302 



d"- 



^ = .v-\-iy, ri^=.v — iy 



and letting 



i„ = 1 ; J,, 6,, . . . = i 

 it may be shown that [see appendix formula (Ij)] 

 cos m (p^ P^ (cos ^,) ( — )" 2"» b 



(n—m)! „ \rj 



the differentiation being performed with respect to the end point of 

 the vector j\ i.e. with respect to [r, I'K f/>). Thus 



/ 4jtA'\ 0/ (—Y2r"b,nA'" ,„ /I 1 \\ 



V 3 ^V ÖrV,v.=:l.... («-m)/ n 1 V^ ^JA^l 



1 1 



where -^ has been subtracted from — so as to secure absolute 



convergence of expansions that follow. Now 



1 1 » r" (n—m)! 



---=2 — — tVt-^ ^ n (''^^ ^) ^r i'OS®,) COS m {<f~^,). 



T^ R^ n=iRi''+^o,n{n \-m)! 



m=0,...u 



When this is summed with respect to R^, ©1,^1 terms in sin m^^ 

 and all terms with an odd m drop out. Hence 



m m 



Een=( l+— -tI yosd-~l 2 -—^2r^P::{cos»)cosiicp~ 



\ 3 AV OrVm.n {71— m)! ^^fj, V/r=i 



where 



' ~ {v-\-(x)f 72/4-1 . • . • I ; 



Now it may be shown that [see Appendix formula (2^)] 



P!!^{cosd)cosiicp\ ( — )'» {v^yi)! r''—^P^—n^{cosd)cos{m+yi)(p 



m f f 



V ) 2"» (r + |w— n-)-m)./ 6,„_|_^ 



which when substituted into the expression for jE^e,, just found gives 



/ 4^ AX\ 

 E,,^[\^~-y^P,{cos&)- 



y/ \n+rn ^"* i^ -\- l^V i^ — ^) ^'n S^ P!^^,1'^ (cosd-) COS {m-\-ll) (p 



b,n+^ (n — m) !{v^ii — n-\- m) ! 



(««0 4" ^ ~h ^) -^n Pn {''OS &) COS mrp 



— ^ ^^ — ■ . 



n.m €* 1 



