RESOLUTION OF ALGEBRAICAL EQUATIONS. 147 
is (Cor. 1, Def. 9) impossible, unless Y and Y be be the same power 
of Y, that is, unless m be unity. Therefore 
o 
A=p+l1: 
an equation of the form (4). Should \ be unity, it is plain, referring 
to the manner in which the first of equations (6) was obtained, that 
A, Y°'=B,.Y;; 
and consequently (Prop. X.) the surd Y is of the form shown in 
{2) Prop. X.; so that, if Y be not of that form, A cannot be unity. 
In this case, also, s cannot be 2; for were s equal to 2, X could have 
no other value than unity. 
Cor. 1.—Should F, (~) not be equal to a term in (2), then no such 
equation as (3) admits of being formed. For, since T disappears 
from 1 os (x), the continued product of the terms in the first hori- 
zontal line of (2) is equal to that of the terms in (1): both products 
being F,,,(). Hence F,(x) has a common measure with some 
term in the first line of (2), which term (on the principle pointed 
out in the Proposition) may be assumed to be ;X,. Let L be the 
H. OC. M. of F,() and 1X,. Since the roots of the equation, 
_F.@) =0, are (Prop. VII.) the unequal cognate functions of f (p), 
obtained by assigning definite values to those surds in f (p) which are 
also present in F, @), and taking the cognate functions without re- 
ference to the surd character of the surds so rendered definite, L, 
which is of less dimensions, as respects «, than F, («), cannot (Cor. 4. 
Prop. VI.) involve, in the coefficients of the powers of x, merely 
such surds as occur in F,(2). But the only surd not in F, (~), which 
can possibly appear in L, is Y, ; because, with the exception of Y; , 
all the surds in ;X, are found in F, («). Hence Y, cannot be absent 
from L. But if such an equation as (3) subsisted, all the powers of 
Y; in L might be eliminated from L, without any surds being intro- 
duced into L, except such as are found in F,(z). Hence no such 
equation as (3) can be formed. 
Cor. 2.—Should no equation such as (3) subsist, any function in- 
volving merely such surds as are in F, (x), together with Y,, is in 
a simple form. For suppose, if possible, that y (p) is such a fune- 
tion, and that it is not in a simple form. Then an equation such as 
(1) Prop. I. must subsist; all the surds occurring in it being found in 
yy (p). One of these must be Yj ; else all the surds in the equation 
