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 quartic equation; hence, supposing the product so expressed, and equating it to zero, we have the required sextic equation. In the same manner can be found the sextic equation having the roots (a - b)^2, (a - c)^2, (a - d)^2, (b - c)^2, (b - d)^2, (c - d)^2, which is the equation of differences previously referred to; and similarly we obtain the equation of differences for a given equation of any order. Again, the equation sought for may be that having for its n roots the given rational functions [phi](a), [phi](b), ... of the several roots of the given equation. Any such rational function can (as was shown) be expressed as a rational and integral function of the order n - 1; and, retaining x in place of any one of the roots, the problem is to find y from the equations x^n - p1 x^(n - 1) ... = 0, and y = M0x^(n - 1) + M1x^(n - 2) + ..., or, what is the same thing, from these two equations to eliminate x. This is in fact E.W. Tschirnhausen's transformation (1683). 14. In connexion with what precedes, the question arises as to the number of values (obtained by permutations of the roots) of given unsymmetrical functions of the roots, or say of a given set of letters: for instance, with roots or letters (a, b, c, d) as before, how many values are there of the function ab + cd, or better, how many functions are there of this form? The answer is 3, viz. ab + cd, ac + bd, ad + bc; or again we may ask whether, in the case of a given number of letters, there exist functions with a given number of values, 3-valued, 4-valued functions, &c. It is at once seen that for any given number of letters there exist 2-valued functions; the product of the differences of the letters is such a function; however the letters are interchanged, it alters only its sign; or say the two values are [Delta] and -[Delta]. And if P, Q are symmetrical functions of the letters, then the general form of such a function is P + Q[Delta]; this has only the two values P + Q[Delta], P - Q[Delta]. In the case of 4 letters there exist (as appears above) 3-valued functions: but in the case of 5 letters there does not exist any 3-valued or 4-valued function; and the only 5-valued functions are those which are symmetrical in regard to four of the letters, and can thus be expressed in terms of one letter and of symmetrical functions of all the letters. These last theorems present themselves in the demonstration of the non-existence of a solution of a quintic equation by radicals. The theory is an extensive and important one, depending on the notions of _substitutions_ and of _groups_ (q.v.).