newson: projectivk traxsi-ormaiions. 79 



The position of the conic which produces the involutoric trans- 

 formation of the group is such that it is its own reflection on the 

 line OX. It therefore has the line OX for one of its axes. It is 

 clear that there can be but one such conic in the system inscribed 

 in the quadrilateral ABB 'A.' 



Property 4. — The group Qt ^ contains one involiitorie transformation 

 wliieli is its own inverse: tlie eliaraeteristie aiiliarnuniie ratio of tlir 

 ijrooltitoric transforniation is — i. 



The group Gj contains two very noteworth}' transformations, 

 viz.: those determined by the conies AB.' and A'B. The tangents 

 from all points on 1 to AB ' intersect 1' in B.' Hence the degen- 

 erate conic AB ' transforms all points of the line 1 into a single 

 point B. The characteristic anharmonic ratio of this transforma- 

 tion is given by k=:(ABPB)=o. The other degenerate conic A' B 

 transforms all points of the line 1 to the point A. Its characteristic 

 anharmonic ratio is given by k=(ABPA)= 00. Strictly speaking 

 these are not transformations at all in the generally accepted mean- 

 ing of the word; I shall call them pseiido-transfor /nations. 

 These two pseudo-transformations form an inverse pair: this is 

 evident analytically, and also from the fact that the lines A'B and 

 AB ' are situated symmetrically with respect to OX. 



Prof. Lie, in defining his transformations analytically, expressl}' 

 states that the determinant of the transformation must not be zero. 

 This condition excludes just these two transformations called 

 pseudo-transformations. For Lie's equation of transformation is 

 written 



ax + b 



If the determinant 



a b 

 c d 



cx-f-d 



be 

 =0, then d:^ — . Substituting this 



a 

 value of d in the equation, we have 



a(ax-j-b) a 

 ' c(ax-j-b) c 

 which shows that ever}' point x of the line is transformed into the 



nxed pomt 



c 



The inverse of the above transformation is given by the equation 



— dx,+b 



x= . 



ex J — a 



The determinant of this transformation equated to zero also gives 



be , . . . 



d== — . Substituting this value of d in the last equation we have 



3. 



— b(cx, -a) b 



a(^cXj — a) a 



