RESOLUTION OF ALGEBRAICAL EQUATIONS. 155 
where B is an expression of the same character with A. And so on. 
1 h H 
ALEAT Cpe Bl vO. OT S 8 s 
Substitute for V V;, V Vi, V Vi. the value, Y , AY, BY , 
&c. Then, when the function is rendered integral, the number of 
surds present in it, (principal and subordinate being both reckoned), 
will be less, by at least one, than it was before. Let modifica. 
tions of the three different kinds described continue to be made as 
far as possible. It is obvious that a limit will ultimately be reached ; 
and if the function be then rendered integral, it will be an integral 
function in a simple form, containing no surd such as Y in (1), and 
having no two surds similarly involved in it. 
Cor.—In f (p), a function which has. been made to undergo the 
modifications described in the Proposition, let Y be a surd, not subor- 
dinate to any other in the function; its index being a Also, let T 
and ¢ be two surds, with the common index = which is not equal to = 
subordinate to Y, but neither of them subordinate to any other surd 
in f(p); and suppose that the form of Y is, 
= 
Vea es EON (4) 
a? 
where m is a whole number, less thang; and H is an expression in 
which the surds T and ¢ are similarly involved.. Asin the Proposition, 
we can choose ¢ and w, whole numbers, such that 
ecs+m=we. 
1 
Y= TK)? 
where K is put for HT”” ; that is, K is the product of an expression 
which is clear of the surds T and ¢, by one in which T and ¢ are simi- 
larly involved. Hence again, as in the Proposition, we can eliminate 
the surds T and ¢ from K, introducing in their room a single new surd 
V; one of the surds T and ¢, as ¢, disappearing from the function 
altogether. And, since T and ¢ are not subordinates of any surd in 
J (p) except Y, the function, after being subjected to this change of 
form, may still, if necessary, be made to satisfy the different conditions 
described in the Proposition. So that, upon the whole, an algebraical 
function of a variable p may be exhibited as an integral function in a 
simple form, with no two surds similarly involved in it; nor with any 
