RESOLUTION OF ALGEBRAICAL EQUATIONS. 131 
let the cognate functions of f (p), obtained without departing from 
such definite values, (obtained, in other words, by proceeding without 
reference to the surd character of 1 , yz, &c.,) be, 
Then if 
(e—9, ) (a@—$2 )...@—on) = 29 + By a" + Bo a". &e., 
the coefficients, B, , By , are equal to expressions which are rational 
as respects all surds except y1, ¥2, &c. In other words, no surds 
not included in the series yi, yz, &c., enter into these coefficients. 
The proof is the same as in the Proposition. : 
Cor. 4. In the case supposed in the preceding Corollary, it may be 
shown, as in Cor. 1, that, if the unequal terms in (10), (the definite 
values of y,, y2, &c., being understood to be adhered to), be, 
and if f (p) be in a simple form, and we write 
(a — >) (%~—%z ) 200 tee cae (2—%.)=X1, 
where the number of terms, ¢1, %a,.........» ¢¢, is less than ¢, these 
terms being terms in (10), X; cannot involve, in the coefficients of 
the powers of x, merely the surds ¥;, yg, &c. For, if X; did in- 
volve merely these surds, %; would be a root of the equation, X; = 0; 
and therefore (Cor. Prop. V.) all the expressions, $1 , $2,...... Pes 
would be roots of that equation; the definite values given to 71 , y2 , 
&c., being adhered to in all the expressions, 9) , $2 ,......... sé But 
these expressions are, by hypothesis, unequal. Therefore the equa- 
tion, X; = 0, has ¢ unequal roots: which, since the equation is of a 
degree lower than the ¢", is impossible. Therefore X; cannot involve, 
in the coefficients of the powers of x, merely the surds y1 , yz, &e 
Cor. 5. In the case supposed in Cor. 3, let the unequal terms in 
the series (10), be, $1, $9,...... vee, 4 3 and let 
(w—$, ) (T—$2)...... (c—¢t )=at +6, xt 14+ box? + &e. 
Then the coefficients 6, , b2, &c., are equal to expressions involving 
no surds which do not occur in the series 7; , yg, &c.; and, if f (p) 
be ina simple form, each of the unequal ‘terms, $1, ¢25...... >?» 
recurs the same number of times in (10) ‘he proof is the same as 
in Cor. 2. 
