ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 31 



but that because of the relations 



— G 1 + F 3 = G l; 

 C 2 + G 2 = F x , 

 F 2 - G 3 = C 3 = (6A 3 - 7A 2 ) -=- , 



we are reduced to the use of the pairs E 2 , F 2 and E 2 , Gr 3 , which by addition give the 



quadrics 



. + 7' y 2 + (6-6')z 2 + (0 + 0>z + (9 + 9')zx + 4 xy , 



■ + (7 + 7> 2 + (~6> 2 + (0 + 0')yz + 9 zx + (4-4')oy. 



Further, since the difference of these two is C 3 , i.e., (6A 3 — 7A 2 )-^0, it is immaterial 

 which we use along with A 2 , A 3 , F 1? G 2 for the purpose of dialytically eliminating 

 y 2 , z 2 , yz, zx, xy. Taking the former of the two, therefore, we have the set 



7V + (6-6> 2 + (0 + 0')yz + (9 + 9>k + 4xy , 



-2y 2 + 8 z 2 + (5-5')yz + zx + Ixy , 



07/ 2 + (-3)z 2 + (8 + 8')yz + 6 zx + Oxy , 



. + z 2 + 7 yz + 4: zx + lOxy , 



Qy 2 + . + 6 yz + 12 zx + 9xy , 



E 2 + F 2 

 A, 



from which there results the eliminant 



7 



6-6' 



+ 0' 



9 + 9' 



4 



2 



8 



5-5' 







7 



5 



-3 



8 + 8' 



6 













7 



4 



10 









6 



12 



9 



The extraneous factor contained in it is readily ascertained to be 0, by trying to express 

 the determinant as an aggregate of products of complementary minors, one minor of 

 each product being formed from the elements of the last two rows, e.g. 



A 10 

 12 9 



-01, 



7 10 

 6 9 



= -04', 



Of course the cyclical substitution gives two other similar forms of the result. 



(34) When the h's vanish, the ten determinants 



0', 1, 2, 4', 5, 6', 7', 8', 9, 10 

 vanish also, and this eliminant of the fifth order degenerates into 



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



6 







9' 



4 



8 



-5' 







7 



3 



8 



6 











7 



4 





E 



