OF ARTS AND SCIENCES. 217 



6. Let 3> be any real improper orthogonal matrix. Then if w 

 denotes a matrix whose determinant is equal to — 1, and whose con- 

 stituents are all zero except those in the principal diagonal which are 

 severally equal to ± 1, the matrix 



is a real proper orthogonal matrix. For 



— $$ 

 = 1. 



Moreover, 



|<£| = |o,|.|$| = l. 



Therefore, we may put 



* = (t+y)' 



for a proper choice of the real skew symmetric matrix Y. Whence, 



since co 2 = 1, we derive 



(I - YV 



