OF ARTS AND SCIENCES. 219 



If now 



lfr = [1 +C 1 <f>- 1 + C2< />_1 2 + ....+ Cml (0_ l)-»-l]J 



where 1, c,, c 2 , etc. denote the coefficients of x in the expansion by the 

 binomial theorem of (1 + a:)* ; then —1 will not be a latent root of if/, 

 and we shall have 



V = <}>, 



\p . tr.ip = 1. 



Therefore, proceeding as in (4), it may be shown that we have 



M > 2 



* - (^y 



for a proper choice of the skew symmetric matrix Y. 



This proof may be extended to any imaginary proper orthogonal 

 matrix for which the nullity of <£ -f 1 is equal to the multiplicity of 

 the latent root — 1. 



For any matrix <j> whose determinant does not vanish (that is, of 



which zero is not a latent root), a matrix & can always be found such 



that 



$ = e». 



Let t> be determined by Sylvester's formula as a finite polynomial 

 in powers of <£ ; thus let 



We then have 



tr. & =f(tr.tf>) ; 



whence, if <f> is orthogonal, it follows that 



& . tr. & = tr. 0- . !>. 



Let 



& -- tr. & A & -- tr. & 



~2 



= e , -±— = 0; 



