( 759 ) 



d£I dn'- c dn'- {ii — K) dv 



Ir /■'- {en V — en {n — K))" i 



and we find for that part of the snrfoce limited by tlie two circles 

 with the arguments -|- ?/ and — u: 



u '2iK' 



£2 r rdv dn'- V dn" (u — K) 



l-M 



4:b- J J i k'- {en V — en {n — K))' 







To perform the integration we start from the identity 



2/A' ' 



J' dv sn (u — A' ) dn (n — A' ) n:v 



~r-^ '~-^~-^ = 2K'Z{H-K) -f - = 

 I en V — en {n — K) K 







= 2h {E'-K) + W"- K' B {u), 

 which furnishes first 



2jA'' 



'dv dn' {u — K) ^' fv* 



ƒ 



i en V — en {v — K) ir ,i 



Moreover 



2/ A" 2rA' 



Jdv dn'' v — dn- {u~K) ' Cdv 



-. ^^ — — -^ - F — en y 4- 2 F K' en {u—K). 

 I en V — en {u--K) ^ i 



u 



A dash before the integral sign indicates that the path of integra- 

 tion does not pass through point v = IK' . 



Out of the two last equations follows by means of addition 



2iA'' 2/A ' 



' rdv dn- V k'f{n) ' rdv 



~ . z T^ = -^^ + k^ - en V + 2k"- K' en {u - K), 



^ I env — c?i {n — A ) en ti ^J i 







an equation which, if we ditFerentiate with regard to u and then 

 divide by k' en u, passes into 



2ïJf' 



dv dn- V dn'' {u — K) 1 d n'{u)\ 2k- K' 



r . - .. . > , . 



^ i k- {en v — en (w — A ) )' efi u du \en uj dn* u 

 u 



+ 



1 d (f{u)\ . 2k' K' . kl'-K' . E'—K' 



2 du \c?r uJ dn* n en" u dn^ u ' en' u 



Now integrating according to z< between the limits and u we 

 tind finally 



Si u , dn^ n 



jr^^-r- i^' - ^ ) H- E'A{n) + K' -^ B{u) . 

 46- en- n en- u 



If the given circles have the radius R = l then b is equal to 

 en u and we can write 



