Section III., 1908. [ 273 ] Trans. R. S. C. 



XXIV. — Some New Symmetric Function Tables. 

 By Professor W. H. Metzler. 



(Read May 26, 1908). 



1. If A represents the determinant of the nth order | «in | , 



then let A2, A» An-i denote the 2d, 3d, . . . . (n - l)th 



compounds of A respectively; and let | A — -c j, | A — •'-• | , • • • 

 I ^„-T ^ I <^^6note the pohmomials in .r obtained by subtracting a: from 

 each constituent along the principal diagonals of A, A2, . . . . 

 An-i res]^ectively. 



It is known'that if aj, «,,... a^ are the roots of | A — -^ i =0, 

 then the roots of the equation | A^. x | =0 are the products of the a's 

 A: at a time (A- = 2, 3, . . . n — I). Let the roots of | Ajr ^'1=0 be 



denoted by a^, aj, a^ ; b^. b^, 6,, .... ; Cj. c.^, c^, .... etc. 



according as k has the values 2, 3, 4, etc., respectively. 



2. It follows from the foregoing' that to every symmetric function 

 of the roots of | A^ ^' | =0, which we have expressed in terms of 

 the p's,^ there is a corresponding relation between the sums of the 

 principal minors of A. For instance, we see from the tables (B, I) that 



2 b\ =p\-2 p, p, + 2 p, p, - 2 p„ 



which may at once be translated into determinant language by 

 observing that 2 b" denotes the sum of the squares of the principal 

 minors of order three, and that Pi denotes the sum of the principal 

 minors of order i (i = 1, 2, 3, , . 6). 



3. In the accompanying tables the symmetric functions of the 

 roots of I A;^^ I = (A; = 1, 2, 3, . . . 7) are given in terms of the 

 p'n. All the functions of the form 2 k^ A-j k^ l\ . . . k^ were obtained 

 by first expressing them in terms of symmetric functions of the a's,the 

 numerical coefficients of each -term being actually calculated, and 

 then by means of the ordinary symmetric function tables in terms 

 of the coefficients {p^). These were then used in connection with the 

 ordinary tables to calculate the other symmetric functions. 



' Metzler — Compound Determinants, Am. Jour. Math. Vol. XVT. No. 2, 1894. 

 Rados — Zur Théorie der adjugirten Substitutioneri, Math. Annalen liand 

 48, IbO?. 

 '^ px, pi, pn repn Bent the coefficients in | /\- x | = 0. 



