SLOCUM. — FINITE CONTINUOUS GROUPS. 247 



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

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

 /x such that -Efj, = Ta; and, the «'s being also an arbitrarily chosen sys- 

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

 symbolically, 



J a J a = J a J^fi. = -^ cii 



or 



where 



fi if{x, a), a) = Fi(J{x, a), fx) =/, {x, a) 

 0'=1, 2), 



a, = ^, (/x, a) = *, (31 (a, a""), ") ; 



that is .to say, the composition of two arbitrary transformations Ta and 

 7'a of the family gives again a transformation Ta of the family. This is 

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

 tioned above, page 244, — the ^g^'s are independent functions of the ;u,'s, 

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

 functions aj, a^, as defined by (19 a), are independent of the /a's, since 

 the Jacobian 





1. H"^ f^2\ e^V 



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

 are infinite. Infinite values of the /x's, however, are expressly excluded 

 from consideration. For //.^ = Aj (< — 7), and since t and } cannot be 

 infinite, if /x^ is infinite A^ 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 Ta of the family (1) belongs 

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



(19, b) 



For ai rj: 0, and ao an even multiple of tt/, [j.i becomes infinite. More- 

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

 mation of this family Ta for which the /x's 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 



