670 DR THOMAS MUIR ON THE 



(7) Now, ns has been already explained, it is the introduction of the differential 

 coefficients of the Jacobian which seems to give unnecessary complexity to Salmon's 

 form, and what we have therefore to seek for is three quadrics of a simpler character 

 to take their place. The further suggestion then arises that in a similar manner 

 Sylvester's form may be more complicated than is necessary, and that possibly there 

 is a cubic more suitable than the Jacobian to complete our set of ten. 



Success has attended both quests, there having been obtained an eliminant of the 

 6th order simpler than Salmon's, an eliminant of the 10th order simpler than Sylves- 

 ter's, a new eliminant of the 7th order, and a full elucidation of the relationships 

 connecting them all. 



(8) The three quadrics for the simplification of Salmon's form have been obtained 

 as follows : — 



Multiplying the first given quadric 



a^ 2 + b^f + Cjf + y x yz + 2g x zx + 2\xy 

 by | h 2 c s | , the second by - 1 Z^Cg | , and the third by | 6 x c 2 1 , and adding, we have 



|aA c 3 l^ 2 + 2 \fA c z\-y s + 2 \gA c s\- zx + %\Wh\- x y- 



Acting similarly with the multipliers \f 2 c s | , - \f 1 c 3 | , j/^ | , we have 



\ a if&\-x 2 + \\Uz\-f + i\gJzh\- zx + ^\KUz\- x y- 



Now in each of these two deduced quadrics there is only one term free of x, and it so 

 happens that the coefficient of this term in the one case is exactly - 2 times what it is 

 in the other. We are thus easily able to eliminate the two terms. In fact, multiplying 

 the former of the deduced quadrics by y and the latter by 2z, and adding, we have, 

 after division by x, the new quadric 



Otfi + 2\J h b 2 c 3 \.f + 4|fc/ A |* 



+ ( 2 l^A c 3l + 4 IV 2 c 3l}-2/* + ^\ a iUz\- zx + \ a A c s \- x y- 



Proceeding with the given quadrics u 2 , u 3 , u x as we have just dealt with u x , u 2 , u z , we 

 obtain in like manner 



+ \aAc 6 \-y* + {2|7w* 3 l + 4 l/i.9 2 « 3 l}-^ + 2\b 1 g 2 a 3 [xy; 

 and similarly from u s , u r , u 2 



2\ 9l aJ),\-x* + 41/^Al-y 2 + 0-z 2 



+ 2 \ c AK\-y z + \aA c s\- zx + { 2 l/i« 2 J 3l + ±\oAh\}- x y- 



These new quadrics, from their mode of formation, are seen to be not mere aggre- 



