FIRST ORDINARY MEETING. H 



Prom these relations we find 



&c. 



Henee if Wq = ^m each of the nth equalities becomes equal to 1, 

 and the points Rqj ^n ^^^ principal points of the system. 

 Thus, if n = 2, d^d^ ^^-Af-i, and dJ^d!^ =f\f\. 

 Also, since A R^ =/'i — dl^, BRi =/2 — d^, we have 



and the values of the principal focal lengths become 



11. Now let R, E,' be the principal points, F, F' the principal foci 

 of a thick lens ; so that we have 



■^ + 4' = 1 (4) 



p p 



Fig 6, in which X is the point (/, /'), 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 x 

 axis will then be equal to R'N' on the y axis. 



If distances are measured from the nodal points N, N', equation 



f f 



(4) becomes 1 , ^ !> in which y, p are measured from N, and 



p p 



f, p' from N' ; and the conjugate points are determined as in Fig. 7. 



13. These figures make the existence of self- conjugate points 

 manifest. Thus in Fig. 7, if S is such a point, we have 



FS. F'S =//', FS + F'S = FF' f= 2h. 

 JTence FS, F'S are the roots of s^ _ 2As +//' = 0, and the self- 

 conjugate points are at equal distances from F, F'. 



14. Fig. 8 exhibits the construction when one of the self conjugate 

 points is taken as origin. 



From the similar triangles PP'P, S'P'X, and also PSP, FSF, we 

 obtain the relations 



PSP ' _ PP _ PP _ SP 



"ST ~s^"~FF~sr' 



