newson: continuous groups. 93 



choice of 8. Consequently any transformation of the group can be 

 generated from the infinitesimal transformation of the group. 

 (See Vol. IV, page 170, of this Quarterly). 



Prop. J. The i(!'oi/p G contains one injinitesinial /raiisforniafio>i 

 whose cliaretcteiistic anhcvi/ionic ratio is (/-f 8). Any transfoiniation 

 of the group or the whole gi'Oiip itself may he generated fro/n the injini- 

 tesinial transformation. 



The foregoing properties of the most general form of a one- 

 termed group in the plane are almost identical with the properties 

 of the most general form of a one-termed group on a line. Both 

 sets of properties depend upon the variation of an anharmonic 

 ratio parameter. 



We proceed now to examine certain properties of these one- 

 termed groups of transformations and their relations to the conies 

 K and K' which ar'e used to construct the transformation. Since 

 four points and their four corresponding points completel}' deter- 

 mine a transformation, we should be able to construct the conies K 

 and K' when the invariant triangle ABC and one other pair of cor- 

 responding points are given. We first show how to do this. 



The conies K and K' belong to a range of oc ' conies touching 

 the lines x, y, z, and 1. If we take any conic K of this range S 

 and consider the transformations formed by taking K with all con- 

 ies of the range, we shall have a system of 00' transformations 

 which may be represented b)^ T (KS). Each transformation of 

 this system transforms an\' point P of the plane into points P', P", 

 P"', . . P" . We wish to find the locus of these points P', P", 

 P", . . P" . The tangents from P to K intersect I in Q and R. 

 The tangents from Q and R on the line 1 to the conies of the range 

 S form two projective pencils of rays. The intersection of corres- 

 ponding rays are the points P'. P", P'", . . P" , which therefore 

 lie on a conic through O and R. This conic also passes through 

 the points A, B, C; for the segments AAj, BB,, CC^ are conies of 

 the range S, and the tangents from Q and R to AA^ intersect in A. 

 Hence this conic which we shall call K passes through A and like- 

 wise through B and C. 



Hence if we have given the invariant triangle and any pair of 

 corresponding points P and P', we can construct K and K', the 

 conies which determine the transformation, and therefore construct 

 the whole transformation. The points A, B, C, P, and P' deter- 

 mine a conic ^ which cuts 1 in two points Q and R; connect P 

 with Q and R; these two lines and the lines x, y, z, and 1 all touch 

 a conic K of the range S. The lines joining P' to Q and R touch 



