RKTTGER. — NOTE ON THE PROJECTIVE GROUP. 497 



A oc^ of the singular transformations jTwill move a given point p of 

 general position on the line ar^ = + c, 3:3 = ^-, to a given point pi of 

 general position on the line Xi == — c , x^=^ k . Nevertheless, we can 

 find one non-singular transformation that will do the same, namely, 



re 1 = — Xi , 



3-'. 2 = A Xi + Xo , 

 it" 3 = x^ . 



For clearly, by a proper choice of A, this transformation T^ has the 

 same effect when applied to a definite given point as the transformation 

 7' for any given values of M and N (N:i: 0). 



Example 3. 



^sPi^ X3P2, X1P1+ 2x.2po. 



The oc^ of non-singular transformations T^ have the form, 



x\ = e"^ Xi-j- ~ (e''3 — l)xs, 

 x\ = e-''3 3^2 + ^ («"'''' — 1) ^3, 



L C3 



3; 3 = ajg . 



The oc^ of singular transformations have the form, 



x\ = — a;i + Mx^ , 



ar'o = x<2-\- Nx^ , (iVrt 0) 



a: 3 = a^3 . 



By means of the latter a given point p of general position on the 

 plane x^ = k can be transformed into a given point p^ of general posi- 

 tion in that plane. But it is easily seen that Ci, c.,, and c^ of 7^ can be 

 chosen in oc^ of ways so that T^ will produce the same effect. 



Example 4. 



XsPi , X3P2 , 2 a-i JO2 -f 3 x.po + x-sps . 



The cc^ of non-singular transformations T,. have the form, 



x\ = e^''^ Xi -\ (fi^'3 — e<^3) Xg, 



c>. 



VOL. XXXIII. — 32 



