316 PROCEEDINGS OF THE AMERICAN ACADEMY. 



equation of T has no negative root. For in this case, since I 1 + T\ ^ 0, 

 we may put 



whence we derive 

 and by (8) • 



If £n tii etc., are the roots of the characteristic equation of T, the 



1 — 4 



roots of the characteristic equation of S are -' for i =. 1, 2, etc.; 



and since no negative number is a root of the characteristic equation of 

 T, the real roots of the characteristic equation of S are in absolute value 

 less than unity. Therefore, by Theorem IV, there is a real polynomial 

 f (S) satisfying the equation 



1 + S = e t{S K 

 From equation (9) we derive 



l — S=l + ASA- 1 = A(l + S)A- 1 = a** 3 *' 1 = e tU3A '^ = e f '" 5 '; 

 and therefore, if U= f (- S) - f (S), 

 ~ 1 - S 



1 + s 



But, by equation (9), 



_ e t(-S) e -l(S) _ gf(-3)-f(5) _ gtf 



U= f (- S) - i(S) = i{A SA- 1 ) -i(-A S A' 1 ) 



= A(f (S) - f(- S)) A~ l = -A UA- 1 ; 



and therefore, since £7 is real, e& v , for any real scalar £, is a transformation 

 of T". Consequently, T is the second power of a transformation eb* 7 of 

 r", and is generated by the infinitesimal transformation e^ u of T". 



The characteristic equation of a transformation of the first kind may 

 have negative roots. But in such case, the numbers belonging to each 

 of these roots are all even.* 



* Cf. note, p. 313. 



