NI''.\VS()N: I'KOJ I'X'TIVl': TRANSI'OKMAIiONS. 8l 



P into p.. Then -.-:^ . .^^ ^a. Also let Ti' be another trans- 



' AP AP, ' 



formation of the same S3'stem which transforms P^ to P„; then 



—— — — -^b. Eliminatinij; the fraction , - from these two 



APj AP2 AP, 



equations, we have . -^ — — -=:a-l-b^c. Bnt this relation is the 



^ AP AP2 



result of a transformation T,' which transforms P to P.,. T,r be- 

 longs to the same system and is equivalent to the combination of 

 T;^ and T,^. Thus TjiTi^^T,:. In the same way it may be proved 

 that the combined effect of any number of transformations of the 

 system is ecjuivalent to some single transformation of the same 

 system; and that the characteristic constant of the resultant trans- 

 formation is equal to the sum of the characteristic constants of the 

 component transformations. 



Here again we have the fundamental property of a continuous 

 group of transformations. The variable parameter of the one- 

 termed group is the characteristic constant. A group of this kind 

 will be designated by G [ ; sometimes it will be written G^ when it 

 is desired to call attention to the invariant point of the group. 



Tlicornn 6. — TJic totalHy of thr projeclivc tratisfoi-iiiatioiis icJiich 

 leave one and oiilv o)ie poijit of a line //17'arian/ for/iis a o)ie-ie!iiied 

 group Gj , lidiose fundaniental property is tliat tlie eonibined effect pro- 

 duced by t7c<o or more of t lie transformations of tlie group is equivalent 

 to that of a single transformation of the same group. Tlie characteristic 

 constant of tlie resultant transformation is equal to tlie sum of tlie char- 

 acteristic constants of the component transformations. 



Other properties of a group of this kind G ,' are easily established. 



We shall show first that every transformation T., of the group 

 G,' has an inverse belonging to the same group. For if T.J^ be the 

 transformation of the group that transforms P to P,, we have the 



relation-—- -r-=^-= a. The transformation of the group that 



AP AP . 



transforms P, back to P gives the relation-r^;:^ — -=constant := 



' ^ APj AP 



— a. This latter exists whenever the former exists. 



We can see in this case just as in the former case that the two 

 conies K and K' which determine a pair of inverse transforma- 

 tions are so situated that one is the reflection of the other on the 

 line OX. 



Property i. — The transformation of the group G ,' mav he ar/'anged 

 in pairs: eacli pair consists of a t/-ansfo/-;nafi<'ii ami its i/n'crse. 



