1889.] A. Mukliopadhyay — TJllliptic Functions and Mean Values. 227 

 Substitute 



a% (t2_r2) tan2 a) = 62 (a^-r^) cot« 5, 



so that for 



Again, from the above transformation, 



sec2u.^co=-^(^— j — 



gg jZ-rg (jZ cos2 g + a2 sing 9) 

 see 0) = ■ ^g-p _ ^.iy^i-Q , 



whence 



da,= -a5 (a2-r2)* (J^-r^)* «2js_,2 (j« cosM+a^ sin^ 

 Making these substitutions and patting 



we get 



^ g (62 -r2)'^ Tsi 



6 (62-02)* (a2-r2)* J (1-^^ 



, (62 -c2/ (a2-r2)* (1-^' '^i^' e;(l-e2 sin2 dji 



But since 



I 1 1_ 1 



(1-^82 ti2 siu2 9) (l-e2 Siu2 6, /32-1 1-^2 e2 siu2 9 (82 - 1 1 - 1'2 sin2 d 



we have 



X2 M 

 „ (l-;82t-2sin2 6l) (l-e2sine)* 



/„ (l-;82e2 sin2 61; (1-62 siu2 ^ 



