Some results in the theory of elimination

In two recent papers, I have given expressions, in the form of determinants, for the eliminants (i) of two quantics in one variable and (ii) of three quantics in two variables. My present purpose is to show how from these expressions may be derived others which give the eliminants as a sum of terms. In the case of two quanticsϕ(x), ψ(x) , each of then th degree in one variable, I obtained the result Δ = ARE, where E is the pure eliminant, Δ is a determinant of ordern , whose elements are denoted by (as , rt ), where (a, r ) is written for [ϕ(a)ψ(r) ―ϕ(r)ψ(a) ]/(a ―r ),a 1 ,a 2 , . . .an ;r 1 r 2 , . . .rn being two sets of arbitrary quantities, and A and R denote respectively the determinants whoses th rows are │1,as ,as 2 ,as 3 , . . .as n -1│ and │1,rs ,rs 2 ,rs 3 , . . .rs n -1│.