RESOLUTION OF ALGEBRAICAL EQUATIONS, 39 
¢,, be F, (py). Then the surds occurring in F (p) are all of lower 
orders than ae NE &c.; hence they are all of lower orders than 
the zt. But we are at present reasoning on the hypothesis that the 
law sought to be established in the Proposition holds for all functions 
which do not involve surds above the (n—1) order. Therefore, since 
F (p), containing only surds which occur in f (p), is in a simple form, 
it follows that the function F; (7) also is in a simple form. Now, if 
we refer to equation (1), we find that the surds involved in P, and all 
the subordinates of those surds whose powers constitute the factors of 
yi, occur in the function F, (py). Therefore, by Prop. III, we ‘can 
deduce from (1) the equation, 
Y= ER, 
& bemg a constant quantity. But this is an equation such as — 
(1), Prop. I. ; all the surds appearing in the equation being surds which 
occur in f (p). Such an equation, however, is directly at variance 
with the hypothesis that / (y) is in a simple form. And hence 4, 
cannot but be in a simple form. Consequently the law sought to be 
established in the Proposition holds good for all functions which do 
not involve surds above the x‘! order. 
Since, therefore, the law holds good for functions involving only 
surds of the first order, and since, on the hypothesis of its holding 
good for functions involving only surds of orders not higher than the 
(m-1)", we have shown that it must hold good for functions involving 
only surds of orders not higher than the x", it holds good universally. 
Proposition V. 
If f (p), an integral function of a variable p, in a simple form, be a 
root of the algebraical equation, F (z) = 0, in which the coefficients of 
the powers of % are rational functions of p, then ¢;, any one of the 
cognate functions of f (7), is a root of the same equation. 
For take ¢, the indefinite expression which includes all the cognate 
functions of f (p); and let F (4), F} We (p)h, F (¢,), developed by 
the ordinary process of mvolution, and arranged so as to satisfy the 
conditions of Def. 8, be, 
F (¢) = A + ALY, + AcYot...... + Ac Ye, 
F{f(pyt}= A+ AVit AeVet .... Vi 
F (fi1)= A + ArU, + AgUe + ...... + AU; 
