314 PROCEEDINGS OF THE AMERICAN ACADEMY. 



r a w\ — l 



jv+Wi'-i = e i- e 



is a transformation of r'.* Moreover, 



^wV=i_ r e wV=iy = 1 . 



and therefore, for any integer m, 



f e WV-iy,n + l _ / g 2JFV-l\ 



= e' 



Whence we have 



1 .v 



T =. e v ' e WSl '~ x = (e im+1 ) 2m+1 /gWt'^iym+i 



§3. TAT=A. 

 If £7 is real and satisfies the equation 

 (5) U=-AUA-\ 



then, by Theorem VI, every transformation of the group e^ u , with real 

 parameter £, is a transformation of Y" : in particular, the infinitesimal 

 transformation e^ v of the group is a transformation of T". 



If the real infinitesimal transformation e SiU = 1 -f- S £ U is a transfor- 

 mation of T", then first £7is real; moreover 



A + h£(UA + AU) = {\ + h'CU)A(l + S££7) = e*S u A e*& = A ; 



and therefore UA + ^1 £7= U, that is, £7 satisfies equation (5). There- 

 fore, for any real scalar £, the transformation e^° generated by e^ u is a 

 transformation of T". "Whence it follows that every transformation gen- 

 erated by an infinitesimal transformation of T" is the mth power for any 

 exponent m, of a transformation of this family ; in particular, every such 

 transformation is the second power of a transformation ofT", and is there- 

 fore of the first kind. 



Conversely, every transformation of the first hind can be generated by 

 an infinitesimal transformation of T". For let T be any transformation 

 of T", then 



* Since the transformations of r" form a group, if T and 7\ are transformations 

 of T', so also is T T l . 



