the Spherical Aberration in a Lens- System. 89 



Substituting for — — — + - as above in terms of s and v 

 r u 



the coefficients of 2® 1 / j u 2 and S^/h? are easily seen to be 

 equal and each may be written 



The whole expression for the longitudinal aberration 

 is thus 



+2( ,- 1 )Ce 1+ e,){(, + i)(i-iy + <i-'-±i)e- 1 ,)} 



where ©! and © 2 have the values given above. 



Some writers give the spherical aberration as a correction 



to - instead of v. Assuming this correction to be 



-(A/ + B/), 



the longitudinal aberration for a thin lens is 



l/Q -Ay 3 -B/)-0 or (A?/ 2 + B</> 2 + A 2 */V 



to this order of approximation. Comparing with the above 

 the result for a thin lens (allowing for the necessary changes 

 in notation) is seen to agree with Messrs. Herman and Dennis 

 Taylor's formula as given in the latter's ' System of Applied 

 Optics ' (Appendix, pp. 6 and 13). 



In applying the formula to any lens system a relation 

 between successive y's is needed. 



If di is the distance between the zth and (7 + l)th surfaces, 

 P»N,-, Pi+iNj+i the successive ordinates, Aj, A»+i the vertices 

 of the surfaces, it is easily seen that 



d , l yf__ i yj±i 



y»+i _ -, , ^ + A,-N,--A t - + iN t - + i * 2 r t - 2 r i+1 



2 r» 



to the 2nd power of the y's. 



