REAL TRANSFORMATIONS. 35 
vision contains an infinitesimal transformation. If either in- 
finitesimal transformation be repeated » times the cross-ratio 
of the resultant is given by k=e*’*; by a proper choice of n 
this may be made any transformation of the group for which 
§is finite. Let 6/=2x—6; since e*4’ =e' @-—9) — e2rt 18 — e— 
it follows that any transformation of the elliptic group 
eG,(AA’) may be generated by repeating either infinitesimal 
transformation of the group. 
THEOREM 21. The one-parameter group eG; (AA’) contains one 
identical, one involutoric, two infinitesimal transformations; it con- 
sists of two subdivisions; the group may be generated by either of 
its infinitesimal transformations. 
41. The Parabolic Group pG,(A). All real parabolic 
transformations of the points on a line which have the same 
invariant point A form a one-parameter parabolic group, des- 
ignated by pG,(A). The parameter of the group is t and the 
law of combination of parameters in the group is expressed 
by t,=t-+t,. This group contains a transformation corres- 
ponding to each number in the real number system. The 
identical transformation of the group is given by t=0; the 
transformation corresponding to t= © is a pseudo-transfor- 
mation of the group. 
This group contains two subdivisions: Subdivision I con- 
tains all transformations for which t is positive; subdivision 
II, all for which ¢ is negative. The boundaries of the two 
subdivisions are the identical and the pseudo-transformations. 
The resultant of two transformations belonging to the same 
subdivision is a transformation belonging to that subdivision. 
The inverse of every transformation in one subdivision is a 
transformation in the other subdivision. 
The parabolic group pG,(A) contains two infinitesimal 
transformations, viz. : those corresponding to t= +4, where 3 
is an infinitesimal. Each subdivision of the group contains an 
infinitesimal transformation; and each subdivision may be 
generated by its infinitesimal transformation, but not by the 
