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

 Hence 



w 



PF PN 



ft/ ~ FX " FN" 



the relation on which is based the definition of nodal points. 



It would seem preferable, however, after having proved the 

 existence of nodal points*, to reverse these steps, and from 



FH = FP t0 deduce i? = YK' &C ' 



21. Again, if o> at N gives &/ at N', 



ft)/ _ ft//' 

 KN~R'N' ; 

 6) __ ft)' 



•'• 7 /_ 7 ; 



that is, the apparent magnitude of co at F is equal to that of 

 ft/atF. 



22. If « at S gives ft/at S, then (fig. 7) from the similar 

 triangles SNS, SFX, XFS we have 



ft>_ NS_ f_ _SF 



co' ~ N'S " SF ~ / ' 

 In like manner, if co at S' gives &)" at S', we have 



ft) _ SF' _ f 



ft)" ~ / ~ SF* 

 Hence from these two relations we have 



a) ft)' &)" 



/'~SF = SF* 



23. If &) at K gives ft/ at F, and w at F' gives &/' at 

 K', then (fig. 7) 



© _ NK _ FX _ /' 

 ©' " N'F ~~ FF ~ 2A ; 

 and 



_© NF _ FF' _ 2A 

 co" "~ N'K' ~ FX ~ /' 



II. 



24. The geometrical method of the preceding sections may 

 also be extended to the case of reflection at one or more 

 spherical surfaces. A few examples will suffice to illustrate 

 the method. Thus, for a convex mirror F and F' are coin- 



* Vide Helmholtz, Optique phy&iologique, p. 75. 



t 



