tlie A.viid Dioptric System. 



:i29 



Again, all points on a oiven ray incident on one spherical 

 refracting surtace have images through which the corre- 

 sponding emergent ray passes. Hence, the image of a point 

 is known when we know the emergent rays corresponding to 

 two incident rays through the object-point ; and the line of 

 an emergent ray is known when we know the images of two 

 points on the incident ray. This is obviously at once exten- 

 sible to the in-rays and out-rays of a dioptric system. 



The properties^ arrived at in the two preceding paragraphs 

 form the basis of the geometrical treatment adopted in this 

 article. 



Given the a,ms of a dioptric system, ami two extra-axial 

 points P, Q, icith their imafies V, Q'_, to construct the out-ray 

 correspondinq to any given in-ray. 



Let PA, QB, P'A, Q'B' (fig. 3) be perpendiculars on the 

 axis. Let PP^ QQ^ cut the axis in a, /3. Then a is the 

 vertex for A and y8 the vertex for B. 



Fio-. 3. 



Fig. 4. 



R 



P 



Q 





\/3 





A 





A' 

 R' 



B 



^ 



6' 

 Q' 





R 









S 









\ 





^^ 





\ 





A 





\/ 



/ 



A' 



B 



V 



Q' 







A 







/ 



^K 



^'^.--^^ 



P 



/ 





^ 



Q, 







s' ~ 



Let any in-ray cut the planes of A and B in R and S. 

 Let Pa meet the plane of A' in P' and Sy8 meet the plane of 

 B' in S^ Then R' is the image of R and S' that of S. 



Hence R^ S' is the out-ray for the in-ray RS. 



To construct the image for any object-point. — Take any two 

 in-rays passing through it. The intersection of their out- 

 rays gives the required image. 



Now modify the above construction by supposing the 

 object-point at infinity. Draw RS parallel to the axis (fig. 4) 

 and find R^ S' as before. Let it cut the axis in Fg'. 

 All in-rays parallel to the axis pass through F^^ after 

 emergence. 



Thus F2 is the second focal point of the system. Starting 

 with an out-ray parallel to the axis, and reversing this con- 

 struction, we get Fi t\iQ first focal point of the system. 



