28 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
(5) are identical. If U, be identical with a term im (4), necessarily 
distinct from Y,, let that term be Y,; and let the two terms, A, Y, and 
B, U,, in (3), the latter removed to the left hand side of the equation, 
be written, Y, (A, — B,). Make all other such modifications as are 
possible. Then equation (8) becomes, 
(A 2B) EW (kes iB eh ne SAV 2+ BaW, tases = Olgacp tes (6) 
where all the terms, Y,, Y,, U,, &c., are distinct ; so that the expres- 
sion on the left-hand side of equation (6) satisfies the conditions of 
Def. 8. Therefore, by Cor. 1, the coefficients, A,—B,,...... 5 ae eee 
&c., vanish separately. But, since the terms Aj, B,, &c., are all (by 
hypothesis) distinct from zero, this shows that there are, in fact, no 
such terms in (6) as those which we have written, A, Y,, — B, U,,. 
Hence the terms in (4) are identical, taken in same order, with those 
in (5). Also, Y, being identical with U,, we have seen that A, is 
equal to By. 
Proposition I. 
If f (p) be an integral function of a variable p, not in a simple 
form, then an equation, 
Ye PY Ne, YX, tah a 
must subsist ; where Y,, Y,, &c., are surds, principal or subordinate, 
occurring in f(p), of the same order, and with a common index = ; 
Ay, A», &c., being whole numbers, less than s; while Pis an expression 
involving only such surds, occurring in f(p), as are of lower orders 
than the surds Y,, Y,, &c. 
For, since f(p) is not in a simple form, an equation such as (1), 
Def. 9, 
A+ BU + CV +...... of OY ceiiesteee + ET = O,:2.... (2) 
subsists ; all the surds involved in the equation being surds present in 
SF (p). Let 
Nebpy 
Ng, = PNG ne Tibteisitstece Ye : @e0cesrecses ree OSB eos 007980 205 002 CO (3) 
be an equation such as (1), with this difference, that the indices of the 
surds Y,, Y,, &<., are not assumed to be equal to one another; but 
A, is less than the denominator of the index of Y,, A, less than the 
denominator of the index of Y3, and so on. Of the terms, U, V, 
veseeey Ty In (2), let those which involve among their factors surds of the 
