TAKER. — REAL LINEAR TRANSFORMATIONS. 81 



Finally, we may show in precisely the same way that c"^" is a real 



solntioQ of eqnation (1) ; and therefore, since (e™^")^ = c"^^", it follows 

 that if/ is tlie second power of a solution of equation (I), and is thus of 

 determinant +1, Wherefore, t/^ is a transformation of group G.* 



2 



By takincc m sufficiently ^reat, —6Ci may be made as nearly as we 



please equal to zero, and therefore i// = e^^" may be made as nearly as 

 we please equal to the identical transformation. Wherefore, <fi- = ^i^ 

 can be generated by the repetition of an infinitesimal transformation of 

 group G. 



The converse is of course also true ; that is to say, every transforma- 

 tion generated by the repetition of an infinitesimal transformation of 

 group G is the joth power (for any positive integer p), and therefore the 

 second power, of a transformation of this group. This may be shown as 

 follows. The transformation (ft, if infinitesimal, may be put equal to 

 1 + 8« . X) where Bt is infinitesimal. If <^ is a transformation of group 

 G, 8t and ^ are both real, and we have 



(I + 8t . x) ^ (I + Bi • X) — $ ^ "1^ = ^ ' 

 That is, neglecting terms involving the second power of 8t, 



xfi + Ox = 0. 



If we put ^ = X ^~S ^ is real, and the preceding equation becomes 



fi5 4-0^0 = 0; 

 that is, ] 



The transformation resulting from m repetitions of the infinitesimal 

 transformation 



</)r= 1 + 8<.x= 1 + s< . ^n 



is the transformation (1 + Si! . ^ O)'", which, if m is infinite, is equal to 

 e'" ' •^", or e'^" if we put t = m8t. The repetition of an infinitesimal 

 transformation of group G results in a transformation of this group ; and 

 therefore, as we have in fact already seen, e'^" for any real quantity t is a 



* The matrix e'" is also a transformation of group G. The totality of trans- 



formations c'" for all possible real values of m constitutes a continuous one term 

 sub-group which contains the identical transformation. 



VOL. XXXII. — 6 



