30 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
c iL 
oe A 
X,, as m* , such that X, either has no power of the surd M as one 
1 
of its factors, or a power of M” distinct from the c®. Both of these 
r 
alternatives are included in the assumption that M” isa factor of X,, 
¢ being a whole number, which is not equal toc, but may be zero. 
Hence, if equation (7) subsist, we have 
c-r 
BEL KOM WES OEE gh idee ee seni (8) 
a 
where X is what X, becomes when the factor M’ is rejected; and x, 
r 
is what X, becomes on the rejection of the factor M*. Since ¢ and 
y are whole numbers, different from one another, and each less than 
the prime number A, we can choose whole numbers, m and m, such 
that m (c—r)=nrX+1. Then 
oe 
(BH, X ) M M*=(4CH,X) . 
1 
—n 
vM (BH, XY M (eCH, XyOOe (Oates 
But this equation will be readily seen, when the expression on its 
right hand side is rendered (Cor. Def. 5) integral, and made to satisfy 
the conditions of Def. 8, to be of the inadmissible form (8). Conse- 
quently 4, cannot be zero; and therefore the coefficient of X, in (6) 
is not zero. In like manner it can be shown that the coefficients of 
all the other terms, such as X,, in (6), are distinct from zero. Again, 
the coefficients of the terms, H, X,, X,, &c., in (6), when rendered 
integral functions, and made to satisfy the conditions of Def. 8, in- 
volve no surd of so high an order as those whose powers constitute the 
factors of X,, X,, &c. This will be plain if it be considered that the 
differential coefficient of the logarithm of any power of a surd does 
not involve, when arranged so as to satisfy the conditions of Def. 8, 
the surd in question. For instance, 
2 
3 
Ah tog log a+ ym'$ _ vp—p 
dp ~ Bp(l=p) ’ 
where the differential coefficient obtained is clear of the surd 
