RESOLUTION OF ALGEBRAICAL EQUATIONS. Da 
A, B, &c., are rational, the coefficients A, B, &c., must vanish sepa- 
rately. Also, should f (p) be of the form, 
Ff (p) = A+ A, Vi + As Ve + ...... +A. Ve 
where each of the terms, V,, Ve, &c., no two of therm identical with 
one another, is either some power of an integral surd, or the con- 
tinued product of several such powers, while the expressions A, Aj, 
&c., involve only surds distinct from those whose powers constitute 
the factors of the terms V,, Ve, &c., then [it being understood, as 
before, that f (p) is in a simple form and equal to zero | the cofficients 
A, A,, &c., must vanish separately. 
Cor. 2. If f(p), a function of a variable p, be in a simple form, 
and if 
Pose Aye ACY, BYR U ABU, Yo 4 BLU liq) 
where A,, B,, A,, B,, &c., none of them being zero, are rational; A 
and B also being rational ; and each of the expressions, Y,, U,, Y., 
U,, &e., is either some power of an integral surd occuring in f (7), 
or the continued product of several such powers; the expressions, 
A+A,Y,+ &., B + B, U, + &c., having been arranged so as 
severally to satisfy the conditions of Def. 8; then the surd parts, 
RV Nc nce Re Lemeiuciesse tet cema sce arucun cme. c, (AN) 
are identical, taken in same order, with the surd parts, 
Wie Ost reek Re eet acy sc ast ac reewet GO) 
and, U, being the part identical with Y,, the rational coefficient B, is 
equal to the rational coefficient Aj. What we mean by identical with, 
as distinguished from equal to, may be shown by an example. The 
surd Jp? —1 is equal to the product of the two surds, Jp +1, 
/p—l1. But the expressions, ,/p2—1, /p+1 ,/p—1, are not 
identical ; because the only surd which appears in the former is not 
found in the latter ; and the surds which constitute the factors of the 
latter do not appear in the former. The truth of the Corollary may 
thus be shown. Should any term in (4), as Yi, be identical with a 
term in (5), as Uj, let the two terms, A,Y, and B,U,, in (3) , the latter 
removed to the left hand side of the equation, be written as one term, 
Y, (A, -- B,). No other term in (5) can be identical with Y,, for 
then it would also be identical with U,; but since the expression, 
B+B,U,+, &c., satisfies the conditions of Def. 8, no two terms in 
