Ll 1 S PROCEEDINGS OF THE AMERICAN ACADEMY. 



V vV Mr A ' •' J£ 



and adds to an arbitrary function /(a-/ . . . x,'\ the increment 

 and, therefore, to r, adds the increment 



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



^■' = 2, A^, (a/ . . . xj) (i = 1, 2 . . . n). 



If the theorem stated by Lie, page 37o, " Continuierliche Gruppen," 

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

 family T„ belonged to the family A,,, it would then follow that every 

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

 transformation ; for then taking the as arbitrarily, we diould have 



En = T a . 



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

 tions M\ (a, a (0) ), i)/ 2 (a, a (01 ) are infinite, there is no equivalent transforma- 

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

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

 transformation T„ considered above, for which a x ± and a 2 is an even 

 multiple of -n-i, cannot be generated by an infinitesimal transformation of 

 the group. 



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* 



f 



XJ= >•, U (x x ... O^f- (t = 1, 2 . . . r) 

 l c x t 



generate a family of transformations 27„. 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',, A'. : , . . . A'- independent if tliey satisfy no linear relation 

 ' i -^i/+ • • ■ + e r A',/EE 0, with cunstant coefficients c. 



