OF ARTS AND SCIENCES. 215 



e l = e-[ : 2i h v~i{fl\e)-ff{6)) 



+ -ih i v=l(f?{e)-f?{0) ) + .... 



- (/, (61) +/J> ((?) +/? (0) + . . . . +/« (6) +/* (0)) 



+ a, +1 v^i/;Vi c> - ^^/if+iW + • • • • 



+ K V~\f? (0) - K V=lf? (6). 



The matrix 6 X is then skew symmetric, and its latent roots are 0, 



±<V + i V— 1 3 etc -5 #1 is moreover real, since f^ +l — f^\ ,('')• 



y^+ 2 (^) — /T+2(0)' etc -> are P ure 'y imaginary scalar multiples of 

 real matrices. We also have 



<£ = e 9 = e 9 i. 



If w = 0, that is, if zero is not a latent root of 6, we may proceed 

 in a precisely similar way to find the matrix Q x . 



3. Siuce no integer multiple of 2 -k \/— 1, other than zero, is a 

 latent root of 6 X , hence it follows that no odd multiple of tt V— 1 is a 



latent root of ~ ; therefore, the matrix 



ij/ = e" 



has no latent root equal to — 1. For the latent roots of \j/ are con- 

 tained in the expression e n , for H equal successively to the distinct 

 latent roots of $ x . 

 We have 



^ = \e*) = e s = <£. 

 Since 6 X is real, ip is real ; and ip is orthogonal, for we have 



\p . tr. \p = e- . e tT - - 



°i _$ 



— e-' . e * 



= 1. 



