newson: continuous groups. 87 



Theorem g. The general projeetive group G ^ may be decomposed 

 into 00 5 three-termed sub-groups caeh of whieh has for invariant figure 

 two tines, their point of intersection, and another point on one of tJiese 

 lines. 



In a similar manner three-termed groups of the type just dis- 

 cussed may be put together so as to form four-termed groups; and 

 this may be done in three different ways. First, suppose that the 

 line y is made to revolve about the point C; it thus assumes 00^ 

 different positions. Belonging to each of these positions is a 

 three-termed group, and by the principle of Theorem 5 these form 

 a four-termed group G^.a,! whose invariant figure is composed of 

 two invariant points and the line joining them. 



In the second place, the point B may be supposed to assume all 

 positions on the line x; corresponding to each position of the point 

 B is a three-termed group, and the totality of all these three-termed 

 groups is a four-termed group G^j,, whose invariant figure is com- 

 posed of two invariant lines and their point of intersection. 



Again the point C may be made to move along the line y; to 

 each position corresponds a three-termed group, and the totality of 

 all these three-termed groups is a four-termed group G^ ,.> whose 

 invariant figure consists of the invariant line y and the invariant 

 point B not on the line y. These three types of four-termed 

 groups are the only possible ones that can be compounded out of 

 three-termed groups of the kind G.j. We shall designate these by 

 the symbols G^.;i, G^_i„ G^.,.. 



Theorem to. There are three types of four-termed groups whieh 

 may be compounded out of three-termed groups cf the kind G^ {and hence 

 out of two-termed groups of the kind G (ABC)). Their invariant fig- 

 ures are respectively 1 100 points and their join; two lines and their inter- 

 section: a line and a point not on the line. 



If oqI four-termed groups of the kind Gj^ ^ be taken such that 

 their invariant figures have common the line x and the point C on 

 X, these form a system of oo^ transformations all of which leave 

 invariant the linear element x, C. Hence these form a five-termed 

 group. Again, if we take c»^ four-termed groups of the kind G^ i, 

 such that their invariant figures have common the line x and the 

 point C, we have the same five-termed group as before. But if we 

 take four-termed groups of the kind G^^. we can not put them to- 

 gether so as to form a five-termed group. This kind of a five- 

 termed group with an invariant linear element is the only kind that 

 can be built up out of two-termed groups G (ABC). Two such 

 five-termed groups will generally have a two-termed group in com- 



