374 Lord Rayleigh, On the minimum aberration [Apr. 19, 



The general expression of the aberration for parallel rays is * 



while r, s, and f are connected by 



7=(-M'-i) • ■••<^>- 



Writing for brevity R, S, F respectively for r~\ s~\ /~\ and 

 II W C 

 taking G = r- , so that — S= R, we get 



^ V / sJ \f SJ 



= G {R' (1^ + 2) - RG {2fi + l) + ,^G'] 



Since /u, > 1, both terms are of the same sign, and the aberra- 

 tion can never vanish. If / and y be given, the aberration is 

 least when 



that is, when 



2.(/. + 2)(/.-l) 

 ^—^(2^ + 1) J ^^^- 



The corresponding value of s is 



so 



that 



—-m^^ <^). 



which agrees with the result of § 130. 



When this condition is satisfied, the second term of (4) gives 

 for the minimum aberration 



_ ;;/ • ^ - > (^^ ~ ^) (R\ 



'J-f 8(^-l)M/^ + 2) ^^^' 



which is applicable to all values of /*. 



* Parkinson's Optics, § 129. 



