its Circular Components in the Faraday "Effect." 4G9 



Since a real or virtual image is always produced by any one 

 or more surfaces and in an infinite number of ways, s i . . . s n 

 may be treated as independent variables. If we take points 

 on infinitely near rays of like order from 0, they will be at 

 equal distances from the intersection or focus, and we may 



8 (N + N/ + N 1 / ' + N 2 ' + N 3 '' + . . . N; + N,/') = 0, 



whereN 1 = N/4-N ] ' 7 &e. 



Taking now the variation with respect to s { . . . s n &c. 

 we have 



+ (N 1 ' , + N./) + ,..(N"..i + N»))=0, 

 or since Si . . . s n are independent 



8U^+NiQ+-.-(N ff .- 1 + N.)} =Q &c _ 



Bs 1 ' 



But N/' + N 2 ' + . . . N„ is independent of s 1 on the above 

 assumptions and so on, hence 



flNo+N/) _ n S(N"„- 1 + N. ) n 



12 ^ i __ h 

 X X x 



l_S/o_ _l_8Jl 



Xq o»i \i 0$i 



g ^=cos(90°-i)=sinj 



But N =?, .*,'=:*-!., Ac. 



Hence 

 But 



and -^-' = cos (90°-r) = sin r, 



where i and r are the angles of the ray respectively with die 

 normal to s v Hence 



1 . . 1 . sin i X n 



— sini= — sinr or = — . 



X X x sin r X l 



Since X X t 



r o T i 



we have, if r = r { also 



sin i _ ?j 

 sin r ~~ v, ' 



