2IO 



KANSAS UNIVERSITY QUARTERLY. 



Thus we may say: 



The effect of the two-teniicd group of clations upon the coucentric 

 syuiuietry is the systeui of all involutions that leave the same center 

 invariant. 



From the form of the above equations we also deduce the fact: 



The effect of t/ie one-termed group of elations upon tJie concentric 

 sxminetry is the system of all involutions lohose axes are parallel to the 

 axis oj the one-termctl group of elations. 



Fi y. Z. 



We illustrate this theorm in Fig. 2. The central symmetry with 

 the centre O transforms any original triangle ABC into A^B^Cj. 

 An elation with the axis e transforms the triangle A^B^Cj into the 

 triangle AgBgCo, such that it forms an involution with the^triangle 

 ABC whose axis is S is determined by the points of intersection of 

 the pairs of sides AB and A^^Bg, BC and BgCg, CA and CgA,. 



The purely geometrical proof for this is easily found by means 

 of the theorem of Menelaos concerning the transversals of a triangle, 

 so that we may leave it to the reader. 



