128 Mr. Ivory on the figure requisite to maintain the equilibrium 
For this purpose the following theorem is premised, viz. 
Theorem. If m, m ' , m" denote any positive integer num- 
bers, such that m-\- m' m" is less than i ; then 
Cos. Try . C (i) df*'dzr'=o, 
the integral being extended from [j! — i to p! = — 1 , and 
from 73-'— o to nr'— 2 t r. 
As expressions of this kind have been very amply discussed, 
and as the theorem follows very readily from the properties 
generally known, I shall not stop to give the demonstration. 
It follows from the theorem that the equation, 
C(4)c? i>t d 'a' n 
0 , 
cannot be true, if ~ contain any even power of the quantities 
ju/, V l — . Sin. to', V i — ft/ 2 . Cos. ®-', above the square ; or 
if it contain any product of two or more of the squares of the 
same quantities. Wherefore the most general value of 
consistent with the above equation, is 
= A -j- B ( i — [a ! 2 ) Sin. 2 m' -j- C ( l — /a! 2 ) Cos. 2 u . 
It may be observed, that this expression would not be more 
general by adding an absolute quantity, as D : for, since 
( i — ^' 2 ) Sin. 2 w'+ (i — p /a ) Cos. 2 73-'= l , 
such a quantity would blend itself with the other terms. But 
the same value of W }H likewise satisfy all the equations, 
^ d f/ d 'sj __ n 
('j 
R '*~ 2 
in which i is an even number. For, because 
i — 2 
-jzg — ( A^+ B(i — y' 9 ) Sin.* w' -j- C ( i — - [aI 2 ) Cos. 2 
R ' 
