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XXVL — On the Eliminant of a Set of General Ternary Quadrics. 

 By Thomas Mum, LL.D. 



(Read June 19, 1899.) 



(1) The process of dialytic elimination was first applied to a set of general ternary 

 quadrics by Sylvester in 1841,* his method being to form from the three quadrics 

 u u u 2 , u-i ten cubics, viz. : — 



xu v yu v zu x ; xu 2 , yu 2 , zu 2 ; xu 3 , yu 3 , zu z ; J(u v ic 2 ,u 3 ), 



and then eliminate dialytically the ten quantities 



re 3 , y % , z 3 ; x 2 y, y 2 z, z 2 x ; xy 2 , yz 2 , zx 2 ; xyz. 



The actual work of obtaining the eliminant as a determinant of the 10th order he 

 did not perform ; and, indeed, with the notation which he used the labour would 

 have been very irksome. 



(2) In the same paper he also dealt with three simple special cases of the problem, 

 but proceeded in a totally different way, the eliminant in each case being obtained 

 as a determinant of the 6th order. Here his aim was to obtain from the three 

 quadrics other three, and then eliminate dialytically the six quantities 



x 2 , y 2 , z 2 ; yz, zx, xy. 



The mode of obtaining the three subsidiary quadrics varied in each case, and the 

 possibility of finding such a triad in every case was not discussed. 



(3) Eighteen years afterwards, when the first edition of Salmon's Modern Higher 

 Algebra came to be published, Sylvester's general method was not given, but another 

 was announced in the following wordst : — 



" We can now express as a determinant the eliminant of three equations, each 

 of the second degree. For their Jacobian is of the third degree, and therefore its 

 differentials are of the second. We have thus three new equations of the second 

 degree, which will be also satisfied by any system of values common to the given 

 equations. From the six equations, then, u, v, w, dJ/dx, dJ/dy, dJ/dz, we can eliminate 

 the six quantities x 2 , y 2 , z 2 , yz, zx, xy, and so form the determinant required." 



* Sylvester, J. J., "Examples of the Dialytic Method of Elimination as applied to Ternary Systems of 

 Equations," Cambridge Math. Journ., ii. pp. 232-236. 



t Salmon, G., Lessons introductory to the Modern Higher Algebra (1859), p. 38, § 55. [The paragraph appears 

 unaltered in all the subsequent editions.] 



VOL. XXXIX. PART III. (NO. 26). 5 L 



