RESOLUTION OF ALGEBRAICAL EQUATIONS. 153 
But, by (12), 
Ag-1 — Bros = Aron — 2 BAc-3- 
r Xe-1 —BAc-3 
Sool ibe) PANE: : 
which is the general form that includes all the equations in the 
series (11). 
Cor. 6.—The (co + 1)™ equation in the series (11) is, 
Bore Nok 2hBN 
YY,= PY, a 
By comparing this with the first of equations (11), we get 
o 
BIB) Ag =B Ag 2 
Yo a) Pa AG . 
But, by Cor. 1, in connection with Prop. III., this is impossible 
unless 
12 Ce UR ananame e 
w being a whole number. Therefore equation (14) must subsist. 
: Proposition XII. 
A given algebraical function of a variable p can always be expressed 
as an integral function in a simple form; the followimg conditions 
being at the same time satisfied: First, that there shall be no surd 
in the function, of the form, 
1 
CES A ete. oe) 
where T is a chief subordinate of Y, with the index a which is not 
equal to = ; and m is a whole number, not zero, and less than o ; and 
H is an expression clear of the surd T; secondly, that no two surds, 
V and Vj, principal or subordinate, shall be similarly [see Prop. IX. ] 
‘avolved in the function. 
For, should the given function, when rendered integral, be not in a 
simple form, an equation such as (1) Prop. I. must subsist ; all the 
surds in the equation being surds which occur in the function. Sub- 
stitute, then, in the function, for Y,, wherever it occurs in any of its 
powers, its value as furnished by (1) Prop. I. Then, when the func- 
tion is rendered integral, the number of surds present in it, (principal 
