312 PROCEEDINGS OF THE AMERICAN ACADEMY. 



hy an infinitesimal transformation of V is the w'^ power for any expo- 

 nent m of a transformation ofT'; in particular, every such transformation 

 is the second power of a transformation of V, and is therefore of the first 

 kind* 



Conversely, every transformation of the first kind can he generated by 

 an infinitesiinal transformation of T\ For let T be any transformatiou 

 of r', then 



(7) T=A TA-\ 



Let Fand TTV — 1, respectively, be the real and imaginary parts of the 

 polynomial l7=^f{T), satisfying the equation T= e^. Since ^is real, 

 both Fand IT are polynomials in T, and therefore commutative. Let 



T^ = e^, To = e'^'^-K 



Then since V, and therefore Ji = e^, is real, and since 



r m e^ = 6''+'^^^"= e^e'^^^ = T^ T^ 



is also real, it follows that Tq = Ti~^ T is real; and therefore, by 

 Theorem I, 



21,2 = (e"-^^) 2=1. 

 "Whence, 



But, since F= ^ (T") is a polynomial in T, we have by equation (7) 

 V=<i>{T) = cf>(ATA-') = A4>{T) A~^ = AVA-'; 



and therefore, since Fis real, e^'' for any real scalar ^, is generated by 

 the infinitesimal transformation e ^ of r'. In particular, 



2r 



is generated by the infinitesimal transformation e of V. 



A transformation TofT' is a transformation of the first kind if the 

 characteristic equation of T has no negative root. For then, by Theorem 

 II, there is a real polynomial U=f{T) satisfying the equation 7^= e^. 

 In this case F= U, jr=0; and C/" satisfies equation (4). Therefore, 

 T is the second power of the transformation e^^ of r', and is generated 

 by the infinitesimal transformation e ^ of T'. 



* See p. 308. 



