ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 29 



Possibly there may be found modifications of this expression which are at least 

 equally compact. There certainly will be unlimited variety if the number of terms be 

 not restricted as here, since for almost every one of the terms a substitution of two or 

 more similar terms is possible. The only terms, indeed, which cannot be replaced are 

 0000, -220048, ±4488. 



(32) Leaving now the subject of these eliminants of high order, — a subject which, 

 as we have seen, originated with Sylvester, — let us ascertain what is possible in the 

 direction of attaining the eliminant in the form of a determinant of an order lower than 

 the sixth. 



In one of the special cases already referred to it has been shown * that from the three 

 original equations in x 2 , y 2 , z 2 , yz, zx, xy we were able to deduce a set of three in 

 yz, zx, xy, a set in x 2 , y 2 , z 2 , and a set in x, y, z, and thus to obtain expressions for the 

 eliminant in the form of a determinant of the third order. That is to say, instead of 

 having in our equations all the possible facients of the second degree, viz., both those of 

 the type x 2 and those of the type yz, we succeeded in confining ourselves to equations 

 having only one type of facient. 



In the same case it was also shown that there could be deduced a set of four equations 

 in x 2 y, y 2 z, z 2 x, xyz ; a set in xy 2 , yz 2 , zx 2 , xyz, and a set in x 3 , y 3 , z 3 , xyz : and that in 

 this way expressions could be obtained for the eliminant in the form of a determinant 

 of the fourth order. Here, where the facients are of the third degree, there are 

 four types of them, x 3 , x 2 y, xy 2 , xyz ; and Sylvester, just as in the case of the 

 facients of the second degree, used the whole of them and thus saddled himself 

 with a determinant of the tenth order. The reduction to the seventh order made 

 in § 1 was due, it may be noted, to the elimination of one of the four types, the set 

 of facients implicitly retained being x 2 y, y 2 z, z 2 x, xy 2 , yz 2 , zx 2 , xyz. 



We shall now see whether the processes applied to this special case can be extended 

 to the general problem at present before us. 



(33) When the set of facients does not possess the cyclo-symmetry apparent in each 

 of the sets just spoken of, it is scarcely reasonable to expect that the resulting eliminant 

 will be simple or elegant in form. If, therefore, we seek to obtain the eliminant as a 

 determinant of the fifth order — a course which would necessitate the use of a set of 

 facients like y 2 , z 2 , yz, zx, xy, or x 2 , y 2 , z 2 , yz, zx, — we must be prepared for more or less 

 irregularity and complexity. It will be found, nevertheless, that this fifth-order form 

 is full of interest. 



Taking the set y 2 , z 2 , yz, zx, xy, we examine our collection of derived quaclrics having 

 co-efficients of the third degree, viz. : — 



[Table 



* Muir, T., " Further Note on a Problem of Sylvester's in Elimination," Proc. Roy. Soc. Edin., xx. pp. 371-382. 



