98 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
/ 
§32. Let 7; be one of the particular cognate forms of the generic 
expression # under which the simplified expression 7; falls. Then, 
because, by Prop. IT., all the particular cognate forms of & are roots 
, 
of the equation / (x) = 0, 7; is equal to one of the m terms 71, 72, 
etc., say to r,. I will now show that the changes of the surds 
/ 
involved that cause 7; to become 71, whose value is r, , cause 72g to 
receive the value r, 4 1, and rg to receive the value 7,4, and so on. 
This may appear obvious on the face of the equations (23) ; but, to 
prevent misunderstanding, the steps of the deduction are given. Any 
changes made in 7; must transform C into (’, , one of the m terms 
vA 
Cy, Cz, etc. In passing from 7; to 71, while Cj becomes C, , let r, 
, , ! 
become 72, and 7 become p; , and pz become pz, and so on. The 
change that causes ('; to become C, transforms C2 into C,;41, and 
Cz into Cs;42, and so on. Therefore, it being understood that 
Pm+1, Cm+1; etc., are the same as p, , Cy, etc. respectively, 
/ / 
, 
ry = piCs + pols 414 ete., 
and 72 = PmC's+ piCs +1 + ete. ; 
which may be otherwise written 
! t 1 
11 = Pm+2—s Cy + pmn+3—sCe + etce., 
ay (24) 
12 = Pm +1—2 C1 + Pm42—8 C2 + ete. 
Therefore, form (24) and (23), 
, 
Ci(Dm +2—2 — Pm 42-2) + C2 (Pm +3—s — Pn +3—s) + ete. = 0. 
’ ! 
Therefore, by §13, pm +9— s= Pm+2—z,Pn+3—s = Pmt 3— mere 
Hence the second of the equations (24) becomes 
/ 
12 = Pm +1—2 Ci + Pm+2—2C2 + ete. = rz 41. 
Thus vr, is transformed into 7, +1. In like manner 73 receives the 
value r, , 2, and so on. 
§33. By Cor. Prop. VI., the primitive n root of unity being one of 
those involved in 7; , x — 1 is a multiple of m. In like manner, if 
the primitive a” root of unity be involved in r; , a — 1 is a multiple 
of m, and soon. Therefore, if ¢; be the primitive m™ root of unity, 
t; is distinct from all the roots involved in 71. 
