PEOPESSOE DONKm ON THE EQUATION OF LAPLACE’S FUNCTIONS, ETC. 57 
representing a function of which the form may be considered to depend in an arbi- 
trary manner upon the value of n. Now we know that every such term as the above 
satisfies the differential equation, and it may be asked, what more is required to prove 
the correctness of the solution I answer, that the point in question is whether 
2.r”(sin^)~”^sin^^sin^^ (A.) 
is equivalent to the most general interjpretation of 
(sin ^)“’'*^sin ^ ^ sin ^ ; 
and, if not, whether the most general interpretation would satisfy the Differential Equa- 
tion (35.). 
The expression (A.) gives, in fact, no new result, being an evident consequence of the 
conclusions of the former sections. 
PosTSCEiPT {Added May 29, 1857). 
Mr. Cayley has been kind enough to communicate to me direct verifications of the 
equation (27.), art. 14, and of the identity referred to in art. 16. Assuming a formula 
established in Mr. Cayley’s paper “ On certain Formulae for Differentiation,” &c.*, the 
former of the two theorems just mentioned is easily obtained, the latter not without a 
good deal of trouble. The investigations of Section II. may be compared with some of 
those in the Introduction to Muephy’s ‘Electricity;’ but I have not at present examined 
the latter with a view to such comparison, and therefore merely mention the subject. 
* Cambridge and Dublin Journal, vol. ii. p. 124, equation (2.). 
