80 PROCEEDINGS OF THE AMERICAN ACADEMY, 



that is, & is skew symmetric. If now 



where and ly are real, both 6 and rj are skew symmetric, that is, 



e = -d, ^ = -7). (3) 



Since O is real, 9 O and rj O V — 1 are the real and imaginary parts 

 respectively of i9- O = x-' -^^^ since the latter is a polynomial in the real 

 matrix <^i, its real and imaginary parts, $Q and rj H V — 1? are polyno- 

 mials in <^i, and are therefore commutative. Consequently, by virtue of 

 a theorem given above, 



on 





_ ea + r,n |/-i _ en ^n i- 1 



Since <^i is real, and since 6Q and therefore e^" is real, it follows 



that e' ^~ is real. Therefore, by what precedes, since r]Q, is real, 



(y,a\'^2^ 1. 

 Whence we have 



and therefore by (2) 



,o ,2 2 9.0 



If now we put 

 where m is any positive integer, \]/ is real, and 



Moreover, siuce 12 == li, and since, by (3), ^ = — ^, we have 



and therefore 



* For any positive integer p, 



Therefore 





