PEOFESSOE DONKIN ON THE EQUATION OF LAPLACE’S FUNCTJONS, ETC. 4'i 
9. In fact, the process of art. 2 gives 
^n^n— 1 ^n^n — 2 • ■ • ^n^n— I • ■ • ^n — m + \ ‘ ^n — m ' 
where satisfies the equation 
and in general we can only arrive at an integrable form by taking m—n. But when i 
is an integer not greater than n, we shall get an integrable form by taking m so as to 
satisfy the condition [n — Suppose ^ positive and less than n^ and take m-=n — i ; 
we then have for the solution of (19.), 
— 1 ' * • 1 * ^ 
where m,- satisfies the equation or 
■ . 
sin — ^cos 6 
the integration of this is easy, and the result is 
C u$ 
M..= C,(sin ^)'+a(sin ; 
so that, in the case considered, the integral of (19.) may be expressed in the form 
w„=(sin ^)~"^sin sin (sin ^)^‘^Ci + C 2 j(/^(sin . . . (21.) 
^for is equivalent to (sin ^)""^sin 0 ^ sin 0^ (sin 0)' (see art. 3)^ 
It is evident that the expression (21.) contains two independent arbitrary constants, 
whatever be the value of i. And i being supposed a positive integer, the second term 
will always involve a logarithmic function ; hence the form ( 20 .), which involves no such 
function, cannot be the complete solution of the equation (19.), and therefore we shall 
not really limit its generality by putting € 3 = 0 . Moreover, since the coefficient of 
r”cosf<p in the development of (18.) cannot contain any logarithmic function, it follows 
that, for the purpose of that development, we must put €2 = 0 in the expression ( 21 .). 
Comparing the two forms thus obtained for the coefficient of r”cos«^, we obtain inci- 
dentally the following theorem, namely. 
sin 0-g-^ sin 0 
= 6 '(sin 0Y\ 
( 22 .) 
c being a constant, of which the value will appear afterwards. 
The preceding investigation shows that the assumption ra-~'0 = 0 (instead of the 
general value C(sin 0)~”') is liable in certain cases to limit the generality of the result. I 
shall return to this point presently, but proceed now to complete the development of 
the expression (18.). 
10. Since the coefficient of r”cos?'(p must contain 0 and 0' symmetrically, and satisfy 
both the equation (19.) and a similar equation in 0\ it is evident that it must be of the 
form f{0)f{0i). Hence, by means of the conclusions of art. 9, we arrive at the following 
