XXXIII. A memoir on the conditions for the existence of given systems of equalities among the roots of an equation

It is well known that there is a symmetric function of the roots of an equation, viz. the product of the squares of the differences of the roots, which vanishes when any two roots are put equal to each other, and that consequently such function expressed in terms of the coefficients and equated to zero, gives the condition for the existence of a pair of equal roots. And it was remarked long ago by Professor Sylvester, in some of his earlier papers in the ‘Philosophical Magazine,’ that the like method could be applied to finding the conditions for the existence of other systems of equalities among the roots, viz. that it was possible to form symmetric functions, each of them a sum of terms containing the product of a certain number of the differences of the roots, and such that the entire function might vanish for the particular system of equalities in question; and that such functions expressed in terms of the coefficients and equated to zero would give the required conditions. The object of the present memoir is to extend this theory and render it exhaustive, by showing how to form a series of types of all the different functions which vanish for one or more systems of equalities among the roots; and in particular to obtain by the method distinctive conditions for all the different systems of equalities between the roots of a quartic or a quintic equation, viz. for each system conditions which are satisfied for the particular system, and are not satisfied for any other systems, except, of course, the more special systems included in the particular system. The question of finding the conditions for any particular system of equalities is essentially an indeterminate one, for given any set of functions which vanish, a function syzygetically connected with these will also vanish; the discussion of the nature of the syzygetic relations between the different functions which vanish for any particular system of equalities, and of the order of the system composed of the several conditions for the particular system of equalities, does not enter into the plan of the present memoir. I have referred here to the indeterminateness of the question for the sake of the remark that I have availed myself thereof, to express by means of invariants or covariants the different systems of conditions obtained in the sequel of the memoir; the expressions of the different invariants and covariants referred to are given in my ‘Second Memoir upon Quantics,’ Philosophical Transactions, vol. cxlvi. (1856). 1. Suppose, to fix the ideas, that the equation is one of the fifth order, and call the roots α,β,γ,δ,ε . Write 12 = ∑ ϕ ( α — β ) l , 12.13 = ∑ ϕ ( α — β ) l ( α — γ ) m , 12.34 = ∑ ϕ ( α — β ) l γ — δ ) n &c., where ϕ is an arbitrary function and l, m , &c. are positive integers. It is hardly necessary to remark that similar types, such as 12, 13, 45, &c., or as 12.13 and 23. 25, &c., denote identically the same sums. Two types, such as 12.13 and 14. 15. 23. 24. 25. 34. 35. 45, may be said to be complementary to each other. A particular product ( α — β )( γ — δ ) does or does not enter as a term (or factor of a term) in one of the above-mentioned sums, according as the type 12. 34 of the product, or some similar type, does or does not form part of the type of the sum; for instance, the product is a term ( α — β )( γ — δ ) (or factor of a term) of each of the sums 12. 34, 13. 45. 24, &c., but not of the sums 12. 13. 14. 15, &c.