86 PROCEEDINGS OP THE AMERICAN ACADEMY. 



is generated by an infinitesimal transfornaation of the group, if and only 

 if the Xi , . . X,. fulfil relations of the form 



where (Xj, X^ denotes the alternant X^ X^ — X^. Xj, and the coefficients 

 Cj^s ^'*e quantities independent of the x's.* 



In Volume XXXV. of the " Proceedings of the American Academy of 

 Arts and Sciences," jip. 239 et seq., 1 pointed out an error in the demon- 

 stration of what Lie calls the first fundamental theorem,! upon which 

 he bases the demonstration of his chief theorem. This error consists in 

 neglecting conditions imposed at the outset upon certain auxiliary quan- 

 tities [xi, ,U2 • • • Mr? introduced in the course of the demonstration. Thus 

 in the " Continuierliche Gruppen," pp. 372-376 (and substantially in 

 "Transformationsgruppen," III. pp. 558-564), Lie proceeds as follows: 

 Being given at the outset a family with an 00"" of transformations 7\, de- 

 fined by the equations 



X ^ ^=^ Ji \pCit • • • x„, cii , . . . cf,.j 

 {1 = 1,2. . . n), 



containing the identical transformation, and such, moreover, that the x"s 

 satisfy a certain system of differential equations, he defines by the intro- 

 duction of new parameters /x a family of transformations ^ij_, 



X i =^ J*i{X I, . . . X ^, ^i, . . . fl^J 



{i = \,2. . . n), 



each of which is generated by an infinitesimal transformation. Lie then 

 establishes the symbolic equation 



* Transformationsgruppen, III. 590; Continuierliche Gruppen, 211, 305, 390. 

 t Transformationsgruppen, III. 503; Continuierliche Gruppen, 376. 

 I If the equations defining tlie families of transformations 7',, and E^ are, 

 respectively, 



x'.=/^{t, . . . x^, a, . . . a^) 



(; = i,2 . . . »), 



and 



x'.= F.rx\ . . . .r , Ml ■ • • M,.) 



{■, = 1,2 .. . n), 



the symbolic equation T^ E^ = T,^ is equivalent to the simultaneous system of 

 equations 



