WITH ATPLICATIOX TO SPHERICAL HARMONIC ANALYSIS. 
207 
the spherical harmonic of degree n will only depend on the Fourier coefficients 
cip, for which p is equal to or smaller than n •+ 1- When cr is even we secure the 
same advantage by developing k'^ and K"” in terms of the sine functions, and we 
write in that case : 
k’^ = hi sin 6 d- &.T sin . 
K"" = (Blsm 0 + yS.t sin 2d + . . . 
hi sin p6 . 
^l sin pd + . . . 
and 
sin pd = . b^q: + ... 
To calculate finally a coefficient such as B;^ we proceed in the usual way, thus : 
I sin^id djx = + <^) •. 
1)1 + 1 {)i — a)\ 
Therefore 
B 
cr 
^4^ ^ - 1):! ()i - 3) :: 
2 (» + ^);: („ + 2)!: 
2n + 1 (a - a) 11 (n - 4) ' ; 
^ 2 (a + (7 - 1) ! I (// + 3) 1! 
when ii is even and cr is even 
nSl when n is odd and cr is even. 
Similarly 
A 
<r 
/». 
2// + 1 (a - cr) I! (« -3)1! 
2^ ' (n + <r - 1)1! (a + 2)11 
271 + 1 (a - <7 - I) 1 ! (71 - 2) 1! 
2 ^^ (71 + a)!! (w + 1)1! 
ttYI when 71 is even and cr is odd 
when n is odd and cr is odd. 
Combining equations, we find that if denote the coefficient of tlQl cos acjj in the 
development of F, and Jd, the coefficient of sin acf), 
2/i/ -j- 
1 
(a. 
— 
cr)!! 
(a - 
-3)!! 
2tl 
a 
-1) 
!:(h 
_L 
1 
2)!! 
2)).+ 
1 
— 
a 
-1) 
!! (a 
— 
2)!! 
■2C 
(a 
+ 
C7)!! 
(a + 
1) 
f 1 
2)1 + 
1 
CT 
-1) 
!! (a 
— 
3)!! 
2tl 
(a 
+ 
<7)!! 
(a + 
2) 
1 1 
2)1 + 
1 
(a 
— 
<7)1! 
(a- 
1 « 
2tl 
(a 
• + 
<T 
- 1) 
!!(a 
+ 
3)!! 
77 o^Yl (p), 'when n is even and cr odd and p odd 
7> = 1 
7=" + l 
77 <<l^p ( l')> n is odd and cr odd and p even 
;, = 0 
p = /' +1 
77 ^iMl (p) wdien n is even and cr even and p odd 
p=i 
p=n hi 
77 hlSl (p) when n is odd and cr even and p even. 
p=- 
To obtain hi, substitute a and /3 for cl and h. 
Tlie coefficient tl is introduced because the ([uantity here designated by is not 
uniformly accepted as the standard form for a spherical harmonic. For some purposes 
