504 EQUATION of the assumption. In illustration, we observe that it is allowable to express a function of p and q as follows, that is, by means of a rational symmetrical function of a and b; this can, as a fact, be expressed as a rational function of a + b and ab; and if we prescribe that a + b and ab shall then be changed into p and q respectively, we have the required function of p, q. That is, we have F (a, /3) as a representation of / (p, q), obtained as if we had p = a + b, q = ab, but without in any wise assuming the existence of the a, b of these equations. 20. Starting from the equation x n - 2x n ~ l -{ . . . =*x-a.x-b. &c. or the equivalent equations p = 2a, &c., we find a n -p i a n ~ 1 + . . = 0, b"-p L b-i + . . =0; (it is as satisfying these equations that a, b . . are said to be the roots of x n p 1 x n ~ 1 + .. =0); and con Tersely from the last-mentioned equations, assuming that a, b . . are all different, we deduce p 1 = Za, p i = 2ab, &c. and x n -p ] x n - l + . ..=x-a.x-b. &c. Observe that if, for instance, a = b, then the equations a" - p^" 1 + . . = 0, b" - pfi 1 - 1 + . . . = would reduce them selves to a single relation, which would not of itself express that a was a double root, that is, that (.r - a) 2 was a factor of x n -p l x n ~ 1 + &c.; but by considering b as the limit of a + h, h indefinitely small, we obtain a second equation na"- 1 - (n- )p l a n - + . . . =0, which, with the first, expresses that a is a double root ; and then the whole system of equations leads as before to the equations p l = 2, &c. But the existence of a double root implies a certain relation between the co efficients; the general case is when the roots are all un equal. We have then the theorem that every rational symmetri cal function of the roots is a rational function of the co efficients. This is an easy consequence from the less gene ral theorem, every rational and integral symmetrical func tion of the roots is a rational and integral function of the coefficients. In particular, the sums of the powers 2 2 , S 3 , &c., are rational and integral functions of the coefficients. The process originally employed for the expression of other functions 2,a a bP, &c., in terms of the coefficients is to make them depend upon the sums of powers : for instance, 2a a ^ i = 2a a 2a /3 -2a a+ ^; but this is very objectionable ; the true theory consists in showing that we have systems of equations Pi =2. Ps where in each system there are precisely as many equa tions as there are root-functions on the right-hand side e.y., 3 equations and 3 functions 2abc, 2,a-b, 2a 3 . Hence in each system the root-functions can be determined linearly in terms of the powers and products of the co efficients : f 2a& = p. 2 > 2 2 i + 32abc , and so on. The older process, if applied consistently, would derive the originally assumed value "Sab, =;>. from the two equations 2a = p, 2a 2 = ;; 1 2 - 2p ; i.e., we have 21. It is convenient to mention here the theorem that, x being determined as above by an equation of the order 71, any rational and integral function whatever of ;r, or more generally any rational function which does not be come infinite in virtue of the equation itself, can be expressed as a rational and integral function of x, of the order n ], the coefficients being rational functions of the coefficients of the equation. Thus the equation gives x n a function of the form in question; multiplying each side by x, and on the right-hand side writing for x" its fore going value, we have x" +1 , a function of the form in question; and the like for any higher power of x, and therefore also for any rational and integral function of x. The proof in the case of a rational non-integral function is somewhat more complicated. The final result is of the form 7!. = I(x), or say <(#) - {j/(jc)I(x) = 0, where <, }/, I are rational and integral functions; in other words, this equa tion, being true if only/(x - ) = 0, can only be so by reason that the left-hand side contains / (x) as a factor, or we must have identically t^x) ff(x)I(x) = M(x]f(x). And it is, moreover, clear that the equation ^~. = I(x), being satisfied if only /(x) = 0, must be satisfied by each root of the equa tion. From the theorem that a rational symmetrical function of the roots is expressible in terms of the coefficients, it at once follows that it is possible to determine an equation (of an assignable order) having for its roots the several values of any given (unsym metrical) function of the roots of the given equation. For example, in the case of a quartic equation, roots (a, b, c, d), it is possible to find an equation having the roots ab, ac, ad, be, bd, cd (being there fore a sextic equation) : viz., in the product (y - ab)(y- ac)(y - ad)(y - bc)(y - bd)(y - cd) the coefficients of the several powers of y will be sym metrical 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 differ ences previously referred to; and similarly we obtain the equa tion 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 <() , <f>(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; and, retaining x in place of any one of the roots, the problem is to find // from the equa tions x n -p^ 1 ... = , and ?/ = M oj"~ 1 + .M 1 ;c"~ 2 + . . , or, what is the same thing, from these two equations to elimi nate x. This is in fact Tschirnhausen s transformation (1683). 22. 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 4- 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 differ ences of the letters is such a function ; however the letters are interchanged, it alters only its sign ; or say the two
values are A, - A . And if P, Q are symmetrical functions