V > 



of Refraction at one or more Spherical Surfaces. 489 



determined as Helmholtz does (Optique Physiologique,^. 74), 

 or as follows: — 



Let Oi be the centre and/i,/^ the principal focal lengths 

 of Si, &c. 



Then (fig. 5) 



*>i = QiRi == gig _ /i = d\ 

 In like manner, 



w 2 _ J2 d 2 a. 



cox d x /V 

 From these relations we find 



o>o c/ ^i /'l/'a 



^3 _ f\fifz _ d / 1 d / 2 d / 3 n 



«o~«MA~/'i/VV ' 

 Hence if ft> =ft>n, the nth equality becomes =1, and the 

 points R , R« the principal points of the system. Thus, if 

 n = 2,d d 1 =f 1 f 2 ,d\d' 2 =f\f 2 . 



Also, since AR 1 =f\—d\, BH 1 =f 2 — d x , we have 



4+^i=/ / i+/,-«5 



and the values of the principal focal lengths become 



^_ /1/3 j?/__ / 1/ 2 



J ~~f\+f2~e> J -f\+f2-e' 



11. Now let R, R/ be the principal points, F, F' the prin- 

 cipal foci of a thick lens ; so that we have 



f -+ f -^T=l (4) 



p p v ' 



Fig. 6, in which Xis the point (/,/0> exhibits the method 

 of finding the conjugate of a given point. 



12. Conjugate points will be nodal points N, N' when 

 on the x axis we have NN / = RR> / . This will evidently 

 happen when (fig. 6) the line through X makes FN = FX. 

 RN(=/ / — /) on the # axis will then be equal to R'N 7 on the 

 y axis. 



If distances are measured from the nodal points N, W y 



f f 

 equation (4) becomes J — 1-^=1, in which /', p are mea- 

 p p 



sured from N, and/, p' from N'; and the conjugate points 

 are determined as in fig. 7. 

 Phil. Mag. S. 5. Vol. 18. No. 115. Bee. 1884. 2 K 



