212 



DR THOMAS MUIR ON THE 



But the left-hand member of this may be partitioned into 



x i x 2 x a • 



a 



h 



9 



a, 



h 



b 



f ... 



tV( 



9 



f 



c ... 



X. 



and 



ni hz 



hi 



hi hi 



23 



the latter of which evidently vanishes. Consequently if we partition the right-hand 

 member in the same way we deduce 



x x 



x 2 x 3 ■ ■ ■ 





a 



h g . .. 



x } 



h 



b f ... 



X, 



9 



/ c ... 



X. 



41 4 2 S3 





a h g ... 

 h b f ... 

 g f c .. . 





+ 



£ 



£ 



& ■.. 







n2 



^ ... 



x : 



21 





^23 



JCc 



^31 



*32 



■ . . 



X, 



Had we used the multipliers £ x , £ 2 , £ 3 , . . . the same process would clearly have given 

 us 



A. £- A. . . . X. fl!_ .7!. . . _ 



41 42 S3 ■ • " 





a A <7 ... 

 h b f ... 

 9 f e ... 



#i 



#1 



x. 2 x 3 ... 





a 



ft ^ ... 



& 



A 



& / ... 



6 







/ i ... 



e 





hi 



h\ • " 



£ 



hi 



- 



hi • • • 



£ 



hz 



hi 



• 



Is 



We are thus led to two equations whose right-hand members are seen to be equal : 

 hence the left-hand members are also equal — which is what was to be proved. 



(5.) Cayley's solution of the problem of automorphic linear transformation of a 

 quadric # differs so seriously from this that a thorough investigation of the discrepancy 

 between the two seemed imperatively called for. 



Writing the substitution of the preceding paragraph in Cayley's manner we have 



( a h + l n g + l l3 . . .){x 1 ,x i ,x 3 ,. ..) = ( 



^1 + ^21 9 + hl •••)(£n6iAf')i 



•21 



h + l, 



9 + hi fH 



f+h 



23 



32 



h+l 



9+hz f+l 



12 



f+l, 



32 



23 



and consequently 



(x 1 ,x i ,x 3> . . .) = ( a h+l u g + l 13 . ..)" l ( 



h + hi 



9 + hi f+h 



f+l 



23 



32 



h+l 



9+hs f+hi 



12 



^ + ^21 9 + l , 



& f+l 



31 



) (41 > 42 > 43 » • ' • ) ' 



32 



23 



Now if, for shortness' sake, we put this in the form 



(4,4,4 ...)=(A)- 1 (A')(£, &,&>•■•), 

 Cayley's corresponding result will be found to be 



(x, , x 2 , x 3 , . . . ) = (D)-X A')(A)-KD)(^ , £ , £ 3 , . . . ) , 



* Grelle's Journ., 1. pp. 288-299 ; or Collected Math. Papers, ii. pp. 192-201. 



