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



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 + FS = FF=2/*. 



Hence FS, F'S are the roots of f-2hs+ff = 9 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 PPT, S'PX, and also PSP, 

 FSF, we obtain the relations 



PSF _ P T _ PP _ SP 



ST ~ S'X "" FF ~ SF* 



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

 of fig. 8 we have, § 7, 



FK . FF'=FS . FS = FF. FK'. 



Hence 



FK=F'K'=«§L 



Also, if T, T are conjugates such that FT = F / T / , then 

 FT 2 = FT . F'T'=//'. 



It thus appears that the middle point of FF / also bisects the 

 lines KK', SS', UW, K'N, TT' and (vide § 28) W. 



16. Helmholtz's method (§6) may be applied as follows to 

 a system of lenses. 



Let there be any number of lenses L 1? L 2 , . . . whose prin- 

 cipal foci are (F ly F^), (F 2 , F^) . . ., and whose principal 

 planes cut the common axis in (A, A'), (B, B') . . . 



Let (R , R x ), (Rj, R 2 ) ... be pairs of conjugate points such 

 that R F! = Bo; RiF / i = B / i, RiF 2 = di, ... In like manner let 

 (P , Px), (Pi, P 2 ), ... be any other set of conjugate points 

 such that R P = p , 'R 1 Pi^=p\ J . . . 



Then, § 7, 



do . B'i _-. 



"T" / J- ? 



Po Pi 

 ^+ B >=1 ,&c; 



Pi P2 



from which, by eliminating p x = — p' 1 ,p 2 = —p' 2 , . . ., we get 

 an equation of the form 



P0 Pn 



where /=RoF,/' = R»F'; F, F' being the principal foci of the 

 system. 



