98 PROCEEDINGS OF THE AMERICAN ACADEMY. 



we juit /• -- 1, we have the group 



q, xq, ;>, X p + lt yq, 



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

 Tlie symbol of the general infinitesimal transformation is 



C T = (a, + thx + a t ay)q + u/, + <> i x)p. 

 Hence, 



Ux = a s -f a 4 x, 



U-x = a :i a 4 + a 4 s x, 



U"x = Wa^"- 1 + a 4 "x, 

 where U"x denotes U(Ux), etc. Similarly, 



Uy = a x + a 2 x + a A uy, 



1 Z 'J = a l a i « + a 2 a i <* X + «■»"' ""# + «S r/ 4 ^ + «2 «3 » 



V s y = a 1 « 4 2 «' 2 + rt 2 « 4 s «'-x + a 4 *a*y + a 3 a?(a + l)x+ o,a 3 a t (a + 1 |, 



Efy = a^-V" 1 + a„a 4 "- 1 «"- , x+ a 4 Vty + a 2 a 4 »- 1 (o B -*+ « n - 3 + ...a + l)x 

 + o 2 a 3 a 4 "- 2 («»- 2 + «^-8 + . . . + „ + 1). 



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

 infinitesimal transformation of the group is defined by the equations 



x = x e + - (c — 1), 



«4 , rt 3 , n t 



(1) 



a 4 (« — 1) 



+ —-T7 TT ( e - « <" - ] +") + —- ( e — !)• 



a or 4 (« — 1 ) « a± 



Let this transformation be denoted by T,,. It transforms the point /* 

 with coordinates (x, y) into the poinl P' with coordinates (.>•'. ;/). Lei 

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

 formation {h x 4- /a.t -f b 4 ay) 7 + {hi -f />, x)p) transform /' into P" with 

 coordinates (x", y"). 7), is then defined by the equations 



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



