220 PROCEEDINGS OF THE AMERICAN ACADEMY 



from the preceding equation it follows that 



O • o = e . O ; 

 therefore 



<f> = e& = e e o + & = e 9 o e". 



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



(e o) 2 = e 2e o = e » + tr -» = e» . e te - d = <j> . tr. <f> = 1 ; 



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

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

 is skew symmetric, e e is a proper orthogonal matrix. Moreover, e 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 0, e^ can be repre- 

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

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

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

 is a latent root, and such that 



e 9 = e i. 

 Therefore, in either case the theorem is true. 



May 1, 1893. 



