OO KANSAS UNIVERSITY QUARTERLY. 



mon; for the common invariant figure is a triangle. If the two 

 lines or the two points of the linear elements coincide, the two 

 groups have in common a four-termed group. 



Tlicoreiii II. The oo^ transfonnatioiis wliicli leave a linear element 

 invariant form a five-termed group . The general projective group con- 

 tains 00 3 such five-termed groups. 



If the point C be made to move along the line x, to each posi- 

 tion of the point belongs a five-termed group. The sum total of 

 the transformations belonging to all these five-termed groups forms 

 a six-termed group whose invariant figure is a straight line. It is 

 clear that to every line in the plane belongs a six-termed group of 

 this kind. The general projective group therefore contains oo* 

 such six-termed groups. 



In like manner if the line x be made to revolve around the point 



C, to ever}^ position of the line x belongs a five-termed group. 



These co^ five-termed groups make up a six-termed group whose 



invariant figure is a point. The general projective group contains 



00^ of these six-termed groups. 



Theorem 12. The 00 "^ transformations which leave a line or a point 

 invariant form a six-termed group. The general projective group 

 co7itains 00 2 sub-groups of each kind. 



This completes the enumeration of the sub-groups of the general 

 projective group, which can be built up out of the two-termed sub- 

 groups of the kind G (ABC). We have a list of nine kinds of 

 groups, as follows: 



G^; Gg p, Gg.i; G,; G^.a> G^.,,, G^.c; G3; G^,. 



So far we have shown how to build up these groups of higher 

 orders out of groups of lower orders. The reverse process might 

 have been followed. We might have started with the general pro- 

 jective group and decomposed it into groups of lower orders. 

 This we proceed to do briefly. 



A transformation of the kind T is determined by eight para- 

 meters, six co-ordinate parameters and two anharmonic ratio para- 

 meters. When all eight of these vary they generate the general 

 projective group. When two or more of the co-ordinate parameters 

 are fixed quantities and the rest of them variables the various sub- 

 groups are generated. In order that a point of the plane shall 

 remain invariant it is necessary and sufficient that two of the co- 

 ordinate parameters shall be fixed; the variation of the other six 

 parameters generates a six-termed sub-group Gg.p. In like manner 

 two conditions or parameters determine a line; the variation of the 

 other six parameters generates a six termed group G^.j. The gen- 



