OF THE HIGHER DEGREES, WITH APPLICATIONS. 87 
§17. Proposition III. The unequal particular cognate forms of F, 
the generic expression under which the simplified expression 7, falls, 
are the roots of a rational irreducible equation ; and each of the 
unequal particular cognate forms occurs the same number of times in 
the series of the cognate forms. 
As in §16, let the entire series of the particular cognate forms of # 
be the terms in (5), the equation that has these terms for its roots 
being X = 0. By §16, X is rational. Should X not be irreducible, 
it has a rational irreducible factor, say /’(x), such that 7) is a root of 
the equation /'(x) = 0. By Prop. IL. because 7, is in a simple 
state, all the terms in (5) are roots of the equation F(x) = 0, 
while at the same time, because /’ (x) is a factor of X, all the roots 
of the equation are terms in (5). And the equation / (x) = 0, 
being irreducible, has no equal roots. Therefore its roots are the 
unequal terms in (5). Should # (x) not be identical with X, put 
X = {F(x)} fe @h. 
Because X and /'(x) are rational, ¢ (x) is rational. Then, since 
g (a) is a measure of X, and the equation # (x) == 0 has for its 
its roots the unequal roots of the equation X = 0, the equations 
F (x) = 0 and ¢ (x) = 0 have a root in common. Consequently, 
since # (x) is irreducible, it is a measure of g (a). Therefore 
F (z)(2is a measure of X. Going on in this way we ultimately 
get X = § F(a) i ; which means that each of the particular cognate 
forms of & has its value repeated JV times in the series of the particular 
cognate forms. 
§18. Cor. 1. The series (6) consisting of those particular cognate 
1 1 
. . . n § . 
forms of FR in which certain surds Z, ,U, » ete, retain the deter- 
minate values belonging to them in 7), each of the unequal terms in 
(6) occurs the same number of times in (6); and the unequal terms 
in (6) are the roots of an irreducible equation whose coefficients 
1 1 
: . : . n s 
involve only surds found in the series z, ,w, , etc. Should X ‘ not 
be irreducible, by which in such a case is meant incapable of being 
broken into lower factors involving only surds occurring in X’, let it 
have the irreducible factor X”. That is to say, X” involves only 
surds occurring in XY’, and has itself no lower factor involving only 
surds that occur in X”. We may take 7 to be a root of the equation 
X” = 0. Then, by Cor. Prop. II., all the terms in (6) are roots of 
that equation, all the roots of the equation being at the same time 
terms in (6). And the equation X” = 0 being irreducible, has no 
equal roots. Therefore its roots are the unequal terms in (6). Put 
