RESOLUTION OF ALGEBRAICAL EQUATIONS. 37 
zero. Also, since Y, is a factor of y,;, y; must be zero. Therefore 
oe he Py. § 
Tn the next place, should P not be zero, the expressions y°, Y , y;°, 
developed by the ordinary process of involution, rendered integral, 
and made to satisfy the conditions of Def. 8, are of the forms, 
y =A+A,0 +A,72 + &., 
Y=A+A,V+A,T + &c., statehstatel stata (3) 
yi = A+A,V,+ A, Ti+ &.; 
where A, A;, &c., are rational ; and each of the expressions, », f, &c., 
is either some power of an integral surd, or the continued product of 
several such powers; the expressions V, T, &c., being what », ¢, &c., 
become in passing from ¢ tof (p); and V;, Ty, &c., what v, ¢, &e., 
become in passing from ¢# to d;. In like manner, the expressions, 
Q, P, rey satisfying the conditions of Def, 8, are of the forms, 
Q = B+ B, m+ BI + &e., 
729 cin) SU SHY) DRE BAN Da ea th RN (4) 
Pe ai UB aN aN Ba li is See 
where B, B,, &c., are rational; and each of the expressions, m, J, 
&c., is either some power of an integral surd, or the continued pro- 
duct of several such powers; M, L, &c., being what m, 2, &e., become 
in passing from ¢ to f (p); and M, , L,, &e., what m, 7, &c., become 
in passing from ¢ to ¢;. From (1), (3), and (4), we have, 
A+A,V+A2QT + &.—B+B,M+ BeL + &...... (5) 
But the surds occurring in the expression on the left hand side of 
this equation, being necessarily subordinates of some of the surds, 
Mais Mars, ash 3 » Ya, are all present in f(p). Those occurring in the 
expression on the right hand side of the equation are likewise all 
present in f(p). Therefore, since equation (5) subsists, the surd 
parts, V, T, &c., are (Cor, 2, Def. 9) severally identical, taken in 
some order, with the surd parts, L, M, &c.; which also (Cor. 1, 
Def. 9) implies, that, if V be the surd part identical with M, A, is 
_equal to B,;; and so on. But since V is identical with M, and 
A; equal to B,, and T is identical with (we may suppose) L, and 
Ag equal to Bz , and so on, the equation, 
A + Ay Vi + Ag T; + &c. = B+ B, My; + Bo Lyi + &e, 
