98 PROCEEDINGS OF THE AMERICAN ACADEMY. 



we put r = i, we have the group 



q, xq, p, xp + ayq, 



which is a sub-group of the general projective group. 



The symbol of the general infinitesimal transformation is 



U= (ai -\- a.x -{- a^ai/)q + {a-^ + a^x)p. 

 Hence, 



Ux = a^ -{- a^ X, 



U"x = a^Ui -\- a^ a;, 



where ZZ^ re denotes U{Ux), etc. Similarly, 

 Uy = a^ + a2X + a^ay, 



U'^y = «! «4 « + aoa^ax •\- a^^ u"y -\- 02 a^ x -\- a2 a^ , 

 U^y = aiai^u" + a2a^a'^x-^ a^ v? y ^ a2a^ {a -}- l)a;+ a2a^a^{a-\- 1), 



U'y = aia4»-^«"-^ + a2ai''-'a''-^x + a^Vy + a2a4""^(«""'+ «""'+ • • • « + l)a; 



+ 02«3«4""^(«""^ + «"^ + ... + « + 1). 



Therefore,* the transformation of the group generated by the general 

 infinitesimal transformation of the group is defined by the equations 



X = xe -\ (e — I), 



(1) 



a^x 



«4 (« — 1) 



a., ^3 / « «4 «4 -1 , ^ , ^1 / " ^4 1 \ 



« a4^ (a — 1) a a4 



Let this transformation be denoted by Ta- It transforms the point P 

 with coordinates (ar, ?/) into the point P' with coordinates (x', y). Let 

 the transformation 7^ of our group (generated by the infinitesimal trans- 

 formation (bi + b2X -\- b^ixy) q + (bs -\- b^ x)p) transform P' into P" with 

 coordinates (x" , y"). Ti, is then defined by the equations 



* Lie, Differentialgleichungen, chap. 3, § 3. 



