10 PROCEEDINGS OF THE CANADIAN INSTITUTE. 



8. The principal foci F, F' of a system of two surfaces S^, S2 con- 

 stituting a lens may be found as follows :-— 



Let (Fj, F'l), (F2, F'2) be the principal foci of S^ and 83, which 

 cut the principal axis in A, B, respectively, so that AF^ =f\, 

 AF'i =/'i, ... In Fig. 4 take the points X^ {fi,f\) referred to A 

 as origin, and Xg {f'^jf^^ referred to B. Then since parallel rays on 

 emergence from the system come from F2, F2 is the image of F in Sj. 

 Therefore the line joining X^ and Fj on the y axis will cut the x 

 axis in F. 



Again, since parallel rays on incidence go to F'^ and thence to F', 

 F' is the image of F'^ in 82- Therefore the line joining X2 and F'j on 

 the X axis will give F' on the y axis. 



The principal foci of any system of surfaces may be determined 

 in like manner. 



9. In the case of a lens the distances AF, BF' may be readily 

 found as follows in terms of f-^, fi,... 



From the similar triangles FAF2, XjF^Fj (Fig. 4) we have 



AF F'lXi , . AF /i 



— ^ ^ that IS — "^' 



AF2 F'lF./ /2-e /\+/2_e' 



where AB = e. 



Also from the similar triangles F'BF'^, 1L^^^)S\ 

 BF' _ F2X2 BF' _ f. 



BF, ~ YWl' """/V^l^/'x+Z.-e- 

 These values can also be deduced from the relation of § 3. Thus, 

 taking the x axis of the figure, 



F,F . F',F2 =fj\, &c. 

 10. In the system referred to in § 6 the lengths of the images 

 (wi, W2 • • •) which an object wq at Rq produces at B,, B2 • • • ^^7 ^^ 

 determined as follows : — 



Let Oi be the centre and /,, f\ the principal focal lengths of Sj, 

 .&c. 



Then (Fig. 5) 



Wo OiRo FjE. d f 

 In like manner, 



ii)^ d f 



