32 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
be distinct from zero. Hence some equation such as (3) must of 
necessity admit of being formed. Now suppose that the indices of 
the surds, Y., Y,, &c, in (3), are, = ae &c.; and that s, is not equal 
tos. By raising both sides of equation (3) to the s power, we may 
easily [compare the manner in which equation (9) was deduced from 
(8) ] transform (3) into an equation, not involving the surd Y,, 
where P, is an expression such as P; 8, beg a whole number less 
than the denominator of the index of Y,; 6,, a whole number less 
than the denominator of the index of Y, ; and soon. By continuing 
this process of reduction as far as necessary, we ultimately arrive at 
an equation such as CW 
Cor. Let each of the terms, Y,, Y,, &¢., be either some power of 
an integral surd, or the continued product of several such powers ; 
while A,, A,, &c., are algebraical expressions, distinct from zero; 
and A is an algebraical expression not assumed to be distinct from 
zero. Then, if 
A+ A, tbs, a Bs, NG as sence es Age Gheccmnt | Pe (12) 
an equation of the form, 
RGR EINCONE Sisehy Sonsacenedecco, macros ronnen ue nsee ioe (133) 
must subsist; where P is an expression involving only such surds as 
are present in the coefficients A, A,, &c., or are subordinates of some 
of the surds whose powers constitute the factors of Y,, Y,, &c.; 
and Y, is a term inthe series, Y,, ......... , Y.3 m being either unity 
or zero. For, in the same way in which we eliminated X, from 
equation (4), we may proceed to eliminate successively the terms 
Y,,....++, Ye, from (12). The result of the elimination of Y, is, 
d } log (2 s+) 
Cie ald +A,Y, Dugas &e. = 0...... (14) 
Here (see remarks in the Proposition) the coefficient of Y,, when 
made to satisfy the conditions of Def. 8, involves no surds except 
such as are found in A, or A,, or are subordinates of the surds 
whose powers constitute the factors of Y, and Y,. Hence the co- 
efficient of Y, in (14) is an expression such as P in (13). Should 
this coefficient vanish, we have A, Y, =A, Y,, & being a con- 
