92 PROCEEDINGS OP THE AMERICAN ACADEMY. 



(Xi, X.) = Xi . The cauonica! form of tlie finite equations of the group 

 pi, ^iPi is 



(9) .'. = .„. + !-(,.= _,), 



3? 2 -— ■''2 • 



These equations define the transfornoation T^ of (?2 • Similarly, the 

 equations defining the transformation Tf, of Cg are 



(10) x^ = .^.'=+ I (.'.-!), 



The transformation Tf, T^, obtained by the composition of the transfor- 

 mations 7], and 7\, is defined by 



C2 = «2 + ^2 + 2 /; TT V— 1 > = 02 («1 ) ^2 1 ^1 ) ^2)? 



where ^' is an arbitrary integer. Consequently for G^ there is more 

 than one system of functions </>. Provided a., + h is not an even multi- 

 ple of TT V— 1; every system of values of Ci and c^ is finite. For 

 a., + b.2 = 2 KTT V — 1, C2 is finite, but the denominator of Cj becomes 

 zero. In this case, however, that system of values of r^ corresponding 

 to ^- = — « is finite. Consequently the parameters c^ and C2 can always 

 be chosen finite, and therefore G2 is continuous. 



For the group G2 , whose infinitesimal transformations are j}^, 

 x.,p2 -f pu T,^ is defined by 



x\ = x^ -\- a.,, 

 (14) 



a:'2 = X2e''= + -(e"^-l), 



(In 



