100 Prof. A. Schuster. [May 30 r 



are obtained in the form of a series having a finite number of 

 terms. 



6. To find 



+1 +i 



| Q£sin pddfi and | QJcos pOdfi, 

 -i -i 



we may either express or the trigonometrical functions in terms of 

 a series of powers of sin 6. The second alternative leads to results 

 which in general are more convenient. 

 If we put 



_ p (j» + X-8)ll . 

 A M (i> - A) ! ! ' 



b = i ; Bl = ?; ife ?r^|±l; b 3 = ^- 1 2 ;|| +2 



B 



1 (p+A.-l)!! 



we find if tr be even, p odd. ami » even, 



q (T — 3 . <r — 1.<t+1.o-+3 q or— 5.0T— 3.<r — 1.0-+1.0- + 3.0- + 5 

 •h - 3 . ;i + 4 w - 3 . n - 5 . n + 4 . ?< + 6 



and if tr be even, p even, and n even, 



= ff , (w + ^iiC ^j)::, f B I-<r+1 _ B °-3.o-i.o-+i.o- -s 



(»-or- 1) ! ! {n + 3) ! ! t n - 4.7? + 5 



cr-5.o--3.o- - l.o-+l.o- + 3.cr + 5 



rt-4.»-6.% + 5.» + ? 

 If 71 + 0-+^ be even, the integral is zero. 

 Similar equations are obtained for 



Q°" cos pOdfi. 



7. The final results are expressed as follows : — 



If gl denote the coefficient of E^cos o-<p in the series of spherical 

 harmonics when R" = tJ t Qj,. & n d is ;l numerical coefficient, o being 



