44 PROFESSOE DONEIN ON THE EQHATION OF LAPLACE’S FUNCTIONS, ETC. 
In the following pages I have treated the equation (2.) by a very simple method*. 
The result bears a general resemblance to Professor Boole’s first solution, and I conceive 
that the two forms must be capable of being identified by the assumption of a proper 
relation between the arbitrary functions; but I am not able at present to show this 
identity. 
Some further investigations are added, wliich it is unnecessary to notice beforehand. 
d 
1. Putting Jc for ^ in equation (2.), we have 
|(^sin^)^^ +^+^^(^^+lXsin^f jw„=0 (3.) 
In the case of %=0, this becomes 
- ^ K 
( where t = log tan ^ j . Hence 
Wo=Ci(^tan2j +C^[tan:^J (4.) 
If, however, k=0, we should have 
Wo=Ci+C2logtan^, (5.) 
* It may be worth while to give a preliminary illustration of this method (which contains no novelty 
except in detail), by applying a s imi lar process to the well-known equation 
d~u 
This may be written in the form 
» /y_ .(»+l) \ ^0 
' V or J 
now 
hence if we put m=^D— and then operate on each side with have 
Assuming 0 as the value of the right-hand member, and then putting t;=^D— ^<1 
sively, it is obvious that we shall ultimately have 
so on succes- 
(D=-1-F)^=0; 
hence, observing that D— -=x^J)x~‘, we obtain finally. 
f 
u=x”^-\ (Cjsia^.r-l-C2C0si'a:), 
which agrees with the result obtaiued by Professor Boole’s general method (Philosophical Transivctions, 
1844). 
