284 Theory of the Jacobian Elliptic Integrals, 



where n is a positive integer, and F(\) is the incomplete 

 elliptic integral 



f 



IX 



J v 1 — k' 2 iW?\ 

 Considering now the equation (62), which may be written 

 f *" rf * « «c> *o? (-)' v I ^ 



Jo J—J a e = ^ K - j2S o (Si+lj» p -W' 



V C0S V _C0S 2 



we have 



«6 



W cos- ^ — COS 2 - 



In this and above, (66)— (70), K/ ( cos^ ) denotes an incomplete 

 integral equal to K ; when (/> = 6. 



""' (7r-cos- 1 Ar)(7r+cos-U)K'+fVK / (cos|)#=:42 ,./~ ) " (P^) ? (79) 



Jo \ & / (Z?i+ 1J' 



and putting 7r — 6 for # in the above, 



-*{»+,i„-nK + j]'+K'(co.*),/*.jr ( -&M l . . . ,80) 



Proceeding in this manner, it follows that 



olifl + l)" o (2n+l)' ? 



where r is a ]>ositive integer, can be expressed in a similar 

 manner to the above in terms of Jacobian elliptic functions, 

 and their integrals with respect to a function of their 

 amplitude. 



The sum of all such series can therefore be concisely- 

 expressed. 



sin 



