306 



PROCEEDINGS OP THE AMERICAN ACADEMY 



through which the ray is bent, or i" — r" =^v — ^\„ . , or subtract- 



sin I 



ing, 



■.v\^ 



sin (/ — r) 



sin I 



F^] 



sin i — sin (i — r) cos r _ 



|_cos r 

 which, with the above values, gives n' 



BUI I COS r 



sin I 



i" 

 ; but ?»'=-, 



Sub- 



■r)' 



sin I — COS/- sin (i- 



stitutiug this value in the equation /'=:/ cos r 2,_ gives /'=/ 



(n' — \) sin/-sin(/-7-)cosr^ ^j^.^ ^^^^j^^ ^^ ^j^^ ^^^^^^ distance if 

 ^ ' sin (i — r) 



the rays on emerging remained in the plane of the section of the lens. 

 But they pass into a plane inclined to this i — r, hence the observed 

 focus^y"'' will be such that when projected on the plane of the section 

 it will equal/', or /" cos (t — r) =/'. Hence finally /" = / 



/ -, , sin J — sin (j — r) cos r rri • i i. , i 4.^ „..,%: 



(n — 1) — — -. ^ ~ -. This last step may be open to cnti- 



^ '' sni {i — /•) cos (« — r) i J 1 



cism, but the close agreement with observation seems to justify it. 



In Table IV., this formula is compared with observation, as the law 



for the vertical slit is compared in Table III. The columns in the two 



tables correspond, and it will be noticed that the agreement is very 



close. 



TABLE IV. 



The principal practical application of these results is to photographic 

 lenses. It will be seen that a single lens, even if perfectly corrected for 

 spherical and chromatic aberration, is still subject to this defect. Con- 



