218 



DR THOMAS MUIR ON THE 



x y 



a h g x 



h I Ay 



9 f e ! z 

 we know that the requisite automorphic substitution in its implicit form is 



ax+(h + v)y + (g-fx)z = a £+(h- v )ti+(g+fA)£< 



(h- v )x+ by + (f+\)z = (h+v)£+ 



cz 



(g + fl ) X + (f-\)y + <^Z (^_ M )^ + (/ + X)^+ cf 



}■ 



A.x = 



Consequently, if we denote the determinant of the coefficients of x, y, z by A, we have 

 on solving for x 



a£+{h- v )q + (g + f,)g h+v g-p 



(h+ v )i+ b v + (f-\)£ b f+\ 



(ff-v)£+(f+Vv+ c£ f-\ c 



a£+2hr) + 2gt; h + v g-fx 

 (h+ v )£+2b n +2f£ b f+X 

 (g- /i )£+2fy+2c£ f-X c 



(2a-a)£+2hri + 2gg h + v g — y. 

 (2h-h=v)£+2b n + 2ft b f+X 



c 



(2g-g + ri£+2fy + 2c£ f-X 

 £- A£ + 2 



a h + v g — ij. 

 h b f+X 

 9 /-X c 



a h + v g — i* 

 h b f+X 



f /"A c 



+ 2 



h + v 



b 

 /-A 



9-H 



/+X 



c 



Now all the determinants here have their last two columns identical with the corre- 

 sponding columns of A. It follows, therefore, that if the determinant adjugate to A be 



^12 1^3 



-^21 -^22 "^23 

 ■^31 "^32 "^33 



the result which we have just reached may be written : — 



Similarly 



and 



A.y 



A.z = 



a h g J* \ h b f 



2 ^12^22^32 )£+ (o 12 22^32 _ ^ )~ 



a h g J \ h b f ) 



2 ^11^21^31 



+ 



\ 9 f C ) 



/n L 13 L 2:) L 33 \ c / 9 L 13 L 23 L 33 \ , /oL^LggL. 



\ a h g r V h b f r^\ g f c 



*. 



We have thus the following rule for the formation of the coefficients a,,, of the linear 

 substitution which transforms the quadric 



az 2 + by 2 + cz 2 + + 2fyz + 2gzx + 2hxy + .... 



into itself: — 



