506 Corrections in a Newton' s-rings System. 



From the first of these we have 



and therefore l 3 = N sin r, 



Hence 



7?7 3 2 -f7i 3 2 = l-/ 3 2 = l-N 2 sin 2 r. . . . (v.) 



From the second or third of (iv.) we obtain 



m s sin -yjr + n 3 cos ty = N cos r sin yjr. 



Substituting this value of n 2 in (v.), we obtain, if tan 2 yjr 

 be negligible compared with unity, 



m 



2 



3 — 



1 — N 2 sin 2 r = cos 2 i. 



If, then, <fr be the angle between the ray in the film and a 

 line parallel to OY whose direction-cosines are therefore 

 0, 1, 0, we have from the usual formula 



cos (£ = w 3 = cos i. 



It seems, then, that the correction, which is of the first- 

 order in the plane of XY, and only vanishes there at normal 

 incidence, is of the second order in the YZ plane, vanishing 

 there for all angles of incidence. These results are in 

 substantial agreement with those of Wangerin *. 



If it be desired, therefore, to eliminate first-order cor- 

 rections, the analysis points to the following experimental 

 precautions, (a) Either the light must be used at normal 

 incidence, or (/3) if obliquely incident light be used, mea- 

 surements must be restricted to those rings for which ^ is 



small compared with unity. But, in this case, it is only to 

 the measurement of diameters in the plane of incidence to 

 which this restriction applies. Even with oblique incidence 

 the diameters of the rings measured perpendicularly to the 

 plane of incidence are only subject to a second-order 

 correction. 



* Loc. cit. 



