the Colours of Mixed Plates. 869 



geometrical optics. Part o£ the light incident on the curved 

 meniscus would be reflected within the liquid, and if the 

 angle of incidence were greater than the critical angle, the 

 reflexion would be total. Part would also be refracted into 

 the rarer medium, and after a second refraction emerge 

 again into the liquid. If these disturbances finally travel in 

 parallel directions, such as those shown by H and K in the 

 plane of the diagram, the path-difference between them may 

 be readily evaluated and shown to be 



t(l— fi sin i)(fju cos i — \/l— /-t 2 sin 2 z) + 6\ . . (1) 



In this formula, i is the thickness of the film, /jl the 

 refractive index of the liquid, and i the angle of incidence 

 on the meniscus of the light which is twice refracted. 8 is 

 the correction necessary on account of the change of phase 

 in total reflexion. If i be nearly equal to zero, both pencils 

 emerge nearly parallel to the direction of the incident rays, 

 £ has the value which obtains at nearly grazing incidence, 

 and the expression for the path-difference reduces to 



(/*-l>-A,/2 (2) 



The light scattered through small angles thus interferes 

 under a difference of path which is the same as that of the 

 legnlarly-transmitted pencils less half a wave-length. This 

 agrees with what we should expect on the simple diffraction 

 theory, the scattered light being of a colour complementary 

 to that due to the interference of the regularly-transmitted 

 pencils. For larger angles of scattering, the difference of 

 path between the interfering pencils given by (1) steadily 

 falls off in magnitude, and finally becomes zero when the 

 angle of incidence on the meniscus is just equal to the 

 critical angle, as the pencils then become coincident and 

 $ is also equal to zero. We should thus expect to observe a 

 series of maxima and minima of intensity in the scattered 

 light in different directions, which is exactly what is found 

 in experiment. The deviation of the interfering pencils 

 within the liquid film is given by 2(r — i) where //- sin i~ sin r, 

 and when this is equal to (ir — 2a) where a is the critical 

 angle, the path-difference vanishes, and we should expect 

 the scattered light to be achromatic. The angle of scattering 

 6 on emergence from the film is given by the relation 

 sin 6 =/jl sin 2(V— i). It is worthy of note that, as the correc- 

 tion 8 for the change of phase in total reflexion depends on 

 the plane of polarization of the incident light, the positions 

 of the maxima and minima in the scattered light should be 

 slightly different for light polarized in and at right angles 



