164 PROCEEDINGS OP THE AMERICAN ACADEMY 



Thus, if (f> is any real proper orthogonal matrix, a veal skew 

 symmetric matrix can always be found such that 



<£ = e s , 



where e denotes the base of the Napierian logarithms ; e e is of 

 course defined as the exponential series, which is convergent, and 

 for which an expression may be obtained by Sylvester's theorem. 

 This function of 6, for all real skew symmetric matrices, gives 

 only real proper orthogonal matrices. Therefore by taking succes- 

 sively all possible real skew symmetric matrices, all possible real 

 proper orthogonal matrices may thus be found. 



POSTSCRIPT. 



Rea\ improper orthogonal matrices are of two kinds: of the first 

 kind are those real improper orthogonal matrices of which unity 

 is not a latent root, or of which the multiplicity of the latent root 

 \inity is even ; real improper orthogonal matrices of the second 

 kind are those of which the multiplicity of the latent root unity 

 is odd. 



If 4> is a real improper orthogonal matrix of the first kind, a 

 real skew symmetric matrix 6 can always be found such that 



<j> = -e*. 



This function of 8, for all real skew symmetric matrices, gives only 

 real improper orthogonal matrices of the first kind. 



A real improper orthogonal matrix of the second kind cannot be 

 represented as a function of any real skew symmetric matrix. 



Worcester, Mass., July 12, 1892. 



