PEOFESSOE DOKKIN ON THE EQUATION OF LAPLACE’S FUNCTIONS, ETC. 63 
16. Professor Boole, in the paper above referred to, obtains the following expression 
for the coefficient of r"cos?'(p ; namely. 
where 
1.2....(re — i).1.2...(ra + i)’ 
d 1 
(29.) 
Here the numerical coefficients are the same as in the form just given. It follows, 
therefore, that the expression on the right of (29.) must be equivalent to 
(sin ^)“”^sin sin A ^tan^V 
This equivalence, and that expressed by equation (27.), art. 14, are instances of theo- 
rems by no means obHous or easy to verify directly. (See art. 5.) 
For actual calculation. Professor Boole’s form would be preferable to either of mine ; 
for though the number of operations is much greater than in the form (26.), their result 
may be assigned with much greater facility. But considered merely as an analytical 
expression, the form (26.) is the simplest of the three. 
Section III. 
17. It was shown above (art. 9), that the expression 
^^„=(sin^)-'^(^sin^^sin^) ^Ci(^tan|j -fC^^tanlj ^ 
fails in certain cases to represent the complete solution of the equation 
namely, when i is an integer^ and not greater than n. (Similaiiy, the expression 
-V-e’ 
dx) ^ 
It is desirable, therefore, to investigate the solution of the equation in a more general 
manner. Putting, as before, sin^^+7^cos^=^5■„, the equation to be integrated is 
u = C^e'''-\-Qie fails to give the complete solution of 
when ^=0.) 
(30.) 
Let then (ra-„zir_„zr„+(w"— ^■^)ra'„)D=0, 
and ang value of v which satisfies this, will give a value of satisfying (30.). Now 
is identically the same as c7„_iTO-_(„_,^ + (n— 1)^ (10.), art. 2 ; hence the equation 
becomes 
similarly, putting we get 
MDCCCLVII. 
I 
