tlic AxiaJ Dioptric Si/sfem. 339 



Let Qi be another old i)oint in Dj. Draw Qi /) cutting 

 D, Do in R. 



Draw Up' cutting C D^ in Q/, the new point for Q. The 

 proof depends on the fact that the anharnionic ratio of 

 four colHnear points is unaltered by the transformation, for 

 by construction 



0{CD:PjP/}=0{CDip/}=E{CD,p/}=R{CDiQiQi'} 

 .-. {CD, P,P,'} = {GD,Q,Q/} .-. {CD,PiQ,} = {CD,Pi'Qi'}. 



We can thus find the new point for any old point in a 

 double-line, if ojie old point in that line along with its new 

 point are given. 



Section TV. 



We shall now give a brief analytical treatment of the 

 general transformation of Section III. 



Taking CAB A' B' cc /3F QF' Q' to denote the same 

 points as before, let us take A B C as the fundamental 

 triangle for areal coordinates, and let the coordinates of the 

 various points be as indicated here : — 



A. A'. B. 



/3. 



P. 



Q. 



P'. 



Q'. 



I -a 



a—\ 



2 U 



The equation to Pa is 

 ' 1 . V 



1 J_ 

 1-6 1- 







1 1 



1-16 1-2^ 







1 



Vi 



_1_ 



1 q-\ 







p-1 







= 0, or xoi,p-\-yp-\-zu = 0. 



i ^_^ 

 I l-aa-l 



That of the line CA^ is 



x-\-ya = 0. 



The intersection of these is P^. Hence 



Xii yi : z^=ao(,: a: p{aci— 1) 



The equation to Q/8 is xPq-\-yq-\-z = 0. 



That of CB^ is x+yh = 0. 



Hence we have for the coordinates of Q^ 



x,.:y^:z, = hp:l3:'j{hp-l) (5) 



Z2 



(2) 

 (3) 



