56 Mr. Ivory on the Attractions of an 
Laplace has proved that every one of the coefficients in the 
series for j- will satisfy an equation in partial fluxions which 
is thus generally expressed for Q^, viz. 
(0 
>( 0 , 
* 0 ' + i)-Q (,) + < 
r d ■ 
0. 
L dp J ' I— 1 *. 
This is a fundamental equation in his investigation, and it is 
necessary for effecting the expansion here proposed: but we 
shall refer to Laplace's work for the demonstration of it.* 
It is plain that when it is considered as a function of p 
and the cosines of <p and its multiples, may be thus repre- 
sented, viz. 
Q(i) _ H (°) _|_ (j _ . H (0 . cos. <p + (l — ,**)■* . H (,) . 
COS. 2 (p -f- &C. 
the general term of the series being (l — (x 1 )' 2 ’. H^.cos. n<p y 
which ought to satisfy Laplace’s equation in partial fluxions : 
now', having actually substituted that quantity in the equation 
mentioned, and having divided all the terms by cos. n <p, I have 
found, 
(z — n) (z'-f- n -j- l) . ( l — f)* . — 2 (/z -f 1) [x (1— 
.fjL" .^ = 0: 
and, after having multiplied all the terms by (1 — the 
result will be equivalent to this equation, viz. 
diiW 
a . <; ( 1 — ul i ■ . 
(z-»)(/+»+i h<">+ 
J $ / a N «+l dW U) I 
whence it follows (equat. 2.) that 
B<”> . — ( ° , where 
du. n 
Mec. Cel. No. 9, Liv. 3c, and No. 11, Liv. 2d. 
