146 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
A 
Yi 4 Py Mi ; 
Ya = fy PY, Veediccatewatelelstaiaistotausteielstens elaine (5) 
Ys pay ks P3 oe 
and so on; where *,, ‘,, &c., are stb roots of unity; and P,, 
P, , &c., are what P becomes when T is successively changed into 
oy, De zy T, &e.; Yi, Y2, &c., being what Y becomes when T is 
successively changed into 2; T, z; I, &c. By eliminating Y, betwixt 
equation (3) and the first of equations (5), Y, and Y», betwixt (3) 
and the two first of equations (5), and so on, we get 
a 
Yeap: 
. (6) 
ye pipe 
beds 
No 
Tne hae) 
and so on. Hence generally, 
AY ey athe CMe (7) 
where n is any whole number whatsoever; and p is the greatest 
multiple of s in A’; and Q is an expression which involves only 
such surds as occur in F,(#), exclusive of Y; none of the surds 
which it involves having T as a subordinate. Now equation (7) 
has been fourd on the hypothesis that F,(«) is equal to a term in 
the first line of (2). But, by the same course of reasoning, an 
equation such as (7) may be established, should F, (a) be given equal 
to a term in any line of (2). And equation (7) includes the form 
(3). Therefore, when F, («) is equal to a term in (2), whatever be 
the line of (2) in which that term occurs, an equation such as (3) 
subsists. In order to establish equation (4), we observe that equa- 
tion (7), when x is taken equal to o, becomes, 
rp 
Y= OV eas 
Let ip =m, m being less than s. Then 
Y—QY =O. 
But, since F,(#) is a function in a simple form, this equation 
