144 RESOLUTION OF ALGEBRAICAL EQUATIONS, 
Also, equation (7) becomes, 
es+m 
b HT ~Q: 
which (Cor. 1, Def. 9) is only possible if the number, es+-m, be a 
wultiple of o. Hence o and s are not equal to one another. 
Cor.—If an equation such as (3) subsist, the form of the surd Y is 
that given in (2), and s and o are unequal. 
Proposition XI. 
In the series, in Prop. VIII., 
x —= f (p), Fa (%), --.. 2.4 Ba(@) or KX, 
let the factors BY oe continued ona F.41 (#) is generated, be, 
F(a) “Ee (@)s, Be (2); 2-9 Pe Cs a 
where ie («), "F, (a), &e., are what F, () becomes, on substituting 
successively for Y, a surd in F, (”), not subordinate to any other surd 
in the function, and having the index us the values z Y, 2° Y, &c.; 2 
being an st root of unity, distmet from unity. Also suppose, that, 
besides Y, there are no surds in F, (x) which disappear from Fe (x), 
except subordinates of Y. Let T, a chief subordinate of Y, with the 
index a disappear from F,+1 (2), in which case T is not. a subor- 
dinate of any surd in F, (#) except Y. When T is changed into 2, T> 
Z, bemg a ot? root of unity, distinct from unity, let the terms in (1) 
be transformed into, 
ie a 
ry , gXy1 ’ 31 Dotere ey sXy ° 
Generally, if ,X, be what “F, (a). becomes when T is changed into 
z. T, 6 being a whole number in the series, 1, 2, ....., c—1, the ex- 
pression, ,X,, comprehends s (e—1) particular forms : 
7D. D.C aan <p. CR 
TAS 54 BND a c7% - 9 SBe os 
ee en aX, 4 coer 9 2. aae 6 
Then, if F, (#) be equal to a term in (2), an equation, 
N 
ce al ea a A NOGA al heal (SH) 
must subsist; where Y; is what Y becomes when T is replaced by 2, T ; 
