88 REPORT— 1902. 



published sixteen years later (which, however, throws no fresh light on 

 the subject). The same principle is fundamental in the other new method, 

 sketched in the Berlin memoir of 1748. But with this difference. By 

 the one method the common linear factor, corresponding to the common 

 root, was eliminated between the two equations, and from the resulting 

 identity the system of linear equations which led to the resultant were 

 obtained. In the other method the resultant is formed of the products 

 of all possible differences of the roots of the two equations, such as (a— a), 

 ii a, b, c . . . ", /3, y . . . are the roots — one of which must certainly 

 vanish — and the crux of the solution lies in the expression of this product 

 in terms of the coefficients of the two equations. Since each equation 

 may also be expressed as a product of factors such as (x — a), the resultant 

 equation is found to be the product of the equations formed from one 

 equation by substituting in it for x the roots of the other equation. And 

 this leads at once to the consideration of certain symmetric functions of 

 the roots, and to the necessity of evaluating them in terms of the co- 

 efficients. This process is only sketched by Eulei',' and thus his proof 

 that the resultant, when the coefficients contain a second variable, attains 

 the degree mn (m and ii being the degrees of the equations) in this vari- 

 able is not conclusive, although the first step, of proving that it is of the 

 7nth degree in the coefficients of the equation of the iith degi'ee and of 

 the nth degree in the coefficients of the equation of the mth degree, is 

 correctly taken. This method is sometimes known as ' Euler's first 

 method,' - or, more properly, as ' elimination by symmetric functions.' 

 It was also employed with far greater success by Cramer (170-4-1752) 

 in the Appendix II. to his 'Introduction a I'analyse des lignes coui'bes 

 algebriques.' ^ In fact, it is from Cramer's work that the impulse to 

 investigate all possible symmeti'ic functions of the roots of an equation 

 dates. Up till this time the only ones discussed were the products taken 

 one, two, three ... at a time, known as early as 1629 by Girard"* to be 

 equal to the successive coefficients of the equations, and the sums of the 

 powers of the roots which Newton had investigated.* 



To Cramer, also, belongs the credit of devising a suitable notation.'' 

 It is in all essentials the same as Leibnitz's, but was probably invented 

 independently.*^ He writes the two equations from which x is to be 

 eliminated thus : 



A . . . x"~[l]x"-' + [r-]x^-'-[V]x"-+ . . . [1"]=0 



B . . . {0) + {l)x + (2)x'^ + {S)x'+ . . . +(m)a;"'=0 



n 



where [IJ, [P], [1'*], . . . [1"] are functions of y of degree, 1, 2, . 

 and (0), (1), (2), . . . (in) are functions of y of degree ?», m — 1, m — 2, 

 . . . ; and calls the resulting equation in y, C. C is, then, the product 

 of the equations formed from B by successive substitutions for x of the 

 roots a, b, c, «tc., of A ; each term of it therefore consists of two factors : 

 ' I'un, facteur-premier, est le produit de quelques coefficients de B ; I'autre, 

 facteur second, est une fonction des racines a,b, c . . . de I'cquation A.'** 

 The ' facteurs-premiers ' ax-e easily found by combining n at a time the terms 



' Acad. Berlin, annee 1748, p. 245. 



" EncyTi. der Math. Wis.sen., Leipzig, ]8!)9, Bd. I., p. 2i5. 



^ Geneva, 1750, pp. 660. '' Cantor, vol. ii. p. 71S. 



* Arithmetica Universalis, 1st edition, p. 251. 



" Analyse, App, I., p. 657. ' Cf. Cantor, vol. iii. p, 586, » Loc. cit., p, 660. 



