674 



DR THOMAS MUIR ON THE 



will appear later. Meanwhile it should be noted that the passing from one determinant 

 to another by means of six of the cycles above given, say from 



H 



9i 



*1 



a i 



<?2 



K 



a 3 



9z 



K 



to 



h K A 

 h h /» 

 \ \ A 



is, on account of the identity of | hji^ | and I&A/3I. much more readily effected by 

 using only two cycles, viz., 



the subscripts 1, 2, 3 being considered invariable. The notation employed for the 

 members of the six derived cycles of determinants enables us to remember them as one, 

 viz., 



(13) With these preliminaries, let us now return to the consideration of the sub- 

 sidiary quadrics necessary for obtaining the eliminant of the 6th order. 

 The process employed in § 8 for obtaining M 2 is given by the equation 



from which we deduce 



or, what is the same thing, 



\ u A c z\-y + 2 K/ 2 c 3 h = x - M 2> 



\u x \y+2f# c 3 1 = a?-M 2 ; 



«i« 2 + \tf- + c/- + 2f x yz + 2g x zx + 2\xy \y + 2f x z c x 

 a#?+b0 2 +c 2 z* + 2f$z + 2g#x+2h 2 xy b 2 y+2f 2 z c 2 

 a.^ + b^ + c 3 z 2 + 2/0Z + 2g 3 zx + 2h 3 xy b$ + 2/ 3 z c 3 



zM.-,. 



Simplifying the determinant by diminishing each element of the first column by y 

 times the corresponding element of the second column, and by z 2 times the corresponding 

 element of the third column, we have 



a x x % + 2g 1 zx -f- 2h x xy b x y + 2f x z c x 

 a^ 2 + 2g$x + 2h 2 xy \y + 2f 2 z c 2 

 a 3 x 2 + 2g z zx + 2h s xy b$ + 2/ 3 z c 3 



= xM a 



