SLOCUM. — FINITE CONTINUOUS GROUPS. 247 



transformations E^, the first fundamental theorem would then be proved. 

 For taking the a's arbitrarily, we could then find a system of parameters 

 ll such that E^ = T a ; and, the a's being also an arbitrarily chosen sys- 

 tem of values of the parameters a of equations (1), we should have, 

 symbolically, 



Ta T a = Ta Eu. = J a j 



or 



where 



(*' = 1, 2), 



«* = $* 0*, «) = <£* (^"(a, « (0, )> «) ; 



that is to say, the composition of two arbitrary transformations T a and 

 T a of the family gives again a transformation T a of the family. This is 

 precisely the step taken by Lie, who assumes that because, — as men- 

 tioned above, page 244, — the <f> t 's are independent functions of the /a's, 

 each transformation of (1) belongs to the family E^. But, although the 

 functions a l5 a 2 , as defined by (10 a), are independent of the ps, since 

 the Jacobian 





is not identically zero, nevertheless, for certain values of the a's, the /x's 

 are infinite, [n finite values of the lis, however, are expressly excluded 

 from consideration. For Li k = \ k (t — t), and since t and t cannot be 

 infinite, if fx k is infinite X k is infinite ; and, by supposition, the A's are 

 arbitrary but definite constants in the integration on page 244. So we 

 cannot assume that every transformation T a of the family (1) belongs 

 to the family E^. Thus, solving equations (19, a), we have 



hi 



1 — e a "- ' 

 (19, J) 



/Xo = — do. 



For a x =}r 0, and u 2 an even multiple of W, /xj becomes infinite. More- 

 over, this transformation of the family (1) is distinct from any transfor- 

 mation of this family T a for which the lis are finite. 



On page 375 of the " Continuierliche Gruppen " Lie points out that 

 every transformation of the family E^ is generated by an infinitesimal 

 transformation. The infinitesimal transformation in question is repre- 

 sented by the symbol 



