872 PROCEEDINGS OF THE AMERICAN ACADEMY. 



if <f> satisfies the above equation, | <£ J tj: 0. In what follows it will 

 be assumed that the determinant of the skew symmetric matrix fi 

 is not zero. 



Since the product of any two matrices satisfying the equation 

 </> Q </> = U is again a solution of this equation, the matrices satisfying 

 this equation form a group. This group can be generated by the 

 solutions of the equation given by Cayley's expression.* For every 

 solution of the equation </> 12 <£ = O is given by the product of four 

 of Cayley's expressions. See (2) and (10). 



The group of solutions of the equation <£ fi <f> = Q can be generated 

 by the matrices of the group which differ infinitesimally from the 

 matrix unity ; for, see (2) and (5), every solution given by Cayley's 

 expression can be formed by the product into itself an infinite num- 

 ber of times of a solution (given also by Cayley's expression) which 

 differs infinitesimally from the matrix unity. 



The group of matrices satisfying the equation <£ CI </> = fi is identi- 

 cal with the group of linear substitutions which automorphically trans- 

 form the alternate bilinear form 



A A 



(Q) (x u x 2 , , , . ) (y 1} y 2 , . . . ), 



in which the a?'s and y's are cogrediant. Therefore, this group can be 



generated by the infinitesimal transformations of the group, namely, 



those differing infinitesimally from the identical substitution. 



2. The proof of these theorems relating to the group of solutions 



of the equation </> O </> = Q may be made to depend upon the proof 



in the special case in which the skew symmetric matrix Q is also 



orthogonal (i. e. if CI 2 = — 1). 



Thus, let 



<f> 12 <p = 12. 



If ft 2 -}r — 1, let w denote any fourth root of —ft 2 expressible in powers 

 of fi 2 ; then to is symmetric and is commutative with £2. Therefore, 



v. Q O 



or or 



Q 



If ij/ = u> cf> uT , O' = — g , this becomes 



CO 



^n f i}/ = n', 



* This has been proved otherwise by Frobenius Crelle, 1878, who shows that 

 every solution of the equation <p fl cp = fl which is not given by Cayley s expres- 

 sion is given by the limit of this expression when the symmetric matrix T 

 becomes infinite. 



