84 KANSAS i;N1\ i;kSI IN' (U:AR ikkia . 



of our ^roup detinitious. Thf rangt.' of conies S touching the sides 

 of the quadrilateral AA ' B ' B is composed of three distinct sub- 

 divisions. These are separated from one ailother'by the three line 

 conies of the range, viz. : O 00, AB, ' AB.' The transformations of the 

 group G^„ also arrange themselves in three corresponding sub- 

 divisions. The hyperbolas which touch 1 between B and O give 

 rise to transformations whose characteristic anharmonic ratios vary 

 from o to I. Let us call this sub-division I. The hyperbolas 

 touching 1 between O and A give rise to transformations for which 

 this ratio varies from i to 00. These constitute sub-division II. 

 The ellipses touching 1 between A and B give rise to transforma- 

 tions whose ratios are all negative and vary between — 00 and o. 

 These constitute sub-division III. 



The combination of any two transformations of sub-division I 

 gives rise to a transformation belonging to the same sub-division; 

 for the product of two proper fractions is a proper fraction. The 

 inverses of all transformations in sub-division I are in II. The 

 combination of any two transformations in II gives a transformation 

 also in II; but the inverses of those in II are in I. Sub-divisions 

 I and II contain each an infinitesimal transformation. Sub-division 

 III contains no infinitesimal transformation; the combination of 

 any two transformations in III gives one in either I or II. The 

 involutoric transformation divides subdivision III into two parts: 

 all the transformations in one of these parts are the inverses of 

 those in the other part. 



In Thcoric dcr ']'raiisfoniuitioiis^^ni[>prii. Vol. I, page 163, Lie and 

 Engel exhibit x,=ax, where o<a<i. as an example of a group 

 containing no inverse transformation. This group is identical with 

 subdivision I above. However valid their reasoning may be from 

 an analytic point of view, it is hardly proper from a geometric 

 point of view to consider either sub-divisions I or II, or the combi- 

 nation of I and II, as a complete group. 



In a similar manner the continuous group G^ is divided by the 

 identical and the pseudo-transformation into two parts. Each of 

 these parts has an infinitesimal transformation and the character- 

 istic group property. The transformations in one part are the 

 inverses of those in the other. 



• ^.\ Two-termed Groups of Projective Transformations. 



Having shown how to construct one-termed groups of the pro- 

 jective transformations of the points on a line, and having developed 

 the chief properties of such groups, we now proceed to arrange and 

 classify the groups themselves. 



