48 PEOFESSOE DONKIN ON THE EQUATION OE LAPLACE’S EHNCTIONS, ETC. 
. 1 . 
d 
In other words, if we put ij„=(sm ^)“"(sin ^ ^ siu 1, then 
(1 - 2r cos y + • • +«« i.L..n + 
(From this it is easy to deduce the relation Vn~(2n — l)cos^.i;„_, — by means 
of the known relation between the coefficients in the above series and those in the 
development of (1 — r^)(l — 2r cos 
7. Adopting the notation of the last article, we have 
gp sin sine (sin e“'’di(sm 
=s4i(i+«-f)r- 
i+l 
=(sui^)'(l — 2rcos^+r^) « . 
Hence this theorem, which will be useful afterwards. The coefficient of :r“r in the 
’ 1 . 2 ... .71 
i+I 
development of (l“2r cos 2 is 
d 
(sin "*^(sin 6 ^ sin ^)”(sin 6J (17.) 
8. Let us next consider the development of the expression 
{1 — 2r(cos^cos^d-sin^sin^'cos(p)d-r^}‘‘‘^ (18-) 
Since the coefficient of r” is necessarily expressible in the form 
Jo d-^', cos 9 +^ 2 cos 2^+... cos n(p, 
and must also satisfy the equation (2.), art. 1, we find, on substitution in that equation, 
that ji must be a solution of the equation 
sin^^) d-K’^d-l)(sin^)*— i"jw„=0 (19.) 
I proceed to consider this equation in a general manner, without reference, in the first 
instance, to the problem immediately in hand. 
Since the equation (19.) only differs from (3.), art. 1, in having — instead of we 
may apply the process used in the solution of that equation. We thus get 
i^„=(sm^)“”^sin^5^sin^^ ^Cj^tan^ -fQ^cot ^ . . . (20.) 
Now, although this expression appears to contain two arbitrary constants, it will be 
shown presently, that when i is an integer and not greater than n, it really contains only 
one. This is seen immediately to be the case when ^=0, and may be easily verified in 
other simple cases, such as ^=l, w=2, &c. 
