226 



PKOFESSOR A. G. GEEENHILL ON THE 



- \/^-t^i 



s" (nv) 1 



is' (nv) s (3nv) s (nv) 



+ iyQ 



is' (nv) 



{s(Snv) s(nv)}* 



s (2nv) s (nv) _ s (4m>) + s (2nv) 2s (nv) 

 is' (nv) 2is' (nv) 



/r> s (4nw) s (2nv) 

 2is' (nv) 



No 



w 



so that 



and taking 

 so that 



& 

 _x/Q 



(3m') s(nv) 



M 



?'' (m;) 



(2nv) s (nv)} 2 



we obtain from (17), (21), (22), 



p =: 4?V (2m') 



_ ,s" (2nv) 

 ~ if' (2nv) 



.s (2nv) 



and ABEL'S integral 



~ A- 



d* _ H ' v ( nw ) 



v/Z J 1 ,s (3nu) - (nv) 



. f 

 -I 



2P 



c?, L _ T , v 



"" 



on taking 



k = - Y - '-- ' / x - 2P (nv) 

 s (3m>) 6- (nv) 



= P (2nv) P (4m;) = a P (2nv) 



Putting 2nv = w, 



(22). 



(23), 

 (24), 



(25), 



(26), 

 (27), 



(28), 



(29), 

 (30), 



(31); 



(32), 



(33). 

 (34), 



