Theory of the Jacobian Elliptic Integrals. 211 



The value of each 



4 o\2»-l 2/i + :y ' (2«+l) :! 



__ 1 v 1 /^_ + M V 4 i 2 [ — — ) 

 ~ i*o HV2n-l^ 2n + d) «l(2« + l) 3 T (2« + I) 3 (2» + 1)/ 



If -t ^_+_l_'- 



8 1(2m + 1/ (2rc + l)' 2 2m + 1> 



V 





1 



-f 



1 



4% 



(2w + l) 3 



16 



7 

 32" 



j + 



1 

 16 ; 







i 1 MvE^=( 1 / l K'E'V^=:^c7,+ 1 i; . . (53) 

 Jo Jo ™ lb 



AVe may employ the result 



|" (l-V)^ ^#^=0, or » ^, (54) 

 J-i d/jb' dyf 2n + l n — r\ 



according as n and m are unequal or equal, provided r be a 

 small enough integer. E. //., we deduce, if r = l, 



which coincides with a previous result, 

 and also 



y 



_ dVcn\ __ 2n(« + l) 



V ^ j ^ <*> ^~ (2/i- i) (2tiH- l) 2 (271 + 3) ' 



[so a previous result. 

 U r = 2. 



JV ^ j <//* 2 ~ V ^~ (2»i-l) (2«+i) 2 ("2w + 3) 



or 



Jo *' r »v^ x ^~ 4V(2*-l)(2* + l) 2 (2rc + 3) (05) 



and further particular theorems follow in a similar way, the 

 expansion for E / giving rise to an equal number. 



Some applications to incomplete elliptic integrals will now 

 be treated. The elliptic integrals in question occur in the 



