GEOMETRIC THEORY. | 53 
O until it coincides with 1; we thus 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 l’ by means of a conic K’ touching AA’ and CC’. If we 
now join the points of the first range on / with the corre- 
sponding points of the third range on 1’, these joins all touch 
a conic K” which determines 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 extended to any 
number of transformations. 
For the geometric proof of the second group property, see 
exercise 2, page 62. 
67. Cross-ratio of the Resultant in G,(A). The trans- 
formation produced by the conic K” has one invariant point 
PIG. 5: 
