Corrections in a Newton' s-rings System. 505 



Again, 



i' — <£ = ijr — tan ty = sin yfr=p, . . . (iii.) 



to the order considered, which, combined with (ii.), gives 



r' z=r+p. 

 Hence 



sin i' = N sin r' = N sin (p + r) = N(p cos r + sin ?*) . 



Similarly from (iii.)? 



sin i' = sin (p + fa) =J9 cos <£ + sin fa 



and therefore p cos <£ + sin (£ = N(p cos r-f sinr), 



which, solving for cos <£ and reducing, gives 



, . , oc . . f +/W-sm 2 i\ 



cos 9 = cos i -f -o-sm z < 1 — ¥. . > . 



J* v. cosi J 



Hence coscp is no£ equal to eosz, but is subject to a small 

 first-order correction. It is important to notice that the 

 correction vanishes to the first order for normal incidence. 



Turning now to the case (fig. 2) where the ray is incident 

 at P', a point in the YZ plane, the direction-cosines of the 

 ray after refraction at the plane surface of the lens are 



Z 2 = sinr, w 2 =cosr, ?? 2 = 0. 



If the tangent at P' makes an angle -yjr with OZ, the direction- 

 cosines of the normal at P' are 



p' = 0, q' = cos-yfr, r' = cos (90° + ^)= — sin yjr. 



Then if fa and fa be the angles between the ray and the 

 normal before and after refraction at P', Z 3 , m 3 , n 3 the 

 direction-cosines of the ray after refraction at P', 



cos fa = m 2 q f = cos r cos-\|r, 

 cos fa = m z cos ty — n z sin yfr. 



From the general formulae for refraction, where yu, and yt! 

 are the " absolute " refractive indices for glass and air * 



(so that ^ =N), 



fll 2 — fl% =( /J cos fa — /Ub' cos fa)p' , 

 /jLm 2 - u'm 3 = (/a cos fa — p cos fa)q', ]> . . , (iv.) 

 fin 2 —fi'n% = (fi cos fa — fjf cos fa)r' . 

 * Heath, < Geoui. Optics/ p. 21. 



