368 Mr Greenhill, Integrals expressed hy [Mar. 8, 



p dx 1 f / Jl^-K.l + X.x j\ 



joV^~ v(i+«-i+^) r^ '"^ vv i + Kx.i + xx ' 7 



/ /1 + K.1 + \.X 



+ SLvg: sn . / , c 



* {y l + Kx.l + \x' 



X. 1 f / /l +/g.l+\.^ 



- arg sn ^y 

 where 



— arg sn . / ^ ^ , c 



^ \fK + \/A- _ \J K — \J\ 



^ — 77:. ^ ^^ 1 c — 



V(l + « . 1 + A) ' V(l + /^ • 1 + ^) ' 



More generally (M. Roberts, Tract on the Addition of Elliptic 

 and Hyper-elliptic Integrals, p. 53) putting x = f, 



ff dt 



Jo^a-pf + qt*- 



+ 



,sl{l-pf + qf-pt+f) 



" du 



° dv 



2' VK - (i? - 4) w'' + 2 - 2p + 5} ' 



where i + T^ = ?t 



t-t~^^v 

 = h {II + V), suppose ; 

 fdt 



and 

 Also 



o^/il-2yf + qt'-pf + f) 

 = 1{V~U). 



V{4 + j3 + V(8-4g+/)j^ /4+j3-V(8-4g + ;p^ 



2 



arsf sn 



V{4-;7 + V(8-4g+/)} 



s/{'i-p + ^{8-4<q + f)]t /f4-£-V(8-4£+/ 



1 + f ' V (4 - ^j + V(8 - 4g +/)j ' 



