22 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
by some power of an integral surd, or by the continued product of 
several such powers. Take Y, one of the surds of highest order 
present in any of its powers in the function ; and arrange the terms 
in N and D according to the powers of Y not exceeding the (m—1)", 
+ being the index of the surd Y. Then 
3 m-I 
ataY+a.Y +...t@n.1 1 
m-1 ? 
Be BiG HORN ft A 
where the coeflicients, 6, a, b,, a,. &c., may involve powers of any surd 
in f (p), except Y. No powers of Y higher than the (m-1)" are writ- 
5 ° m-+2. 
ten; because, for instance, if there were a term AY ** in the nu- 
merator, A being an expression clear of the said Y, it might be 
St (y= 
written, (AY )¥. But Y may be written so as to involve only 
m-+2 
the subordinate surds of Y; and hence the term AY: may be con- 
2 
sidered as contained in the term, a, Y . Assume 
and, when the expressions, 6+6,Y + &€., e+c¢Y+&c., are multiplied 
by one another, let the product, arranged according to the powers of 
Y not exceeding the (m-1)™, be, d+ d,¥ + &c.; where d, d,, &¢., are 
clear of the surd Y. Then 
1 
ore mex aaa ela gel youd 
Determine the m unknown quantities, c, ¢,...... > ma, by the m 
simple equations, 
Hs Gy eh TR De hal 
Then the function may be written, i 
f (p)yscteaYtaY +&e. ; 
where the coefficients, c, ¢,, &c., are clear of the surd Y. Again, 
let a surd of the highest order present in any of its powers in the 
coefficients c, c,, &c., be V; and its index 1. By the process already 
exemplified, we may find, for each of the coefficients, c, c,, &c., an 
equivalent expression such as 
n-1 
h + AV + arin + eoorce + Ia aV ’ 
where h, i, &e., are clear of the surds Vand Y. Let it be remarked, 
that, in consequence of our having commeneed with Y, a surd of the 
