318 PROCEEDINGS OF THE AMERICAN ACADEMY. 



c^ S for any real scalar ^, is a transformation of T" ; and therefore 



is a transformation of T". Whence it follows that 



for any real scalar {, is a transformation of T".* Moreover, we have 



and therefore for any integer m 

 "Wherefore, 



7^= e^oe^l = Cgn'o~\2m+l /'g2«i+l''i\2!n+l_/'gI7pg2-/H-l''l\2m+l __ /'gf7„+2m+l''l\2m+l^ 



§4. r-i^r=A 



If Z7is 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 F'". 



If the real infinitesimal transformation e^^^ =l + S{£/^isa transforma- 

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



A + h'C{- UA + AU) ={l -htU)A{\ + hKU) = e-^^^AJ'^''=A, 



and therefore — UA+A 1/^=0, that is, U satisfies equation (5). 

 Therefore, for any real scalar ^ the transformation e^^ generated by e^^^ 

 is a transformation of V". Whence it follows that ever?/ transformation 

 generated by an infinitesimal transformation of V'" is the m'^ power for 

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

 transformation is the second power of a transformation of T'", and is 

 therefore of the first kind. 



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

 and T(i) are transformations of T" aud TTH) = T'^^) T, then T Tii) is also a trans- 

 formation of r". 



