130 KANSAS UNIVERSITY SCIENCE BULLETIN. 



These two sets of symbols represent exactly the same things, but 

 from different points of view. " 



Loxodromic, hyperbolic, elliptic and parabolic transformations. — 

 We have already shown that the transformation T : xi = "1^ is re- 

 ducible to one or the other of the normal forms : 



x i ^A j = k x-A 1 1 = 1 +t 



x, — A x — A X! — A x — A 1 



Transformations of the first type are subdivided into three kinds, ac- 

 cording to the values of k. When k is real,* the transformation is 

 called hyperbolic ; when k is of the form exp i$, i. e., when | k| = 1, it is 

 called elliptic ; for all the other values of k it is called loxodromic. All 

 transformations of the second type are called parabolic. These dis- 

 tinctions will be of much use to us. 



Known subgroups of 11$. — From § 4 we know that He contains cc 2 

 subgroups H4(A), one for each value of A. EU also contains oc 4 sub- 

 groups H_>(AA'), one for each pair of values of A and A'. It also con- 

 tains go 2 groups Hj(A), one for each value of A. The oo 4 parabolic 

 transformations in H fi do not form a group. This follows from the 

 fact that the resultant of two parabolic transformations is not gen- 

 erally a parabolic transformation. 



One-parameter subgroups of Hi{A). — In the group H>(A) the 

 law of combination of parameters is U= t + ti; the parameter t may 

 be written in the form, /'exp 10. Thus we have 



r-2 exp id) = r exp id -j- r\ exp idy. 

 If 6i = 0, then 6-2 = 6. Keeping 6 constant and letting r vary, we get 

 r-2 = r 4- ri. 



We have here the conditions for a one-parameter subgroup of 

 H-j(A). All transformations in H 2 (A) for which the t's have the 

 same amplitude form a subgroup of H2(A). It is clear that there is 

 one such subgroup for each value of 6. 



If t = rexpi0 be represented in the usual manner by points in the 

 complex plane, we see that there is a transformation in H_>(A) cor- 

 responding to each point in the complex plane. The transformations 

 in H-.'(A), corresponding to the points of the complex plane on a line 

 through the origin and making an angle with the axis of reals, con- 

 stitute a subgroup Hi(A)#. 



Properties of H\{A)6. — The group Hi(A)0 contains the identical 

 transformation corresponding to r = o. Two transformations cor- 

 responding to values of r numerically equal, but with opposite signs, 

 are inverse to one another. Hi(A)0 also contains the pseudo- 



* Klein (Modulfunctionen, Band I, S. 164), and Poincare (Acta Mathematica, Tome I, p. 5), 

 call the transformation hyperbolic only when k is real and positive. The change here made is 

 ustitied by the simplification it eil'ects in the statement of the results. 



