§6 KANSAS UNIVERSITY QUARTERLY. 



(x>" transformations which leave the triangle invariant. By Theorem 

 5 these oo^ transformations form a two-termed group Gg or 

 G (ABC). 



Since there are co'' different triangles in the plane, it follows that 

 there are oo'' such two-termed groups. Hence the general projec- 

 tive group Gj^ is composed of oo^ two-termed groups G^', thus 

 Gg^ oo^Gj,. No two of these two-termed groups can have a trans- 

 formation of the kind T in common; for if two transformations T 

 and Tj are identical the eight parameters of the one must be equal 

 to the eight parameters of the other. Now the two anharmonic 

 ratio parameters of one of the transformations may readily be 

 equal to those of the other; but if the transformations leave differ- 

 ent triangles invariant, all of the six co-ordinate parameters of the 

 one can not be equal to those of the other. Hence T and Tj can 

 not be identical. (Later it will be shown that for particular posi- 

 tions of the triangles two or more groups Gg may have common 

 many transformations of the type S.) 



Theorem 8. The general projective group G^ is composed of oo^ 

 tivo-tcrmcd sidy-grotips G.^. Each of these two-icrmcd sub-groups has 

 an invariant triangle. No two of these tiao-termed groups can have a 

 transformation of the kind T in common. 



We now proceed to show that there are other sub-groups of the 

 general projective group G,^ that can be constructed out of these 

 two-termed sub-groups of the type G (ABC). Suppose the vertex 

 A of the triangle ABC to move along the side AC. It may assume 

 00^ different positions on the line and thus form oo^ triangles of the 

 type A„BC. To each of these triangles belongs a two-termed group 

 of transformations. Consider any two transformations taken from 

 different groups of this series. These two transformations both 

 leave invariant the points B and C, and the lines x and y; and they 

 are together equivalent to a third transformation which leaves the 

 same figure invariant and therefore belongs to some one of these 

 00^ groups Gg. Thus we see that the co'^ transformations leaving 

 the lines x and y and the points B and C invariant form a three- 

 termed group Gg, which is made up of ooi two-termed groups. 

 Thus G.^^oo^G.,. It is easily seen that the general projective 

 group G^ contains oo^ such three-termed sub-groups. Two three- 

 termed groups, whose invariant figures contain no geometric ele- 

 ment in common, contain no transformation in common. But it is 

 possible to chose the invariant figures so that they shall contain a 

 common triangle; the two three-termed groups then contain a 

 common two-termed group. 



