AUTOMORPHIC LINEAR TRANSFORMATION OF A QUADRIC. 



219 



Denote the discriminant by D, and form the determinant R adjugate to 



a h + v g—v 

 h-v b f+\ 

 g + fX y_\ c 



or 



then 

 and 



A . a n = 2(r th col. ofR)(s ih row o/D) when rd£s , 



A . ttrr = 2(r th col. o/R)(r th row o/D) - A . 



This is manifestly just as simple as Cayley's rule for the formation of an orthogonal 

 substitution, into which rule it is seen to pass when a = b = c = . . . = 1 and f=g = h = 

 . . . = 0. 



(8) The most elegant proof of the rule in all its generality is got by using the theory 

 of matrices as follows : — 



Taking the substitution in its implicit form, and solving for x , y , z , . . . we have 

 (x,y,z, . . . ) = ( a h + v g-n . . . ) \ a h- v g + n ...) (i, »?,£,... ). 



h-v b f+\ 



g+v- /-A c 



h + v b f-\ 

 g-H f+\ c 



Now the first matrix here is equal to 







( 



hi 

 A 



^21 



A 



L 3 i 

 A 





^12 



A 



L 22 



A 



L32 



A 





^13 



A 



^23 



A 



^33 

 A 



and the second can be expressed as the difference of two, viz., as 



(2a 2h 2g ...)•( 



2h 2b 2/ 

 2g 2/ 2c 



a 



h + v 



g-n • • • ) 



h — v 



b 



/+x ... 



g+v- 



/-A 



c ... 



On performing the required multiplication we thus have 



( 2 n 21 -^31 



a h g 

 2 12 -^22 ^3 



a h 

 o n ^23 L 33 



a h 



2 L ii 



L21 



L 31 . . 



h 



2 L 12 



b 



L 22 



/ • • • 

 L 32 . . . 



"h 



2 L 13 



b 



^23 



/ • • • 



L 33 . . . 



u h 



b 



/ • • • 



O^tl 



L 21 



L 31 . . . 



g 



0-^12 



/ 



^22 



c . . . 

 L 32 . . . 



g 



Q L 13 



/ 



L 23 



c . . . 

 L 33 . . . 



9 



/ 



c . . . 



? ( 1 



and this difference being expressed as one matrix we obtain the matrix of the substitu- 

 tion. A glance suffices to show that it is identical with the matrix 



