PROFESSOR A. G. GREENHILL ON THK 

 31. ^=7, 14;= .'.*/*; 



D 7 = (1 - 2m) :! a'- - m ( 1 -- 2-H?) ( I -- 3w) - t ' ( 1 - w) 3 = (l) ; 



and on putting 



I - 2<m = ^, (l -- 2wi) = (1 -- "<) & 2/8 = 1 + (2), 



the relation becomes 



]>~<f 4- pfj - P - <1 = (3), 



an elliptic-function addition equation, connecting 



t> = p (u) and 7 = p (n -f fw) (4), 



-3;;+ 1) (5), 



and 



p(&) = 0, p(&)=l (fi). 



Thence 



a-m 



The connexion with KiEi'Kin'ss results in 'Math. Ann.,' 32, p. 87, has been given in 

 L.M.S., 27, p. 4:58 ; here 



f = 86=:< + 1 H' J " 1 > s (10), 



" 



a transformation of the Third Order; and 



suppose, when- w is the real half-period of the. elliptic function P=JO(M) of the 

 elliptic argument //, and now 



f, = f(-i w ), J =f(-i) (13), 



bt 



f 2 =l, p-2p-|) + l = (14), 



