92 PROCEl DINGS OP THE AMERICAN ACADEMY. 



(A",. A V. The canonical form of the finite equations of the group 



E • 



These equations define the transformation T„ of G. z . Similarly, the 

 equations defining tin- transformation T h of 6\ arc 



n i 



X .. = X .. , 



The transformation 7' '/' . obtained by the composition of the transfor- 

 mations '/', and 7'. is defined by 



(11) ' ' ' «k + -(<*-l)* + 5(*-l). 



and, if this is equivalent to a transformation /' of the group, we have 

 also 



(12) 



/', =x 1 e f « + -(c'»- 1), 



Therefore 



X ., — x % 



(18) = </>, (//,. a,, /.,. 6,), 



".. 4- &j i 2 kit y— 1 . = </>2 ("i. "■_ . Jj . 6j), 



where i is an arbitrary integer. Consequently for (? x there is more 

 than one system of functions <j>. Provided o s -I 6j is nol an even multi- 

 ple of 7r y/— 1. every system <»f values of '-, and c s is finite. For 

 n, + l>.. - 2 k 77 V — I, - is finite, but the denominator of <\ becomi - 

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

 to k = — k is finite. < lonsequently the parameters Cj and c 9 can alwaj - 

 be chosen finite, and therefore '»'. is continuous. 



For the group Cr t , whose infinitesimal transformations arc p fJ 

 / y i /'i. /' i- defined by 



r g< • "'(-• -1). 



