FIRST ORDINARY MEETING. 9^ 



6. The following proposition which is employed by Helmholtz 

 (vide Optique Physiologique, p. 72), I have modified by changing 

 his notation and applying the rule of signs (§ 1), in order to exhibit 

 the result of the elimination in a symmetrical form. 



Let there be any number of spherical refracting surfaces whose 

 principal foci are (F^, F\), (F^, F'^}, &c., and which cut the common 

 principal axis in A, B, C, . . . Let CR^, R^), (R^, R^) ... be pairs 

 of conjugate points, such that Ro^^i = ^O) ^i^'i = d'l, ... In like 

 manner let (Pg, Pj), (Pj, Pj) ... be any other set of conjugate points,, 

 such that RqPq = Pq, RiPi = p'l, . . . Then by § 4 



Po Pi 



l^ + ^^ = l,&c. 

 Pi Pi 



Also by the rule of signs (§1) we have p-^ = — p\, p^, = — p'2, . . . 



Hence, on eliminating these quantities, the position of P„, the point 



conjugate to Pq with reference to the system, is determined from an 



equation of the form 



•^+^ =1 (3) 



Po Pn 



where /= R^F, f = R„F', F, F' being the principal foci of the 



system. 



The values of— for 2, 3, 4 . . . refractions are, respectively, 

 "o 



di did^ d^d^d^ 



do + d'l d^d^ + d'ld^ + d\d'^ d-^d^d^ + d\d^^ + d'-^d'.^dz + d\d\d\ 



/' /' /' 

 and the corresponding values ^^ -jr , ~^ ^ -^ ■> ^^^^ 



cZ'i d\d'i d\d\d'2, 



^1 + d'l d-^d^ -f- d\d2 + d\d'2 d^d^^ + • • • 

 7. The construction of § 5 (Fig. 2) applies to equation (3), and 

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



dd' ==//', i^ + ^ = l. 

 -^-^ ' D D' 



The latter, it may be observed, also follows from (3), since Rq, R„ are^ 

 any conjugate points. 



