56 PEOFESSOE DONEIN ON THE EQUATION OE LAPLACE’S FUNCTIONS, ETC. 
Omitting the second arbitrary function to save space, we should have (observing that 
r must be considered to enter the arbitrary functions in an arbitrary manner) 
w=(sin ’^‘^'•^sin ^ ^ sin f{T, 
'tan 2 ) (37.) 
23. There is no diflEiculty in the interpretation of this expression. It is only neces- 
d rp 
sary to recollect that c'^ (p{x)-=<p{cx), and we have the follouung result : — let ^—7 = 5 , then 
f sin ^ ^ sin tan 
(38.) 
T 
where f is to be put = after all other operations, and the function must be supposed 
to be developed as if the two symbols ^ sin sin tan^ were commutative, the 
powers of the former being always prefixed to those of the latter. 
It is easy to derive from the form (38.), particular expressions which we ah'eadv know 
to be solutions of (35.). For example, supplying the other arbitrary function, it is 
evident that if we i^ef{x, y)-=.\x'^y\ and F(.r, y) the same, we get 
M=r”(sin^)“”(sin^^sinfilj (tan-j cos^^, 
which, we know, satisfies (35.). 
Again, we may take/(.r^, y)=e', or 
p sine A sin 9 
^^=e do ,1^ 
which has been already shown (art. 6 ) to represent 
(I — 2r cos 
These verifications afford, I think, ground to believe that the form (37.) is a true solution 
of the equation (35.). And here I leave the subject, hoping that it may attract the 
attention of some mathematician more able than myself to bring it to a satisfactory 
conclusion. 
Note on the last paragraph. 
Eeceived December 6, 1856. 
If the expression sin ^ ^ sin e'^'^tan^^ be supposed to be developed and 
d 
arranged according tP powers of the term ^ sin ^ ^ sin each term in the result will be of 
the form 
