SS KANSAS UNIVF.KSITV Ql'ARTERLV. 



Tlirorriii /2.--77if tohilitx of tlw project iiw transformations of tlir 

 pt>ints on a line form a thi-ec-termcd group. 



This general projective group does not leave any point or points 

 of the line invariant. It is easy to see that this three-termed group 

 G3 falls apart into 00' two-termed sub-groups, each of which leaves 

 one point of the line invariant. The properties of these two-termed 

 sub-groups were discussed in the last section. Since each of these 

 two-termed groups falls apart into 00' one termed groups, we see 

 how the general three-termed group falls apart into 00- one-termed 

 sub-groups. We are now in position to arrange a scheme for the 

 complete classification of the projective transformations of a line. 

 03= oc'Gg— ooi( oo'Gj + iG; )= oo^G^+oo 'G;. 



The chief properties of the general three-termed group may be 

 readily collected from the discussions of the preceding pages. G.^ 

 contains one and only one identical transformation: this is determ- 

 ined by the degenerate conic Ooo perpendicular to OX. This 

 identical transformation is contained in every two-termed and also 

 in every one-termed sub-group. In this respect it is peculiar 

 among all the projective transformations of the line. 



G.J contains od- pseudo-transformations; these are so distributed 

 that everv two-termed sub-group contains 00' of them, and everv 

 one-termed sub-group contains two of them. 



Every infinitesimalh' narrow conic touching 1 at </0, and also 

 touching 1,' determines an infinitesimal transformation. There are 

 00- such conies; hence G.j contains 00- infinitesimal transforma- 

 tions, oc ' of these conies touch anv line as AA ' perpendicular to 

 OX; and onlv one of them can touch two such lines as AA ' and 

 BB.' These 00- infinitesimal transformations are therefore dis- 

 tributed so that ever\' one-termed sub-group conlams one and onl)' 

 one of them. There are two \arieties of these infinitesimal trans- 

 formations and the\' give rise to two varieties ot one-termed groups, 

 G , and G \ . 



As before remarketl the general projective group (j.j breaks up 

 into 00' two-termed sub-grou})s. These two-termed sub-groups 

 are all very much alike and have like properties; two of them how- 

 ever call for special attention, viz. : the grouj* w hich leaves O 

 invariant, and that one which leaves the point at inhnity invariant. 



The two-termed sub-group Go,,, which leaves the point O 

 invariant, is made up of one-termed sub-groups each of which 

 leaves O and senile other point as A invariant. Fig. 2 shows the 

 construction of a transformation which belongs to the one-termed 

 sub-group G,^,,. This transformation is determrlied by the degen- 

 erate conic Oy. All transformations \\hich leave the point O 



