ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 673 



In the case of expressions of the latter kind, a great saving of trouble is effected by 



o 



using the symbol of cyclical summation, 2- Thus M 3 may be written in the form 



O 



22j\a 1 b$ z \.xhj + 4X|W 3 I^ 2 + {kA^I + Sl/i^l}-^ 21 . 

 and the Jacobian in the form 



- 2 I <h9A\-& + 2 {| OiVs I - 1 «i/A \}- x2 y 







+ 2{| a AA l+l VA \Y x v 2 + (I a A c 3 1+ 2 l/i^A IHf- 



(12) Further, as the twenty determinants of the 3rd order formed by taking every 

 possible set of three columns out of the six, 



«i h c i /i ffi h i 



a 2 \ C 2 f% 9-2 K 



a 3 "3 C 3 J 3 d h \ ' 



will be often in evidence, it will be convenient to have temporarily a short notation for 

 them ; and as they, like all other expressions connected with the investigation, appear 

 in cycles of three (when not cyclically symmetrical), it is desirable that this should not 

 be lost sight of in choosing the symbolism. Now the twenty, by virtue of the cyclical 

 change, break themselves up into the following groups — 



I <*A C 3 



I «A/ 3 1 



I «A#3 I 



I a AK I 



I aJzh I 



I «l?2^3 I 



1/rfA ! 



& 1 C 2#3 



\ c A 

 We 

 \9A 

 l 19jz 

 & A/ 3 



C \ a 2J3 



c -P"l9?, 



c Afs 



C A#3 

 C l/ 2 #3 



and may therefore be fitly denoted as follows 



1 «A C 3 1 







by 



[0] 



/tf A 1 







by 



[O'l 



1 a i9 2 h s 1 ' 



1 &A/s 1 » 



1 C l/^3 1 



by 



[1]> [21, [3] 



1 a Aj 3 1 . 



1 & l C 2^3 1 . 



1 C i a 2J3 1 



by 



[4], [51, [6] 



1 VA 1 . 



1 &1&/3 1 . 



1 C A#3 1 



by 



[4'1, [5'], [6'1 



1 Oi^/s 1 1 



1 Ws 1 > 



1 c i a 2^3 1 



by 



[7], [81, [9] 



1 &10 A 1 • 



1 c A/ 3 1 » 



1 Ox/aft 1 



by 



[71 [8'1 [9'] 



! «i&A 1 > 



1 V2/3 1 1 



1 C l«2^3 1 



by 



[10], [11], [12] 



The reason for repeating the numbers [4], [5], [6], [7], [8], [9], with an added dash, 



