676 DR THOMAS MUIR ON THE 



On looking back it will be seen that tins is what would be obtained on changing the 

 rule of § 13 so as to read ' y ' for e as ' and ' x' for ' y.' Indeed, as there is no reason 

 for treating x differently from y, an alternative form of the eliminant might thus have 

 been anticipated. Both forms are secured by putting ' x or y ' for ' x, and 'y or x ' 

 for f y' in the said rule. 



(15) It might be expected that by subtracting the second derived quadric from the 

 first, we should obtain a better form than either. It will be found, however, that 

 though the quadric so deduced is simpler, it is absolutely useless for the purpose in 

 view. The result in fact is 



- [9Ja? + [5]f + 2[8'}yz + 2[Q'}zx, 

 or 



- \c 1 a, 2 h z \.x l + \\c % \\.7f + 2\cji 2 f z \-yz + 2\c 1 h 2 g 3 \-zx, 



which is easily seen to be an aggregate of multiples of the original quadrics, viz., the 

 aggregate representable by 



It thus appears that only one independent triad of derived quadrics is in this way 

 obtainable, but that two of the interdependent triads seem equally advantageous for 

 the purpose in view. 



(1G) The question now arises as to the connection of the differential-quotients of the 

 Jacobian with either of the triads here proposed to displace them. 

 From § 6 we have 



5 - - 3[1K + {[7] + [7']}y + {[6]-[6'}.*« 



+ {[0] + 2[0']}l/z + 2{[9] + [9']}.*b + 2{[4]-[4']}ay, 



and the second member of the first of the said two triads is 



- 4[l]x 2 + 2[6]-* 2 + [0}yz + {2[9]+4[9']}-ct + 2[4]-*y; 



so that the difference between them is 



PJ* + {[7] + [7']}y - {[6] + [6']K 



+ 2[0'}yz - 2[9'}zx - 2[4']-a^. 



Now this difference, as might be expected, is found to be an aggregate of multiples of 

 the original quadrics. In fact, taking the aggregates 



\Ui9A\ i-e-> [I]-* 2 + [T]-y 2 - [G> 2 + 2[0']-yz, 



Im^/,1 i.e., - [7}y 2 + [G]-z 2 + 2[9'}zx + 2[4']-ay, 



