360 
Proceedings of Poyal Society of Edinburgh. [sess. 
On the Eliminant of f(x) = 0 , 
By Thomas Muir, 
LL.D. 
(Read December 7, 1896.) 
1. In a paper “ On the Existence of a Root of a Rational In- 
tegral Equation,” published in the Proc. Lond. Math. Soc., xxv. 
pp. 173-184, the author, Professor E. B. Elliott, says (p. 184) that 
it is unfortunate, for the simplicity of the argument of his paper, 
that a proof of a certain property of this eliminant, viz., that 
when two linear factors have been withdrawn from it, there is 
left a perfect square — “ is one which direct algebraical methods 
have not yet supplied.” 
In the course of the following year the want referred to received 
attention, a demonstration being given by Mr W. W. Taylor, in a 
paper entitled “ Evolution of a certain Dialytic Determinant,” 
which was read before the same Society (see Proc. Lond. Math. 
Soc. } xxvii. pp. 60-66). 
I purpose here giving another demonstration, which I think has 
the merit of bringing out more clearly the character of the consti- 
tution of the eliminant, and in which is followed, at the same time, 
that direct and expeditious course most likely to be taken by a 
student familiar with the theory of determinants. 
2. Taking the case used in the second of the above-mentioned 
papers, viz., where 
f(x) — ax 6 + bx 5 + cx^ + ( Jx 3 + ex 2 +fx + g , 
and where, therefore, the eliminant is 
