ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 37 



established by obtaining for them equivalent expressions which bear the symmetry 

 on their faces. 



Of such expressions three at least are useful in the process of separating E from its 

 co-factor. These are, in the case of y 3 , 



K±lTT + ±4'8), 1Q111 ^- 789 , 12 3 + 4W . 



The first arises from the theorem repeatedly used in the note to § 36, — a theorem which 

 ensures the identity of 



2 12 + 67, 3 TO + 4'8, 1 TT + 5'9 , 

 and therefore gives 



2 12 + 67 = K2lll + 24'8). 



The second and third have essentially the same source, for from it we obtain 



02l2 + 067 - (10 TT-57)T2 + (5 T2-89)7 , 



= TO TT T2 - 7 8 9 ; 



and 



0'212 + 0'67 = 2(13 + 6'9') + 6'(4'5'-29') , 



= 12 3 + 4'5'6' . 



The similar expressions for the extraneous factor 67' — 310, which occurs in the case 



of y 4 , are 



t^-iTT *aq'\ 10TTT2-4 5 6 12 3-7'8'9' 

 ^2,lll-Z4»;, q , - t ; 



and it may be noted in passing that there is a third triad of such equivalent expressions, 

 viz. 



K±4'8 + ±48% 456 ~ 789 , 4W + 7W 



which are got from the two previous triads by subtraction. 



(40) To each of these triads, however, a fourth member may be added, as there exist 

 symmetrical expressions of a quite different kind which can be proved equal to 

 2 12 + 6'7, 67'- 3 10, 67 + 59' respectively. 



The origin of this is an identity in compound determinants, viz. 



7i£> I 7 2 & I I 7s£i 



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Ti&fs I I 7i&Aj I I I I «i^2?3 I I ai&73 



I I 7i&£$ I I 7i&«3 



I I fefs I I A^s 



