210 



DR THOMAS MUIR ON THE 



we have from the second 



(6 l1 6 2 ,6 3 , . . . ) — ( 



and therefore by substituting in the first 



( l n l 12 l n 



hi hi ^23 

 ^81 ^32 ^33 



)( 'n hi hi • • • ) l (?1 ' ?2 ' 53 > 



hi 22 ^32 

 ^13 ^23 ^33 



hi hi hi ■ • • 



) V£l > £2 ' fc.3 > 



hi hi hi • • • 





^13 ^23 ^33 





• • • ) = 0*1 , 



^2 » *^3 > * * " ) > 



which is all that was required. Indeed, it may be seriously questioned whether up to 

 the present date this be not the best way of formulating the theorem for the construc- 

 tion of an orthogonal substitution. For example, in the case of two variables, we have 



(x lt x 2 ) = { 1 X)(l -xr 1 (&,&), 



l-x i||x 1| 



=( 1 



l-x 



x)( i x )(&,&), 



II 1+X 2 1+X 2 

 -X 1 



1 + X 2 1+X 2 



= (1-X 2 2X )(£,£>■ 



1+X 2 1+X 2 

 -2X 1-X 2 



1 + X 2 1+X 2 



(2) The next step in advance was taken by Hermite,* but he also made use of an 

 intermediary set of variables, his mode of defining them being that each one of the set 

 was to be the arithmetic mean of the corresponding members of the two given sets — 

 that is to say, t was defined by the equation 



0r=K*r + £). 



Strictly speaking there was not much new in this, for it was in entire correspondence 

 with Cayley's intermediary equations, and very probably was derived from them. 

 Thus, from the first equations of the two sets we have by addition 



2hA + (h2 + l n)0-2 + (hs+hi)0z+ ■ ■ • =*! + £> 

 i.e. 2d 1 = x 1 + g 1 . 



His after-procedure, however, was different from Cayley's, and apparently lent itself 

 more readily to generalisation, both in his own hands and in Cayley's. 



(3) An entirely new departure was taken by VELTMANN.t He dispensed with inter- 

 mediary variables, laying down at once the connecting equations 



* Hermite, Ch., "Sur la th£orie des formes quadratiques temaires indeiinies," CrellJs Joum., xlvii. pp. 307-312. 

 Sec also xlvii. pp. 313-342 ; and Gamb. and Dub. Math. Joum., ix. pp. 63-67. 



+ Vei.tmann, W., " Die orthogonale Substitution," Zeitschr. fiir Math, it. Phys., xvi. pp. 523-525. 



