316 PROCEEDINGS OF THE AMERICAN ACADEMY. 



equation of T has no negative root. For in this case, since 11+ T\ -^ 0, 

 we may put 



1 — T 



whence we derive 





and by (8) 



^^^ ^^^ -l + A TA-' -1+ T-'- T+1--^' 



If ^15 C25 etc., are the roots of the characteristic equation of T, the 



1 — ^. 

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



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

 T, tlie 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^<^'. 

 From equation (9) we derive 



1- S= 1 + ASA-^ = A(1 + S)A-'' = e^<«w" = e^<^«^"'' = e^'-^); 



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



1 c 



2^— Z — gf(-S)g-f(5) _ gf(-S)-f(5) — gJ7 



1 + S 



But, by equation (9), 



U= i(- S)- f(S) = f(A SA-^) -f(-A SA-^) 



= A(f (S) - f(- ^S-)) A-' = -A UA-' ; 



and therefore, since U is real, e^^, for any real scalar ^, is a transformation 

 of r". Consequently, T is the second power of a transformation ei^ of 

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



