12 PROCEEDINGS OF THE CANADIAN INSTITUTE. 



15. If F is the image of K, and K' of F', then on the asjaxis of 

 Fig. 8 we have 



FK . FF' = FS . F'S = FF' . F'K' 



Hence 



//' 

 FK = F'K' = —• 



2A 



Also, if T, T' are conjugates snch that FT = F'T', then 



FT2 = FT . F'T' =//'. 

 It thus appears that the middle point of FF' also bisects the lines 

 KK', SS', RN"', E'N, TT' and {vide § 28) YV. 



16. The method of § 6 may be applied as follows to a system 

 of lenses. 



Let there be any number of lenses L^, Lg, . . . whose principal 

 foci are (F^, F'^), (Fj, F'j) . . . , and whose principal planes cut the 

 common axis in (A, A'), (B, B') . . . 



Let (Eq, Ri), (Rj, Rj) . ■ • be pairs of conjugate points such that 

 RqFi = 5o, RjF'i = b\, R1F2 = ^1, . . . In like manner let (Pq, Pj), 

 (Pj, P2), ... be any other set of conjugate points such that RqPo. ^ 

 ^0, RiPi = /i, . . . 



Then (§ 7) 



- + — = 1, &c. ; 

 Pi Pi 



from which by eliminating jOi = — ^'1, ^2 = — Pit • • • we get an 



equation of the form 



- + 4- = i' 



Po Pn 



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

 system. 



1 7. The principal foci F, F' of a system of lenses may be deter- 

 mined geometrically as in § 8. 



Thus, let there be two lenses L^, Lg, whose principal foci are 

 (Fi, F'l), (F2, F'2), and principal points (Rj, R'j), (Rg, R'2). Then 

 (Fig. 9), since parallel rays on emergence come from F2, F2 is the 

 image of F in L^. Hence the line joining X^ and Fg on the y axis 

 gives F on the x axis. 



