AUTOMORPHIC LINEAR TRANSFORMATION OF A QUADRIC. 



and by reason of the identities 



215 



A H G 



a 



h 



9 



A 



H 



G 



h 



b 



/ 



A 



H 



G 



9 f c 



= D, 



= 0, 

 = 0, 



the first of these parts 



a , h 



= D. 



9 



cof (a), coi(h — v), coi(g+/u) 

 a , h+v , g—fx 



+ D. 



— V 



cof (a), cof (A — v) , cof (g + /ui) "^ ''cof^), cof(h — v), coi(g+/u.)' 



Consequently by removing the expression which is common to both sides we have only 

 to prove that 



D- {— j/ cof (h — v)+/jlcoI (g + /u)} + 



Now, as will be seen from a later paragraph, 



cof(a) = A + X 2 ; 

 cof(h-u) = R + \fx- 



A 



H 



G 





v - 



-M 



— v 





X 



ft- 



-X 





a h g 



cof a , . . . . 



= 0. 



X » M > v 

 9,f, C 



The first part of the vanishing expression is thus 



X fi. V 



= d(-kH + /U G + 



and the second part 



A H G 



a h g I . 

 h b / L 



— v 



V -fX 



. X 



M —X 



A 



A+X 2 



H+ * x -;rf-: 6+A+ * £? 



H+V+^-' B + „* 



^ X u 





By reason, however, of the trinomial elements in the second square array this can be 

 partitioned into 



