( 23 ) 



III. — On the Eliminant of a Set of General Ternary Quadrics. — (Part II.) 



By Thomas Muir, LL.D. 



(Read December 4, 1899.) 



(26) Of the various determinant forms thus far obtained the most promising is that 

 of § 8 or that of §14; and to these it is desirable now to return in order to obtain an 

 expression for the eliminant in the ordinary non-determinant notation. In doing so it 

 will also be well to make a slight change in the coefficients of the three quadrics, viz., 

 to write f g, h for 2f 2g, 2h, as in this way the diversity in the cofactors of the deter- 

 minants occurring in the last three rows of either form of the eliminant disappears. 



Using first the result of § 8, we have therefore as the eliminant of 



the determinant 



where 



a x x 2 + b t y 2 + c x z 2 + f$z + ffjZX + \xy = v 

 a 2 £ 2 + hgj 1 + c# 2 4- f 2 yz 4- g$x + h 2 xy = > 

 a s x 2 + b.^ 4- c 3 2 2 4- f0z + y. d zx + \xy = ) 



a x 



\ 





h 



A 





9x 



*1 



a 2 



h 





o-i 



A 





9>2 



K 



a. A 



h 





C 3 



A 





9 S 



i h 





[5] 



- 



[31 



[8] + [81 



[6] 



[0] 



-[1] 



. 





[6] 



[0] 





;9]+[9'i 



[41 



[4] 



"[2] 





• 



[5] 





[0] 



[7] + [7'] 







[0] 



— 



1 <h h 2 C 3 1 













[1], [2], 



[3] 



= 



! O^jAg | 





1 \KA 1 



, 1 c iA9z ' 



> 





[41, [51, 



[6] 



= 



| Ojft^g | 





i x c. 2 h s \ 



, | ^^2/3 



, 





[7], [8], 



[9] 



= 



1 oA/s 1 





1 W 8 \ 



, I.C^g | 



j 





[7'], [8'], [9'] 



= 



1 hffA 1 





1 c AA 1 



; 1 a iJ29z 1 





Now, as the minors formed from the first three rows of this determinant of the 6th 



order are the set of twenty to which belong the thirteen determinants appearing in the 



other three rows, it follows that if we take the expansion in terms of minors formed 



from the first three rows and their complementaries, we shall obtain for the eliminant 



an expression consisting of terms each of which is the product of four of the twenty 



determinants [0], [1], [2], .... Doing this, and bearing in mind the existence of 



triads due to the cyclo-symmetry, we have as a preliminary form of the development 

 VOL. XL. PART I. (NO. 3). D 



