PBOFESSOE DONKIN ON THE EQUATION OF LAPLACE’S FUNCTIONS, ETC. 51 
Hence it is easily found that when n—i is even, the term independent of fi, f/f in (25.) is 
Ci . l^ 3 ^ 5 ^ . . 1)^(2? + 3 )^ . . 1 f ; 
and when n—i is odd, the term involving is 
3^.5^ ..(w— ^)^(2^4-l)^(2^^-3)^ . 
If these expressions be equated to those obtained in art. 11, we find that whether 7i —i 
be even or odd the value of Ci is 
' 1.2.3...(re— i).1.2.3...(w + 2y 
the factor 2 being omitted when f=0. 
13. We have then the following result : — 
The coefficient of r” cos i(p in the development of 
(l — 2r(cos S cos ^'+ sin 6 sin & cos ^ is 
, U.32.52.... (22-1)2 
'■ 1.2.3. ..(m- 2). 1.2.3. ..(n + f 
(sin ^ sin ^') ”0” '0'" "(sin ^)2’(sin , 
(26.) 
where 0= sin sin 0'= sin ^ sin ; and with respect to the numerical coefficient, 
it is to be observed that when 2=0 the factor 2 is to be omitted, and the numerator 
considered to become miity. The extreme values of i are of course 0 and 22 , and the 
term 1.2. 3... ( 22 — 2 ) is to be taken =1 when 2 = 22 . 
14. It is now easy to ascertain the values of the coefficients 22 - 0 , a,, &c. in the form 
(23.), art. 10. But first it is necessary to prove that 
^sin^^sin^^ ^tan^^ =1.3.5...(222 — l)(sin^)’”. . . . . (27.) 
We have proved this already (art. 9, equation (22.)), except in so far as the numerical 
coefficient was left undetermined, so that we have only to establish its value. Now the 
expression on the left of the above equation may be Avritten 
(sin ^)“‘ M sin sin ^^^tan ; 
and if we put cot so that tan ^=\/ 1 and (sin the equation (27.) 
becomes 
v/ FT 
1.3.5...(2w-l) 
2n+ 1 ) 
(i + 0~ 
and since it is only the value of the coefficient that is in question, it will be enough to 
prove this when ^=0, 2 . e. to show that the coefficient of f”- in the development of 
is 
1.3.5...(2ra-l) 
1.2.3. ..n 
{t- 
Now, expanding the numerator by the binomial theorem, the ab(n^e 
