analytical and geometrical Methods of Investigation. 10 
f V^ - - j • • 
= as = — a, or x -|- a must be a divisor of x n 4- a n . 
XV. Resolution of x 27? — 2 la n x n f-a zn into its quadratic factors 
l A 1. 
Now, from the equation x n =a n j Z=±= </ I — 1 }= A =±= B v 7 — 1 " 
Let x=m n £ n ^ 1 .*. me ^ ~~ l = A + B V — 1, ms x/ ~~ l = 
A— ByZA>=y (A*-f-B’), s— { 
A 
(ay— !,)-*{ 
B 
V A a + B 2 
- =v 1 ~r 
VA^B 1 
but (Art. XI.) these equations are true, when instead of £ are put 
+z, •••• generally sQ 4-2. 
±50 + 2 
■A 
Hence, the general value of a: is as * 
±20 + 2: 
[ 
of a? are 71 
±Z ±0 + 2 ; 
v — 1 — - — v — 1 
<ZS 
M 
, and the values 
or x 2n — 2 la n x n -\ -a 
x — a \ £ 
f 
a\e n 
l 
n 
+ 8 •* r+o/ x&c - 
+ s n J 4 " a ) x \ x* 
XVI. Such are the analytical processes according to which 
the resolutions of x n =r= a n , x 2n =iz2la n x”-\ -a 2n are effected; and 
gjll* » &c - &c - ma y be ob - 
thence the fluents of 
a"’ x rn - 
tained, by resolving the fractions &c. into a series of partial 
fractions, of the form ~ - +B 
2«jc-t-« a +|3 2, 
Since the above resolution of x n =i=a n into its quadratic factors 
would, it appears to me, be strictly true, if such a curve as the 
circle had never been invented, nor its properties investigated, 
it is erroneous to suppose that the theorem of Cotes is essen- 
tially necessary for the integration of certain differential forms. 
P 2 
