1882.] Mr Greenhill, Note on Prof. Cayleijs paper, &c. 223 



(3) Note on Professor Cayley's paper on the elliptic function so- 

 lution of the equation x 3 + y 3 - 1 = 0. By A. G. Gkeenhill, M.A. 



In this paper, read before the Society on May 23, 1881, 

 Professor Cayley expresses the x and y of any point on the cubic 

 curve 



x s + y 3 -l = 



in terms of elliptic functions of u by means of the relations 



2rsnwdnw — (I +cn w) a 

 2r sn u dn n + (1 + en w) a ' 



_ m (1 + en u) {1 + r 2 + (1 - r 2 ) en u] 

 ^ ~ 2r sn u dn u + (1 + en uf 



the modulus being sin 15°, and r = $3, m = 1/2. 



The values of u for x — he finds are given by the equation 

 m 6 (1 + en iif = {1+ en u + r 2 (1 - en u)}\ 



the real root of which gives 



_ r 2 + 1 — m a 

 en u — 5 z ■ „ i 

 r 2 — 1 + m 2 



and therefore ?« = \K. (Legendre, t. I., § 24.) 



The other values of en u are obtained by writing com 3 and &>W 

 for m 2 , co being an imaginary cube root of unity. 



But the corresponding values of u will be found to be * 

 fa) A" or -IK+^iK', 

 and §eo 2 A or -±K-±iK'. 



For this cubic equation in en u is a particular case of the more 

 general equation 



sn u dn u sn a dn a 

 (1 + en uf ~ (1 + en af ' 



* Quarterly Journal of Mathematics, Vol. xviii. p. 66, 

 " Ou the reduction of the elliptic integrals 



[ dz , [ zdz .. 



J [!»-l)s/(z*-b*) an J t*-l)V(*-P) ' 



