248 PROCEEDINGS OP THE AMERICAN ACADEMY. 



r r n £j f 



^j ^j ^jf = 2^- ^i h ^ji (^1 • • • a^«) -^ > 

 and adds to an arbitrary function y(rri' . . . a-/) the increment 



and, therefore, to a:^ adds the increment 



8xi = iji(xi . . . x„) • St. 

 This shows its relation to the simultaneous system on page 245, namely, 

 dxf r 



dt 



r — '^3 \- ^a (^/ • • • ^n) (i = 1,2 . . . n). 



1 



If the theorem stated by Lie, jjage 375, " Continuierliche Gruppen," 

 was true without exception, namely, that every transformation of the 

 family T„ belonged to the fiimily E^, it would then follow that every 

 transformation of the family T„ could be generated by an infinitesimal 

 transformation ; for then taking the a's arbitrarily, we should have 



^ ^M = To.. 



But, for a system of values of the a's for which one or both of the func- 

 tions Ml (a, a*"'), M2 (a, a*°') are infinite, there is no equivalent transforma- 

 tion of the family £J^ ; and, consequently, such a transformation cannot 

 be generated by an infinitesimal transformation of the group. E. g., the 

 transformation T,, considered above, for which oi iji and a^ is an even 

 multiple of tt i, cannot be generated by an infinitesimal transformation of 

 the group. 



1 

 In demonstrating the second fundamental theorem (the chief theorem) 

 Lie assumes the results of the first fundamental theorem. He shows 

 that a system of r independent infinitesimal transformations* 



Xif= 5, $,, (xi... x,,)^ (^• = 1, 2 . . . r) 



1 dx^ 



generate a family of transformations K„, with r essential parameters, 

 which contains the identical transformation, and is defined by the 

 equations 



* Lie terms the infinitesimal transformations or symbols of infinitesimal 

 transformations A'^, X2, . . . Xr independent if they satisfy no linear relation 

 eiXif+ . . . + Cr Xrf^ 0, with constant coefficients e. 



