580 PROCEEDINGS OF THE AMERICAN ACADEMY. 



r 



the c's being quantities independent of the a:;'s. In which case, from 



(6) x'i =fi(xi, . . . x„,ai, . . . a,.) 



(i = 1, 2, . . . n), 



(7) x"i —fi (x\, . . . x'„, bi, . . . b^) 



{i = 1, 2, . . . «), 



we shall obtain (by the chief theorem of Lie's theory) 

 (J) X j =y,- (a^i, . . . x„y Ci, . . . c^) 



where 



(10) Cj = cpj (ai, . . . a„bi, . . . b,) 



= 1,2, . . . r). 



Whence it follows that T^ T^^ is also a transformation of the family, 

 which, tlierefore, constitutes a group, — denoted in what follows by G.* 

 The equivalence between the transformation T^ and the transformation 

 resulting from the composition of T,, and Tf, may be denoted by writing 



T, = T, T,, 



In general there is more than one system of functions cp^{a, h), gjjC^' ^)' 

 etc., such that T^ = T^ T„ if Cj = cpj (a, S) for^' = 1, 2, . . . r. For finite 

 values of the o's and b's it may happen that every value of some one (or 

 more) of the c's is infinite. In this case, from what precedes, it follows 

 that, while both T^ and T^, are generated by infinitesimal transformations 

 of the group, the transformation T^ T„ resulting from their composition 

 cannot be generated thus, and the group cannot properly be said to be 

 continuous. But, if each of one (or more) of the systems of values of 

 the C'S, is finite for the assigned values of the o's and b\,T^ T^ can be 

 generated by an infinitesimal transformation of the group. f A transfor- 

 mation of G which cannot be generated by an infinitesimal transformation 

 of G may be termed essentially singular. 



In what follows I shall modify the preceding notation, restricting the 

 use of the symbols T^-, T^, etc., to denote, unless otherwise stated, trans- 

 formations of group G with finite parameters, and therefore generated, 



r r 



respectively, by the infinitesimal transformations Sa^- X^, %bj Xj, etc. 



* Lie: Transformationsgruppen, I. p. 158. 



t Rettger : American Journal of Mathematics, XXII. p. 62. 



