OF ARTS AND SCIENCES. 221 



Note on Symmetric Orthogonal Matrices. 



Every symmetric orthogonal matrix is a symmetric square root of 

 unity, and therefore, if $ is a symmetric orthogonal matrix, an or- 

 thogonal matrix xs can always be found to satisfy the equation 



in which w is a matrix whose constituents are all zero except those in 

 the principal diagonal which are severally equal to ± 1. 



If 4> is real, cr may be taken real, and heuce it follows from the 

 theorems of § 1 and § 2 that, for a proper choice of the real skew sym- 

 metric matrix Y, and of J a matrix similar to w, we may put 



l-Y 



" = ^rqrY, 



•\2 



Therefore, 





August 17, 1893. 



[\ + YV _ n - Yy n + Yy 



Vr^^/ "" Vl + Y/ '^ \1 - Y/ ' 



