15. Returning to equations, we have the very important theorem that, given the value of any unsymmetrical function of the roots, e.g. in the case of a quartic equation, the function ab + cd, it is in general possible to determine rationally the value of any similar function, such as (a + b)^3 + (c + d)^3. The _a priori_ ground of this theorem may be illustrated by means of a numerical equation. Suppose 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 and c, d to be 3, 4 (viz. a = 1, b = 2 or a = 2, b = 1, and c = 3, d = 4 or c = 3, d = 4) or else a, b to be 3, 4 and c, d to be 1, 2; and it therefore in effect determines (a + b)^3 + (c + d)^3 to be = 370, and not any other value; that is, (a + b)^3 + (c + d)^3, as having a single value, must be determinable rationally. 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 before, a^2 b = 2 determines a to be = 1 and b to be = 2, but if the roots had been 1, 2, 4, 16 then a^2 b = 16 does not uniquely determine a, b but only makes them to be 1, 16 or 2, 4 respectively. As to the _a posteriori_ proof, assume, for instance, t1 = ab + cd, y1 = (a + b)^3 + (c + d)^3, t2 = ac + bd, y2 = (a + c)^3 + (b + d)^3, t3 = ad + bc, y3 = (a + d)^3 + (b + c)^3: then y1 + y2 + y3, t1y1 + t2y2 + t3y3, t1^2 y1 + t2^2y2 + t3^2y3 will be respectively symmetrical functions of the roots of the quartic, and therefore rational and integral functions of the coefficients; that is, they will be known. Suppose for a moment that t1, t2, t3 are all known; then the equations being linear in y1, y2, y3 these can be expressed rationally in terms of the coefficients and of t1, t2, t3; that is, y1, y2, y3 will be known. But observe further that y1 is obtained as a function of t1, t2, t3 symmetrical as regards t2, t3; it can therefore be expressed as a rational function of t1 and of t2 + t3, t2t3, and thence as a rational function of t1 and of t1 + t2 + t3, t1t2 + t1t3 + t2t3, t1t2t3; but these last are symmetrical functions of the roots, and as such they are expressible rationally in terms of the coefficients; that is, y1 will be expressed as a rational function of t1 and of the coefficients; or t1 (alone, not t2 or t3) being known, y1 will be rationally determined. 16. We now consider the question of the algebraical solution of equations, or, more accurately, that of the _solution of equations by radicals_. In the case of a quadric equation x^2 - px + q = 0, we can by the assistance of the sign [root]( ) or ( )^1/2 find an expression for x as a 2-valued function of the coefficients p, q such that substituting this value in the equation, the equation is thereby identically satisfied; it has been found that this expression is x = 1/2{p [+-] [root](p^2 - 4q)},