90 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
UL) 
Pm +1= pr That is to say, the coefficient of a or a is the same 
as that of w. In like manner the coefficients of all the terms in (8) 
are the same. Therefore one group of the terms that together make up 
the coefficient of #7 —1 in (12) is properly represented by —(p + p;)C1. 
In the same way another group is properly represented by —(p + p2)Co, 
and so on. Hence 
F (x) = xt — {p + (p + pr) C1 + (p + po) Co + ete. ad — 1 + ete. 
And by (11) this is equivalent to (10). The form of (x) has been 
deduced on the assumption that the series (7) contains more than one 
term ; but, should the series (7) consist of a single term, the result 
obtained would still hold good, only in that case each of the expressions 
C1, Cg, ete., would be a primitive n** root of unity. 
§21. A simplified expression will not cease to be in a simple state, 
if we suppose that any surd that can be eliminated from it, without 
the introduction of any new surd, has been eliminated. 
§22. Proposition V. In the simplified expression 71, one of the 
particular cognate forms of R, modified according to §21, let the 
1 
surd 4 a of the highest rank be not a root (see §1) of unity. Then, 
1 
if the particular cognate forms of & obtained by changing , Pa in rl 
successively into the different m* roots of the determinate base 4), be 
Tl) 5 SUD iaeeer te » Tm; (14) 
these terms are all unequal. 
For the terms in (14) are all the particular cognate forms of R 
1 
obtained when we allow all the surds in 7; except 4 Pe) to retain the 
determinate values belonging to them in 7}. Therefore, by Oor. 1, 
Prop. IIL, each of the unequal terms in (14) has its value repeated 
the same number of times in that series. Let «w be the number of 
the unequal terms in (14), and let each occur c times. Then we = m. 
Suppose if possible that w= 1. This means that all the terms in 
(14) are equal. Therefore, 7; being the expression (1), 
mr =n1+re+ .... + etc. = 91. 
pees 
Therefore the surd ay can be eliminated from 7, without the intro- 
duction of any new surd ; which, by §21, is impossible. Therefore w 
is not unity. But, by §1, m is a prime number. And m = we. 
Therefore c = 1 and wu =m. This means that all the terms in (14) 
are unequal. 
