S2 KANSAS INIVKKSirV (J^AR•|■KR1,^ . 



Two conies of the system S' degenerate into riglit lines; viz.: 

 the line Ojo parallel to AA.' and the line AA ' itself, excluding the 

 segment included between 1 and 1.' The degenerate conic Ooo has 

 for tangents all lines parallel to AA.' The transformation determ- 

 ined by Ooo gives therefore an identical transformation. The 

 characteristic constant of this identical transformation is given by 



the relation -t^ r^i^'='i<^ ^o. 



AP AP 



Pi'opci'tx 2. — The ,^roiip (i ,' couiains our iiloitical tiajisfoiinatioti 

 whose cliai-actcristii cotisfani is zero. 



The transformation determined by the degenerate conic Aoo A ' 

 must now be examined. Tlie tangents to Aoo A ' from all points 

 on 1 intersect 1' in A.' Therefore ever}' point on the line 1 is trans- 

 formed to A. This is a pseudo-transformation; its characteristic 



constant is given h\ the relation . .^ —-~z=^ — oo. But on the 



AP AA 



other hand the tangents to Aoo A ' at A form a pencil of lines cutting 

 1 ' at all points. Therefore A is transformed into every other point 

 on the line. This is also a pseudo-transformation: its character- 

 istic constant is given b\' -.~r .^=^ = co. These two iiseudo-trans- 



^ - AA AP ' 



formations form an inverse pair. 



J^ropcrty j. — T/w group Ci ,' contains two psciidfl-transformations 

 whic/i form an inverse pair. T/ie e/iaracteristie constants of these t7vo 

 psciido-transfonnations arc oo and -oo . 



If T,, and T .^ be a pair of inverse transformations, the charac- 

 teristic constant of their resultant transformation is given by 

 a-f-( — a)=o. Hence their resultant is the identical transformation 

 of the group. 



J^ropcrty p.. — The combined effect of any transformation of the i;roi/p 

 and its inj'crsc is equivalent to tlie identical transformation of tiic x'r''i'p- 

 77///.V T,T ,=To. 



As stated above the degenerate conic Ooo produces the identical 

 transformation of the group G ,' . The infinitesimally narrow 

 hyperbola of the system S' which touches O at dO, a point on 1 

 infinitesimally near to O, produces an infinitesimal transformation. 

 The point dO may be either to the right or to the left of O, hence 

 the infinitesimal transformation may be either positive or negative. 

 Let d represent its characteristic constant. If the infinitesimal 

 transformation be repeated an infinite number of times, the result 

 will bo equivalent to a finite transformation of the same group 



