WITH APPLICATION TO SPHLRICAL HAPMONIC ANALYSIS. 
185 
But 
^vhe^e the special case n = 0, for which the integral vanishes, is excluded. 
Hence 
r+i ^ . 
I Qn = -i, if n he even ; = 0, if n l)e odd ; 
AlPn 
+ 1 
and finally 
r + V-w 7 (n + a-l)l\(n~2)l] ... , , 
= 2,7 , It <r and n be even; 
= 0, if cr he even and n odd. 
For odd values of cr, it is more convenient to use formula (Gg), which leads to : 
r+ 1 
Q;dM = 
> -1 
0i + a- 1) (n - 2) 
(n — a) {n + 1) 
By repetition of the same process, we get for even values of n idtimately 
r+L.. , f"/;, + cr — 1)! 1 (/; — 2)!! <t'.\ 1 7 
Qn = -—7- - - - 77 . - Z T V~" Li—r->- 
1 ni — cr ) . ; + J J ; ' (T — O ) : ; f j:»t — Z K : ' i 
>);;(2 
As the expression under the integral sign of the I’ight-hand side vanishes, the 
value of the integral on the left-hand side is zero. 
o 
W1 len n is odd we am ultimately led to 
]-P 
(7i, + CT - I);i (n - 2)i: (cr + I): 
(a~cr;i:(u + i):: (^ 7 - 2 ) 1 : (2cr-i)::j„ 
+ H" 1 7 
But 
dp = (2cr — 1 ) ! ! f siiP dd/x = (2cr — 1) 1 1 [ siiH'*"’d = , ] \\,, (2cr — l)! I tt. 
-1 .'0 ( (T + J j : ; 
-1 
Collecting the results, we may put generally 
djji = ccr 
(n -f (7 — 1) !! (71 — 
(71 — cr) ; ! (77 + 1 ) 
if 77 + cr be even, 
where c is equal to 2 or n according as cr be even or odd. When (n + cr) is odd, the 
integral vanishes. 
The results of this paragraph allow us to represent a quantity which is constant 
over a sphere, in terms of a series of tesseral functions v'hich are all of the same type, 
the type being arbitrary. It follows that we may express cos crX and sin crX in terms 
of a series of tesseral harmonics of type cr. 
VOL, cc. —A. 2 B 
