[ ^'^ ] 



III. 



THE SYMBOLICAL EXPEESSION OF ELIMINANTS. 

 By EEV. W. E. WESTEOPP EOBEETS, M.A. 



Read April 11. Ordered for PublicatioQ ArRii 13. Published October 18, IDIO. 



The object of this paper is to show how the eliminant of any two binary 

 quantics may be expressed in a syrdbolical form by the aid of certain 

 operators. 



Let u and v be the two c[uantics whose eliminant we desire to express 

 symbolically, and let us suppose that it is of the wi"^ degree, and v of the %"' 

 in X and y; and further, let the roots of the equation u = 0, be »,/v/i, 

 x^liii, . . . x„,/y,n ; and those of v = 0, be ?,/.;„ ?„/,,o, . . . g„/,,„ 

 we may then write 



(1) u {x, y) ^ A^"> + A,x"'-'y + . . . + A,„y„, 



^ (xy, - yx,) (xy.. - y.v.) . . . (p:y„, - yx,,,), 



(2) V {x,y) ^ B,x« + JB,x"-hj^ + ...+ B„y" 



^ (^jji - ySi) i^m - 2/?«) . . . {xrjn - yl.), 

 where u and v are written without binomial coefficients. 



It is well known that a binary quantic or a covariant quantic can be 

 derived from a certain term called the source of the quantic as well as from 

 the leading term of the quantic by certain operative processes, which we 

 now proceed to discuss. 

 If we write 



d d d d d d 



(3) ; 



d d d ^ d ^ d ^ d 



dyi dUi dy„, d,u f?i)2 dt)„, 



thus giving to the well-known operative symbols a wider meaning than 

 that usually attaching to them, and consequently a wider application, the 

 reader will readily perceive the truth of the following equations : 



dAr = (m + 1 - /■) A,..u dB, = {)i + 1 - s) Bs.u dA, = 0, dB, = 0, 



(4) 



[AA,. = {)•+ l)Ar,u AB, = (.s f 1) ;?,„, AA,„ = 0, AB„ = 0. 



R.I.A. PROC, VOL. XXVItl., SKCT. A, [lOj 



