TABER. — ON SINGULAR TRANSFORMATIONS. 579 



generation of a finite transformation by an infinitesimal transformation is 

 not applicable in this case. Moreover, for t ^= oo, the resulting trans- 

 formation of (7i(") has properly no inverse. 



For assigned finite values of the a's, the transformation T^ of the 

 family defined by equations (1), if not illusory, belongs to the group 

 G^i*"' with a single parameter generated by the infinitesimal transformation 



whose symbol is 2^ cij Xj. Thus the totality of transformation with finite 

 1 



parameters of the family (1) separate into anoo'""^ of groups Gi'-"K In 

 consequence of what has been said relative to the group G^i'"', it follows 

 that each transformation of the family (1) with finite parameters is 

 paired with its inverse, and we have 7^, ~^ = T_a. 



As the «'s approach certain limiting values, of which some are 

 infinite, the transformation T„ may approach a definite finite transforma- 

 tion Z' as a limit. This transformation may be equivalent to a 

 transformation 7|, of the family (1) with finite parameters bi, b.2, . . . b,. 

 In this case T is generated by an infinitesimal transformation of the 



r 



family, namely IjbjXj, but not otherwise.* 



1 



The composition of two arbitrary transformations T^, Tj of the family, 

 defined, respectively, by the equations 



(6) x%=fi(xi, . . . x„,ai, . . . a^) . 



(i=l,2, . . . «), 



(7) a:",=/(x'i, . . . x',„b,. . . b,) 



('■=1,2,. . . n), 



gives a transformation which may be denoted by T^ T^, and is defined by 



(8) x'\ =f, (f, (x, a), . . . /„ (X, a), b^ . . . b,) 



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



This transformation is not, in general, a transformation of the family. 

 It will, however, be assumed throughout this paper that 



Let fli = a^ <, 02 = oo^ . . . ar = art. It is obviously necessary to distin- 

 guish between the equations of transformation which result from assigning definite 

 finite values to the o's, and then increasing t without limit, and those which result 

 when fli, do, . . . Or (without preserving the same ratio) approach severally cer- 

 tain limiting values some of which are infinite. The transformation which results 

 in the first case has properly no inverse. It transforms every point on any one of 

 the path curves of the group G-^ia) into invariant points of such curves. The trans- 

 formation which results in the second case if non-illusory may possess an inverse. 



