OF ARTS AND SCIENCES. 215 



^•ih,V~i{fJ{e)-ft{e))'] 

 + K V^/l^' (^) - K V~iff (0). 



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

 ±/«^i + i V— I5 etc.; ^1 is moreover real, since fj^^^^^O — f^\■^{0), 

 y^* {0) — y^aC^)' etc., are purely imaginary scalar multiples of 

 real matrices. We also have 



(ji = e^ = e^i. 



If ?Hq =: 0, that is, if zero is not a latent root of 0, we may proceed 

 in a precisely similar way to find the matrix 61. 



3. Since no integer multiple of 2 tt V— 1) other than zero, is a 

 latent root of 0^, hence it follows that no odd multiple of tt V— 1 is a 



n 



latent root of -^ ; therefore, the matrix 



h 



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

 tained in the expression e^, for JI equal successively to the distinct 

 latent roots of Oi- 

 We have 



,/,2 = (e2J _ gg ^ 0. 

 Since Oi is real, i(/ is real ; and if/ is orthogonal, for we have 



ij/ . tr. ijy = e^ . etr- 2 



01 d, 



= e^ . e 



a -^^ 





= 1. 



