8 
SYNOPSIS OF THE THEORY OF EQUATIONS. 
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It follows that it is possible to determine an equation (of an assignable order) 
having for roots any given (unsymmetrical) functions of the roots of a given equation. 
For example, in the case of a quartic equation, roots (a, 6, c, d), it is possible to find 
an equation having the roots ab, ac, ad, be, bd, cd, being therefore a sextic equation; 
viz. in the product (y — ab) (y — ac) (y — ad) (y — be) (y — bd) (y — cd), the coefficients of 
the several powers of y will be symmetrical functions of a, b, c, d, and therefore 
rational and integral functions of the coefficients of the original quartic equation. 
In connexion herewith, the question arises as to the number of values (obtained 
by permutations of the roots) of given unsymmetrical functions of the roots; for instance, 
with roots (a, b, c, d) as before, how many values are there of the function ab + cd 4 , 
or, better, how many functions are there of this form; the answer is 3, viz. ab + cd, 
ac + bd, ad + 6c; or, again, we may ask whether it is possible to obtain functions of a 
given number of values, 3-valued, 4-valued functions, &c. 
We have, moreover, the very important theorem that, given the value of any 
unsymmetrical function, e.g. ab + cd, it is in general possible to determine rationally 
the value of any similar function, e.g. (a + b) 3 + (c + d) 3 . 
The d priori ground of this theorem may be illustrated by means of a numerical 
equation. Suppose, e.g. that the roots of a quartic equation are 1, 2, 3, 4; then if it 
is given that ab+cd= 14, this in effect determines a, b to be 1, 2 (viz. a = 1, 6 = 2, 
or else a = 2, 6 = 1) and c, d to be 3, 4 (viz. c = 3, d = 4, or else c = 4, d = 3); and 
it therefore in effect determines (a + b) 3 + (c + d) 3 to be =370, and not any other 
value. And we can in the same way account for cases of failure as regards particular 
equations; thus, the roots being 1, 2, 3, 4, as above, a 2 b = 2 determines a to be =1 
and 6 to be = 2; but if the roots had been 1, 2, 4, 16, then n 2 6 = 16 does not 
uniquely determine a and 6, but only makes them to be 1 and 16, or else 2 and 4, 
respectively. 
As to the d posteriori proof, assume, for instance, t x = ab + cd, y x = (a + b) 3 + (c + d) 3 , 
and so t 2 = ac + db, y 2 = (a + c) 3 4- (d + b) 3 , &c.—in the present case there are only the 
functions t x , t 2 , t 3 and y x , y 2 , y 3 —then y x + y 3 + y z , t x y x +t 2 y 2 + t 3 y 3 , t x z y x + t 2 2 y 2 + t 3 2 y 3 will 
be respectively symmetrical functions of the roots of the quartic, and therefore rational 
and integral functions of its coefficients, that is, they will be known. 
Imagine, in the first instance, that t x , t 2 , t 3 are all known; then the equations 
being linear in y x , y 2 , y 3 , these can be expressed rationally in terms of known functions 
of the coefficients and of t x , t 2 , t 3 , that is, y x , y 2 , y 3 will be known. But observe 
further, that y x is obtained as a function of t x , t 2 , t 3 symmetrical as regards t 2 , t 3 ; 
it can consequently be expressed as a rational function of ^ and of t 2 +1 3 , t 2 t 3 , or, 
what is the same thing, of t x and t x + t 2 +1 3 , t x t 2 + t x t 3 + t 2 t 3 , t x t 2 t 3 ; but these last will 
be symmetrical functions of the roots, and as such expressible rationally in terms of 
the coefficients: that is, y x will be expressed as a rational function of 4 and of the 
coefficients, or, t x being known, y x will be rationally determined. 
We may consider now the question of the algebraical solution of equations, or, 
more accurately, that of the solution of equations by radicals.