154 J- C. Decius. [J- F- I- 



Such changes of unit magnitude form the representation of a group 

 of continuous transformations. A group of continuous transformations 

 may be represented by the equations: 



X' =^ ^'ix\x\ ■■■ x"; X', X2, •■• X"), .. .. 



i = 1,2, ■■■ n ^^-^^ 



in which the barred quantities are the new values of the variables, which 

 were originally x', x^, ■ ■ • x", after the transformation induced by the set 

 of continuously variable, independent parameters, X'ij = 1,2, • ■ • m). 

 The usual postulates for a group require that the X' can assume such 

 values as will : 



1. Induce the identity transformation 



x< = ^<(x^ • • • x"; Xo' • • • Xo"). 



2. Induce the resultant of two or more successive transformations in 



It may then be shown that the equations of transformation (1.1) 

 determine a set of linear partial differential operators 



which represent the independent infinitesimal transformations of the 

 group (the subscript zero for d^'/dX' implies that the derivative is to be 

 evaluated for those values of the parameters, X', which induce the 

 identity transformation; without loss of generality these values may 

 henceforth be assumed to be zero). The most general infinitesimal 

 transformation is represented by 



U = a'U, (1.3) 



in which the a> are arbitrary constants; any finite transformation is 



* The summation convention of tensor notation is used throughout. Thus I r--. I t— . 



implies 2 I —. I —. ; this is true only if the index appears in one place as a superscript, in 

 another place as a subscript. 



