AUTOMORPHIC LINEAR TRANSFORMATION OF A QUADRIC. 



229 



if (X , Y , Z) = ( a h g ) (x, y, z), 

 h b f 

 c 



9 f 



— in other words, we have a condensed expression for the product of a ternary quadric 

 and its discriminant. For, on performing the substitution, the left-hand member 



x y 



z 



a h 



9 



h b 



/ 



9 f 



c 



A H G 

 H B F 

 G F C 



a 



h 



9 



X 



h 



b 



f 



y 



9 



f 



c 



z 



and therefore 



x y 



n 



y 



where Q is the product of the preceding three square arrays in reverse order. But by 

 actual multiplication 



n = ( A . . X « h g ) 



. A „ . I i h b f 



• • A , g f c 



«A hA gA 

 hA bA /A 



gA /A cA | , 



-which proves the theorem. 

 Again, we have the identity 





P 





(7 





X 





T 



V 





V 







a h 





a g 



a r 



h 9 | 



1 h r 





9 r 







h b 





h f\ 



h q 





b f\ 



1 b 9. 





f 1 







a h 





a g 









9 f 





9 c 







j a h 









r q 























a g h r 



9 

 b 



f 

 2 

 VOL. XXXIX. PART I. (NO. 7). 



+ 



a g h r \ or 



+ 



a g h r 



2 M 



