536 Mr Hargreaves, The Harmonic Expression of the 

 Also 



\p cos X cos 8 ( X P-i, s -i ~ Xp-i,b+i) =f% sin r sin sf . cos X cos 8 



(sin X sin 8 + cos \ cos 8 cos -^-J ( ty 



= - sin s Y (sin X sin 8 + cos X cos 8 cos ^ *' + f* 1 s x 



Jo Jo 

 (sin X sin 8 + cos A, cos 8 cos i/r)p cos 6-i/rc% 



%.•=*!>• °°s X cos S^^-^^) (9) 



From (8) and (9) we get 



*P C0S X cos S f(« ~ l ) X^-m-i " (« + 1) %^- 1)S+I } 



= ( s2- .P)Xp,s+£%,-i )S sinXsin8 (10) 



which is wanted below. 



Again 



d . . 

 (fy l sm Y cos s ^ (sin X sin 8 + cos X cos 8 cos ^>)p-i} 



= (cos r cos s Y -s. sin f sin s r ) (sin X sin 8 + cos X cos 8 cos ^-i 

 - (p - 1) cos X cos 8 sin 2 ^ cos s Y (sm X sin 8 + cos X cos 8 cos *)»-* 

 Hence, writing 



cos^cos^- 5 .sin^sin^ = i{(,+ l)cos(, + l)^-( 5 -i) cos(s _ 1)l/r}; 

 and integrating 



C'i'i 



(p - l)cosX cos 8j o sin-^r cos^r (sin X sin 8 + cos X cos 8 cos if>-<ty 

 or by (10) = i K* + 1) Xp-m+i - (« - 1) ^-i,^}, 



^(^-l)cos 2 Xc^8j%in^cos^(sinXsin8+cosXcos8cost>'- 2 ^ 

 " fiut eyidently = (P - * 2 ) ^, -1%^. sin X sin 8. 



cos- X cos 2 Sj^ cos 2 ^ cos s r (sin X sin 8 + cos X cos 8 cos f )*-* <ty 



= Xp.* - 2 %p-i )S sin X sin 8 + Xp _ 2s sin 2 X sin 2 8. 



p,ev! re q u^ Plymg ^ * ^"^ «* ^ to the 



P(p-l)%,- 2;S cos 2 Xcos 2 8 = (^- s2) ^ )S _^ (2 ^_ 1) ^_ ii . sinXsinS 



+ £» O ~ 1) %p- 2 , . sin 2 X sin 2 8, 



