32 



DR THOMAS MUIR ON THE 



which by translation of the last row or column to the first place is seen to be axi- 

 symmetric. We thus have the following theorem # : — 

 The eliminant of the equations 



a x x 2 + \y 2 + c x z 2 + f x yz + g x zx = \ 

 a 2 x 2 + \y 2 + c 2 z 2 + f 2 yz + g#x = OV 

 a 3 x 2 + b 3 y 2 + c 3 z 2 + f 3 yz + g 3 zx = ) , 



or the expression which equated to zero gives the condition that the loci of 



a x x 2 + b x y 2 + g x x + f x y + c x = 0\ 

 a 2 x 2 + b 2 y 2 + g 2 x + f 2 y + c 2 = I 

 a 3 x 2 + b 3 y 2 + g 3 x + f 3 y + c 3 = 0) 



have a point in common, is 





1 a A9s 1 



1 «A/ 3 1 



1 a A C 3 



«A#3 1 



1 «l/ 2 03 1 



\a>A c z 1 



1 a iJ2 C 3 



^A/s 1 



1 «A C 3 1 



1 hfrfz 1 



1 Ws 



a A c s I 



1 a iJ2 C 3 1 



1 h°iffs 1 



1 /l C 2#3 



(35) Let us now pass from the eliminants of the fifth order to those of the third. 

 Taking the original set of three quadrics, and a derived quadric which is known not 

 to be an aggregate of multiples of these, say F 2 of § 33, we have 



a x x 2 + \y 2 + c x z 2 + f x y% + g x zx 

 a. 2 x 2 + & 2 2/ 2 + " "< 



+ 



r c 2 z + J $% + g%zx + 

 a 3 x 2 + b 3 y 2 + c 3 z 2 + f 3 yz + g 3 zx 

 5y2_ S z 2 + (8 + 8') yz + 



h x xy 

 h 2 xy 



+ Vw 



6zx + Oxy 



and therefore by elimination of x 2 , y 2 , z 1 



or 



a 2 



c i Av z + 9i zx + K x v 



C 2 fiV Z + 92 ZX + K x v 



c 3 f& z + g$ zx + K x v 



•3 (8 + 8')yz+ 6zx + Oxy 



-3 



A 

 A 

 A 



! + * 



V z + 



9x 

 9 2 

 9z 



-3 



zx + 



b a 



-3 



K 



xy = 0. 



Expanding each determinant here in terms of the elements of the last row and their 

 complementary minors, we change the equation into 



{0(8+8')+37-56}^ + {06 + 34-512}^ + {00 + 310-59}^ = 0, 



* The result obtained by Lord M'Laren in his paper on " Symmetrical Solution of the Ellipse-Glissette 

 Elimination Problem," in the Proc. Roy. Soc. Edin., xxii. pp. 379-387, is the particular case of this where f v f 3 , g 2 , g 3 

 are made to vanish and a 1} a 2 , a 3 are put equal to b 2 , b lt b 3 respectively. 



