82 PROCEEDINGS OF THE AMERICAN ACADEMY. 



-ea 



transformation of group G. But then e^ , for any positive integer p, is 



a transformation of group G ; and since (c^ y = e'^", any transforma- 

 tion generated by the repetition of an infinitesimal transformation of 

 gioup G is the pth power, for any integer p, of a transformation of this 

 group. 



It may be shown that any transformation of group G that cannot be 

 generated by the repetition of an infinitesimal transformation of group 

 G is tlie (2 p +l)th power of a transformation of this group for any 

 integer j). 



For any transformation of group G that can be generated by the 

 rejjetition of an infinitesimal transformation of group G, the numbers 

 belonging to each negative root of the characteristic equation are all 



even.* 



Postscript. 



Let (3 denote the group of all linear homogeneous transformations, 

 real and imaginary, of determinant +1 whose invariant is the real quad- 

 ratic form of non-zero determinant, 



J = (n^ Xi, X., . . . x")- ; 



and, as above, let G denote the sub-group of real transformations of 

 group ®. 



A transformation of group (3 can be generated by the repetition of 

 an infinitesimal transformation of this group if — 1 is not a root of the 

 characteristic equation of the transformation, or if — 1 is a root of 

 the characteristic equation, provided the numbers belonging to — 1 are 

 all even.f 



As stated above, if a transformation of group G can be generated 

 by the repetition of an infinitesimal transformation of this group, the 

 numbers belonging to each negative root of the characteristic equation 

 of the transformation are all even. These conditions, though necessary, 

 are not always sufficient. Thus, for n = 2, the form jf is transformed 

 automorphically if 



/>*' ' 'Y* 'vi' ■ — ^v» • 



* For definition of the numbers belonging to a root of the characteristic equa- 

 tion of a transformation, see Tiiese Preceedings, Vol. XXXI. p. .336. 

 t Proceedings of the London Mathematical Society, Vol. XXVI. p. 374. 



