370 Mr Greenhill, Integrals expressed by [Mar. 8, 



Again (Legendre, Fonctions Elliptiques, 1. 1. p. 178). 



Jod-Jc 



dcf> 



.= 9, 



sm' 



^dos 



(p)i Jois/iji-x^ .x*-k'^) 



72 if x^ = l—k^ sin^0, 



V(2 + 2A;') 



arg en 



{\-¥dn^<^f^-^/k' l->^k' 



(1-Fsin»»(l-V't-')' V(2 + 2^') 



+ arg en 



( (l-Fsin^(/))^ + VA;^ 1 + \/k' 



and 



1(1 - F sin^</))^ (1 + V^^') ' V(2 + 2A;')J J ' 

 d<^ f^ 2cZa; 



o(l-FsinV)^ j^V(l-«'''.^'-^'") 



V(2&' + 2F0 _ 



— arcf en 



Let 



+ are: en 



Z= 



(1 - F sin^0)'' - ^Jh' _l-^J¥_ 

 (l-FsinV)* + \/^"' l + \/k' 



— 3 



2 



(1 - F sin^)' (1 + VA;') ' V(2 + 2^^') 



■^ d6 



o(l-FsinY)* 



if «^ = l-Fsin> 



(Legendre, t. i. p. 180). 



Put k' = n\ 



and x^ + w^ = ocz ; 



therefore a:; + n = a;^ /y'(2: + 2?i), 



ir — /I = «2 \/(^ — 2^). 

 2a;^ = V(2 + 2%) + V(^ - 2n), 



2a;-2(^^= -,^ — -^^— + 



V(i? + 2?0 V(-^-2?i)" 



