of Refraction at one or more Spherical Surfaces. 491 



17. The principal foci F, W of a system of lenses may be 

 determined geometrically as in § 8. 



Thus, let there be two lenses L 1? L 2 , whose principal foci 

 are (F v F',), (F 2 , F' 2 ), and principal points (R x , R^), (R 2 , R' 2 ). 

 Then (fig. 9) ; since parallel rays on emergence come from F 2 , 

 F 2 is the image of F in L 2 . Hence the line joining X x and 

 F 2 on the y axis gives F on the x axis. Again, since parallel 

 rays on incidence go to F^, and thence to F', W is the image 

 of F\ in L 2 . Hence the line joining X 2 and F\ on the x axis 

 gives F / on the y axis. 



In the construction, of course, any pairs of conjugate points 

 may be employed instead of the principal points. 



18. In the system of § 16 the images ft> 1? ft> 2 , . . . which an 

 object o> at R produces may be determined as follows: — 



Let (fi,f\), (f2yf / 2)j • • • be the principal focal lengths of 

 L 1? L 2 , . . . Then since (§ 20) in a thick lens the ratio of the 

 object to the image is that of their respective distances from 

 the nodal points, we have (fig. 10), 



coo "NA ~B fl F 1 -B -/V 

 In like manner we have 



/2 ~d 2 



Ol / 2 



=ft vkV =^077-77-; &c 



O0O1 / 1/ 2 



Hence if a> = co n , R , R„ become the principal points of the 

 system, and 



dodi • • • = /1/2 • • • 



d'idV • • —f\f f 2' • • 



19. The equation for the system of lenses being - + — =1, 



referred to the principal points, the corresponding equation, 



f f 

 when the nodal points are origins, becomes - +^=1, in 



which f f , p are measured from N, and /, p' from W. 



20. The lengths of object and image at various pairs of 

 conjugate points may now be compared. 



Thus (fig. 7), if &) at P gives ft/ at P', we have* 



ft> + ft) / _ PR _ PF 

 ft/ ~PF~RF + 



* Vide Croullebois, LentiUes ipaisscs, p. 32. 

 2K2 



