256 Abhot and Fowle^ Jr. — Aberration of Prisms » 



^P- cos(,^--a^') («> 



BH = BDcos(<^-8<^") (1) 



But yasin.(<^'+ 8<^') =:sin. {<^ + 8<j!)) (9) 



^ , 1 COS <f> . , 1 sin <f) / co8^<f>\-,„ , - 



whence 8<^' = - . f, 8<^- — . — ^( I — , — ^, )8<^^ . . . (10) 



fX COS <fi 2/x COS <^ \ />t'cos'<^ 7 ^ ' 



Similarly /x sin . {<}>' — B<f>') — sin . (<l>-8<t>") (11) 



whence 8«^" = Bcf> (12) 



Substituting these values, expanding and reducing we have, 

 e cos^<f) i / , tan d>' cos <f> \ I /sind)C0S< 



/W, COS (^ { \ fl GOS<f>/ 2fx\ COScf) 



tan <!>' . cos> \ ] 



and 



( / tan <f>' cos rf> \ I /sind). 



= ]2t^(tan<^ ^. ttI+o ( ^ 



( \ /A cos<^ / 2/x \ cos' 



<i>' 



''^^.'-^)\h. (U) 



For the rays incident at the angle (<^ — 8^) we must give h(f> 

 the negative sign in these expressions. Hence if v^ and v^ 

 are the focal distances for the rays incident at the angles 

 {(1> + S(j>) and {<I> — B^) respectively, we may express v^ and v^ a& 

 follows : 



v^ = u + Al+BB<f> + CB<f>''-\-T>B<}>'+ .... (15) 



v^ = u + Al-B8cf>-^C8cf>'-DS<fy'+ .... (16) 



where A, B, C and D are factors independent of 8^ ; 



whence y^-v, = 2B8<^ + 2D8<^'+ . . . (17) 



or, neglecting powers of 8<^ higher than the second, the total 

 longitudinal aberration Av is 



Av=:2B8<^ (18) 



^ / tan di' cos d) \ I /sin <A . cos <f> 

 where B = 2u(tm<b — . ^ ) + — ( , , — 



\ fX C08 <f>'/ 2fJi\ cos'<^ 



tan^' co^X - (19> 



