AUTOMORPHIC LINEAR TRANSFORMATION OF A QUADRIC. 



217 



D 



rx 



M 



V 



i 



h 



9 



I* 



b 



f 



I* 



f 



c 



■MXa + jjih + vg) 



"i 



J, 



= -D 



X 



M 



V 



a 



h 



9 



h 



b 



f 



9 



f 



c 



The sum of the three parts is thus 



d(Vh- m g 



X 



V- 



V 



a 



h 



9 



h 



b 



f 



9 



f 



c 



), 



an expression differing merely in sign from that to which it has to be added. The 

 desired vanishing result has thus been reached.* 



(7.) It will be remembered that in the case of an orthogonal substitution, viz., the 

 case where f= g = h = . . . = and a = b = c = . . . . = 1, Cayley was enabled to give a 

 simple rule for the formation of the coefficients of the substitution, the original wording 

 of the rule being t 



" Les co-efficients propres a la transformation de co-ordonnees rectangulaires 

 peuvent etre exprimes rationnellement au moyen de quantites arbitraires l rs sournises 

 au conditions 



L rg = — t sr \_r ip sj ; l rr = L . 



Pour developper les formules, il faut d'abord former le determinant A de ce systeme, 

 puis le systeme inverse L rs , . . . et ecrire 



Aa rs =2L rs [r±s]; Aa rr = 2L rr -A 



ce qui donne le systeme cherche." 



Such a rule was a practical impossibility when the problem for the general quadric 

 came to be solved, because of the very complicated character of the results to which 

 both he and Hermite were led. It will now be seen that as a consequence of the 

 simplification above effected this impossibility disappears. 



Taking once more, for shortness' sake, the quadric with only three variables, viz. 



* Since this was written I have ascertained that the simplification here given of Cayley's solution was known to 

 Feobenius, whose paper of the year 1877, "Ueber lineare Substitutionen und bilinear e For men " (Grelle's Joum. 

 lxxxiv. pp. 1-63) is a carefully written and methodically arranged exposition of the theory of matrices with applica- 

 tions. It would appear not to have received due attention from subsequent writers. 



The simplification is also explicitly referred to in one of a series of valuable papers by Dr Henry Taber in the 

 Proc. Lond. Math. Soc. (1890-93). 



t GrelUs Joum., xxxii. pp. 119-123 ; or Collected Math. Papers, i. pp. 332-336. 



VOL. XXXIX. PART I. (NO. 7). 2 K 



