OF ARTS AND SCIENCES. 219 



If now 



..(^■)v...+.-.(i^r>. 



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

 binomial theorem of (1 + x)^ '■> then —1 will not be a latent root of i/^, 

 and we shall have 



"A" = ^7 



\f/ . tr. if/ = 1. 

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



^ \1 + 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 <^ + 1 is equal to the multiplicity of 

 the latent root — 1. 



For any matrix <^ 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 & be determined by Sylvester's formula as a finite polynomial 

 in powers of ^ ; thus let 



We then have * 



tr. ^ = /(tv.cfi) ; 



whence, if c{> is orthogonal, it follows that 



& . tr. d- = tr. t'> . &. 



Let 



^ + tr. {y ^'^ — tr. {^ 



2 " ^'" 2 "^ 



\ 



