88 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
A’ = (X") (X”). Then, by the line of reasoning followed in the 
Proposition, X’” has a measure identical with X’. And so on. 
Ultimately X’ = (X”)¥, 
§19. Cor. 2. If 72, one of the particular cognate forms of R, be 
zero, all the particular cognate forms of # are zero. For, by the 
proposition, the particular cognate forms of R are the roots of a 
rational irreducible equation / (x) = 0. And re, one of the roots of 
that equation, is zero, but the only rational irreducible equation that 
has zero for a root isa = 0, Therefore F(x) = « = 0. In fact, in 
the case supposed, the simplified expression 7; is zero, and R has no 
particular cognate forms distinct from 7}. 
§20. Proposition IV. Let W be the continued product of the 
distinct prime numbers v, a, etc. Let w; be a primitive mn» root of 
unity, 4, a primitive a» root of unity, and soon. Then if the equation 
F (x) = a4 + bat—-1 4 boxt—?2 + ete, = 0 
be one in which the coefficients 6, b2, etc., are rational functions of 
#1, 4, etc., and if all the primitive mt® roots of unity, which, when 
substituted for w, in F(x), leave F (x) unaltered, be 
Ole, Cen as sss (7) 
the series (7) either consists of a single term or it is made up of a 
cycle of primitive nt roots of unity, 
A r~? as—1 
5 SA) Ere og. 3 (18) 
that is to say, no term in (8) after the first is equal to the first, but 
As ; 
wo; = 4. Also, if (let it be kept in view that m is prime) the cycle 
that contains all the primitive n™ roots of unity be 
2 n—2 
ee oe ea 3 (9) 
and if Cy be the sum of the terms in the cycle (8), the form of F (z) is 
F (a) = x4 — (pC + polo + .... + PmCm) xt-1 + (10) 
(C1 + q2C2 + ete.) at1—? + ete. 
where each of the expressions in the series C1, C2, C3, etc., is what 
the immediately preceding term becomes by changing w; into 
B ; f 
1, Cy through this change becoming C;; and 71, po, qi, ete., are 
clear of 4. 2 
For, assuming that there is a term wz in (7) additional to 1, we 
may take w2 to be the first term in (9) after w that occurs in (7) ; 
me 
; B é : ; 
and it may be considered to be w, , which may be otherwise written 
oie Then, if #’ (x) be written g (w1), we have by hypothesis 
