in contributing to the expense of publication. Teinitt College, Dublin, June 21, 1871. CONTENTS. CHAPTER I. PAGE Jacobia proof of Abel's theorem . .... 1 CHAPTER II. Applictvtion of the general theorem to the case of elliptic integrals . S CHAPTER III. Application of the general tlieorem to the first class of hyper-elliptic integrals ......... 13 CHAPTER lY. On the periods of the elliptic integrals . . . . .S7 CHAPTER V. On the periods of the first class of hyper-elliptic integrals . . 4U CHAPTER VI. Cases where the first class of hyper-elliptic functions depends on elliptic integrals . . . ... .5.? CHAPTER Vn. On the rectification of the lines of curvptiire of an ellipsoid . . 69 CONCLUSION. General remarks ........ 80 APPENDIX. Note (A) 81 Note (B) . . , 81 ERRATA. Page 72, line 2, Jor P (- .-, ^^^^^ , &^, .", «') Page 82, line 3,/o)- Mathematica, rearf Materaatica. „ „ line 9, /or (o' + P'y) oij, rencj {a' + ^'ij) dy. TRACT ON THE ADDITION OF THE ELLIPTIC AND HYPEBELLIPTIC IN^TEGEALS. CHAPTER I. JACOBl'S PEOOF OF ABEL'S THEOREM. 1. The functions which I intend to consider in this tract are comprised under a remarkable class of transcedents, namely, those ■whose differential coefficients are algebraic. The solution of the problem of their addition depends in general on the determination under an algebraic form of the integrals of a system of differential equations of the first order. Suppose, for instance, that we had no previous knowledge on the subject, we could derive the fundamental property of the function of x whose differential coefficient is - from the addition of two such functions. Let/iz) =1 — : :t J , X then the differential equation — - -i ^ = has a transcen- dental integral /{•r,)+/'v-^s) = '^'; ^ ^^ ^ ^^^ algebraic integral .t^x^ = c'. The constants in either form are deter- mined by the supposition that we are acquainted with simul- taneous values of x, and x^. If when x^ = l, x^ becomes a, we have/\.r,) +/(.r,^ =f,^'', -f.-^'s = °h ^^ o° *^^*^ equations the whole theory of logarithms can be based. Again, let F[x)=l , , and the addition of two func- tions of the form F(x) depends on the determination under B 2 JACOBl's PliOOF OF ABEL'S THEOKEM. an algebraic form of the integral of the differential equation -^L- + ^°, = 0, which is easily seen to be 5±^ = c, and l + x/l+a;/ ' -^ l-a.a;, if a is the value of a;^ when x^=0, the transcendental equation F{x,) + F{x^)=F{a) connects a;, and a;, by the algebraic relation ^'"^^^ = a, and the whole theory of circular fiuic- 1-x^x^ tions can be derived from this property. After this illus- tration of what is meant by the addition of transcendental functions, I proceed to the consideration of Jacobi's proof of Abel's theorem, which contains the solution of the problem of the addition of a most important class of transcendents. 2. The analysis employed in this investigation involves some properties of rational fractions, which, in the first place, I proceed to demonstrate. Let {x-x^){x-x^)...{x-xj = (i){x), let also -^ [x) denote any rational and Integral function of x ; then, by the known theory of the decomposition of rational fractions, tM = A + 2^^tM,—L- m {x) (j}' {x^) ' X — X, '' where .<4 is a rational and Integral function of x, If the degree of >fr [x) exceeds m, or the number of the quantities a;,, cCgj i>' [x] denotes, as usual, the first derived function of ^ (a;), and the sign of summation 2 extends to the quantities Now also dx^dx^...dx,^ (<^'(a;,) {f(^.)}' d" dx,dx„...dx^. ■yjr {x)] _ yjr (x) h {x)i {.^(a:)f' 80 that operating with -5—^ -5— on (1) we find i^jx) ^ d-jA) .^dr i fix, } 1 {4, [x)Y dx,dx,...dx^ ^ dx, l{x - X,) {0' (x,))'J ' wh ence JACOBI'S PEOOF OP ABEL'S THEOBEM. ±{x) _ d"'A , ^ I jrjx,) {4>{x)Y dx^dx...dx. + 2 + 2 {x-x,f{'[x,)r 1 d fix,) {x-x,)dx, [j>'{x,)Y- It follows from this last formula, that if [4> H}^ ij be developed by descending powers of x, the coefficient of - is SO d_ fjH^Ll dx, l{-{x,)Yi' Now let f (x) =A„ + A^x + A^x^ +. . .+ A^^^^x^' then ^(^) £" + 0.-- +-^ x" {ix)r- IJFirFW' and the right-hand side of this equation being developed by descending powers of x, the coefficient of - does not appear : 80 that when f (x) is of the degree 2»i - 2 in a;, d_ f{x,) _ 3. I now propose to lay before the reader the manner in which Jacobi has determined the algebraic integrals of the following system of equations: x^dx, V{/K)} " V{/(^J} x„dx„ xdx^ g 3 +...+ = V{/W} xr'dx, . xr^dx. Tr + +...+ ■'dx. ^ =0 (2), V{/K)}'"V1/K)} VI/lo'J} ■where /(«) denotes a rational and integral function of x of the degree 2wi-l. If the quantities a;,, a!^...a;„, which JACOBl'S PROOF ' {x^) ' ■"' where yjr [x) is a rational and integral function of x of the degree 2m -2. Whence, by differentiation with respect to t, we find d r 1 ^,'1 ^ ^f{ia.-x,)^ix,)} d_ 'JWJx,)] rf< LV(ar-«,) dt\ '[x^) dx, '{xy the latter summation extending from x^ to a;„, ; but we have d I _ 1 dx^ ' (a?,) ~ (aj, - a;,) 0' (a;,) ' , 1 ^ r 1 dx.l '^^^''''^ V(a.-a;,) ^< LvKa.-a'.)} dt \ Id yjr (a;,) 2 dx, {f (a;,); Similarly TI+S d_ Via, -a;J rf< 1 d ^K) i+S 1 ^ a, -a;, this last summation extending to oJj, a;,...a!„^ t.e, to all with the exception of x^. Hence, taking the sum of all m similar equations, V(a, - ^,) dt LV'(a, - K,) (?< J 2 ^, IfK)}" fK)<^'K) V{(a.-a^.)(a.-^;)} JACOBl'S PROOF OF ABEL'S THEOREJL. But yjr (x) is of the degree 2m — 2 in a;, so that we have _, 1 d { \ dx' V'(a,-x,) dt (V(ar-*i) ^^ Now = S ±i .(3). c?a;. 1 (£x^ 1 1 ^ V(a,-a;,) J< (V(a -x,) rf< j a-— x, df 2 [a^-xj di f (xj («.)} - A^ V{ («/)} = A'B^ - A^'. 5. Since ^ (a^)={A^t+B^Y, by taking any m roots of/ (a;) we derive m equations linear in the sum, in the sum of products in pairs, and so on, of the quantities x^, x^...x^ ; so that 2a;„ ^x^x^^ &c. are all quadratic functions of t ; and if B {x) is a rational and mtegral function of x, . , ; '; + -rrr^i +...+ _,, , "' ( is therefore a rational and integral function of <, so that CR (a;,) dx^ rJR (a;,) dx^ fB{xJdx^ iV{/>,)} ^iv{/K)} ^"^ivi/Kj} is a rational and integral function of t. Thus since XX X we find /• a;,'Va!, /' x'"dx., f xj'dx^ [, 6. For the sake of precision of our ideas we shall sup- pose that the roots of/ (a;) are all real, and that the quantities 03,, x^...x^ occupy an interval terminated by two roots be- tween which none of the remaining roots intervenes. 8 JACOBl'S PROOF OF ABEL's THEOREM, It is important to observe that the equation gives us, according to the roots of f{x) which we select, a variety of algebraic expressions for t in terms of a;,, x^. . .a;„,, 7. By putting 1 dx, 1 dx^ we find 1 _^JL 1 +«»'„. V{/(a;J} + ^TT—— m"," m = Tdt., 11. 1 ^ 1 1 ^ y [1 + nx,) 0' (a;,) (I + nx^ ^ (a;J 1 + nx,„ <^' (a;„,) Now the left-hand side of this equation is equal to {zA (l + »ja;,)(l + «xJ...(l + «a;J' but from what has been already demonstrated we infer that the denominator of this fraction is a quadratic function of <, so that dx, 1 dx„ \ dx \ 1 + nx^ ^lf[x^ (1 +«^,) VA ■■■ 1 + ''a;^, V/la;,,.) is the differential of a logarithmic or circular function of t. If is a root of /(a;), {l + na;,)(] +na;.J...{l + ?ia;,,j is the square of a linear function of i, and this latter differential expression becomes the differential of an algebraic function of t. ( 9 ) CHAPTER II. APPLICATION OF THE GENERAL THEOREM TO THE CASE VI = 2. 8. In this case we shall assign to /(a;) the form x(l-x){l-¥x), whence o, = 0, a, = 1, Wg = tt; : /c is supposed to be < 1. To consider the differential equation ^*. , ^ V{a;, (1 - a;,) (1 - FxJ} VK (1 - ^.) (1 " ^\)} where we suppose that a;, and x^ are comprised between the roots zero and unity of /(a;), we shall put a;, = sin^^, aj, = sin"^. By these substitutions the above differential equation is transformed into dd ^ d _ V(l - A' sin" 6') V(l --^^ sin'^) To apply the general theorem to the equation in this form wc shall put A {0) = V(l -*' sin' 61), whence d0_ A(g) d _ A (0) rf< ~ sin" ^ - sin' ' dt sin" ^ - sin" ^ ' and since V{<^ (a,)} =sln^ sin<^, V{^ (a,)} =cos^ cos^, ,,,, u A(g)AW 10 APPLICATION OF THE GENERAL THEOREM we deduce from the supposition that when 6 = 0, ^ becomes a, the following equations, in consequence of (5), d ,. „ . ,, sin sin^A(^) A (o-) T- (cosb' cosrfi) = ^^ — ■ ., . ^"Trrrj dt^ ^' am 6 — am (f) am a d f A (<9) A {4>) ) _ sin0 cos j>A {6} - sing cos6>A ((^) ^ ^^^ It \ ¥ J ~ sin''' d - sin' - co )=F[lc,tr). 9. Let G'f/t,^)=r5^J^,then and when 6 = 0, t vanishes also. By integration with respect to t, and remembering that when = 0, ^ = a,'we deduce from the equations given in the previous section the following : sin 6 sin (f> sina = — t, C036 cosrf) = ~' i + coscr, sin cr ' A [0) A (<^) = - k'' cot at + A (o-). By substituting in the last two of these equations the value of t derived from the first, we find cos o- = cos ^ cos (^ — sin^sin<^A(a), A (^) A ((/))= A [a) + 7c' sin^ sin^ coso-, whence we deduce easily A {0) A { + sin o- sin A (6), A () (1 - /c'^ sin''^) (1 - k'"' sin''^)}, and the above system is transformed into the following d(p d\{r dx _„ Bin" sin'ylrdylr sin^x^X _(\ 15. Let us designate / —^. by the symbol L ((/>), or by i («, k', /c", (i) if wc wish to put in evidence the elements 14 APPLICATION OP THE GENERAL THEOREM «;, «', «", and let M{ and fjr become respectively ' cr and T, we are led at once by integration of the above system to the transcendental equations L{) + M{y{r) + Mix) - M{a) - M{t) =0) 16. Since V{-<^ («,)}= sin sini|f sin^, and '^{(j) (aj} = C03(^ cosi|f cosx, if we put d^_ A(0) dt ~ {sm'(j) - sin^i/r) (sin''^ - sin'^jj;) ' di^ _ A (i/^) c?< "" (sin^yjr - sin'''5(;) (sin'^i/r - sin''^) ' cosi/rA {yfr) \ (sin'''0 — sin'''i|r)(3in"^ — sin''';)^;) (sin'''i|r — sin'^;i^) (sin''i|r — sin''^) sin) or N[>c, «', «", ,^) denote J* ^^ ■ if the transcendental system (6) holds, then N{cl>) + N{f) + N{x)-N{a)-N{T)=^t. 18. Since -=- (sin (A sin-i/r 8inY) = -:- at ' ^' "'r f^ / „ , , , smtr coso-A (t) - smr costA (ct) TT (cosd) cosiir cosy) = . .—~-.--r, .-t;-,-^— ^ , dt^ ^ ^ ^' smc7 sinT(sinV-sInV) ' by integration with respect to t and remembering that this quantity vanishes for ;;^ = 0, we deduce, on putting , , sin cr C0SC7A (t) - sinr cost A (cr) V (o-, T = .-\-^ — i-i sm cr- sm T ' t = sin (j) sin i/r sin;;^; sin a sin t COS* COS\/r C0SY = -: . + COS a COST, ^ sma- sinr ' and by substituting in the last of these equations the value of t derived from the former, we find C0SC7 cosT=cos<^ cosi|r co3;)(;-sin^ sm>^ sin;!^V (cr, t)... (8). The transcendental system (6) is unaltered if wc inter- change inVsinV sin o- cos cr A (t) + sin t cos r A (o-) 19. If we eliminate % between the equations (8), (9), there results sinV sin^'r [v (<^, i/^)]"- sin^<^ sin'-'i|f [v (f, t)]''' = co8'^^ cos^ yfr - cos' a cos'^t (10). 16 ArPLICATION OF THE GENEfiAL THEOREM If we multiply the first equation of the system (7) by cos^ cosi/r cos;;^, and the second equation of the same system by sin^ sini^ ^^"^X^ ^^^ ^^^ *^^ results, we derive, in con- sequence of (8), the following sin2'>|f 8in2%A((^) sin 20 sin 2;;^ A (■\/r) (sin'''(^ — sin^i/r) (sin''^ — sin'^ji;) (sin'^ilr— sin''';)^)(sin'''i/r — sin''^) sin 2(f) sin2i/rA (v) + n—^ • 2 .N , • . • 2 ,x = 4 coto- COtT. (sm ;)(; - sm''^) (sm ;;^; - sm •^j li X — i^'j ^^^ equation gives tan^ tan^ tan. tanx^ ^^^^ _ ^.^ ^^ ^^,^ ^^_ ^„^^ ...(11). By solving for smx^ cos;;^; from the equations (8), (9), we find cos''^ cos''' 1^ — cos'' o- cos'V sin0 sini|f coscr cosTv(o-,T)+sinCT sinr cos|'' coscr cosT+sin^ sini/r sino- sinTv(0,-<|(-) v(o")T) '^'^^^ ~ sin'V sinV [v (, ^jr)]" + cosV cosV " The denominator of the last expressions can be written under the equivalent form sin''<^ sm'yfr [y (cr, t)]''+ cos'^ cos^'yjr, and, on dividing one by the other, we find _ tano- tan TV (0, ^ff■) - tanc^ tan^/ry (p-, t) ^ l + tan^tau'i/f tano-tanrv {4>i'f) v(o-, t) ' and from this equation the following are easily seen to be true : (p = tan"' {tano- tanry (i|r, ;^)} -tan"' {tani/r tan;;^v (o", t)}, f = tan"' {tano- tanry (x, (f>)} - tan"' {tan^ tan^y [a, t)}, X = tan"' {tano- tanry (<^, f)} - tan"' {tan<^ tan>/ry (o-, t)J, and any one of these is necessarily involved in the other two. TO THE CASE »w = 3. 17 20. If X — i"") ^^^ transcendental system i () + M{yjr)+M{^7r) - M{a) - M{t) = 0, can be replaced by the algebraic system ^{(1 - k') (1 - /e'") (1 - «"")} tan<^ tan-f tano- tanT= 1, V (, y}r) V {/r - sin'^x) sin-\|r sin;;^ cos cr A (o") (sinV — sin''T|r)(sin'''cr - sin";;^) sin^ sinr ' sincr sin r cos;)(;A (x) sinr sin;)^; coserA ( u u >< ^ J3 II :§: 1 i O a -^ a :. "a > t ^ a .2 .-s ^ ^ ^ o ^ O ^ ■-3 =* a _ * a ITS .2 c a c6 a '^ a ^ O ^ a> o .£; t- cr' o Pi '^l'^ Si I 1— I _'a I 1—1 .a I e8 3 ^ .9 i a" ^ o » o 2 .£; =! Oh bC'^S > ■e- -^ -^ a a '« s I "a °"a ;s > CO o o "a I i-H --^ Ts T I rs -^ ^ "a :s 'f m I > X m O o X a > I X |a ^a I ?< "a > b .s 'm I "se I > ^1^ a a CO b "a "a b b a > h a + + TO THE CASE M = 3. 19 §;§ 'b IN - • p CO '^ 5 g d- na -p OJ 'a 1 . a S ^1 N ■ b are sim <, thex'e 1 'b 1 : S i : ^3 ;i • ^ CO '^1 .S i '^ J, 1 O tH ' j5 1 i th O M s? , — 1_ -1 1 \ : _g ] : 05 i r 05 _, 3 -di 1 1 T— 1 -*3 ^ 1 1—1 S -e-i ? '?< "« ■"-' >~, I- • ■*-' 1— N 1 O V 05 3 .s' o m .9 'co ^H ■« _^ •3 v,i ~ii ~« ^ R b- 1 > 1 1 3 -a 1 i 1 1 ^c S.s 2 ^ CI _a .4-) CC t— 1 > b CO o o C "m 05 S S- 1 C5 'sj 1 ■©- Ts "a 1 1-H 1 c i _d 'ot '"!<; 1 1— ( 5^ 1 1 1— ( .S t- const 10 sin 1 .s 'cc N .s 'm 1 .a 03 b ■5 ■" 1-H ^ ro 1 1 -^ "« 1 > 'm SID S ? 1 05 '^ ' -"a 1 "h 1 •s "i :4 .s 1-H CO -° :=^!-§ 1-H *CD _^d 'So a > 'S ?^ 05 C5 ■r^ > > > 'a J 55 ^ 1 — -e- Xtl CO ion, det( oducing 1 s O o O O ■"« 1 05 -ij 'S .2 O .5 CO -^ _d 'S I 1-H "b^ s c 51 TO ^ — ' "s -^ "S- ■e- "■« "m -.::: -O C _d O r^ 1 ^ "m 'm J .2 i i >< «^° "E- -6- ;^ X 1 ■^ ?5 t—i ^1 2 "d 05 1 1-H 1> „ 1 "^ _g ■©- a ■fO -\ II > ■^^ "•« -■« > .s =■ to 8 b .c 1 -Sis 1 s rH to (M + + II . 73 X r^l 20 Application of the general theorem •6- > ?1 -■'2 "s *0Q 1 'j^ cn 1 1 ■ "ii ■e- '5j f- '^ M » ~ — ' ^^ ~ ~ve ~ii ~v» i ^ -> exe 1 1 -1 'c j5 __^ 'S *S 'm ■ — - ~" ' — ' t^ _:; "ii "« "i: ?. -^, ^ 'S 1 1 i 1 Ts ' Ts .s ^ ?« '>$:^ ^^ 1 M 60 1 ^'" (A « ii °a "c 's^is Ts 'rs ' , --" 1—1 I'S l^-c 03 2 Cm O e O 1 fa OS 1 1 ■©- ■3 1 al •H '51 J. ! 05 1—1 1 Xi 1 a 1 03 1 >< ■^ 1 •ire 9 '" ^? '^ "^ j*x >:'J, .^^ ''iii. "a ,^ "« 'S ■* -a "a ^ ^. — „i ^ I : 1 1 "a 1 o 1 •e- 31 h :^\ " 1 ^H h OT -^ ,2 *(K 'S 'cn ' _B > > > ^ r: -a > 'a ! 1 % 1 \' b -6- b ^ 1 tZZ ^2 • ^ o H- •^1 _^^ •^ 02 >^l X' ■^ OS Ol -e- ? 'S 1 [3h ?^ ■>• ■6- -e- .S > - 'S 'm *^ 1 + + 1 05 s ^ -S-; -6-' ^ •M r— .s ' .£ — - <1> *a5 ' *x 'S ** •> ^C + + 1 .*■ ^^ ^ a: to THE CASE m = 3. 21 lO 1 ^ a 2 X QQ h C — P-"5S a " tc 'rt * " •■^ a 2 Ss '^ ^^1 o S -^ 58 2 :< ^ "tS .S c3 H ttJ •5 ^ -€- - re . _. _2 ° s.s -^ "^ ^ c ^

* "c ^. 't& id '« cq 1 II ^ 1—1 >^ H ^ « b .s CQ *tn o ^ "ii ^ b 1 '^ C ^fl N ^ 'E 1 1 ^-^ (N 1 1 b 'P N .s :'c 'S c « [ I ! ?— I "b" I'd I >; h i 8i b B > 5* I 'is > ■5d c o *^ B cn r ~\ I t> h I to I t- no b o c (» ai <1> B .2 (3 B O" 0) T3 fl toi .S' b QO B «43 B CS P cr -B o '13 d be s o b d b °d b'^ + b "a + ^ b b o o B <1 cn O u b a t- .s I _^b I'd '£ _B b B IN I "b ^b "b 22 APPLICATION OF THE OENEltAL THEOKEIVt s o 111 b K o I I <1 a: O + t- + b l- + a .= II (N -^ 1 l- h ^ b^ C ■« > b -»<:i "d C- 'm O II <1 + <1 + ■e- -§^1- + t + •e-iS ** 9) -n fc rs =ir 'a: « r-J + + HO r— I (N ^-^ + ? C rs g '^ ^ ^ + a A -■ — " -M ■e- ^ i"a o s -1 j'^ 11 + S -a i^_- o *- a iw + qj O __^ ^ o t <] the uenc ^ -ti c' ^ a a> 03 o a:> _fi ^ o % 1— ( ^ -S A + n3 "TS" 1-H §^ b " + + + 1— I a r^ b II I> c o o h rxi l- O a O 'm 1 b a h cc s I TO THE CASE m = B. 23 I a a. « > + ■e- * ST II <1 a as a .fcp ns +-> CS Oh i'fi 1 a '55 b b vi^ .9 S *s Rh - h + ST + Si 'P Ts S + 'b' 'm S + + is o a b t- + s b o s |g 'm ^b [s 'S Si I O b- .9 I ^b a 'S + b .9 .s b a Ji ~Si + + ^Si + 'I' b ."h h ! -13 n3 cS 13 S o o ■> o X a o a. n! Si + ~Si + "si + s + s T3 03 b a a 0} i -a a c 13 a be a cS -a o s o >^ '-C '^ ^ s o; OJ ^ "a; a: a ^ +- o © "3 ^ ■^ Of OT -*-> h-l cu O a o a .a + s i— < c3 a aj o a c I- o o g " I = + .s " -^ .S ST . S b ^ -e- 's ■— ' a =*- ^ V 4- + .9 '9 -e- b .9 "c + ^ ^^-^ h & c + CI Si X a "a a > w -^ b -A. -9 '."9 •9 -e-,-^ , iM In I 'S .s + ^b "a I 1 ■f- s + T" ig 3 -^ .9 'm CD ."g "S g C 'm s + o o + + r-( _g ^ 1-* ^^ ~ °c ■©- V "g .> ■"a 'm tuo "m g .2 s + •£ + on, TS Cl. II § 0) o S >^ o t- to > .9 O B -e- .9 I :^ j> ^g ^b -13 b > >: t- 1 ii bii a en O I .2 '.S 'S 'to I ^b ■=>: I ■t- ^ h" -9 > b _g I > I Si a > c I I o a S b ^ a TO THE CASE m = 3. 25 b- ,s .S I 13 m O o o a + m O O + ^^ O 03 -*-• — - ■s > bo a ■73 s 03 s C^ 5 2 :-; Si Oh I •si o r= a ;s O ft. a Qi o i I -« o a ns o a § lU CO f^ a - -^ .S (D M a + ^ II Rh b -e- .a .a c6 S > -(-> 'B a o- ft- (U OJ £ is c9 u- na 1— 1 a eS a a a s -^ ■e-i-^ as I b I =£, fi^ be 1 a ,-^ ^ + "I ^ ^ ■ i + o ■©- ^ ^'- I en ^ a tn ^_t b .9 _a -^ _a •©• a I a HI Cm a ■e- + cn O u a b a ■4-* a >n a O c3 O -IJ b -6- 26 APPLICATION OF THE GENERAL THEOREM As in the values of sin'^+sm"Vf+ ain'x ^^^ °^ ^^^'^ sm'y}r-\- sin'^ sin'^+sinV sin'-'x the quantities k, k, k" enter symmetrically, it appears that we might have employed for their determination equations (8), (9) along with either of the equations derived from (13), by interchanging k and k, or k and «"; which equa- tions are thus shewn by actual calculation to be involved in equations (8), (9), (13). 28. Let us put and we find 4.sin^"'-'^cos0A(e) au = ^.}!^L^ {(2m-l)-2m{)-\-F) s\n'e + ['2m+l){F+G) am*0 - (2m + 2){G + H) sm' e + {2m + i]E sin' 6], so that the integration of — ,„ can be reduced to depend on the integration of sliT'-yje s\n""-' Ode sin""-°gjff sin^"-'0d0 ' A{e) ' A{0) ' A [6) ' A(6') ' and if we designate by Q (0) I ~r~/-s^ > | — ,^, de- pends on the functions L{0), 3I{0), N{0), Q (0). If we suppose that the system (6) holds, we have Q{^)+Q{^)-^ Q{x)-Qi<')-Q{'') =/(sin^<^ + sin^,^+sin^x) dt, and by substituting for sin''^ + sin'''-v|r + sln^;^ its value already found in terms of t, the right-hand side of this equation becomes , . ,, . ., , y (a, Tj ., K K K „ sincr siUT 3 on ^v . AT//m f' tan'0 sin'^c?!? 29. If we put N'(0)=^ -^^ , we have P{-\,0) = L(e)-vM[0) + N' (0) (19), TO THE CASE m = 3. 27 but by differentiating tan^A [6), we deduce i tan^A (6) = ^ '^^n'^ + G ^J n^OSsin^e ^^ ^ ' cos' e A {6) - 1^ {F-2G sin' e + 311 sm' 6}, ■whence, by integration, P{-l,e}^ tan eA [6] + 2FM{e) -{2G + H) N{e) + {F-G-^E)N'{e) + 3HQ{e), whence, eliminating N' {6) between this last equation and (19), we find (1-«')(1-0(1-0^(-1,^) = {F+G-B)31{e)-{F-G.+ H)L{d)-{2G + H)N[0) + 3EQ{e)+ta.n0A(0). 30. We find ^ d_ f sing co3(9 V{( 1 - k"' sin'g) (1 - v"" sin'^)} ] " dd'l '' V(l-«'sin'(9) J = {«* (1 + k" + k") - k' [k" + «"^ + «'V'^) + «'V"^} -^. A(y) - k' [k" (1 + «'^ + «"^) + «•■= (/c'^ + «"■■' + «'V"') - h!''k"'\ ^ t V ' A (0) , sin^ _ ( l-^^)K-O(^'--O 1 -d«/c a: ^^^^ l_«^sin^^ ' A(0)' whence P(-«', ^) Is made to depend on i((9), ilf(6'), 7\^(6'), and ^ (^) by the formula (1 - k') («' - «"^) [^ - k'"') P{- k% 6) = {«' (1 + «'^ + k") - k' {k" + k'" + jcV) + k'V"} L [d) - k' {«* (1 + «'' + «'") + k' {k" + k'" + k'V") - k"k"'} Mid) + k' {2^ {>c'-' + k"-' + kV) + >cV} N{e) . . ..ntm ,c sin g cosg V{(1. - «'- sin-g) (1 - k"'' sin^^)} : (20). 28 APPLICATION OF THE GENERAL THEOREM a u a .2 %-» s o s 13 > CO s I + I + + I + "ii an o a >< >« a a c -^ ^ I d ! a ID 03 ^ b h b" Oh •- "a I a I + TO THE CASE m = 3. 29 3 C8 a a £3 a ■©- .S I h <] b" 1 ?5 >< >; -^ 1 <1 + fcT -^ ^ 0) c s e« <1 + -G- G :§; is < Si I 1— I .9 I i-H "a ;6- :> a "a !« li « 5e "a > -^ > + h *s b ?< a -^ .s ■©- a ^ ^4- O CO "45 ■S 1 be rt p II S t4 02. ~s; a o ^ «« Qi ,^ + ! cS ^ >£- 03 -*-► O o A I—* _g -*-» "« _ce *m b A '42 a oT eg a> ^ '« Lh C3 © s| rC o ■^ 03 a ^ ^ +j 03 03 ^ 1— 1 II o 772 CI. 2 -e- to ^ i -ai cS ;-! a O J .f.3 ci > r^ CO y 03 V V o 3 03 30 APPLICATION OF THE GENEHAL THEOREM wliere J 77' 5 «"' "^ than unity, and k" > ^^~t > j^. • Since, when = 0,^ becomes |7r, we deduce /•« {a + p&m^6)de _ f^-^ (a' + ! , M ■J[k"-k') V(«"'-0 „ k , j^' , «", ^ can be expressed by L [k, k\ k", ^tt), i («, k', k", 6), Jlf (k, k, k", ^tt), iJ/(«, k', k", 9). 33. We have also A («, k', /c", 8) d^ «"''(l-«"^sin» fV(«"'-V) V(«"'-0 „ , 1 sm'd4> a:"^ A fV(« k ' k' k" ■ 1/1 \ d(j) TO THE CASE m = 3. ^J{K"^'-^) v(«"'-«'") 31 /«' , «", k > k, it follows that for all values i' — CO to ^ 1 of X increasing from 1 to K ' 1 , 1 — to — K K X is negative; while , to 1 -to — for values of x increasing from / k"^ k" -^ to + co X is positive. Let us put r" f g + I3x) dx r (a + ^x) dx , _ f '~L A-^) ' ^"i. V(-A') ' = i. (a + /3a;) dx V(-Z) ' r^ (a + M Ax , _ r (a + j8x)(fa , _ r (a + /3a;) -*-► s ■?> a CO TO THE CASE m = S. "5 a o o o bD a 1 1 » d 'S N et 1 1 "^ "Ji "— ^ M M -« Se 1 1-1 ''^^ 5a ^fl -^ 'CQ F q-~ ^-^ "a ~!* 1 'S 1 1 ^ e* 03. "^ ~V! + "'""^ ^ "« ^54 iH ■II e» .s CQ N ;« -« 1 i 1 ~« 'se 1-^ > u a IN ^ > Si + 8 <53. + « II B 8 60 s > u ID m Ji O o i n3 'I o 3 n3 Oi u o H CO CO T3 C 'bD a o a e8 o s s g .o .£3 0) OS bo a o S -§ to. a OS ■©- a a > bo a 'e 9 'to w a 'S D. |» 5*1 s< 1> 1 -TS I-- "55 ^ ^ ?» ^ a 'm "a ,. — . '3 02. + i^&^ )« !« ~a 1 ^ 1-^ "« ^ ^ 1 «^ ,, — , "a oa. '« + . — . ,-^ •'si 'v> "Ji ^ 1 1 1 s* N ? Ji^ "^ '« *-— ' N =? !< « ■ "^ > TS *. ^. a "« «c 1 a> 1 ^ (N Si, vT "> A >4i n >«P« ">4 A 11 SifSi l-l* 8 a in 34 APPLICATION OF THE GENEEAL THEOREM 87. If we assume a; + 2 = x;?!, and put Z=z{l- z) (1 - Fz) (1 - ¥Vi (1 - k"'z), we find X+j- r5 = 0, whence ^{-X)= ^ ' , , and if (1—2) ±[i-—Zj we regard v'(— -X") and i\/{Z) as essentially positive, for values of z which are less than unity '^{-X)=j- r-j , and for [I — z) values of z which exceed unity \/{—X) = z-iy If then we express the differential — j-. — ~— in terms of z, we find for values of 2, whicli are less than unity, the formula (a + ffx) dx _ (a + /3) g - g , and for values of «, which exceed unity, the following formula (g + /3a;) dx _ a-(oL + l3)z Vl-X) - V(^) 38. To apply these formulae to the reduction of the in- tegrals /,, /j, ij to the normal form ; namely, h {l + m sm'0)dd V{(1 -/ sin'^) (1 - q' em'e) (1 - r' sin'^6')} ' we must observe that when a; = ± 00 , then s = 1 ; when a; = 0, then 2 = 0; when a;=l, 2 becomes +00; and when x takes the values -j , -j:^ , -ttj , the corresponding values of z are K K K 1 J_ -1 k' ' k" ' /^"^ • 39. As in 7, x ranges from - cc to 0, the corresponding values of z are included between 1 and ; hence TO THE CASE ?>l = 3. 35 and putting z = sin°^ or « = - tan^^, we find Je V{(l-^'sm''6lj(l-A;"^sm^6')(l-A"'9m''6')}"'^ ^' and since i > ^' > k" by assuming « tan^ tan^ = 1, we find T^J_ /■* ((g/g' + jQ) sin'^0-j81J^ 40. As in /j a; ranges from 1 to -775 , the corresponding values of a are + 00 and p^ ; so that the values of z cor- responding to the values of cc, being greater than 5^75 , we have and between the limits Z preserves the sign +. The value of 7, is therefore derived from the expression for 7j (26) by changing the limits of integration in this ex- pression from }d<}> '~M' j„ ^|(i_i'. 3i„=^) (^i_|: sin'^j (i_|:' gin-^)J ' C0S6C {6 1 where « = —j^rr- , whence x = ^_j^..^^,^ ■ 41. As in ig a; ranges from -75 to — , it is expressed in terms of z by the integral f {a-{a + ^)z}dz ■'"i^ V(^) 36 GENEUAL TIIEOKEM tO THE CASE )/l = 3. >-t TS -c uT n e^. P ■$ ^^ Cm ■& O t^ ■4-* TS M c4 +3 ^ a o a> ^ 3 s° t* o O ^ o ^ C3 s 1 o 43 -e -*J 4) ■fi e -§ CO (U •^ 05 13 s 'S QQ > 00 V .- ^ 13 .£3 a c« aj c ^ .. -1^ -g S" ^ fe 'S 1 » -Q ^-1 o £ 'a £ ;ri g &c o Sb c 'C u ■g (U S _o s "o ,o o ■as i)-\-iF{k\ w)- F[k^ a) = 0. The rela- tion between i^ and to can be written under the forms ,1, 71! • 2 IN ^Jil-k!'^ sm'm) , . , cos'v/r = seca), 'Jn—k smV)=— ^ ', and is ob- ^ ' ^ ' COSQJ ' viously reciprocal. Thus it is easily seen that the last men- tioned transcendental equation is equivalent to any of the following equations, see Chapter II., section (9) : cosff = cos^ secta - i sin<^ tanw \/(l — i^" sin'o-), cos^ = cos (7 seco) + i aintr tanw V(l — ^^ sin^i^), V{(l-/fc'sinV)(l-'5:"'8in''Q))(l-^"sinV)} ,,,,., _, -^^-^ 52i '-^ ^ =k^+ k^ COS* sec a coso-, cosw which lead to the following : cos^ = coso- C08a)+ i sin^ sinw ^(1 — ^^ sin"©-), cos (7 = cos(f) cos ft) — i siuo- sino) ^(1 — A'' sin"^), \/((l— /<-'" sin'''^)(l-A;'" sin'''£o)(l-A;'' 8inV))=7i;''' coaw + k' cos) + iK' = F{k,) (1 - k^ sinV)} = k cos(^ cos>^, from which we find k sin(f) sin (7= 1. 43. Let (T, a' be two amplitudes satisfying the equations F{k, i>) + iK' = F{k, a), F{k, a) + iK' = F{k, a), we have then F{k, ^) + 2iK' =F{k, a'), but ^ sinc^ sino- = A; sino- sino-' = 1, 80 that sin^ = sintr' ; also cot^ = I ^(1 — k^ sinV), coto-' = — i \/(l - A" sin'cr), 80 that cos^ = — C0SO-' ; and since coto- = - { ^(1 — k sm'cj>) = ^■ n/(l — k' sin^o-'), there results ^J{^ - ¥ sm'4>) = - V(l - ¥ sinV). 44. These equations are written in a most expressive form by the employment of Jacobi's notation, see Chapter II. If then F{kj (f>) = u, ) = ^(1 - ¥ sin'^). Now the formula for the addition of elliptic functions gives sin am [u + 2K) = — sin am (u), sin am {u + iK) = sin am (m) , and if m, m' denote any integers positive or negative li'mahi (u + '■ImiK' + im'K) = sin«)« [u) ; ELLIPTIC INTEGI!ALS. 89 this equation expresses the property of double periodicity belonging to elliptic functions, and teaches us, that if an elliptic function u is increased by any multiple of UK' as well as by any multiple of 4cK which are its periods, the value of the sine of the amplitude is unaffected by such increase. Admitting imaginary values of ^, sin^, when real, is larger than unity ; premising this remark, I propose to determine the value of F[h^ 6) when sin^=T; this equation can be regarded as resulting from the following system : Ic tan 6 tan0 =1, sin<^ = i tantu, &> = ^tt. Now if F{k, 6) is ti-ansforraed by the first of these equations, it becomes K—F[k, cj)) ; which value becomes K—iF{lc\ w) in consequence of the equation connecting ^ and 0) ; or, finally, K— iK' by taking into account the third equation. Hence sinam (£^±iiv') = y ; the values of cosam{K-tK'), cosam{K+iK') having opposite signs; as appears also from the formulse given above by putting u = K. 45. If the elliptic function of the second kind /V(l - h' Bm'4>) d(f), or E{k, <}>) is transformed by the equation sin0 = itanw, it becomes . rV(l -r^in^ ^^_ Now J cos CO da) -li J — -+V(i-« 9'n)+F[k', w)-E[lc, a,)}. ..(28). ( 40 ) CHAPTER V. ON THE PERIODS OF THE FIRST CLASS OF HYPER-ELLIPTIC INTEGRALS. 46. Foe the determination of the periods of these func- tions I shall employ a mode of investigation precisely similar to that which has been already adopted for the determination of the periods of the elliptic functions. We must revert then to the notation of Chapter III., and to the equations given in that chapter for the addition of the first class of hyper-elliptic integrals. If we put sin 9 = { tan w, L {k, k, k", 6) becomes i[L {k, Jc, k", oy)-M{k, k\ k", m)], and M{k, k, «", 0) is transformed into —t'M[k, k', k", co); so that the transcendental system L {KjK ,K"^(f>)+L{K,K' ,k" ,->p-)-\-i [L {k, k' , k" , Q>)—M{k,k',k", to)] I — L {k, k, k'\ a)—L («, jc', k", t) = 0, M{k, «', /c", v((^,>/r)J If in the quantity y [)) \ sin'^ cos^Q) + 8in'''£o J cosi/r = cGS<7- COST cos<^ cosw + slncT" sinT sin<|) Jsinilr cosi/rA {k, Jc\ 1c'\ to) — i sinw costuA («, k\ k'\ i/r) I sin' yjr cos'(o 4 sin" to (31). 47. If. we put L {k, k', kr, Itt) - M[k, k', k", ^tt) = ©, M{k, k', k", ^tt) = ©', and if to becomes ^ir in the equations (30)., we infer that the transcendental system Zi {k, k, k", )-i-Z' (k, k, k\ ^Jr) ■+{Q—Zi (k, k'j k", a) — Zi [k, k, «", t) = M{k, k', k", )+M{K, «', «", yjr)-i&-M{K, k\ k'\ a) is equivalent to the equations cot (^ cot t/t = t'v (o-, t), cot o- cotT = — i'v (^, V') • • • (33), which must involve the equation kk'k" sin^ sln'^ sino- 8inT= 1 (34), which is obtained by putting <» = ^tt in either of the equa- tions (31). 48. If the quantities cr, t, a'^ t are connected by the equations L {k, k\ k'\ a)+L {k, k, k", t) + t@ -L («, «', «", o-') -X(«,«>",r') = 0, M[k, k\ k", a) + M{k, k, k", t) - 1& - M{k, k; k", a') -M{ic,K',ic",r') = 0, we deduce by taking into account (33), (34), kk'k" sin cr siuT sino-' sinT = 1, coto-' cotT - z'v (o", t) ; G 42 ON THE PEEIODS OF HYPER-ELLIPTIC whence we infer that the system L («, k\ k", ^)-\-L (-«, «', k", ylr) + 2j0 -L («, «', «", o"') — i («, «', /c", t') = M {k, k\ «", 4>) + M {k, k, k", yjr] - 2i& -M[k, k\ k\ a) -JW"(«, «',«", t') = J is equivalent to the following sin0 sin>^ = 8ino-' sinr', cos(/> cosifr = - coso-' cost'. ..(35). We have also the relation 49. If we refer to the value given for the integral 7, in Chapter III. (27), we see, if it be taken between its ex- treme limits, that it depends on 0, and on ©'; and that 50. Let L (at, «', k", ) + -£ («, «', «", i/r) = M, ir(/«:, «', «", ^) + JIf («, k', k", l/r ) = v, then if aincj) Bm-\]r=f{uj v), we have /"(m, »;) = f{u 4 2mt&, v — 2oti0'), TO being any integer positive or negative, and if cos^ cos'^=y^ (m, v), we have / (m, ?j) = (-])"/, (m + 2 jwi0, v-2mi@'). 51. We deduce from the transcendental system (32), and from its algebraic equivalents (33), (34) that the equations kk'k" sin^ sini/r' sino- sinT = ], cot^ cot^' = — i'v (o", t) can be replaced by L («, K, k", 4>)-i- L [k, k, k", i/r') — j0 — L (k, k, k", ) + M{k, k, k", y}r') -f «©' - M[k, k, k", a) — M{k^ k, «", t) = 0, INTEGRALS OF THE FIRST CLAfiS. 43 SO that an amplitude -jr' such that sin^' = slni/r, cos \jr' = — cos yjt satisfies the system L {k, k\ k", Vr') = L {k, k\ k'\ yjr) + 2z0, M(k, k, k", y{r'] = M{k, k\ k", yfr) - 2t0', and V{(1 - «' sin'Vr) (1 - «"" sia">|r) (1 - /c'" sin'V^)} = _ V{(1 - k" 8in>') (1 - /c" 8in>') (1 - k" sm'y}r')]. 52. I now proceed to determine what i [k, k, k", 6), M{k, k\ k", 6) become when sin 6=—^,: this equation can be regarded as resulting from the system k" tan 6 tan =!, sin ^ = t tan (u, a> = ^tt. If we apply these successive trans- formations to the functions of which we are treating, and if we put L (Af, k', «", |ir) = *, JIf («, «', k", \it) =*', _J_ [^ (1-Z:"''siu'a.) a/iIt»" ^'" *" J \ 2^ " 0) 1(1— « sm to) ,^ a/ )( i?" ^'° ^ ) ( ^ "" "Ti sin w 1(1- A sin o)))- we find that L{ic, k, k", d), M{k, k, k", 6) become re- spectively -fll, '-«i2', when sin^=— . If we refer to the value given for the integral 7^ in Chapter III., section 40, we shall see that its complete value depends on Q. and on i2', and that ...(36), 44 ON THE PERIODS OF HYPER-ELLIPTIC From what has been observed in section (51) it follows that when sin 61 = 4, L[k, «', «", 6), M{k, k', «", 6) become respectively $ - ii2 + imiQ, 4>' - iil' — imi&, or 4)-jQ+(4ot + 2)z0, *' - lil' - (4»i + 2) j©' ; where m denotes any integer, positive or negative, the values of cos^ corresponding to these systems having opposite signs. 53. Let us now seek the algebraic forma of the trans- cendental system L {k, K, k'\ 4>)^L («, K, «", >|r) + * - ifl ] — L (k, «', «", a) — L (k, /c', «", t) = M(«, «', «", if,) + -¥ («, «', k", i|.) + *' - iiir - Jf («, «', «", a] — M («, k', k", t) = and for this purpose we shall refer to the algebraic system found by interchanging « and «" in (17) Chapter III.; if in this we put sin;^ = -77, we deduce the following as an algebraic representation of the system (36) : (1 - /c'"" sinV) (1 - «"^ sinV) 4 k" Bintf) amy}r [k"\' {tr, t) + «"" coso" cosry (cr, t)] = (1 -«"" sin" ^) (1 -«"" sin" i|r) — k" sino- sinT [k"''v' {, yjr) + k"'' cos cosi/rv (<^, ■yfr)] = •' -(37). Again, if we interchange k and k" in equation (14) Chapter III., and put amx = —nj we find {k"K" »J{{k"'^- /c") {k"'' — k')} sm(j> sin^lr sincr sinr _ VKI-k"" sm'(ji){i-K"' sin" ylr){\-K"'' sinV)(l-«"' sinV)j ^ (38), which is involved in the system (37). INTEGRALS OF THE FIRST CLASS. 45 64. If we suppose that )+L (at, k', k", 1^) + 2* - 2tI2 - L (k, k\ k", a')- L {k, k, k", t) = 0, M{k, k, «", , a) = ^7r. By means of these transformations L («, k, k", 6) becomes i /■i'f (l-7g" sm'ui)d(o 46 ON THE PERIODS OF HYPEE-ELLIPTIC ns a O 2 3 34 1 1-H 4 1 3 0* 1 3 :a en -^ ' 1 1 1 1—1 ■> >1 O 0) U3 + ■73 to ■-=! C >i '5 >^ 3 c ; "^ '^^ Ts 'm ? (N -a be -M ■-a - " o ^ «« a a ^ W ■bo's ) +L [k, k', k'\ -i/r) + — H —iQ, — L (ac, «', «", a) — L (k, k', ac", t) = 0, M[k, k, k", (f>] + M{k, k, k", ■«/»•) + ' - E' - iil' — M{k, k, k'\ a) - M[k^ k, k", t) = is equivalent to the algebraic equations (l-/<:"'8inV)(l-«"^sinV) + K sin^ sin'^ [^'V (cj t) + /c" cos<7 cosry (o-, t)] =0, (l-«"''8in'(^)(l-A:"'8in-'i^) — «' sin (7 sinr [^V (^, -(/f) + k'^ cos cosi|ry (^, ■^)] = 0, and we deduce also, by means of (14) Chapter III., k'x c"' sm'f){l-K'' siiiV)(l- «"^ sinV)} " '' We thus find, as in section 54, that If i (/c, K, k", ) + L [k, k, k", -,|r) + 2* - 23 - 2in — i [k, k, k", ' - H' - iQ! - imi&, or *-E-^I2+(4m42)^■0, «!>' - E' - iI2' - (4»i + 2) ;©', where to is any integer, positive or negative, and the values of cos ^ corresponding to these systems have opposite signs. INTEGRALS OF THE FIRST CLASS. 49 57. To determine what L («, «', «", 0) and M{k^ k\ k", 6) become when sin 5 = - , we shall regard this equation as resulting from the system sin^ = ttanw, « tanw tan«i)'= I, sina)' = z tanco", a>" = \ir; and if we apply these transformations in succession to the functions L («, «', k'\ 6) and M[k^ k, k", 6), we find, by putting , „^ ,r^'^ sin''io"d{o" •'» y/|(l-«'^sbV')(l-5,8inV') (i- IjsinV')! , „ ,_ fi^ dea" •''' y|(l-«= sin^-a,")(l-^ sin"a,") (l-^, sinV')| ' that X («, «', /c", 0) becomes i@ - S, and that M(k, k', «", ^) becomes at the same time — {&' — S'. The algebraic forms of the transcendental system L {k, k, k", 4>) + L {k, k, k", f) + i@ -S, — L («, k\ k", a) — L {k, k\ «", t) = 0, .¥(«, «', /e", 0) + ^Tf («, «', «", >/r) - 10' - 2' - M{k, k\ k", a) —M(k, k\ k", t) = 0, are found by interchanging k and k" in equations (37), and wc derive from the system L («, «', k", ^)+L («, K, k'\ f) + 2iQ - 2S - L {«, K, k", a-') - L («, «', «", t') = 0, M{K, k\ k", 4>) + M{k, k, k", ^) - 2{& - 2S' -M{k, k, k", a) - M{k, «', k", t') =0, the equations sin ^ sin ^ = - sin a sin t', sf[{\-K' sm'){l-K' sin^-^)} = -V{(l-«' sinV)(l-«'' sinV)} (^1). 50 ON THE PERIODS OF HYPER-ELLIPTIC If we refer to the value given for I^ in (26) Chapter III., we see that its complete value depends on S and S' ; and that As when sin5 = - , there is an ambiguity of sign in the cosine, the values of L [k, k, k", 6), M(k, «', «", 6) corre- sponding to such ambiguity can be obtained in the same way as they have been already obtained for the other similar cases. 58. It is easy to see that if L (k, k\ «", <})) + L [k, k, k", yfr) + 44> — L (k, «', Ac", ff') — L (k, «', k", t) = 0, M{k, «', «", tf>) + M{k, «', «", y}t) 4 4*' - M{k, k\ k", a) - M{k, fc', k", t') = 0, we have sin am yjr = sin a-' sin t'. If we put, as before, L [k, k, k", ) + L {k, k, k", ■^) = u, M [k, k\ k", 4>) + M {k, k, k'\ yjr) = V, sin +im"{^-'iil) + im"'{i& -2),) ■' ^ ' ■' Iv-2m©'+4m'*' + 4m"(*'-/i2')-4m"'(i0' + 2ij ' on, vi', m", m!" being positive or negative integers. Hence, if /i, /x', fi\ fj!" denote any integers positive or negative fin w) = / f""^ •^'^* + ^1^'^ + 2a'"*0 + 4/i"Vi2,| '^ ' ' ■' \v-\ 4/i*' + 4/^'2' - 2ya"i0' + 4/i"'il2' J ' so that /(m, v) is a periodic function of the two variables INTEGRALS OF THE FIRST CLASS. 51 u and V, each of which has four periods, two real and two imaginary, and the nature of the periodicity is indicated by the above equation. 59. It might be thought that the functions E, E' would give rise to another period, but these are connected with the periods 4>, 2, and ', S' in a way which I now proceed to point out. If we refer to the equations (33), we see that they are satisfied by the following values: rf) = Att, Vr = sin"' — , «r = sin"* - , t = sin"* -r, , ' K K K which must also satisfy the transcendental system (32) of which they are the algebraical translation. Hence L («, k', k", ^tt) +L (k, k, «", sin"' — ) + *© - L(k, k', k", am"'- 1 -L(k, «', k", sin"' — | = 0, M{k, k', k", ^tt) + m(^k, k', k", sin-' i) - i& - mL, k\ «", sin"' -\ - m{k, «', «", sin-' 4) = 0, and if we substitute in this system the values already found for the quantities which enter into it, it reduces itself to the following : -S + S = 0, *'-E' + S' = (42), which express relations between certain definite integrals, which depend on the quantities «, «', «". 60. Jacobi denotes by m„ m„ m„ m„ Mj, m,, the complete values of the integrals 7„ /„ Z,, /„ /„ /, as defined in sec- tion 34 ; which he proves are connected by the relations U,~U^ + % = % W,-M, + M5 = 0. 52 FIRST CLASS OF HTfPEE-ELLIPTiC INTEGEALS. The first of these relations is evidently equivalent to the equations (42), which I have just demonstrated ; the second results from the first by writing in it A, ^', k" respectively for k'\ k\ k. 61. The equations (42) can also be demonstrated by a diflferent arrangement. K we refer to the equations (33), (34), we see that the following transcendental system holds : -L ( «, «', k\ sin'' -\-\-L («,*:', k", sin"' — ,) + i® ~L («, «', k'Wit) — L I «, «', /c", sin"' — I = 0, m{k, k\ k", sin"' J) + m(k, «', «", sin- i ) - i& -M{k, k\ «", ^if) -MIk, k, k", sin"' — ) = 0, which, by substituting the values already given for the quantities which it contains, is reduced to 2mi0 -2 + H-* = 0, - imi®' - 2' + H' - *' = 0, m being an integer, which must in this case be zero. 62. The real peliods of I" (a + /3sin"g)t^g j„ V{(1. - «' sb"5) (1 - «" sin'^) (1 - «:"» sin"^)} when multiplied by i are expressed by the imaginary periods of I" {a+^&we)de j„ V{(1 - 4" sin^^) (1 - r sm'e) (1 - k"' sin'^)} ' and vice versa. ( 53 ) CHAPTER VI. CASES WHEEE THE FIRST CLASS OF HYPER-ELLIPTIC INTEGRALS DEPENDS ON ELLIPTIC FUNCTIONS. 63. There are cases in which, if certain relations exist between the quantities «, «', k", the value of JV{(l-«:'^n'6i) (1 - k" aiji'0) (1 - k" sin"^)} depends on elliptic functions and on functions of a lower order, circular and logarithmic, ^(sin^) denotes a rational function of sin^. Supposing, as we have already done, that k" > k'> k, if K = k'k'\ then L («', k", k'k", 6) depends on elliptic functions of the first kind, as I now proceed to demonstrate. If we put \/{k'k") sm0 = t, then de __^ V{(1 - k' sin-^) (1 - k'" sin"5) (1 - kV sin^^)} 1 dt " s/{k'k") V(1 -pf + qi" -p^ + f) ' (l + «'=')(l + «"^) 2«'V'»+(«'^+0(H-«'V'^) where » = t-t, > 1= -irin , and the differential at the right-hand side of this equation can be written 1 it sI[kk') V{('^^)-^('-'^^)-'' Let ^ + - = M, and because the origin of de V{(1 - k" sin"^) (1 - «"" sin^^) (1 - «'V'^ sin'^^)} 54 CASES WHERE THE PIEST CLASS OF HYPER-ELLIPTIC corresponds to m = oc , we must have 2 2< = M-V'(w'-4), - = M + V^w" - 4), z . f' dt io V(i-X+#-K+«') _ 1 r" 1 J tidu , ] or, if we put m'' — 4 = v^, r' ^< 1 r r ^^ + r ^^ 1 Jv '^{v'-(p-i)v'+2-2p-i-q}j and if we put 1 V(«V') . - 1 V(«'«") . ^ Now w.= we have ' 1 + k'k'" '"»-1-/cV" dt 'Jjk'k") f 1 de. de. 2 (!+«'«" V(l-<8in=^,) " !-«'«" V(l-»j/sin''^J Hence, because 0, t, 6^, 0^ all commence together, we find by integrating L («', «", «V', 0) = 1 {j-A_ i.(^^, ^j + _i__ ^(^^^ ^^ .(43). INTEGRALS DEPENDS ON ELLIPTIC FUNCTIONS. 55 64. In a similar manner we find ain^ede V{(1 - k" sm''^) (1 - k"" sin''^) (I - kV aWd)} 1 edt («'«")* V(i -i><' + qt* -p^ + 1') ' and _ r du ' Ju 'J{u*-{p+4:)t(:'+2+2p + q}j^ and introducing as before the amplitudes ^,, 0^y we find (4*)- Hence L{k\ k", k'k", 6), M{k', k", k'k", 6) depend on the elliptic functions F(m^, ^,), F{m^, d,), whose amplitudes are connected with the amplitude 6 of the hyper-elliptic integrals by the equations .. {l + K'K")ame .„ (l-«V')8in0 , , ' 1 + KK sm p ' ^ 1 - AC « sm c' It appears from these relations that 6, ^„ 6^ vanish together, and also attain the value ^tt together. 65. We have also the following forms of the equations connecting 0^ and 0^ with : . cos^ V(l-«V''sin'6) ^ ^«'^'= l + «V'sin^^ ' V(l-»«. sm^.)=-J i + ^v'8in'^^ ^ cos0V(l-«'V"sin"0) ^°'^^ = l-«V'sin'0 ' _ Vf(l-«"sin-g)(l- «"-sin-g)} V(l - »«, sm e,) = r^^"-sin"-^ J (46), 56 CASES WHERE THE FIRST CLASS OP HYPER-ELLIPTIC which lead to ^ = r-r, i an equation connecting tan ^2 1 - « /« directly 6^ and 9^. 66. In consequence of (43) and (44) we see that the trans- cendental system L («', K, kk\ (f)) + L {k, k", k'k", ■^■)+L {k\ k", k'k", x) — L {k, «", k'k", a) — L («', k\ k'k", t) = M{k', k, k'k", 4>) + M{k', k", k'k", A/r ) + M{i^, k", k'k", x) - M[k', k", k'k", (t) — M[k', k", k'k", t) = (47) can be transformed into the following : F{m„ ,) + F{m^, t,)+^K, X,)-^K> <^,)-^K, tJ = F{m„ ,)+F{m„ f,) + F{m„ x,)-F{m„ ,^ are derived from (/) in the same way as 6^ and 6^ In the previous section are derived from 0, and so for the others. Hence the addition of functions of the form aZ (/<;', «", k'k", 0) + ^M{k', k", k'k", 0), where a and /3 are arbitrary quantities, depends on the addition of functions of the form F(m,, 0^] and on the addition of func- tions of the form F{m^, 9^, and the formula requisite for the solution of this problem are given in Chapter II. Each equation of the system (48) has, independently of the other, an algebraic equivalent, expressing an algebraic relation between the trigonometric functions of the amplitudes •^ii V^i) Xi) ""]) '''i ^^^ ^^^ ^^"®'^' ^"^^ ^ similar relation be- tween the trigonometric functions of ^^.j V's) Xii ""a; '''a ^°'' the second; if In each of these we pass from the ampli- tudes ^„ i^r,, &c., ^j, il^,^, &c. to ^, >|r, &c. we arrive at the algebraic equivalents of the system (47) in which the amplitudes ^, i/r, Xi °'i "^ alone appear. 67. There exists an elegant equation connecting (f> and i^ with the amplitudes (/>„ i/r^, j,,^, yjr^ which I now proceed to investigate. INTEGRALS DEPENDS ON ELLIPTIC FUNCTIONS. 57 Let J-K, <^,) + ^K, t.) = F{m^, r.), then, in consequence of the third equation given in section 8, we have ^_sin<^j C08^, V(l-»»,' sin'^,)— sin^, cosi/r^ «/(l— Wj'sin"^,) ' Bin^, — Biny, ' y._ 8in^^ C08^, V(l-wi/ Bin'i/rJ-Bim/r^ cos>^, V(l->»,' sm' ^'"^'^'"^'^ l-^'V-BinV 5 /2 "2 -,' 2- ^ cob".^ (1 - kV sin'^) COB^. 008.^, = j_^-V"8in*^ ' /fM »•»^^,, i.-2^M (l-A;'"Bin»^)(l-«"''Bin» V{(l-< sm>,)(l-< Bm>J} =^^ i-«'V''BinV ' , (1 + «'«") Bin<6 co8. cos<^,= (i-.V'sIn'<^)' ' V{(l-<8in>J(l-m,'sin''t,)} V{(1- k" sin''<^)(l- k"" Bm'4>){l-K" BinV)(l- «"'^ sinV)) . (1 + k'k" sin" <^) (1 - k'k" sin^ t) ' whence we deduce, if we put ^ - {(l-«'V'^Bin*^)(l - *"«"" sm»}^ ' fsin^j cos<^, V(l-»»," 8in>,)-8inV^, co8i|r, >^{l-m^'' s\a')}"" - k"k"^ cos" <^ cos' i/r (sin' - sin'-«/r) ^ X {sm"<^ (1 - K-' Bm'yfr) [1 - k'"' sin'^) ' [ - sin'f (1 - k" sin» (1 - k'" Bm'4>)} and also (sin'c^, - sin'V^,) (sin"<^j - sin'i/rj {(] _ k'^ k") (sin'i^ - sin'f ) (1 - «'V" sin^c^ 8In°^fr)}° = {{I -k'k'" sin* 4>){1-kV s\n*ylr)Y ' so that, by restoring for Z> its value and remembering our ordinary notation, we find that *^°*^. '=°*^= (1 - «'V") (1 - «V" sin> sin't) '"^ '' 68. Suppose now that in the system (47) x = |7r, we have then Xi '^ Xa ~ i"") ^°^ ^^ ^^ P^* the system (48) becomes in this case F{m^,Q+F(m^,i7r)=F{m„^^'), F{m,,Q+F{m^,\ir)=F{yK:), •whence, by the formula for the addition of elliptic func- tions given in section 13, we find tanS; tanS; = - ^^^_^^-^ - - ^{(1 -^ (1 -«'")} ' 1 — KK tan?, tan?;' = - ■v(i-o" v{(i-o(i-o}' BO that, if in the system (47) we put x^i""? ^® have, in consequence of (49), f [V (0> V^)]'- *^"*"'' co8'4> cosV ] [ [v(o-,t)]''-/c'V"cosVco3° ( 1 - «' V" Bin"<^ sin'-i/r j| l-«VsinVsinV = (i-O(i-«"')(i-«'V"0. INTEGRALS DEPENDS ON ELLIPTIC FUNCTIONS. 59 This equation, whicb we have deduced from the formulae for the addition of elliptic functions, can be derived from the algebraic forms of the hyper-elliptic system (8), (9). If, in the equations (8), (9), we put x = i''') w® ^^^ ,. ,. cosd> cosylr , . cos o- COST 80 that [ [V {, W- ^'V" cosy C09> ] ( [v( + q} {v^+l)dv ^1 ~ VK-(p-4)r" + 2-2i> + 2}J • Now we find easily r* u^du = 1 + **! {F{m,, ^,) -E{m„ 6^) - cot^, V(l - K ^^"6^), nj^KK ) r" v^dv = \z4^'' [F{m,, 6,) -E{m^, 6,) - cot5, V(l - K sin^/?J], 60 CASES WHKRE THE FIRST CLASS OF HYPER-ELLIPTIC so that •' 2 («vy V{(1 - k" sin'^) (1 - «"" siB"^) (1 - K'tc"" sin" 6*)} 1 rr {v'+i)dv ~ 2 («'«")* Lir VI*'* - (^ - 4) «" + 2 - 2p + 2} _ r {u''-t)du 1 (1 + «V') {cot^, V(l - m," sin' (9,) + ^K, ^,)} ") II t tt , tS "V 1 + /.V' -^('"■'^') - (1 - k'k") {cot ^, V(l - < sin' e^) + E (m,, ^J} (50), 1/ /I — KK wliich, if we adopt the notation of Chapter III,, expresses the value of the hyper-elliptic integral N[k', k", k'k", 6] by elliptic functions. 70. We find also In the same way de cos^e V{(1 - «'" 8in=0) (1 - «"" sin''^) (1 - /c'V" sin''*?)} _ V(«V') ■ and the differential at the right-hand side of this equation may be written ^{kk") m' + mV(,«'-4)-2(14/cV') X f , iidu ) 1 ^/{u* - {p + 4) li' + 2 +2p + q] ' which can be put imder the form V(« K ) U —1 —KK 2* ft , f //\2 I II a [l+KK ) — KK U X du ^[{.J^^}{,.ii^ + INTEGRALS DEPENDS ON ELLIPTIC FUNCTIONS. 61 VV'O v^+1-k'k" _ V dv (1 — KK ) — KK V B m'-I-zc'/c" 1 r l+/tV' > " (1 + «'«")« - «'«'V ~ «'«" 1(1 + «'«")» - «'«V J ' v'+I-k'k" _ J^ f 1-k'k" _ \ (1 - «'«")' - «;'k'V ~ «'«" i(l - *'«")" - /cW ) ' 80 that we have / dd „ cos'^ V{{1 - *'* sin"^ (1 - k"" sln'^) (1 - /eV'' sin'^)} 2 V(acV') jj. ( (I + k'k'Y-k'k'u'J du "'"2V(«v')|J„ f (i-«vr-«vvj But hy introducing the amplitudes 5, and 0^ and integratmg: we have du r 1 du /^{k'k") (1-«V')(1-«"J(1- *"") {tan^, V(l-< 8in'^^J-i;(rn„^,)}, 62 CASES WHERE THE FIBST CI-ASS OF BYPEE-ELLIPTIC SO that finally ;• de^ j„ cos"^ V{(1 - «" sin"^) (1 - *"" sin"!?) (1 - /cV sin"^)} = 2(1-0(1-0 ^*^°*' ^^^ "'"•' ''°'^>)--^('"" ^>) + tan ^, V(l - m^ sin'^J - ^K, ^,)} + HrT^''^'"''''^^T^''("-'4 ^''^- 71. From the formulae gwen in the previous sections we can derive another case in which hyper-elliptic integrals of the first class depend on elliptic functions. For this purpose let us put 8iu^ = /tana> in the equations (43), (44) ; by this transformation L («', «", «'«", 6) becomes iL {V(l - O, V(l - k\ V(1 - «'V"), the same transformation gives sin^,= t tanto,, sin0,= i tanca . Now by the employment of these latter equations F{m^, e^ becomes iF{s/[l - »n,»), «,}, and F{m^^ 6^ becomes {Fy{l-m^% wj ; so that the equations (43), (44) when transformed by the equation Bind = i tanco lead to L {V(i - O, V(i - «""), V(i - «'V™), 0,} ■ 1 rj,f v{(i- 0(1-0} ) ~2«V'L 1 ! + «'«" ' M INTEGRALS DEPENDS ON ELLIPTIC FUNCTIONS. 63 -T^- r-^^'^ '^^ '-'• 72. To express these results in a manner conformable to our ordinary notation, according to which we write the hyper- elliptic functions as L [k, k\ /c", «), M[k^ «', «", w) where K >k'>k, we shall remark that 1 — k'^k" > 1 - «" > 1 — «"", and if we put for a moment 1 — «'*«'" = f ""'', 1 — «" = f '*, 1 -«"•■' = f, we find P = r+r-fr, and the above for- mulae become ^{f,r,v(r+r-rr),«} = 2 v{(i - r) (1 - ni r ji + v{(i - n (i - ni ' "j -^1 €. U-v{(i-r){i-n}' ' 1 -2V{(i-r)(i-r)} "" Li+v{(i-r)(i-r)} 11 + v{{i - r) a - ni ' 'I ~ i-vui-r)(i-r)i ^ii-v{(i-r)(i-r) ' "'J J ' in which, if we write «, k in place of f , f ', and A;, i' in place of V(l-D> Vll-I")! we have the following, which are expressed in our ordinary notation: 64 CASES WHERE THE FIRST CLASS OF HTPEE-ELLIPTIC (55), Hence hyper-elliptic functions of the form olL{k, k', V(«'+K"-i^O, «}+/3-3^(«, «', \/{k'+k"-k'k"'), m] can be added by the formula for the addition of elliptic functions of the form F ( ** , , a A , and by the formula for the addition of functions of the form .F f , , , , eoj . 73. We now proceed to apply the transformation sm6 = ita.Qm to the equation (51). In the first place we remark that by this transformation j„ cos"^ V{(1 - «" 8iii'6') (1 - k'" 8in'(9) (1 - k"'k"' sin'' (9)} becomes . /■"■ coa' coda * ;„ V[{l-(l-0 8in'^a;}{l-(l-«"'') sin^a)){l-(l-/i:'V'^) sin'^o.}] ' which is equal to J (1 — 2 sin* 6) + sin^w) dot j^ V[{1_(1_«") sin'^w}{l-(l-0 sm'to}{l-{l-K'V"'] aiu'co]] ' To deduce the effect of this transformation on the right-hand side of equation (51), we observe that tan^, i\J[l—m^ sin'^J becomes « tanw, \/{\ — {\ — m^ sin'o),), E (ot„ 6^ becomes i [tantBj \/{l — (1 — "*/) sin^o),} + 2^{V(l-0,».}-E{V(l-m,=),a,J] (28), and F{rn^^ 6^ becomes ^F{^J[\—m^\ wj ; similar re- lations are also deduced by replacing jn,, ^„ w, by m^, ^j, to^ ; INTEGRALS DEPENDS ON ELLIPTIC FUNCTIONS. 65 the right-hand side of equation (51) is thus seen to be trans- formed into 2(1-0(1-0 ^-^^"^^^ ~ '"•''^' '*'«) -^(^(1 -'»•')' "t) + 1 [itW' ^(^(^ - *"'"^' •"') + i^k^ ^^^(^ - '"='>' '"^^J ' which is equal to *j„ V{l-(l-0 8in»«}{l-(l-«;"") sin'o>}{l-(l-«"0 sin"w} ' and by taking into account the expressions for fa> da j„ v'{l-(l-«'") sin"a.}ll-(l-«"') 8in»o)}{l-(l-/e'V'") sin"©} ' /■"' sin'Wo) j„ Vfl-Cl-Ac'") sin^a)j{l-(l-0 Bin"a)}{l-(1-«'V'") sin"©} ' already given (52), (53) we find 6ln*ada j^ ^(1_(1_«'«) sin*a)) (l-(l-«"') sin"«) (1-(1-«'V'^) sin'^o)) = 2(l-0(i-0 ^^^"^^^ " '"•'^' "'^ "^ ^^"^^^ " '"'"''' ""'^^ + 2.V'(1-0(1-0 L ~*141''"' i^(V(l -<), -J 74. In this equation we shall put, as before, 1-k'V^=^"\ ] _ «'■' = p, 1 - k"'^ = f J and it becomes /•(o Bin''(ii><^&> j„ V(l - r 8in'6,) (1 - r sin'o,) (!-(?' + T - f D sin^'o,) = 2Fr r (i+v(i-r)(i-r) ' "*) + ^ (i-v(i-r)(i-n ' "'')] K 66 CASES WHERE THE FIBST CLASS OF HTPER-ELLIPTIC + 2rrv(i-r')(i-n ^ L i+v(i-r)(i-r) Ti+v(i-r)(i-n' •} i-v(i-r)(i-r) ti-v(i- r)(i-r)' 'jj' This equation can be put in conformity with our usual notation by writing k and k respectively for f and f, and by writing k and A' for V(l-f), Vll-D- We thus have re sin* )(l-/c'* sin^w) {1 - {^ + k^' - k'k") sin'w) we must remark that if we put tanX= ,/,t.» > it follows from (56) that «, continues positive from © = to g) = \, and is negative from to = X to w = ^tt. Let a be the value of a^ corresponding to u> = \. Now we have I -I „ V(l - «' sin'ft)) (1 - «'* sin'ft)) (1 - («" + k" - /c''*'^) sb'"©) , „ V(l - «" sin'w) (1 - *:'^ sin'w) (1 - {k' + «"* - «V) sin'co) r^ dm ^j\ V(l - «* sin"®) (1 - «'" sin'o)) (!-(«"+ /e'^ - kV) sin^w) ' INTEGRALS DEPENDS ON ELLIPTIC FUNCTIONS. 67 I t and if we put % = ,,, , rj^ = ,,, we deduce from (54) f^ dm }„ V(l - «' sin'o)) (1 - k" B\a'(o) (1 - («' + k" - k'k") sin^o)) 1 [j^ dco, f da, ) 2kk'\j, V(l-Vsm'«.) j„ V(l-'7/8m''a,.)J' r**_ da jx V(l - «* sin^o)) (1 - «" sm^w) (1 - («" + «" - /cV^ em'to) 2kk' y^ V(l - V' sm^w.) i a V(l - < 8m*a,JJ ' we find consequently /■*•»• da j„ V(l - «» Bin' w) (1 - «'" 8in»e>) (1 - («' + k" - k'k") sin'o.) "^&'j„ V(l-VBiii'o),)- In the same way we deduce from (56) BJa'axZo) , V(l - «» sm'o)) (1 - «" sm^o)) (1 - («» + «'" - «V) sin'a>) 1 fi^ da^ '^kk'{l + kk')}, V(l-'7x'sin''»i)' and from (57) rBin*«adf«i) V(l - /e" sin'ft)) (1 - «" sin"©) (!-(«' + «'" - /eV) sin'w) = i {/ ^^^ " '''' *™'*"'^ '^"^ ■*■ M'(l-l-M') j„ V(l-^/(^')>V'(•^). we see that the hyper-elliptic fiinctions L («, «', k", &) and if («, k', x", d) depend on elliptic functions if it"'(A;"'- k > k, smce ■^ a' c' /It' ' fi is less than c and greater than b ; and we find tfee = J 1-4 - «' (- + 4) sm'^ + kk'k" sin*^} dd "^ V(l - ic" Bin'O) (1 - «" sin'^) (1 - k"" sin'Oy^^^'' so that the arc of the line of curvature counted from the vertex where it meets the plane of the mean and least semi- axes of the ellipsoid, is h ^^, i(«, K\ «", e) - «' (^ + -J) m{k, «', «", e) + ««V'iV(/c, «',«", 5)1. CURVATUfiE OF AN ELLIPSOID. 71 > o o a It o 03 U s o 13 t> > S 9 •S "So ^ .a .a ^ GO (D -S- b .a a 0) 0} hi =,.s b _g X a % o a i +1 m CO QQ O O rt ITS t- o u •e- -C! a a '^ I e o II Si o o u u ■e- b 00 OQ o o o a II II t- -^ OQ EC O O o o 2 „ O M o o ^ I Cm 0) I— I a w b •©--« ai n ^ O o t^ « u a a 00 n o I > f ^ I -" ><; %-i o + m 8 o + s a '8 8 "8 "8" =0^ + ■5- 05 ■e- 9 13 i ^ -^ 60 g. T3 o ■£ § .2^ 2 ^ ^-' ^ o CD e« S m O -js C8 r5 o ID > .a ^ .■ti "o ,£1 eo t-l «s 5j ^ -^ t2 --^ 5 P J IM •M d — ' » TS S 'g ^ * "S — ^-( (D ^ O k<< ■4-' "it - •^ ^ ^ 602 c o uu — ^ 60. S rS » i ^ ! -s •'^ ' (O oj -£ s .a 7 -|:§ 00 72 RECTIFICATION OF THE LINES OF ID -a OS O n:) a (D ns a o o a 3 o C3 a a a C3 n u SjO a > r« ^^ ig + -2 ^ > i-a 'a > « ■S = CQ >_^ "h « ^^ 'b" 1 e< ^ ff 1 -?< "Vi «' ^4^ '-^ > Vt )£ ^ ^ 'W 'Si a 1 1 ■ 05 2? M "^ rO 1^ -*-» ►« Cm > ^« r^ ** ii 1 > 1 ri; "s -« > 1 M ^ ~5e H h > -a r« X rii rii + > ri; ■e- > I f^ > > r« r« I "b rij 1 -je Sii > Si > °Vi 1 > i^ ^ a o C3 £> rS " "Sb 2 a ' ^. <*- § s « 11 o^ o 9 4) ! ° .S " ' _ ™> a S a) ^ ^ -^ o S' ■♦-* DO O a a -^ ^ > d =« ->r >-< -0- 'C (^ ° a a -M ^ § O Cm b 'a I b a • I— ( OQ "?< = _a .9 a ~« I 1—1 a "si I -©• to CURVATURE OF AN ELLIPSOID. 73 or if , yjr, &c., in the same way as 6' is derived from 6, we have 1)kk'k" S /*, this line of curvature does not exist on the ellipsoid unless c^ — ab> Q. Again, in consequence of (54), (55), (57), if k''=k^-\-k!'^-k'k"' the value of ^l resulting from this relation will determine a line of curvature, whose rectification depends on elliptic functions. This value is easily seen to be -77-5 .1 — ^,-^s , •' V(a + c - 0) and as this quantity is intermediate between c and 5, such a line of curvature always exists and is situated on the gauche hyperboloid whose equation is x^ V' z^_ ^ 1 d'6' ^ {a' - h') (c^ - h') d' [6' - F) a" + 6' - V ' Again, in consequence of (58) the value of /* derived from the equation «'"■' = — p ^ will determine a line of curvature, whose rectification depends on elliptic functions: this value is a J i^ZT^ ' *°^ '^^ °°* ^^^^^"^ ^'-S'>0 unless d'—V-a \/{d' — d')>0, or unless the mean semi-axis is greater than the mean proportional between the greatest L 74 UECTIFICATION OF THE LINES OF and least semi-axes, in which case such a line of curvature exists. 81. If in the expression (60) for the differential of the arc of a line of curvature of the ellipsoid, which is situated on a confocal hyperboloid of two sheets (v), we put we have ^« = ^^R^T^^^^^ {V^^^^^d-^"- "" dd X V(l - if sin' 61) (1 - V sin^ 61) (1 - V" sin'' 61) ' i;, 7?', ij" are less than unity and i)' >i}>7]. Since ^ = corresponds to the vertex of the curve which is situated in the plane of the greatest and mean semi-axes, if Sj denotes the arc counted from this point, we have d'-^ X {[a'-c') P{-r,\r,,r,\7)'',e)-[d'-v') L{v,v',v",0)}-{e2)- 82. If , -^j X) °'i ''■ denote five angles satisfying the transcendental system L in, v', v", i>) + L {v, v', v", -f ) + -^ {v, v', v", x) - L {n, v', V", ff)-L [rj, 7)', 7)", t) = 0, M{v, v, v", ^) + ^{v, v, v'\ t) + ^iv, v', v", x) - Mir,, V, V, <^) -^{V, V, V", t) = 0, or any of the algebraic systems which are equivalent to it, and if s,^, s,/,, &c. denote the corresponding arcs of a line of curvature situated on the surface (v), we find in con- sequence of the formula (18) given for the addition of the function P f}* sin^ sinilr sinj^ sino- sinr ^ v'(l-V8in»(l-i7" sin»(l-i7''sin"-'x)(l-V'sinV)(l-97"sin''T) ' CUKVATURE OP AN ELLIPSOID. 75 83. We have the following equation connecting <^ with the corresponding value of /i, which we shall call /i,, si"<^ ^ /f (a'-y)fc'^-M.'') ] V( 1 - ■^^ sin'' ) VI («' - c") (c' - b') j ' whence if z is the s of this point, which is the variable extremity of the arc s^, we deduce 3in0 cVCa'-y) , V(l - V' sin'<^) [a' - e'O V(c' -v')^' Let 2", s'", a", s' be the s's of the variable extremities of the arcs s^ s^, &c ; in consequence of what has been just proved, we have sin-»/r c ^(0" - F) „ V(l -'j' sin^V^) ~ [d'-e) Vlc'-v") ^ ' sinx ^ cx/(a'-5') ,„ Vl 1 - V' sin^ x) («' - O V(c' - v') ^ ' and so on, so that finally we find \ V S\tf T Sy — Sff — Si [[a'-c']ic'-vr 84. By the formula (20), given in Qiapter III., P (— rj'^j 7), t}\ rj", 6) can be expressed by the functions L (77, ,?', V, 6), 31{v, V, v", e), N{r,, v\ v", 6), Q {v, v\ v\ e) ; so that the arc of the line of curvature which we arc con- sidering can be made to depend on these functions by substituting for P[—7)^,rj,Ti\r]'\6) in (62) this expression. The formula thence resulting for the comparison of the arcs of a line of curvature of an ellipsoid, which is situated on an hyperboloid of two sheets, is however rather complicated. 85. To find the lines of curvature of an ellipsoid lying on a confocal hyperboloid of two sheets, which are rectifiable by elliptic functionfl, we shall proceed as before. If we seek the value of v derived from the equation 17 = rj'tj", we find v= — — ^^ j—^ -^^ such a line of curvature is 76 EEC'TIFICATION OF THE LINES OF therefore not possible unless b \/{d' + c') > ac. Again, the V derlv( '2 l-l 2 value of V derived from the condition t)"'' = — -^ — gives 1-7] v = a ^ /(-i — p] ; such a line of curvature is not possible unless h>a . /( ^5 — j^j or unless a' {a' - c') — [a' - Vf > } and will not exist unless the mean proportional between the greatest and least semi-axes exceeds the square of the mean semi-axis. The curve of intersection of the ellipsoid x' 1/' z'' — ; + -V — f, 4 —, 5 = 1 with the surface a a — b a — c x" v" s^ 1 d^c'-h') ' {d'-Vf-d'id'-c^) ¥{d'-c^) a'~b^ will be rectifiable by elliptic functions, and this latter surface will be an hyperboloid of one or of two sheets according as [c^ - by - a' (a" - c'") is greater or less than zero. 86. The value of v derived from the equation ,. . be v"'=v'+v''- vW is — . The rectification of the correspond- ing line of curvature offers results of sufficient interest to justify our entering into the question in some detail. In this case the length of the arc is best derived at once from the formula (60), which gives us the following equation for the element of the arc, as = - / ] --7 ■, - i^.T^ -.,,- } du., and if we put /J, = ^{bc) t, ■j- = a, - = /S, we find ch- bc ^Jibc] L a Y it. If < + 7 = ") ""'e have 2< = m + ^{d'' - 4). For points of the arc of the curve which are situated between the vertex (0) for which /t = 5 and the point (P) at which /4 = V(5c), we CURVATURE OF AN ELLIPSOID. 77 must use the lower sign in the radical entering into the expression for t: and for arcs of the curve counted from this latter point up to the vertex [Q) for which /i = c, the upper sign is to be employed. Consequently the length of the arc counted from the vertex (0) to any point {T) of the arc OP is expressed by the integral Va + ^ v(^«) jl ^' \/a — u if we put ,(^''+;^J-"' du ■ udu \ V(«'^-4) V = ^{ti' - 4), u = f V/3 + -^) siuS,, v = {y0- ^) sin 5.,, _a(c + b) _ a{c-h) "'""^■^T'i^' "'~ a' -be ' and integrate, remembering that when ^ - i, 5, and S.^ both become ^tt, we find (X arc 0T=\ To find the length of the arc OP, let smX = -^qp^ , and for the point P we have ■?, = \, ^, = 0, so that arc 0P=\ The length of the arc counted from P to any point T' be- tween P and the vertex ^ is given by the integral tJ/V< ''''*Ta ^^*TP} - \du-\ udu vK-4: so that we have 'd'+lc ar cpr=i {^(«„.9j-i:("„x)] + ~^Kr^J 78 RECTIFICATION OF THE LINES OF For the vertex Q we have ^^ = ^^ = ^tt, so that arcP(2=i 1^^' {^(«,^^)_^(„,,x)} + ^^K,i,r)] . By taking the sum of the arcs OP and PQ, we find for be the quadrant of the line of curvature for which v = — the a 2 7 expression ^(n^, |7r); so that the circumference of this line of curvature is equal to the circumference of a plane ellipse whose greater semi-axis is , and whose eccen- We find the following expression for the arc QT': + ^'l^K,i7r)-i:(n„^,)) It is easy to see that the angles ^, and S^ are connected L ii 1 x- cos^, c — b by the relation ~ = = . •' cos3„ c + b Let F, V be two points situated respectively on the arcs OP, PQ, and let /i, /j,' be the focal semi-axes of the gauches hyperboloids which determine them ; we shall call correspondivg points two points so related, that for them iiu! = bc: as this relation can be written ,,, . = , ' , we \'[oc) IM see that at corresponding points the angles $^ and S^ have the same values. Supposing that V and V are corre.i2)onding points, we have arc gr'4 arc 07="^ {E[n^, \-^)-E{n., 5J}, arc(2r'-arcOF='^^^ (7?(w„ ^tt) - £(«„ 5,)}. CUUVATURE OF AN ELLIPSOID. 79 If ^Ye subtract from the quadrant OQ the sum of the arc OV and QV\ we find for the arc VV\ which is terminated by two corresponding points, the expression E[n^^^,y, which is therefore equal to the arc of a plane ellipse, whose greater semi-axis is , and whose . . . a[c-h) eccentricity is — ^ — y- • Along the line of curvature, which we are considering, ( for which v = — j the as's of every point aj-e equal to the focal semi-axes of the correspondiog gauches hyperbololds ; so that if £c', x" are the x'b of two corresponding points, we have aV = he. If we put /»" = ¥■ cos' + c" sin''^, /i" = y' cos'yjr + c' sin'i/', for corresponding points, we have ( 8i1 ) CONCLUSION. GENERAL REMARKS. 87. From the formulae given in Chapter II., it is plain that given any number of elliptic fiinetions of the first kind with a common modulus, a single function of the same kind can be "found which is equal to the sum of the given func- tions, the trigonometric functions of whose amplitude are algebraically connected with the trigonometric functions of the amplitudes of the given functions. From the results given in Chapter III. it appears that given any number of functions of the form aL {ic, «', k", 9) + ^M{k, «', /c", 6) which arise from assigning arbitrary values to 6 exceeding two, the sum of such functions can be expressed by two functions of the same sort. In genei-al if i„ V((l-'<:"sIn'''^)(l-«"'sin''6')(l-«:"^sin''6')...(l-A:'-'^'''sin''6')}' where k, k\ /c", ...k''^"' are all less than unity, and a„, a,, ...a„ are arbitrary quantities, the sum of any number (exceeding w4 1) of such functions, which are formed by giving arbitrary values to 9^ can be expressed by n + 1 such functions. If U[9)+ U[d')^ U{6")+...+ U{9^'"-'>)= U{a')+ C/(a")+...+ C^(a'""'), we have (denoting by B the radical) ffl am'"''' ed9 r('' sm"'*'9d9 r»- " sm'"-"' 9dd J„ S J a R J a^-'^'l a = (- 1)"" sin^ 8in<9' sin 6"... sin ^'""' sina' sin a"... sin a'""'", »am''''''">9d9 [»' sm<''*'"^9d9 r»'"*" sin''"^"''^rf5 •and ft> s\n'^"^"'>0d9 r» sin' "^"'>9d0 fH'"*" Jo R Ja' -K ■■■ ja(»^l) E is a rational and integral function of the right side of the last equation. • { 81 ) APPENDIX. NOTE (A), p. 8. Haedenkamp has deduced by the method of elliptic coordinates the form under which Jacobi has presented Abel's theorem, see Crelle, Vol. xxv., p. 178. If f{x) is of even dimensions in x, Jacobi has, by a slight modifica- tion of his demonstration, deduced the algebraic integrals of the transcendental system in this case. Retaining the notation in the text, it follows that fix,) fix) ■^ "'' - k ia the coefficient of - in the development of ---m—m . Now if f(x) = a„ + Oyx +...+ a^mx^ ; this coefficient is — a.^, so that the calculations in the text being suitably modified, we find ■^ + •^=-1'^' or '^^^^" ^i^^MM]; whence, by integration, — ^ = iatm ("r) + e j and if we Bubstitute in this equation for (jt (or) and ^ J[i^ (or)} their values in terms of Xj, x^, ... x„, we arrive at an algebraic integral of the transcendental system, which is derived from the root a^ ; [m — 2) of the remaining roots will furnish the remaining independent integrals. In this case < is no longer an algebraic function of x„ x^, ... Xm, hut becomes a circular or logarithmic function of these quantities, and Xi^-'dx, Xj^-'f&j , , Xm^^'^dXm i + ItTi — ^T +■'•"• is the difierential of a circular or logarithmic function of x,, Xj, ... xm. NOTE (B), p. 68. Legendi-e, I believe, first observed that \ ^^^^ + p^2 + y^ + ^^ + ^) '''i«° P is a rational function of x, depends on elliptic functions. Jacobi subsequently f sin^'-Sde , , ., remarked that J ^(i _ ,. ^^e) (1 - <" sin'9) (I - .V sin'e) ^^^""^ °° ^^ same functions. The remaining conditions given in the text that the first class M 82 APPENDIX. of hyper-elliptic integrals ehonld be expressed by elliptic functions are giren by Kbnigsberger, see Crdle, vol. LXVii., p. 77. In a recent number of M. Brioschi's AnriaK di Mat\ematka, Mr. (Jordan has shewn that if X denotes a sextic func- (■(a + Bx) dx tion whose skew invariant vanishes, the hyper-eUiptic integral jp^ depends on elliptic integrals. This elegant theorem can be thus simply demon- strated. If we put x — —. — ^ , p, q, »', q' can be determined so that when P +iS X is expressed in terms of y, the absolute term and the coefficient of ^ dis- appear. The above integral then takes the form f ■{'•: + ^yfiy ]iy (1 + i\y + iif) (1 + 2\'y + ^V) but from the given condition, we have : 0, (see Salmon's Higher 0, 1, 1, X, M 1, \', / A Igebra, p. 275), whence /i = /x. If we assume then y J/ji = f, this integi-al is of the form (g" + P"t) dt k and if ( + - = M, we have 4« + -j- = 4(a + 2), J< - - = J(m - 2) ; so that — - i /_^!f_ du ] dt _ ! du du ] j«~* W(«+2) + j(5r32)/' ■|-Mj(^r^~4orT2)r and the integral which we are discussing depends on the integrals du r du ! du r l-l{u + 9){u + h) (u + 2)' Jj(m + ff) (a + A) (« - 2) • THE END. PniNTED BT W, METCALFB AND SONS, GREEN STREET, CAMBRIDGE. 104, Gbaftos Street, Dublin. MATHEMATICAL WOEES PCBLISHED BT HODGES, FOSTER AND 00. BY THE EET. GEORGE SALMON, D.D. A TEEATISE ON THE ANALYTIC GEOMETRY OF THEEE DIMENSIONS. 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