250 PROCEEDINGS OF Till! AMERICAN ACADEMY. 



Therefore, the «'s ami u's being taken arbitrarily, we have 



Ea€a = 8Ca 



where 



o»=*»[«a(a,o«»),a] (*=l,2...r). 



But, whereas the parameters </ with whicli we (hah in that part of [his 



paper, pages 240—248, relating to the firsl fundamental theorem were not 



restricted in range, the present parameters a, involved in the transforma- 

 tions 2T„ defined by equations (A), must be finite, at least if the trans- 

 formation <T„ is to be generated by an infinitesimal transformation. And 

 if a and a are chosen arbitrarily, 



a k =^(0, a) (*=1, 2...r) 



may be infinite. 



Thus, in the example chosen, if a y ^0, a x 4 1 0, and « 2 -f a. z = an 

 even multiple of vi, the transformation 



x{ = fj (x, a) = x x -r- (i, 



x 2 = fo (x, a) = e rt 2 a- 2 + Oj I 1 



is finite, while 



fll = 4k, (a, a) = ^ ! J J j |J (* _ 1) + +£(* - i)j 



is infinite; and the transformation (£„ cannot be generated by an infin- 

 itesimal transformation. 



