NKWsrix : PKC)ii:<-ilvi: i ranmokm a rioxs. 



8(1 



invariant are determined b} conies which touch 1, 1.' and O or.: but 

 since these three lines meet in a point, it follows that all these 

 conies must be degenerate, and each must consist of linear seg- 

 ments terminating at O. Ever}' transformation of this kind is a 

 pcrsprctive transfflnnaiioii. The point Q from which the projecting 

 lines are drawn ma\- be an\" point in the ])lane. Q may therefore 

 have 00- different positions: and we see that there are oo- perspect- 

 ive transformations, each of which leaves the point O invariant. 

 These oo- perspective transformations form the two-termed group 

 (ioq. This sidD-group contains all the perspective transformations 

 in the general projective group, and no others. 



Theorem rj. — A// tlie perspeetii'e irausfoniiatiiVis eoiitai)ieJ in the 

 i:;eiieral projeetii'e group f(>n/i a tioo-t ei' Died siih- group. Go,-,: this sub- 

 group contains no ti-ansforniation luhielt is not perspective. 



We now proceed to the consideration of the one-termed sub- 

 groups which com- 

 pose the two-termed 

 perspective group. 

 Each of these sub- 

 groups has the point 

 O and some other 

 point as A for invari- 

 ant points. For oo' 

 positions of the point 

 Q the resulting per- 

 spective transforma- 

 tions leave the point 

 A as well as O in- 

 variant. It is easy to 

 see that these oo^ 

 positions of O must 

 all be on the line AA.' 

 because the second 

 invariant point of a 

 perspective transfor- 

 mation is found b}' 

 dropping a perpen- 

 dicular from O on 

 OX. Thus the oo' 

 perspective transfor- 

 mations obtained b^■ 

 taking the center of 

 perspective at all point 

 one-termed group. 



a line perpendicular to OX form a 



