90 K A \ S A S r NM \- T''. K S T I A" (.» t ' A RT V. K 1 A . 



This conclusion, which is thus deduced from the consideration 

 of the line OQ as a degenerate conic, is capable of a direct geomet- 

 ric proof as follows: If we take on the line AA ' fFig. 5), two 

 points Q and O' as centers of perspective transformation, we can 

 show that these two perspective transformations are together equiv- 

 alent to one whose center of perspective is also a point on ;VA.' 

 With Q as the center of perspective, P and R are projected into P' 

 and R' and transformed into Pj and R,. With i) ' as a new cen- 

 ter, P^ and R, are projected into P" and R". The joins of P and 

 P", and of R and R" meet in Q" . 



According to tlie construction the two triangles P,P'S and 

 Rj R ' T are homologous; because the sides P ,S and R, T meet in O.' 

 P'S and R'T meet in O, also P|P' and R,R' meet at iniinit\' on 

 AA.' Hence the lines joining corresponding vertices must meet in 

 a point. Thus P,R|. P'R.' and ST all meet in O. 



Consider now the triangles PP"S and RR"T: these are likewise 

 homologous, tor the lines PR, P"R", ST meet in C). Hence the 

 intersection of corresponding sides must lie on a line. But PS and 

 RT meet in O, P"S and R"T meet in (_)' : therefore PP" and RR" 

 must meet in Q", a point on AA.' 



There are 00' perpendiculars to the line OX each ot which 

 determine a one-termed sidi-group of the two-termed perspectix'e 

 group. One of these one-tt-rmetl sid)-groups is of the tvpe Ci J . i. e. 

 it leaves one and onh' one jioint in\'ariant. This particular sub- 

 group is determined b\' the line () 00 })eri)endicular to OX. 



'riirorriii /./.- Tlir co- />i-rsf>rctirr /r(iiis/oriii(i//<>//s. /(/r i^'liich thr 

 crnirrs of frrsprctiiu- lir on a line prrfrtiiliciiloi' />> the hisrctor OX. 

 form a onr/rniinf i:;roi/f. 



Let us e.xamine a little closer the j-)erspfctive transformation 



exhibited in Fig. 2. The characteristic anharmonic ratio k is 



determined by tlie range (OAPP,): thus k=(OAPP,)r= 



OP OP, OP AP, ,, OP sinr/ , ,,^ .^«i"Y 



PA ■ P,A = op;- AP • ^^"^ OP, =sin^^ ""^ '^^^^^si,^^ 



(- j .-. ■.- \ ' r~i 



AP,=;\'0-^ — . Hence we have kr=: ^ --^^ . When the pf)int (,) is 

 ' 'sm^r A(J 



at infinit\- on AA.' we have the identical transformation. The 

 infinitesimal transformation of the group is obtained by taking Q 

 on AA ' so far away that the angles at Q are infinitesimalh' small. 

 Two points Q and O ' equally distant from OX give a pair of in- 

 verse transformations. When Q falls at A and at A.' we have the 

 two pseudo-transformations. When Q falls on OX, we have the 

 involutoric transformation of the group. 



