78 KANSAS UNTVKRSITV (,)UARrKKI.V. 



rrof^rrtx i. — 77ic transjorinations oj tlir ^^I'onp (i ^ may he arranged 

 ill pairs; cacli pair cons is ting of a transforination and its inverse. 



We can also show that the group G, contains one identieal trans- 

 formation, i. e. a transformation that leaves every point of the line 

 1 invariant. Three conies of the system S degenerate into right 

 lines, viz: the three diagonals of the quadrilateral ABB 'A.' The 

 segments AB.' BA.' and the line joining O to the intersection of 

 AA' and BB' are to be considered as conies of the system S. The 

 last mentioned is of course the line Ooo parallel to AA; All lines 

 parallel to AA ' and BB' are tangents to the degenerate conic Ooo 

 at the point at infinity on Ooo . These parallel lines all cut the 

 lines 1 and 1 ' in points equally distant from O, and hence all points 

 of the line 1 are imaltered by the transformation. The character- 

 istic anharmonic ratio of this identical transformation is given b\' 

 k:=(ABPP)=i. O is the point of contact of the conic Ooo with 

 the line 1. 



Piuipertx 2. -The group (j, eontains an identieal transjorination 

 7i'//t'se eiiaraeteristie aniiarinonie rati/' is unity. 



If Ti^ and Tu' be a })air of inverse transtormations, we have the 



I 

 relation k^ ^ "i" kk' = i- 1 his gives n^ another propert\' of the 

 k" ir t - 



group (i,. 



J^ropertx J.- - 'Jlie eoinl'ined efl'eet of any transformation and its in- 

 verse is equivalent to tiie identieal transformation of tlie group. J'/ii/s 

 TkT,.=T,. 



Since ever\- transformation in the group d has an inverse belong- 

 ing to the same group, the question arises can an\^ transformation 

 of the group be identical with its inverse. If such be the case, 



tlien must k and k ' .be e(iual. This conditit)!! gives us 1<= , , or 



k-=i: wlu'uce k=_'_i. The first value k=i gives the identical 

 transformation of the group: that this is its own inverse is self- 

 evident. Tlu' value kr=- i gives a very important transformation 

 of the group. To say that this transformation is its own inverse is 

 equivalent to saying that this transformation applied twice in suc- 

 cession will return every point on the line to its original position. 

 Therefore a single application of this transformation has the effect 

 of interchanging P and P; a pair of corresponding points; and so 

 with every pair of corresponding points. Hence this transforma- 

 tion T__i gives rise to an involution, and shall therefore l)e called 

 the inV(diitorie trans/or mat ion ot the group. 



