488 Prof. J. Loudon on Geometrical Methods in the Theory 



f ft ff 

 and the corresponding values for ~, ^-, ^-, . . . are 



d\ d\d' 2 d\d\d' z 



d 1 + d\ d x d 2 + . . . dxdzdz + . . . 



7. The construction of § 5 (fig. 2) applies to equation (3) ; 

 and from the figure we at once deduce, as in § § 3 and 4, the 

 general relations for any system 



dd'=/f, b + J = 1- 



The latter, it may be observed, also follows from (3), since 

 R 0? R w are any conjugate points. 



8. The principal foci F, F / of a system of two surfaces Si, S 2 

 constituting a lens may be found as follows: — 



Let (F 1? F'i), (F 2 ,F' 2 ) be the principal foci of S x and S 2 , 

 which cut the principal axis in A, B respectively, so that 

 AF 1= /„ AI",=/„ ... In fig. 4 take the point X, (/„/',) 

 referred to A, and X 2 (/ 2 ,/ / 2 ) referred to B. Then, since 

 parallel rays on emergence from the system come from F 2 , 

 F 2 is the image of F in Si. Therefore the line joining X x and 

 F 2 on the y axis will cut the x axis in F. 



Again, since parallel rays on incidence go to ¥\ and thence 

 to F ; , W is the image of F'i in S 2 . Therefore the line joining 

 X 2 and F'i on the x axis will give F' on the y axis. 



The principal foci of any system of surfaces may be deter- 

 mined in like manner. 



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

 readily found as follows in terms of / x , / 2 , . . . 



From the similar triangles FAF 2 , XiF'iF 2 (fig. 4), we have 



AF _ F'iXi ,,,. _AF /i 



AF 2 ~F'iF 2 ; tnatls / 2 -,-/^ 1+/2 _^ 



where AB = «. 



Also from the similar triangles F'BF'd X 2 F 2 E" 1 , 



BF_F 2 X 8 BF _ f\ 



bf; - F,IY 1S 'f\-e ~f\ +/,-• 



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

 Thus, taking the x axis of the figure, we have 



Fjr. F'A ■-/,/'„ &c. 



10. In the system referred to in § 6 the images (a 1} a> 2 , . . .) 

 which an object co at R produces at R 1? R 2 , . . ., may be 



