312 PROCEEDINGS OF THE AMERICAN ACADEMY. 



by an infinitesimal transformation of T' is the m th power for any expo- 

 nent m of a transformation ofV; in particular, every such transformation 

 is the second power of a transformation of T 1 , and is therefore of the first 

 kind* 



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

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

 of T', then 



(7) T=ATA~\ 



Let Fand W \/ — 1, respectively, be the real and imaginary parts of the 

 polynomial U = f(T), satisfying the equation T ' = e u . Since T is real, 

 both J 7 and Ware polynomials in T, and therefore commutative. Let 



T x = e v , T = e wV ~\ 



Then since V, and therefore T x = e v , is real, and since 



T=e u = e T+wV ^ = e r e w ^ = T x T 



is also real, it follows that T = T x ~ l T is real; and therefore, by 

 Theorem I, 



T 2 = (e wV ^) ■ = 1. 

 Whence, 



T 2 = {T X T ) 2 = Tf 7 T 2 = T x 2 . 



But, since F= <f> (T) is a polynomial in T, we have by equation (7) 

 V=cf>(T) = <l>(ATA-i) = A </, ( T) A-i = A VA~ l ; 



aud therefore, since Vh real, e for any real scalar £, is generated by 

 the infinitesimal transformation e * of T'. In particular, 



IV 



T 2 = T x 2 = (e F ) 2 = e 



is generated by the infinitesimal transformation e * v of T 1 . 



A transformation TofV 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 T= e u . 

 In this case V= U, W=0; aud U satisfies equation (4). Therefore, 

 Tis the second power of the transformation ea^of T', aud is generated 

 by the infinitesimal transformation e ^ u of V. 



* See p. 308. 



