PEOFESSOE DONKIN ON THE EQUATION OF LAPLACE’S FUNCTIONS, ETC. 47 
5. The expression (16.) may be compared with the following result of Professor 
Boole’s first solution * : — 
where F(|U/, 
and \M=. cos^. I have stated above my impression that this form must be capable of 
transformation into identity with (16.) ; but it is probable that this can only be effected 
by means of some not obvious theorem of differentiation. Examples of such theorems 
will be seen below (arts. 14 and 16). 
Section II. 
6. I proceed to discuss some particular applications of the formulae obtained above. 
If the expression (1 — 2rcos^+r^)“^ be developed in the form 
then satisfies the equation of Laplace’s functions, and does not contain (p. Hence 
must be given by the expression (15.), art. 4, if the constants be properly assumed ; and 
since it is evident that C 2 must be = 0 in this case, we have 
i>«=c„(sin^)“”^sin^^sin^^ .1, 
where c„ is a function of n. 
This form of may be independently verified, and determined at the same time, 
as follows : — 
Put r(sin ^)"*=g>, then 
(1 -2r cos ^+r^)‘*=^ (l+(cot 
and if we put cot ^=t, this may be written 
x/l + f e 
d 
2 
where is to be treated as constant till after all operations. Now in general, if P, Q 
be any symbols whatever, quantitative or operative, we know that iy(Q)p=/’^PQpy 
Hence it is evident that since and — ^=(sin^)^^, the above expression 
is equivalent to 
psinS— sinfl 1 
e ^ . 1 , 
and therefore this is a symbolical form of (1 — 2 pcos^ 4-^^) Restoring the value of ^ 
after development, we find that the coefficient of p” is 
* Cambridge and Dublin Journal, vol. i. p. 18. 
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