June 22, 1893] 



NA TURE 



189 



with « = ('« - \]v ; and the continued fraction expansion em- 

 ployed by Abel is not required, except, perhaps, for the deter- 

 mination of P. The integral I is pseudo elliptic when the 

 parameter v is an aliquot part n of a period ; and then 



p(« - !)£< = pz;, or p(« - m)v = p;«z', 

 expressed by 



S„_^ -}- JT = O, Zjt_-i =■ -1, ^«-2 ~ ^J 



or 



The integral I can now, for odd values of «, be expressed 

 in the form 



2(0 -I- .v)tM" = ;s*<"-3 -f Bzif'-'l -I- C;!!"-'! -^ ... } sjT. 



+ i ;Ps«'''-i) + Qz*'"-') 4- RsS("-5 -f ... } = H VZ -I- «K, 



where H and K are rational integral functions of s ; the circular 

 form of the integral being chosen on account of its dynamical 

 applications. When n is even, a factor c - a of Z can be in- 

 ferred by forming s!" + x; and then \{ z — b, z — c denote the 

 other factors of Z, the value of I can be expressed in the form 



(z + j-)i'V*'i = {3t("-2) + B2!("-« + ... ]^{(z - *) (s - c) 



■y 2{Pii("-=) + Qiil"-*! + ... !V(2 - a). 



The results for « = 3, 5, 7, 9 have been already given in the 

 Proc. London Math. Society, vol. xxiv. pp. 7-10; thus for 



«=3. -^ = 0; «=4, >' = o; «=5, j' = jr= - c ; 



n = 6, y = - c, X = - c{l + c) ; 



« = 7. ;' = - c{l + c),n = - c(l + c)- ; 



« = 8, 7 = - c '-+ ^'■, .r = - c{l + 2<) ; 

 I -f- f 



" = 9,y = - <:{l + -^JS X = - <■(! + f )'^ ( I -h c -^ i--) ; 



10, y ■■ 



c(l + , 



-'■(■ +c) 



(2 + c)[l - c - <r2) (2 4- f) (I - ^ - ci)2' 



But the next case of « = II presented difficulties, which were 

 only overcome by the kind assistance of Dr. Robert Fricke, of 

 Goitingen, and a reference to his article in the Math. Annalen, 

 t. 40, p. 478. It was found that the relation 



^11+-"^ = °> Of ^5 = %, 

 equivalent to Halphen's y^^ = o, or 



{xy - x'^ - y^) {y - xY - xy(y — X - j/-)' = o, 

 could be satisfied by 



X = - c(l + c) [l + c + q), y 



where 



1[<I + I) = ^(i + cY; 



I + c 



The relation between this c and the parameters t and t' em- 

 ployed by Klein and Fricke (" ModuKunctionen, t. ii. p. 440) 

 or the parameters n and W employed by Dr. Kiepert {Math. 

 Ann. I. xxxii. p. 96), was finally found to be 



I -(- 4c: -f 2c' 



5^3 



IOt -I- t' 



c\\ +cY 



2T-= 



= \W- -H 6?) - 16 -f W). 

 Given t and t', or •>; and W, the five roots of the quintic in c 

 will correspond to the five parameters, 



(2, 4, 6, 8, 10) ". 

 II 

 conversely, given c the values of 



p(2, 4, 6, 8, 10) " 



can be found ; as also the values of t and t', or n and W. 

 According to Dr. Fricke's theory (Math. Ann. t. 40) the case 

 of H = 19 should have a .solution similar to that of h= ii. 

 The general problem of the pseuJo-etliptic integral is thus re- 

 duced to the determination of x and y, considered as the 

 coordinates of a point on the curve 



Zn ■¥ X = o, or S„_,„ = 3,„, 

 or 



7,1 = o (Halphen), 

 as functions of a parameter c ; and when this is effected the 



values of p can be found : and thence, in the manner of 



« 



NO. 1234, VOL. 48] 



Kiepert, Klein, and Fricke, the various corresponding modular 

 functions can be determined. In the dynamical applications to 

 the motion of a top or gytoslat, the azimuih ^ can be divided 

 into Iwo parts, -i/y and -i/.^, where, accoiding to the notation of 

 Roulh's "Rigid Dynamics," 



r 1 (; •^ C 



'('1 = i 



A I I 



dl 



r cos 6 



,, +.■= \ 



G - C; 



-'ir 



dl _ 

 - cos e' 



9 denoting the angular distance of the axis of the body from its 

 highest position ; and ifi], if/^ aie thus two elliptic integrals of the 



iU; 1 l.;_J 1 ; .i-_;_ i-_ -^ . l _ i . i i-;_t . : 



highest position ; and ifij, if/^ aie thus two elliptic integrals ot the 

 third kind, having their poles at the lowest and highest posi- 

 tions of the axis, Ihe positions of stable and unstable equili- 

 brium. In i/^ we may put the parameter a — /wj, where w.^ 

 denotes the imaginary halfperiod, and/ is a proper fraction; 



also i/*! is pseudo-elliptic when/ = — , where rand n are in- 



K 



tegers. When « is an odd integer, the value of tfi^ can be 

 expressed in the form 



(I -f cos 9) !"£>"W,-/.' = H^/0 + iK, 



where II and K are rational integral functions of cos 8, of the 

 degree J (» - 3) and A(« - i), and 



denotes sin 8' — 

 dt 



But in if'j Ihe parameter is of the form 

 i = Vi + q'-.j, 

 where w, is the real half-period ; and to deduce a pseudo-ellip- 

 tic expression for 1^0 corresponding lo ij = —, the factors of 



« 

 must be known ; say 



cosfl — cos a, cos 9 - cos 3, cos fl - cosh 7, 



a and $ being the inclinations between which 8 oscillates. 

 Then, when i/*^ is pseudo-elliptic, 



(l - cos8)SV"<+>-^'') = HV!(cos)3 - cos «) (cose - coso)| 



-I- iK' ;^(cosh 7 - cos 8), 

 or 



= H\/!(co«h7- cos 9) (cos 9 - cos o)[ -f «K'v/(cos;3 - cos 9), 

 where H' and K' are rational integral functions of cos 9. By 

 multiplication of these two equations for if/j and if^, we find an 

 expression for 



(sin 9)"«'"('*-/') 



where/ ~ p^ + p.^ ; the values of the secular terms p^ and /2 

 being most readily determined by a differentiation and verifica- 

 tion. Changing the sign of i in if/.,, and denoting if/, - if/^ byx, 

 pi - p., by (/, we should find, as a verification, 

 (sin eye""^-"') = 



[L^[(cos $ - cos 9) (cos 9 - cos o)} + jM;^/(cosh 7 - cos 9)]", 

 where L and M are constants, corresponding to an elliptic 

 integral of the third kind, with a parameter 



l) — a = 0}^. 



The cases of k = 3 and 5 are worked out at length in the paper. 

 The pseudo-elliptic expressions for tp., are immediately available 

 for the construction of algebraical herpolhodes, as the parameter 

 in this mechanical problem is always of the form 



"i + '/") ; 



while the pseudo-tUiptic expressions for -i/.^ can be utilised in the 

 construction of solvable cases of the tortuous curve assumed by a 

 revolving chain. In the herpolhode the case of « = 3 is realised 

 when "the focal ellipse of the momental ellipsoid rolls on a 

 plane at a distance from the centre equal to the difference of its 

 semi-axes ;" and when n = 4, "the distance of the fixed plane 

 is equal to the distance from centre to focus of this focal ellipse." 

 By Prcif Sylvester's theorems on correlated bodies, Ihe molion 

 of the bodies having momental ellipsoids confocal to this focal 

 ellipse, can be inferred immediately. In the equations of the 

 Precession and Nutation of ihe earth, or of the molion of an 

 elongaied projectile in an infinite friclicnless liquid, the function 

 will be composed of four linear factors ; so that in 

 the construction of pseudo-elliptic algebraical cases of this 

 motion, a return to Abel's original method may prove 

 preferable, especially when « is an even number. — On Ihe 

 expansion of certain infinite products (II.), by Prof. J. L. 



