36 RESOLUTION OF ALGEBRAICAL EQUATIONS, 
Wii % and P is an expression involvig only such surds as occur in 
Sf (p); the whole numbers Ay, Az, &c., being lessthan s. Take ¢, 
the general expression which includes (see def. vi.) all the cognate 
functions of f(p); one of its particular forms, distinct from f (p), 
being ¢,. Inpassing from (p) to ¢, let P and Y become respectively 
Q and y; and, in passing from ¢ to ¢,, let Q and y become respec- 
tively P; and y,. Then the equation, 
Ui = eyed sitaarave..aae:: eee) 
subsists ; 4 being an s" root of unity. 
Explanatory remark.—When we speak of Y becoming y in passing 
from f (p) to d, we do not assume that the expression Y is present 
in f(p); but we mean that all the surds which occur in f(p), and are 
also found in Y, must, in order that Y may be transformed into y, 
undergo the same changes which they require to suffer in order that 
JS (p) may become ¢. 
We proceed with the proof of the Proposition. In the first place, 
should P be zero, Ye = 0. Let Y, be of the form, 
1 
Yo = (a+ a 81 + ag Seo + lee sels = ae 3 
where the coefficients, a, a,, &c., are rational; and each of the terms 
S,,8,, &¢., is either some power of an integral surd, or the continued 
product of several such powers; the expression, @ + a, 8; + &e., 
satisfying the conditions of Def. 8. Then, since Y, = 0, we have 
a+ a8; + a82+ &. = 0. 
Now all the surds present in this equation, being subordinates of 
Y, , are (by hypothesis) surds occurring in f(p), a function in a simple 
form. Therefore (Cor. 1, Def. 9) the coefficients, a, a , &c., vanish 
separately. But, if Y, be what Y, becomes in passing from f (p) to 
, and Y, be what Y, becomes in passing from ¢ to ¢; , we have 
1 
Y, ==) (Gy 2 ln S, Set ae + Gy J), 
where Si, &c., are what 8,, &c., become in passing from f(p) to di*. 
Therefore Y. =0. But, in the same way in which, from the fact 
u 
that Y, is zero, we have deduced the conclusion that Y, is zero, we 
may, from the fact that P is zero, deduce the conclusion that Pj is 
