749 





sm sfp 

 X ~ ■stw" 7 co.s j/ co« /? dy d^ d(f dn^ . . . </»<», c?2j 



If we substitute in this 



"i + 2 'V ^'*"* y ^"^'^ i^ = ^^t ^'s — h ^'f 'S"' 7 <'0S ^ := ^, 



r, -j- ^ /// sin y sin i = 7^^ v^ — ^ t</i sin y sin /? =: tj^ \ (6) 



"'i + 2 "/cos y = ^1 "'j — . 2 ty cos y = C,^ 



we get : 



1] = 



- 1 1(^8 ->i) -*""y cos^+{ii^-i]^)siny sm ii+(?,-?,)cosy+K/ j' 



X ^in'* y cos y cos [i d(f dy d{i ds,^ . . . dC,^ dz^ | 



On integration with respect to ^i . . . ^^ ^erms containing odd 

 powers of ^, . . . ?, vanish, so that the only terms left are those with 

 ^\[(ê,'^'i,')'^^n'r^-os\i^{n,' f >i,='>nn='7.mV f {^,' \^ C^')cos"-y^ifpmiycos^+ 

 -\- 2i,^* ((f sin y cos /i — ^ i(f^ sin y cos (j]. 



These terms do not change when $,- is snbstituled for ^j', i;,' 

 '>i5%?i', and i-'j*, so that -j- [S^,'' k/ —h t(p*) sin y cos ^ maybe written 

 for the snm of the remaining terms. After execution of the integra- 

 tions we tind : 



2 r 



>/ -= 7i^r/ m a i \ (3-y') > e~^ V+r« X *"' •^'7' X 



n J 



X sin^ y cos y cos^^dff dy d^ dz^ (5</) 



Let us now replace e^'r" by cos rps -\- isin ifs, and execute the 

 integration with respect to 7, bearing in mind that we seek the 

 value of the integral for lini. s = cc. Then the term with .vm .sv/- ("r>5 .f</^ 

 vanishes, and in the term with sm"^ sq we may replace this expres- 

 sion by its mean value 4. Thus we find: 



— i\e-'k ■r+'r-'< X <sm s (f y^ {'6 - ip"" ) dtp ^z V 2:x 



— 00 



•27r 



I cos'^^ dii = jr 

 u 



