40 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
where A, A,, &c., are rational; and each of the terms, Y,, Yo, &c., 
is either some power of an integral surd, or the continued product of 
several such powers; Vi, V2, &c., being what Y,, Yo, &c., be- 
come in passing from ¢ to f (p); and Ui, Ue, &e., what Yi, 
Y2, &c., become in passing from ¢ to ¢,. The expression for 
F § f(p)} can only involve such surds as are present in some of their 
powers in f (p). And f (p), by hypothesis, is in a simple torm. 
Therefore F ; Fi py, as exhibited above, is in a simple form. It also 
satisfies the conditions of Def. 8. But, since f(p) is a root of .the 
equation, F (7) = 0, F ; tS (p)$ is equal to zero. Therefore, in the 
expression for F ; VA (py, the coefficients A, Ay, &c., must (Cor. 1, 
Def. 9) vanish separately. Hence, F (¢,) = 0; and consequently 
: $1 is a root of the equation, F (x) = 0. 
Cor.—Let f (p) be an integral function of p, in a simple form ; 
and let certain surds in f (p), viz.: yi, y2, &c., (in which series of 
terms, as was noticed in Def. 7, all the subordinates of any surd 
mentioned are necessarily included), have definite values attached to 
them ; and let the cognate functions of f(p), taken according to the 
manner described in Def. 7, without reference to the surd character 
of 71, y2, &e., be 
Maina hc a Mba 
Also let F («) = 0, be an equation in which the coefficients of the 
powers of « are rational as far as all surds except y1, y2, &c., are 
concerned ; that is, the coefficients contain no surds besides y;, y25 
&c. Then, if f(p) be aroot of the equation, F(x) = 0, any one of 
the terms, ¢1, $2. ......, dn, (the definite values of y1, y2, &e., 
being adhered to), is a root of the same equation. For, in this case, 
in the same manner in which the expressions for F } / (p)$ and 
F (1) in the Proposition were formed, we get 
F \ f(o) t= A + AV + Ao Vet &e. 
F (di) = A + A, U, + AgUe4 &e.; 
where A, Aj, &c., are rational as far as all surds except y1, y2, &c., 
are concerned ; and each of the expressions, Vi, V2, &c., is either 
some power of a surd in f (p), not contained in the series, y1, y2, 
&c., or the continued product of several such powers; U;, U2, &e., 
being what V,, V2, &c., become in passing from.f(p) tod: the 
expressions for F } f(p)} and F(¢;) satisfying the conditions of 
