Theory of the Jacobian Elliptic Integrals. 26.") 



order, and to deduce from the theory certain integrals which 

 are otherwise difficult to evaluate. Most of the results of 

 this latter clas-. also, are new. 



We shall employ the two expressions for P»0*) 3 where 

 /jl = co*6. which are due to Mehler, viz. : 



p.OQ-T cos (" + ^*, . . . (i) 



¥ . (li) M'^^t d t- ... (2) 



7rJ e \^2(cos0— cos0) 



and in all future work 6 is considered to he a positive angle. 

 lying between and 77. 



Let F (:) be a function of z, satisfying the conditions for 

 a Fourier expansion between z = z x . z = z 2 . the expansion 

 having one of the two forms : — 



Y(z)=i flB eos(2n + l>, 

 a =o 



GO 



F(s) = 2 a H sin(2« + l)^. 



n = 



Then between the limits tz u 2z 2i Fl ® J has one of the 



F(|)= £a n cos(n + i)<t>, .... (3a) 



?{$) = $ «»sin(n + i)0 (36) 



2 1 



Multiplviim- the first form by - . — , and 



J * V /2(COS0 — COS0) 



integrating from to 0. supposed not to include 2c 1? or 2z 2 , 



-UP.W.. • (4) 



forms 



y 



V / 2(cos^> — cos^) 

 Similarly, the second form give- 



>W V * =2«,P.O*). • • (5) 

 e V2(cos — cos 9) o 



These formulae will be applied to >everal Fourier expansions. 

 The latter will be freely quoted, and are all to be found in 



y 



