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PEOCEEDINGS OF THE AMERICAN ACADEMY. 



is a transformation of T'.* Moreover, 

 and therefore, for any integer m, 



/gWV^\2m+l __ /^2W V—i\m ^W V—T ^ e^^^ 



"Whence we have 



T = 6^e^^'~^ =■ fe^""*'-' \2m+l /gTrV-l\2m+l 



§3. TAT=A. 

 If C is real and satisfies the equation 



(5) 



U=-A UA-\ 



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

 parameter ^, is a transformation of V" : in particular, the infinitesimal 

 transformation e^^^ of the group is a transformation of V". 



If the real infinitesimal transformation e^^^^ =■ 1 + 8 { Z7 is a transfor- 

 mation of r", then first U'ls, real; moreover 



A-\-Zt,{UA + AU) = {l + htU)A{l-^hiV) =6^^'' A 6^^''= A; 



and therefore UA + A U =■ U, that is, C satisfies equation (5). There- 

 fore, for any real scalar I, the transformation e^^ generated by e^^^ 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 he generated by 

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

 of r", then 



* Since the transformationa of r' form a group, if T and Ti are transformations 

 ofr, soalsois TTi. 



