672 



DR THOMAS MUIR ON THE 



Now as M 2 -z is not an aggregate of multiples of the given quadrics, and as 

 2 1 u 1 b 2 g s | y + 4 j 111/293 1 -2 is such an aggregate, it follows that the new cubic is in this 

 respect similar to the new quadric. The simplified eliminant of the 10th order thus 

 obtained is therefore 



a. 



\ 



2\ 



2K 



2K 



2/x 



2/3 



2/s 



2ft 



2ft, 



2ft 



2h, 



h 

 2h 9 



2K 



2/i 



2/ 2 



2/s 



2ft 



2/x 





2ft 



«! 



2^ 



2ft 



2/2 



. 



2ft 



«2 



2\ 



2ft 



2/3 



. 



2ft 



a 3 



2^3 





(*•-! t/oGo 



2 K& 2 ftl 217^3 1 2la i/ 2 C 3 l 4 l A Aftd ^WU'i\ ^ft/kftl + 



1/iftAj 



(10) This is not all, however, for the simplified form thus reached immediately 

 suggests a further simplification by reason of the absence of terms in x s , y s , z 3 from the 

 cubic which has been used to replace the Jacobian. 



Multiplying each element of the 9th row by c 2 and subtracting c 3 times the corre- 

 sponding element of the 6th row, and operating similarly in five other instances, we can 

 transform the determinant into one of the 7th order, viz., 





2 | h x a x | 



. 



1 c 2 »i 1 



1 Vi 1 





21^x1 



21/AI 





KM 



2I/AI 



. 



2IAAI 



IflAl 





2|ftAl 



1 





1 & 2 c 1 1 



2|#2 C ll 



. 



2 l/ft 1 



1 *2 C 1 1 



2IV1I 



n h 



2 1 K a -2 1 





|c 3 a 2 | 



1 h a 2 1 





2|ft« 2 l 



2 l/ 3 «2 1 



U' 2 U 2°2 



1 «3 & 2 1 



21/AI 





2|*A;| 



1 «A 1 





2|ft« 2 l 







| 6 3 e 2 | 



2|ftc 2 | 



. 



2J/AI 



i «3 C 2 1 



2 | Vi 1 





2 1 a x \g 3 1 



2 1 \\c z | 



2 I aj/gCg | 



4 I^Aftl 



4 1 V** 1 



4 1 <hf&3 1 



KVsl+Sl/ift^ 



(11) Before proceeding to a comparison of the various derived quadrics and cubics 

 which may be found useful in obtaining an expression for the eliminant, it is desirable 

 to note that, as the given quadrics are obtainable one from another by performing 

 simultaneously the cyclical changes, 



kj 



any single derived equation not symmetrical with respect to this system of cyclical 

 changes will readily give rise to two others. Thus, when the quadric M 2 was obtained 

 in § 8, the two still awanting necessitated no additional work. On the other hand, the 

 cubic M 3 of § 9 is unique, being simply twice reproduced by the cyclical change. 



