86 KANSAS UNIVKRSfl'V QUARTKKI.V. 



about O until it coincides with 1; we then have a second range on 

 1. Let A and B be the invariant points of the transformation 

 due to K. Now project this second range into a third range on 

 1' by means of a conic K' touching AA ' and CC If we now join 

 the points of the first range on 1 witli the corresponding points of 

 the third range on 1.' these joins all touch a conic K" which determ- 

 ines the projection of the first range into the third. This last 

 transformation is equivalent to the combination of the other two. 

 AA' is one of these joins; hence the transformation determined by 

 K" leaves the point A invariant. This same process may be ex- 

 tended to any nuniber of transformations. 



Theorem g. — The system of o:i~ projective tratis/on/iations that leaves 

 one point of a line invariant forms a ^roiip whose fundamental property 

 IS that the combined effect of tioo or more transformations of the gronp 

 IS etjuivaloit to some si/Zi^le t/-a//sfor///ation of the same group. 



When we come to examine the structure of one of these two- 

 termed groups, it is easy to see that it is composed of an infinite 

 number of one-termed groups. Take for example the two-termed 

 group Go ^ that leaves the point A invariant. It is made up of all 

 the one-termed groups which have A for one of their invariant 

 points. A may be taken with each of the other points on the line 

 and once with itself. Thus we see that Go ^ is made up of an 

 infinity of one-termed groups of the kind G, and one of the kind 

 G,'. What is true of the two-termed group Go.^ is also true of any 

 other two-termed group. 



Theorem lo. — Ei'ei-v tico-ternied group of projective t/-ansfor/nations 

 of a line leaving one point invariant is composed of cc^ onc-ternted 

 groups of the type (j j and one of the t\pe G ,' . G 3= oo^ G , + 1 G 'i . 



Any two such two-termed groups have common a one-termed 

 group. Take for example the two-termed groups Go^ and Gog. 

 Each of them contain the one-termed group G.^„. We can also see 

 that the converse of this is true; every one-termed group of the 

 t}'pe G , belongs to two and only two two-termed groups. 



It was pointed out above that the most general projective trans- 

 formation of the points on the line involves three parameters, the 

 coordinates of the invariant points, m and n, and the characteristic 

 anharmonic ratio k. We have seen that when m and n are fixed 

 quantities and k variable, the 00' transformations form a one-termed 

 group; also that when n is fixed and m and k variable we have a 

 two-termed group. On the other hand when m and k are fixed 

 (Quantities and n variable, the transformations do not form a one- 

 termed group. Also when m and n are variable and k constant 



