224 Mr Greenhill, Note on Prof. Cayley's paper on [March 20, 

 (1 - en u) (A;' 2 + k 2 en 2 u) _ (1 -en a) {k' 2 + tfcn 2 a) 



or 



(1 + cnu) 3 (1 + cna)' 



(l-cna)(A/ 2 + & 2 cn s a) 



• 1+12r2 g-^)(r+ycn^) 



(1 + en uf 



= 1 + 123 



(1 + cna) 3 



/., „ 1 - en w\ 3 /., „ 1 — en a\ 3 



or l + r 2 ^— = 1 + r. 2 — , 



V 1 + cntt/ \ 1 + en a) 



„ 1 — en u /- „ 1 — en a\ 

 or 1 + r 2 _ = &) ( 1 + r* =— , 



1+cnw \ 1 + en a/ 



giving en w = en a, or en toa, or en co 2 a ; and in the particular 

 case considered by Professor Cayley, a = %K. 



The equation 



x 3 + y 3 = l 



leads to the differential relation 



dx dy „ 

 i + ^— a = ; 



(1-aO 1 (i-y 8 )* 



and to reduce to elliptic integrals, put 



(I-*) 3 *»' (l-y) 3_2/l ' 

 then J5 + ^t - o 



where 6 3 = J. 



Again, put 



, , ._ 1 — en u 



x.-b = b*J3, , 



1 y 1 + en u ' 



, . _ 1 - en v 

 Jl y 1 + cni;' 



where the modulus k = sin 15°. 



Then du + dv = 0, 



or m + v = constant. 



For the particular case when the integral is 



* 3 + y 3 = i, 



