220 PROCEEDINGS OF THE AMERICAN ACADEMY 



from the preceding equation it follows that 



therefore 



(ji = e& = e^o + ^ ::z: e^o gC. 



Since Oo is symmetric, e^o is symmetric. We have 



therefore, the first factor in the above expression for </> is a symmetric 

 square root of unity, that is, is a symmetric orthogonal matrix. Since 

 6 is skew symmetric, e^ is a proper orthogonal matrix. Moreover, e^ 

 can be represented by the square of Cayley's expression. For, if no 



integer multiple of 2 tt V — 1 is a latent root of 6, e^ can be repre- 

 sented by Cayley's expression ; if, on the contrary, there are integer 

 multiples of 2 tt V — 1 among the latent roots of 0, a skew symmetric 



matrix Oi, can always be found of which no integer multiple of 2 tt V— 1 

 is a latent root, and such that 



Therefore, in either case the theorem is true. 

 May 1, 1893. 



