244 PROFESSOR A. G. GREENHILL ON THE 



(A 3 + B 3 ) = 2x* -xy(l+y) (5), 



4AM 3 = xy (6), 



so that 



256M*A 2 = 64.x 2 32xy (1 + /) [(I + yf SxJ 



(l-y)-(l+yf] (7), 



Now denoting the elliptic argument by it,, where 



ds 



+ 7P(2r)=I(2r) (10), 



and the preceding integrals can be utilised for the construction of solvable cases. 



The chief interest is in the purely algebraical case, obtained by putting P (2r) = 0. 

 Thus we find for /A = 5, putting // = x = -, in (3), 7, 



'2r> cos *0 = (r + a) v/(2r' s - 4ar- + Gcf-r + 3a s ) (II), 



JH sin I 6 = ( - a) v /(2?- 3 + 4r 2 + Ga-V + 3a 3 ) (12). 



20. The expression of the pseudo-elliptic spherical catenary, discussed in L.M.S., 

 27, p. 127, can be halved in degree by changing to the stereographic projection, with 



tan0 = *, 2 = cos0=| (1); 



i -\- t 



Z = i - *-,- A- - 2 



:4 **( 1 X J -+"-A- I ^"-(l+T = T 1 T i .(3), 



[^ _ 2t (^~'-\ -f ^ + 1 



* \ A A 



\ -ii. Xi. 



a- 

 In a pseudo-elliptic case, with 





