NEWSON : ONE-DIMENSIONAL TRANSFORMATIONS. 131 



transformation of H-_>(A), which is given by r oo. This group con- 

 tains two subdivisions corresponding to positive and negative values 

 of r. These are separated by the identical and the pseudo-transforma- 

 tions. 



The group Hi(A)0 contains two infinitesimal transformations cor- 

 responding to -+- 8r and - Sr. Each infinitesimal transformation 

 generates its corresponding subdivision of the group. It is evident 

 that each transformation of the group H 2 (A) can be generated from 

 one and only one infinitesimal transformation. 



One may, if one chooses, call the aggregate of the transformations 

 in H-i(A), which are represented by points on a half ray from the 

 origin to the infinity point, a group. Every such group contains the 

 identical transformation, the pseudo-transformation, one infinitesimal 

 transformation, and every transformation in the group is generated 

 from the infinitesimal transformation of the group. But the inverse of 

 every transformation in such a group Hi( A)0 would be found in group 

 Hi(A)(0+ 7r). It is better, however, to consider the group as made 

 up of two subdivisions than to regard each subdivision as a separate 

 group. The group He contains oo 2 subgroups H-2(A), one for each 

 value of the complex number A. Each of these two-parameter groups 

 H>(A) contains oc 1 one-parameter subgroups, one for each real value 

 6 between and it. 



Theorem 15. There are two varieties of subgroups of parabolic 

 transformations in He, viz., H>(A) and Hi(A)0. 



One-parameter subgroups of H-i(AA'). — In the group H-_>(AA') 

 the law of combination of parameters is K2 = kki. The parameter k 

 may be written in the form exp(c+i)0; then we have : 



exp(c-2+i)0>=exp j (o+i)0 + (ci + i)0i j . 



If ci==c, then C2 = ci==c, and $%==0 -\- 61. Keeping c fixed and 

 making 9 vary, we get exp(c -f i)02 = exp(c + i)(0 -f #i). We have 

 here the conditions for a one-parameter subgroup of H2(AA') ; all 

 transformations in H>(AA'), for which the k's have the same c in the 

 formula k = exp(c + i)0, form a subgroup Hi(AA')c. There is one 

 such subgroup for each real value of c. 



There is a transformation in H2(AA') corresponding to each value 

 of k, which is a complex number. Let k = exp(c -j- 1)6 be repre- 

 sented by points in a complex plane. We wish to see how the trans- 

 formations in the one-parameter group Hi(AA')c are distributed in 

 the plane. W T e have the equation k = exp(c + i)0, where c is a 

 constant and a variable. Let k=^x + iy. Then we have : 



x -f- iy = exp cO(cosO -j- isin#); (33) 



x = exp cd cos 6 ; 

 hence, y = exp cd sin 6 ; 



therefore, x' 2 -f y 2 = exp2 c0 ; 



