76 KANSAS LXIVI'.KsnV (Jl'ARTKRLN. 



Tlu'orriii 4. — There are 00'* projeetive iransfonnatioiis T of the 

 />(>in/s on a litie: each of these leaves a /^air of points invariant. Among 

 t/iese are 00- projeetive transformations T ' . <-ae/i of li'/iieh ieai'cs one 

 and oniv one point invariant. 



^2. One-termed Groii])s of Piojertive Transformations. 



Our next object is to subdivide and to classify these oo-^ project- 

 ive transformations of tlie points on the line 1. We consider first 

 the quadrilateral ABB 'A.' (Fig. i ). X system of 00' conies ma}- be 

 described touching the sides of this quadrilateral. Call this system 

 S. Each of these conies determines a projective transformation 

 which has A and B for its invariant points. Each conic of the 

 system S touches the line 1 at a different point: and every point of 

 the line 1 is the point of contact of some conic of the system S. If 

 C be the point of contact of any conic K, the characteristic anhar- 

 monic ratio of the transformation produced by K is given by the 

 anliarmonic ratio of the four points A, B, C, O. The points A, B, 

 () are fixed, while the point C is movable. From the continuity of 

 the point system on the line 1 we infer the continuity of the system 

 of 00' transformations which leave A and B invariant. 



Let T)^^ be the transformation of this system which transforms P to 

 P, ; then k,=(ABPP,). Let T]^.^ be the transformation of the same 

 s}'stem which transforms P, to P^; then kg;=(ABP , P^). The 

 two transformations T]^- and T|^^ are together equivalent to a single 

 transtormatifin ot tlu' same s\stem which transforms P to P.,. To 



prove this we liavu k , = ( AHPP, )= A|^ :-^^^. and k.,=:(ABP, P, ) 



AP. AP.. ,,,. . . , , . ■■ 1^ , 



^ - - -p : V3^ Yj • r-lnninatiMg the traction contaming r^ irorn 



AP AP, 

 these two equations we Inave k ^ ko= ^3-^ '.^15— *- = (ABPP._, ). The 



conic of the system S whose tangential anharmonic ratio is k^k^ 

 gives a transformation which is equivalent to the combined effect 

 of T)^^ and Ti^._. This may be expressed symbolically by the equa- 

 tiod Ti^,Ti^-,= Ti^j^._. In the same way it may be shown that the 

 cOjmbined effect of an\- number of transformations of the system is 

 equivalent to some single one of the same system. Thus T-iT^T,. 



invariant points of tlu' transformation, but does not explicitly identify his third 

 parameter witli tlic cliai acteristic anharmonic ratio. This last, however, is implied in 



equation (4). p. 121 of ■ Conl. Gruppen." viz.; -^ -=k^ -. It cannot be to strongly 



X,— n X— n 



emphasized that the three parameters of Lie's equation of transformation mean 

 geometrically the coordin;ites of the invariant points and the characteristic anhar- 

 monic ratio. ' In regard to transformations of tlie type T" our result is in compleia 



accord with Lie's eiiuation (7) p. 1-3. "Cont. Gruiiin)!." viz.: ■= --k. 



X,— m X— m 



