234 PROFESSOR A. G. GREENHILL ON THE 



The normalising factor is 



M = (M 1 M S M 8 M 4 M 5 ) (14), 



but so far this has not made evident any symmetry of results ; it will be noticed that 

 our parameter c here is an elliptic function of (L.M.S., 27, p. 429) 



,, rX 

 (15), 



which is 2" 5 -th of a period out of phase with that required to lead to KLEIN'S results. 



12. ,1= 13, v = ' 2 , y lt , = (L.M.S., 25, p. 251; 'Math. Ann.,' 52, p. 484) 

 1 o 



by the substitutions 



,2 



x = y(l-z), z y=~, z-c(p-}) (I), 



leads to a C, with class p = 2, in which 



2p = 1 - o - c + x/C (2), 



0=1 + 4c + 6c 2 + 2c :! + c 4 + 2^ + c 11 



= (1 + 2c - c 2 - o :i )- + 4c- ( I + c)- (3), 



and we find, using detached coefficients of ascending powers of c, 



, , C, + I 2 - 'J - 33 + 4 + 8 - 18 - 1 1 + (4 + () - 1 5 + 7 + 1 1) V /C , ,v 



'-' 



/T"T 



1 + -2 - 3 - 9 + 1 + 2 - 5 - 3c 7 + (1 + - 4 + 2 + 3c 4 ) /C 



9 /i _i_ ,.y^ v /' 



2(1 + cf ' 



A a = 1 4 1 + 16 + 15 26 28 + 28 + 14 38 20 + 6 5 10 3c'*, 

 B 3 = 1 -- 2 + 4 + 9 7 15 + 12 + 14 7 - 2 + 7 + c" (7), 



Ao = 1 + G + 7 - 2G 64 + 24 + 154 6 222 + 32 + 266 



+ 10 - 10!) + 104 + 143 + 22 + 4 + 32+ 21 + 4o 1!) , 

 B 3 = 1 + 4 - 2 - 25 - 10 + 61 + 27 - 97 - 28 



4. 90 - 7 82 12 + 15 - 15 17 - 4c 16 (8), 



