140 KANSAS UNIVERSITY SCIENCE BULLETIN. 



plane. It is called the real subgroup of He. The equation of 

 RH.s(L) is 



*=St*' (43) 



in which the variable and coefficients are all real quantities. The 

 structure and properties of RH a (L) are identical with those of Hs(C). 



Theorem 26. The three-parameter group of real projective trans- 

 formations of points on a real line is a special subgroup of the six- 

 parameter group of circular transformations of the complex plane. 



Properties of RH%{L). — The properties of the real projective 

 group in one dimension, which we shall now designate by RGs, may 

 be determined in two ways. They may be deduced from the proper- 

 ties of the group H3(C), or they may be derived directly from equation 

 (43). The analytical developments of §§ 1, 2, 3, 4 apply equally well 

 to equation ( 1 ) whether the coefficients a, b, c, d are real or complex. 

 From the known properties of Hs(C), given in §8, we may deduce 

 the following for RG3 : 



The real group of projective transformations RG3 contains three 

 kinds of transformations, viz., hyperbolic, elliptic, and parabolic. A 

 hyperbolic transformation has two real invariant points; an elliptic, 

 two conjugate imaginary invariant points; a parabolic, one real in- 

 variant point. RG3 breaks up into od 1 subgroups RGn>( A) , one for each 

 real point on the line. It further breaks up into one-parameter sub- 

 groups, as follows : oc 2 hRGi(AA'), oc 2 eRGi(AA'), and cr^pRG^A). 

 The natural parameters of hRGi(AA') are A, A' and k, which are all 

 real. Those of eRGi(AA') are A, A' and k where A and A' are con- 

 jugate imaginary and k is of the form exp iO. The parameters of 

 pRGi(A) are A and t, both real. 



Generation of real groups from infinitesimal transformations. — 

 The generation of the real groups of one-dimensional transforma- 

 tions requires special notice. We consider first the group hRGi(AA'). 

 The natural parameter of this group is k, which is always a real 

 number. The identical transformation of the group is given by 

 k = l. The two transformations of the group given by k=±d, 

 where d is a real, positive, infinitesimal quantity, are infinitesimal 

 transformations. The transformations of hRGi(AA'), for which k is 

 between 1 and 00, may be generated from the infinitesimal trans- 

 formation for which k = l-fd; for (1-j-d)" may be made any 

 positive number greater than unity by proper choice of n. In like 

 manner we see that the transformations of the group for which k is 

 between 1 and may be generated by a repetition of the infinitesi- 

 mal transformation for which k==l — d. The transformations in 

 hRGi(AA') for which k is negative cannot be generated from either 



