analytical and geometrical Methods of Investigation. 105 
or 47 r+2 .... 
Hence, %' or 2 tt— -2= — X+ a. Let 2 = 0 .-. X=o 2^== u. 
Again, z" or 6tt — % = — X-{- a. Let z—o .*. X=o 6?r= cl. 
Hence, the arbitrary quantity cl may generally be represented 
by (2/z-J- 1 ) 27 t, or by 4?Z7 t .-. ^•••• w =( 2 ;z 4 -i) 27 r— X, 
or = 4«7r-j-X. 
XIV. I shall now shew, by a purely analytical process, what 
are the divisors of x n =^a n . It seems a very strange and absurd 
method, to refer to the properties of geometrical figures, for the 
knowledge of the composition of analytical expressions. 
A A v /__x 
Let x—m n s n .\a 
*V“ 
—1 
-.me ,\mz 
a” 
r, and m will 
£ZV 1 
l = 1. But (Art. XI.) the values of % 
Z\J— l 
be always positive, if 
that answer the equation s ~ v “^i, are o0,=±=0,=±= 20, d=gd, or ge 
nerally =*= s 9 , s, any number of the progression o, 1,2, 3, See. 
zsd 
v: 
Hence, x=a e n 1 generally 
or values of x are a, as n 
vzr; 
- / 20 
n V - x „ ~n ^~ l 
as n ,as ' 1 , as 
0 . — 0 
V— I . — y/ 
— 29 / 
71 \ Sec. 
■••• x n —a n -(x—a) (x*— a) s n + e n 1 j+a*) (x*—a 
20 —29 
« 1 i - n ^ 1 
-f e “ " ~ 1 J ?z being odd ; 
when n is even, (and of the form 2 p, p odd,) there must be a 
number (s) in the progression (o, 1, 2, 3, &c.) that : 
consequently, there must be a value of x, as 
s0 
2 
V. 
= 
v: 
27 tV_i 
= — a, since (Art. XI.) e' ,or r = _. 1( 
Hence, a quadratic divisor of of— a n will be (x—a). (x+a), or 
x a ; when n is even, and of the form 4 p, p even or odd. 
MDCCCII. 
