emch: projective groups. 21 



special value of the characteristic anharmonic ratio. By any of 

 the perspective collineations belonging to such a system a point A 

 is transformed into A^. Taking another collineation of the same 

 system and A ^ as an original point in it, the corresponding point will 

 be A' '. Whenever now A and A' ' are related in such a manner as to 

 be a collineation of the given system, i. e., subject to the same 

 conditions, and, inversely, if each point A^ can be transformed 

 back into its corresponding A by a collineation of the S3'stem, 

 such a system of collineations is said to be a continuous group of 

 collineations, or simply a group of collineations. By this state- 

 ment it is easy to enumerate ths groups which may occur in the 

 general and special cases of perspective collineations. We ma}^ 

 however, occasionally avail ourselves of a theorem of Lie concern- 

 ing a criterion of groups by means of invariant properties of 

 transformations. The theorem is: 



"The system of all projective transformations of the plane 

 leaving a certain figure invariant has the property of a group. 

 The transformations of the system are inverse by pairs."* 



In enumerating the groups we follow the order of chapters 

 2 and 3. Thus, we have first to consider the general perspective 

 collineations. In Fig. 11 we assume 1 as the axis common to oo-* 

 perspective collineations of the plane and two collineations of this 

 system determined by (CLAAi) and {CjL,A'A"), or (claa^) and 

 (c^la'a") respectivel}', where C and C, are the centres of the two 

 perspective collineations. The transformed point to A in the first 

 collineation is A' and the transformed point to A' in the second 

 collineation A". Taking an other point B on a, the points B' and 

 B" can be constructed, or in general, to each point on a there 

 are coresponding ones on a' and a". 



Now the point-range AB....is perspective to the point-range 

 A'B'.. ..with C as the centre and, again, A'B'.. ..perspective to 

 the point-range A"B"....with C^ as centre. The three ranges 

 have a self-corresponding point, S, on 1. Hence, the point-ranges 

 AB....and A"B"....are also perspective, i. e., AA", BB "... . 

 pass through one and the same point C" which obviously may be 

 taken as a new centre of collineation with 1 as an axis, and A, A"; 

 B, B";....as coresponding points. As is immediately seen, to 

 each transformation in a perspective collineation may be found its 

 inverse belonging to the same system. We have therefore the 



TlicorcDi II. The system of all persprctix'c collnwatiojis h-aviiig the 

 points of a strcu\i^ht line Invariant fon/is a thrcr-ti-rnwd i^roi/f. 



'Lie's " Vorlesiingen," page Hit. 



