294 PROFESSOE A. G. GREENHILL ON THE 



It is now possible theoretically to determine the coefficients in 



r 



\ 



= J. 



M 



, 



" 



11 (8 - a 



. . . - o /r- x 



- 8 * 



sin -i 



U (S-Z 2 )* 



and to construct an algebraical herpolhode and associated top motion complete in 

 44 branches. 



This is as far as we can go at present, as the next cases of p. = 52 and 60 must 

 await the solution of p = 26 and 30, not yet accomplished. 



46.11. ip. = 2n, /x = 8n, v = K + fK'i, f = . 



47i 



X t = C + C^ + x* = (x, + x) (x, + x) (I), 



X 2 = C - C^ + x z = (x, - x) (x, - x) (2), 



G = s (w s ) - s (2nv) = ^/(s l - s s . .s- 2 - * 3 ) (3), 



G! = y(*! - %) + y(s 2 - * 3 ) = 2P (2m.) (4), 



G G t = ^y (2m') = Q (2nv) (5), 



(o+^-?J 2 = C 2 (6), 



o 

 and normalised by M = ^/Gy, 



X,, X 2 = 1 Ca: + x\ x, = I, a; a = o (7), 



x = osn (K - ) " = U (8), 



o M 



and the associated integral 



So 

 a 



M M 3 c/a; 



~2 



I*/ 



~R w _L _1_ T? /V.SM 2 I "R /y.2 1 



= - sin- ~ ,, ^ - *-' y| X 2 (10) 



2i (D x 2 )" 



