OF ARTS AND SCIENCES. 803 



angle of incidence, the next four columns the observed conjugate 

 focus, u. or position of the slit when the telescope was focussed on a 

 point seen tlirough the lens at a distance of .5/,/, 1.5/, and 2/, in 

 turn. The next four columns give the computed value of f, assuming 



that a lens placed obliciuely conforms to the law 1 — = -:;, as well 



^ ^ •' U V f 



as when in the ordinary position. The result justifies this assumption ; 

 for the four values oi f are nearly coincident, and agree well with the 

 mean given in the last column. The phenomena are thus greatly sim- 

 plified, since we have now only to consider the case of the principal 

 focal distances, or that the incident ray forms a parallel beam. 



To represent these results theoretically, let us suppose the slits and 

 lens so small, compared with their distance apart, that we may neglect 

 all aberration except that due to the obliquity of the incidence. Con- 

 sidering first the case of the vertical slit, let Fig. 1 represent the sec- 

 tion of a horizontal plane passing through the centre of the lens. 

 Then let D represent the position of the slit when the emergent rays 

 are parallel ; that is, when AB is parallel to A' G. Now CD=.f' is the 

 new focal length which is to be determined. Caliy the principal focal 

 distance, n the index of refraction, i and r the angles of incidence and 

 refraction of the light on entering the lens, and r' and i' the corre- 

 sponding angles on its emergence. Call also A the angle between 

 the two surfaces of the lens at its edge, or of the two surfaces where 

 pierced by the ray. Then, by the law of re- 

 fraction, sin i ■=. n sin r, and sin i' = n sin r' 

 = n sin (r -|- A) = n sin r -\- n A cos r = 

 sin i -\- n A cos r, since r' = A -\- r and sin A 

 being very small may be regarded as equal 



to A. Again, sin ^' — sin ^ = cos i {i' — ^) ^^ n A cos r, and hence, 

 V — i — n cos r 



A cos i 



Now, in the triangle BCD we have BDC=i — i' — A, BCD=i 

 90 — i, and BD sensibly equal to/'. Again, BC=.fA (n — I), for 



by the formula for lenses — =(«— 1) (— -^-p^, or f= _ , 



but 



2BC r BC 



A ■=: „ - •■• /"= -i-, TT' or BC := fA (n — 1). Since the sides 



R 



/=T(7=n)'-^^=/^4(«-l) 



are proportional to the sines of the opposite angles ^Z) : J? (7= sin 

 BCD : sin BDC, or/ i/A (?i — l) = sin (90—0 = i-i'-^ or/' =/. 



— "~ ^"^ ' ; dividing by A, and substituting the value of —r- given 



above, we have j' =j ■ =/• / , — . -,'. : = J' 



' -^ •' n cos r — cos i •' y /j- — sin- i — cos i 



