AUTOMOKPHIC LINEAR TRANSFORMATION OF A QUADRIC. 



211 



Vl "+" hl X Z + ^13*3 + • • • = ni&l + ^21i2 "+" hlC 3 "+" 



tj^ + i 22 a; 2 + < 23 a? 3 + . . . = t 12 f 1 + t 22 S2 "J" ^3263 + 



t 3i a?j + tg 2 £ 2 -J- t 33 a? 3 + . . . = ^I3?i + * 23?2 "I" ^33?3 "t" 



and affirming that these implied that x l} x 2 , x 3 , . . . , and ^ , £ 2 , £ 3 , . . . , were ortho- 

 gonally related. By way of proof he solved f or x x , x 2 , x 3 , . . . and obtained Cayley's 

 expressions for them in terms of & , £ 8 , £ 8 , . . . 



(4) It does not appear to have been remarked — and it is certainly very important 

 that attention should be drawn to the fact — that the much more general substitution, 

 viz., the substitution for the automorphic transformation of a quadric can be expressed 

 in an exactly similar way. 



Denoting any quadric 



ax-j 2 + bx 2 2 + cx s + 



+ 2hx x x 2 + 2gx 1 x. i -f 



+ 2fx 2 x 3 + 



+ 



by 



x l 



x% x s ... 



a 



h g . . . 



h 



b f ... 



9 



/ c ... 



!«i 



— a form which brings into evidence the discriminant of the quadric, and from which 

 the terms of the quadric are readily obtained by multiplying every element of the 

 square array by the x which stands in the same column with the element, and there- 

 after by the x which stands in the same row with it — we may enunciate our theorem 

 as follows : — 



The substitution — 



ax x + (h + l n )x 2 +(g + l n )x 3 + 

 (h + ya^ + bx 2 + (/+ l 23 )x 3 + 



t# + *3lK + (/+*32K+ ™ 3 + 



«£i + (& + *»)&+ fa + 'si)&+- • 



J 



where, as before, the l's are any ^(n)(n — 1 ) arbitrary quantities and l r 



transforms 



V 1 62 S3 ' • • 



= -/., 



Ou-% CCo 





a h g 

 h b f 

 9 f e 



into 



a h g ... 



& 



h b f ... 



£ 2 



9 f c ... 



£3 



Using with the given equations the multipliers x u x 2 , x 3 , . . . and adding, we have 



x t 



X 2 



x 3 ... 





a 



h+l 12 



9 + hs ■ ■ ■ 



X, 



h + l n 



b 



f+kz ••• 



x '< 



9 + hi 



/+*« 



c ... 



30f 



Si • f 2 €3 





a h + l n g + l 3l . . . 

 h+l n b f+l S2 . . . 

 9 + hz /+ ; 23 c ■ ■ ■ 



ii 

 £2 



& 



