94 
Proceedings of the Royal Society of Edinburgh. [Sess. 
and Laplace’s equation is 
it an ; w 9 2 u 
^ dxj 2 0# 2 2 + ‘ + dx n+2 2 
= P n sin”" 1 0 sin”- 2 <^ . . . sin <f> n _ 2 
+ ^[p w “ 2 sin”- 3 6 sin”- 2 <£ t . . . sin <£ n _ 2 
+ ^[p w sin”- 1 ^sm”- 2 ^ 1 . . . sin <£ n _ 2 4?J 
i=n— 1 
P n 2 sin” -3 0 sin” 4 4 > 1l . . . sin” - * -2 <^ i _ 1 sin”-* -1 sin” - * -2 <£ i+] . . . sin <f> n _ 2 — 
i = i 0 9*L “<ty< 
We shall try to solve it by a function of the form 
U“/oWi(U • • • /«-i(<*i) u i(p> 0). 
We readily observe that we may take 
/o = ef 
and 
/i = cos 
where m 1 is an arbitrary integer. 
The terms containing <^ n _ 2 can then be dissociated from the rest, and 
we find for / 2 the equation 
d% 
7 » — 2 + C0t < t >«-2 /£* - m i + *>2 - 0 , 
d<t > n - 2 2 2 sin- <f > n _ 2 
F being a certain function of (p n ~ 3 , . . . , <p v p, 6. We shall therefore 
introduce the associated Legendre function and take 
f 2 = P ^( C0S ^n- 2 )i 
m 2 being an arbitrary integer, and the equation for / 3 will then be 
^2 + 2 cot - m ' 2( TV~ ^ /a + F 1/3 = 0, 
d<f > n — 3 d(f > n — 3 sin (fan — 3 
the law of formation of the successive equations appearing clearly enough. 
Let us now consider the polynomial C^(2) of Gegenbauer, which is the 
coefficient of a A in the expansion, in ascending powers of a, of 
(1 — 2 aZ + a 2 )~ v , 
and let us denote the function 
It /7m v 
where pt is an integer, by the symbol CT ^(z). This function CT ^ plays 
