PEOFESSOE DONKIN ON THE EQUATION OF LAPLACE’S FUNCTIONS, ETC. 45 
which is the complete solution of (2.) in the particular case in which u„ does not con- 
tain <p, and w=0. 
But putting ^ for k in (4.), and arbitrary functions of <p instead of C,, Cj, we have 
— l-logtan|^-fF(^^— x/— l.logtan^^ ; 
instead of which we may e\idently write 
Wn 
) I) + ^ ‘ tan , 
( 6 .) 
( 7 .) 
as well as other forms. Either of these is the complete value of u„ in the case of n=0. 
2. Returning to the general case, if we put for shortness 
it is evident that 
and also, since 
(sin -l-w(%-}-l)(sin 
f— » fn— 1 » 
( 8 .) 
(sin cos (sin — n cos = (sin -l-7^(w-^-l)(sin 6^—71^, 
if we put 
CT„=sin ^^+wcos 
we have = (9-) 
from which, changing n into —n, and observing (8.), we get 
on the other hand, changing n into n—1 in (9.), 
Comparing the last two results, we see that the operation possesses the following 
property, namely, 
+ + — If (10.) 
Now the equation to be solved (.3.), art. 1, is 
( 11 .) 
Let and let the operation ' be performed on each side; assuming for the 
present that nr“'0 may be put =0 without ultimate loss of generality, we obtain 
(^-«^«+^^+^’')^’=0 ; 
and this, by virtue of (10.), becomes 
(^„_,^_(„_i)+(w — 1 f +/r>= 0 ; 
which, compared with (11.). gives the relation and consequently, since 
w„=2r„w„_,, 
or w„= (sin ^^-}-%cos 
MDCCCLVII. 
H 
