124 KANSAS UNIVERSITY SCIENCE BULLETIN. 



variant and transforms x directly to X2. Let T and Ti be given in the 

 implicit normal forms : 



We eliminate Xi from these equations by multiplication, and obtain : 



:kkl 



A' 



x 2 — A x — A 



The cross-ratio of T2 is therefore k-2 = kki. 



In the same way it may be shown that the resultant of any number 

 of transformations with the same invariant points has its cross-ratio 

 equal to the continued product of the cross-ratios of the components. 



The cross-ratio k may have an infinite number of values, and hence 

 there are an infinite number of transformations leaving both A and A' 

 invariant. The transformations of this system have the property that 

 the resultant of any two of them is another of the same system. 

 Hence they form a continuous group, the parameter of the group be- 

 ing the cross-ratio k, which may be made to vary continuously. We 

 shall designate the group by Gi(AA'). 



Theorem 8. The totality of transformations which leave the same 

 two points of a line invariant form a continuous group ; the cross- 

 ratio of the resultant of any two transformations of this group is equal 

 to the product of the cross-ratio of the components. 



Properties of the group Gi(AA'). — The fundamental property of 

 the group Gi(AA') is that the resultant of the two transformations of 

 the group is another of the same group. Other properties of the 

 group will now be developed. 



The inverse of T, any transformation in Gi(AA'), is also to be 

 found in Gi(AA'). For if T transforms x into xi, then 



x, —A' 



k^ 



■ Xj — A x — A 



If T" 1 be the inverse transformation which transforms xi back to x,. 



then 



m-i . x — A' _l_ *i —A' 



X * x — A k x t — A ' 



Hence the cross- ratios of a pair of inverse transformations have re- 

 ciprocal values. 



The resultant of a pair of inverse transformations is the identical 

 transformation. The product of the cross-ratios of a pair of inverse 

 transformations is unity ; hence the cross-ratio of the identical trans- 

 formation is unity, and the group Gi(AA') always contains the 

 identical transformation. 



The group Gi(AA') contains one transformation which is identical 

 with it own inverse. In this case we have the condition k==-^, or 

 k 2 = I • whence k = ± 1. The value k = 1 gives the identical trans- 



