NEWSON : ONE-DIMKNSIONAL TRANSFORMATIONS. 127 



§ 4. Two- and Three-parameter Groups. 



The group G->(A). — Let us take two transformations of type I, T 

 and Ti, having one, but only one, invariant point, A', in common, and 

 find their resultant. The point A' may bo taken for the origin, and 

 the two equations then reduce to the form 



_i_ = k _^_ and _i_ _ kl _i_. (27) 



Eliminating xi from these equations, we have the resultant 



kk, aa, x (99,\ 



A2 " (kk, A— kA + kA, — A,)x + AA,' V °' 



The invariant points of this transformation are found by making 

 X2 = x and solving the resulting quadratic. The invariant points are 

 thus found to be A = o, and 



A ' 2 = kk, A-kA + kA.'-A,' ( 29 ) 



Putting kki A— kA + kAi — Ai= ' kkl ~ 1)AA ' in equation ( 28 ),. 

 we have : 



-^_=kki— V- (30) 



x 2 — A 2 x — A 2 ^ ' 



From this we see that the resultant of two transformations of type 

 I having one invariant point in common has for one of its invariant 

 points the common invariant point of the components; also, we 

 learn that the cross-ratio of the resultant in this case is equal to the- 

 product of the cross-ratios of the components, viz., k2 = kki. Since 

 (30) is of the same form as (27), we infer that all transformations 

 leaving a single point invariant form a continuous group of two para- 

 meters, G-2(A). The two parameters are the cross-ratio k and the 

 abscissa of the other invariant point A2. 



Theorem 12. All transformations which leave a single point in- 

 variant form a two-parameter group ; the cross-ratio of the resultant 

 of any two transformations of the group is equal to the product of 

 the cross-ratios of the components. 



Structure and properties of Gi{A). — The structure of the group 

 G-_>(A) is evident from the above discussion. It contains cc 1 one- 

 parameter subgroups of type I and one of type II. The point A2- 

 may be taken in turn with every other point on the line to form the 

 invariant points of a group Gi(AA') and once with itself to form the 

 invariant point of Gi(A). From the continuity of the point system 

 on a line and from the known continuity of each subgroup, we infer 

 the continuity of the group G2(A). The transformations of the 

 group G2(A) are not commutative. Since k2 = kki, it is evident that 

 the cross-ratio of the resultant is independent of the order of the com- 

 ponents ; but the position of the second invariant point of T2 is not 

 independent of the order of T and Ti. For if k and ki be inter- 



