jViultiplying these relations with the different factors written on 

 the left, the addition of the.^e products immediately gives the formula 

 in question. 



The second relation 



Ê'Lr^f^^=^^e-d-'é. (30) 



1 2».n.' J ^ ' 



Ü 



may be obtained by introducing (9) into the first member. . . 

 Thus we get 



cc, n,^x)IIn-\{ x) 



'^ 2».nl ~~ 



.00 • „2 OS ' . : 



t I e 4 m" cos I ^xu idu I e~""u"~* sin i xv I 



= ^ z : I e '^ W" COS I Xll ::- \du | e~""U"~^ Sin i XV — I dt 



where 



1 a>^ ?i"u'i / njt\ , / njt 



— ^ COS I xu shi I XV 



V 1 2" . n! V 2 y V 2 



CQS XU sin XV, 00^ ifil-ffih g{fi ^u cos. xv cq^ m-^+Iu^^'+I 



'^IZ ■_ '■ ^ — ; : — - ■— • — —- ^ , 



V 1 2'iK{2/i-)f V i r22^H-'(2/t+l)/ 



HI) un uv uv 



COS a'u sin xv fc'- ■\- e, '^ 



/e- 4- e '^ \ sin xu cos xv /e^ — e 2 \ 



Substituting this value, it is evident, according: to the formulae ot 

 Art. 6, that all the terms of this sum vanish except only the term 

 coi-responding to — 1. 



Hence 



°° ilJx)fIn—',{x) e*-^' r P 4 COS x^i sin .XV :, 



2" — — = II t' — ^^ du dv 



1 2» . n! jr J J "" v 



and because 



CD ■*_ 



J 6' COS XU du ■= V jc (■~~^'^ . ... . . (u) 



^, II,{x)TIn-M ^ ,, r 4 sin XV 



^ — — — — — = — ^ir €?■ I e dv , 



1 2' ■ 



— — = — ^ir €?■ I < 



J« . n! Vn J 



If now we multiply the equation (a) by J.i' and integriate l>et,vveen 

 ' and X, we have 



