38 RESOLUTION OF ALGEBRATCAL EQUATIONS. 
must subsist ; because, in passing from f(p) to d , V becomes V; , and 
M becomes Mj, so that V; and M, are identical, and hence A, Vj, is 
equal to B; M;: and so of the other terms, Therefore from (6), 
(3), and (4), 
$s 
Ry 
i — vee — wee 
Proposition IV. 
If f (p), an Agora function of a variable p, be in a simple form, 
each of its cognate functions is in a simple form. 
It is self-evident that the Proposition is true for all functions which 
involve only surds of the first order. Suppose the law to have been 
found to hold for all functions which do not involve surds above the 
(n—1)™ order : it may then be proved true for a function, f (p), in- 
volving surds of the n'*, but of no higher, order. 
For take ¢, the general expression which includes all the cognate 
functions of f(p); one of its particular forms, distinct from / (p), 
being ¢: ; and suppose, if possible, that ¢; is not in a simple form. ° 
Then an equation such as (1). a I, 
d2 
ne ae Wen 
must (Prop. I.) subsist; all the surds involvedin the equation being 
surds which occur in ¢;. We may write this equation in the form, 
where y, denotes the continued product of the expressions Y , 
wae &e. Let y; in ¢; correspond to y in ¢, and to Y in f(p): 
that is to say, y is what Y becomes in passing from /(p) to ¢, and 
yi is what y becomes in passing from to ¢,. In like manner, let 
P in ¢; correspond to Qin ¢, andto Rin f(p). Let the the surds 
Y., Yi, &c., im di, correspond to Y./: We, &C., " St (p); and since 
the surds Y, ; Y, , &c., have the common index = ~, let their forms be, 
1 2 1 
Ne Ne eae SOOnGG »>Y,= 
Take F (/), a function involving all the surds which occur in the ex- 
pressions R, Vc, Vi, ....6 , Va; and let the particular cognate func- 
tion of F (p), obtained by making the same changes in the surds in- 
volved in F (p) as require to be made in order to pass from f (p) ta 
