1890.] Functions of the Roots of Systems of Equations. 177 



thereby obtained an identity which is fundamental in the subject. 

 This identity involves those symmetric functions which are here 

 termed fundamental, and marks the starting point of the present 

 investigation. 



The memoir is divided into sixteen sections. In § 2 a preliminary 

 algebraic theory is given, and then in § 3 is commenced the theory of 

 the differential operations. A prominent feature presents itself in 

 the very interesting correspondence between the algebras of quantity 

 and differential operation. 



In § 4 is discussed the theory of three identities, formed similarly 

 to the fundamental identity alluded to above, and such that the 

 quantities involved are related in a particular manner. The theory 

 of differential operation proceeds collaterally with that of quantity. 

 The succeeding four sections, § 5-§ 8, are devoted to the results 

 which flow in a direct manner from this discussion. In particular, 

 three distinct laws of symmetry are established, large generalisations 

 of those established by the author in the ' American Journal of 

 Mathematics' (loc.cit.). Of these the first two are of importance, 

 and are examined in detail. A leading idea in these theorems, as in 

 the whole investigation, is the "separation " of a partition; the sepa- 

 ration bears the same relation to the partition as the partition to the 

 number or collection of numbers. The first law of symmetry appears 

 to be of cardinal rank in symmetrical algebra. It involves, at sight, 

 a law of expressibility in the theory of separations which is of a 

 general character. It demonstrates at once the possibility of forming 

 a pair of symmetrical tables of symmetric functions in connexion 

 with every partition of every collection of m numbers (regard being 

 paid as well to order as to magnitude). The necessary tables for the 

 bipartite theory (i.e., of two systems) as far as the weight, four 

 inclusive, are exhibited in § 14. An extension of the Vandermonde- 

 Waring law for the expression of the sums of the powers of the roots 

 of an equation by means of the coefficients is generalised, in a single 

 formula, from two points of view in § 6. In § 9 and § 10 the decom- 

 position is effected of the operations previously encountered in § 3. 

 The linear weight operations are found to break up into as many 

 linear partition operations as the weight possesses partitions. An 

 important theorem is reached when it is established that the annihila- 

 tion of a symmetric function by a linear weight operation necessitates 

 annihilation by each partition operation of the same weight. The 

 weight operations of higher orders, partially examined in § 3, which 

 may be termed " obliterating," from their characteristic property, 

 break up similarly into partition operations which possess an oblite- 

 rating property in regard to products of symmetric functions. In 

 this manner all the differential operations of § 3 are adapted for use 

 in the theory of separations, as distinct from the theory ordinarily 



p 2 



