RESOLUTION OF ALGEBRAICAL EQUATIONS. 33 
stant ; which equation is of the form (13). Suppose that the coeffi- 
cient of Y, in (14) does not vanish; and let equation (14), for the 
sake of simplicity, be written, 
Reereaey EE RNG te dist BY, == Onan, (15) 
In the same manner in which we proved B, to be an expression such 
as P, it can be shown that all the other terms, B, B,, &c., are ex- 
pressions such as P. Hlminate Y, from equation (15), as Y, was 
eliminated from (12). The result of the elimination is 
d log (sy) 
A Sl eR ASC AS — 
Ff + &e. = 0...... (16) 
As above, the coefficient of Y, here is an expression such as P. 
Also, if that coefficient vanish, we have B, Y, = 4B, Y,, &beinga 
constant quantity. And this equation is of the form (13). Should 
the coefficient of Y, in (16) not vanish, we may proceed to eliminate 
another of the terms, Y,, Y,,...., Y,3 aud it will be found that 
the coefficient of Y in the equations that result from such elimina- 
tions can never at any stage become zero, unless such an equation as 
(13) subsist. Suppose then that all the terms, Y,, Y,, ...... SN ge 
can be eliminated in the manner described, without the coefficient of 
Y, at any stage becoming zero. Then ultimately we get 
HA + KATY! 20, 
pac Le BAY 
where H and K, the latter (and consequently also the former) not 
zero, are expressions such as P. And this is an equation of the form 
(18), m being taken equal to zero. Hence an equation such as (18) 
must necessarily subsist. 
Proposition II. 
In f (p), an integral function of a variable py, in a simple form, 
satisfying the conditions of Def. 8, let Y be a surd which is not 
subordinate to any other in the function, its index being }. Arrange 
Tf (p) as follows : 
FM=HAFAY +4,Y +4,Y +8, 
where A,, A,, &c., are expressions distinct from zero, and clear of the 
surd Y; A being also clear of the surd Y; and yn Ms, &c., being 
Vo. V. D 
