318 PROCEEDINGS OP THE AMERICAN ACADEMY. 



e* l , for any real scalar £, is a transformation of T" ; and therefore 



is a transformation of T". Whence it follows that 



e Wi = e u o e$\ 

 for any real scalar £, is a transformation of r". # Moreover, we have 



e 2u o = (e u °) 2 = 1 ; 

 and therefore for any integer m 



(e u ff mJrl = (e 2u o) m e u o = e u o. 



Wherefore, 



T '=■ e u <>e u i =■ ( e u ) 2m+1 (e*^ Vl ) 2m+1 = (e u oe^+ i ' Jl ) 2m+1 = ( e ^ +2m+ll/1 ) 2m+1 . 



Stf,^,, ,„ srrrrnixo-,, , rr.^rrTC. 



§4. 2 t - l ir=i 



If Z7 is real and satisfies the equation 



(5) 11= A UA~\ 



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

 parameter £, is a transformation of T'". In particular, the infinitesimal 

 transformation e ^ of the group is a transformation of T'". 



If the real infinitesimal transformation e* J7 =l + S££7isa transforma- 

 tion of r'", then first U is real ; moreover, 



A + 8£(- UA + AU) = {1 -8£U)A(l + ht,U) = e~^ Ae Ku = A, 



and therefore — UA+A U=0, that is, U satisfies equation (5). 

 Therefore, for any real scalar £, the transformation e$ u generated by e s ^° 

 is a transformation of V". Whence it follows that every transformation 

 generated by an infinitesimal transformation of T'" is the m th power for 

 any exponent m of a transformation of Y'". In particular, every such 

 transformation is the second power of a transformation of T" 1 , and is 

 therefore of the first kind. 



* The transformations of T" that are commutative form a group. Thus, if T 

 and TO) are transformations of T" and T TO) = J"* 1 ) T, then T TO) is also a trans- 

 formation of r". 



