XKWSOX; I'ROIKCTTVF. I'R \ NSrCjRM ATIONS. () 1 



We now turn our attention to the two-termed sub-f^roup of trans- 

 formations that leave invariant the point at infinity on 1. This 

 s^roup will he designated b\' G-i^. The projective transformation 

 determined by a parabola has for its invariant points the point at 

 infinity and some finite point. There are co- parabolas touching 

 the lines 1 and 1'; these determine a two-termed group of trans- 

 formations whose invariant point is the point at infinity on 1. 

 Every transformation belonging to this group is determined by a 

 parabola: for every conic touching the line at infinity is a parabola. 



All the transformations belonging to this group exhibit a peculiar 

 property. Let us take for example a transformation which has A 

 and 00 for invariant points, and whose characteristic anharmonic 



ratio IS K. 1 hen we have k=( Aoo FP, )= ^=r — : :f;, — - = --^ . 



' Poo P,oo AP, 



Whence AP=k.AP,. If we take O and Q,, another pair of 

 corresponding points, we shall have in like AQ=k.AQj. Therefore 

 PO=AQ-APr=k(AQ, — AP,)=k.P,Q,. Thus we see that the 

 effect of such a transformation is to nudtiph' the length of anv seg- 

 ment In- a constant, the constant multiplier being the characteristic 

 anharmonic ratio of the transformation. This is analogous to the 

 mechanical effect of stretching a rubber cord with one end fixed at 

 A. We shall call this effect a Dilation: and shall speak of the 

 group Go „ as the grouj) of dilations. If the multiplier k is less 

 than unit\-. the effect of the transformation is contraction: but if \\e 

 think of contraction as negative dilation, the word dilation is suf- 

 ficient to cover all cases. 



'JlicorcDi /_•,'.- -Tlic 00- pai-alTt>l(is toiirluni:; 1 a)i(i 1' lirtcniiiiu- a fioo- 

 tn-iii(\i i:;roiip ot dilations, infinity Ix'ini::; t/ir inrai-iant fi(>int. 



This two-termed group breaks up into oo' one-termed groups. 

 oo' parabolas can be drawn touching 1 and 1' and some line as 

 A/V ' perpendicular to OX. These determine a one-termed group 

 whose finite invariant point is A. One of these one-termed groups 

 is of the type G ,' . This group has two coincident invariant points 

 at infinity on 1. This group is determined by the svstem of para- 

 bolas touching 1 and 1 ' and having the line Ooo as a common axis. 

 The onl}- tangent to a parabola of this svstem perpendicular to 

 OX is the line at infinity: and it is a common tangent to all of 

 them. The characteristic anharmonic ratio of anv transformation 

 of this group is given by k=(LO cxd PP j )=i . Hence the constant 

 of dilation for this group is unity: that is to sav. ever}' segment of 

 the line 1 is transformed into an equal segment. Again let us take 

 one of these parabolas touching 1 and 1' and having its axis per- 



