250 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Therefore, the a's and a's being taken arbitrarily, we have 



where 



. «, z= 0, [|H (a, a<"'), «] (k=l,2... r). 



But, whereas the parameters a with which we dealt in that part of this 

 paper, pages 240-248, relating to the first fundamental theorem were not 

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

 tions JE-a defined by equations (A), must be finite, at least if the trans- 

 formation QTa is to be generated by an infinitesimal transformation. And 

 if a and a are chosen arbitrarily, 



«* = ** (a, «) (k^l,2...r) 

 may be infinite. 



Thus, in the example chosen, if Oi 4^ 0, ai 4^ 0, and a^ + a^ = an 

 even multiple of 7^^, the transformation 



Xi =z fi (a;, a) — a^i -f- a 



/e«2 — 1 \ 



is infinite ; and the transformation SEa cannot be generated by an infin- 

 itesimal transformation. 



