RESOLUTION OF ALGEBRAICAL EQUATIONS. 145 
and P is an expression involving only surds which occur in F;, (2), 
exclusive of Y; A being a whole number, distinct from zero and less 
than s, satisfying the condition, 
nee AG eG SAIN pee Ee, PRONE 
where wis a whole number. Also, if Y be not of the form shown in 
equation (2) Prop. X, A is not uuity, and s is not 2. 
Let us in the mean time reason on the supposition that F, (x) is 
equal to a term in the first horizontal line of (2). We here make an 
observation to which we shall have occasion subsequently to refer. 
When an expression is equal to some term in a series such as that 
constituted by the terms in the first horizontal lime of (2), any one 
of the terms in the series may be assumed to be that to which the 
expression in question is equal; because any particular term im the 
series stands, in fact, as the representative of all the terms im the 
series, in consequence of the s distinct values which may be given 
to the surd Y,; _— Proceeding, therefore, on the supposition that F, (x) 
is equal to a term in the first horizontal line of (2), we may under- 
stand that Fy (x) is equal to ;X,. Take «®, a power of x in F, (2) 
having some power of Y present in its coefficient ; and let the co- 
efficient of x? in F, (x), satisfying the conditions of Def. 8, be, 
c li 
Dr oe A Ged Aves NA oe OGe s 
where Ac, Ay, &c., none of them zero, are clear of the surd Y; no 
: " n seh’ d 
two powers in the series, ye 4 ay , &ec., being identical. ‘Then, D, 
being the corresponding coefficient in ,X,, and B,, By, &c., being 
what A, A,, &c., beome when T is changed into 2, T, we have 
ee Baty 6 Balert fu. 
But D= D,. Therefore, by Prop. II., the terms, Ag y : AY" » &C., 
J Cc B th 
taken im some order, are equal to the terms, Be iy CN REN a Ge ares 
each to each Hence we may put 
ALY oBs Note 
where By vr is some term in the series, B, Vi ‘Be view &e. 
And, since A, and By involve only surds which occur in F, (a), exclu- 
sive of Y, this equation can easily be reduced to one of the form (3); 
X not being zero, because neither c nor m is zero. Now, from equa- 
tion (2), the following may be derived by Prop. III. : 
