42 Lord Rayleigh's Investigations in Optics. 



or 



u v ^ \r sj 



_&^i/vp-iL i(' + -fyi(< + *)\ . (8) 



2 (. u ' \r s/ u\r uJ v \s v/j K y 



The quantity within brackets in (8) may be written 



4(l_i)_f(l_i) + (£!_^)(l + i + I). 



u \r s/ u\r s/ \v u/\v u cc/ 



On substitution for of its approximate value from (8), 



becomes a factor of the whole expression, and we get 



(1_1){4_£_ ( ^_ C) (1 + 1 + I)1. 



\r s/ lu' u v y \v u cs/ J 



Again, from (5) with sufficient approximation for our pur- 

 pose, 



d c d fi/d , c /2 \ 1 , f . d , f N 1 



— = — — -/ - + -7)+ - = — {fid — c) — —(fjud—c) — ; 



u u u c\r u / r v ' cu x cr 



so that the bracket in (8) assumes the form 



-O^P-Jjf^+i+ia+i}. . (9) 



v y \r s/ leu' v u cr cs J 



From (6) and (7), 



Using this in (9), we get 



so that 



From this we see at once that in the case of an equiconvex 

 or equiconcave lens, for which - + - = 0, the aberration va- 

 nishes if - + - = 0, i. e. if the primary focus be at the same 

 u v 



distance on one side of the lens as the radiant point is on the 

 other. 



