XXIII. A memoir on the symmetric functions of the roots of an equation

Abstract

There are contained in a work, which is not, I think, so generally known as it deserves to be, the ‘Algebra’ of Meyer Hirsch, some very useful tables of the symmetric functions up to the tenth degree of the roots of an equation of any order. It seems desirable to join to these a set of tables, giving reciprocally the expressions of the powers and products of the coefficients in terms of the symmetric functions of the roots. The present memoir contains the two sets of tables, viz. the new tables distinguished by the letter ( a ), and the tables of Meyer Hirsch distinguished by the letter ( b ); the memoir contains also some remarks as to the mode of calculation of the new tables, and also as to a peculiar symmetry of the numbers in the tables of each set, a symmetry which, so far as I am aware, has not hitherto been observed, and the existence of which appears to constitute an important theorem in the subject. The theorem in question might, I think, be deduced from a very elegant formula of M. Borchardt (referred to in the sequel), which gives the generating function of any symmetric function of the roots, and contains potentially a method for the calculation of the Tables ( b ), but which, from the example I have given, would not appear to be a very convenient one for actual calculation. Suppose in general (1, b , c ..↺(1, x ) ∞ = (1 - αx )(1 - βx )(1 - γx )..., so that - b = Σ α , + c = Σ αβ , - d = Σ αβγ , &c., and if in general ( pqr ..) = Σ α p β q γ r ..., where as usual the summation extends only to the distinct terms, so that e. g. ( p 2 ) con­tains only half as many terms as ( pq ), and so in all similar cases, then we have - b = (1), + c = (1 2 ), - d = (1 3 ), &c.;