the Colours -of Mixed Plates. 681 



second set being o£ inappreciable intensity except near 

 the centre of the halo. The position of the dark rings in 

 the first set is given by the angle of diffraction #, where 



x ) 



t(l— a sin i)(fi cosi — v 1— /j? sin 2 i) — 8 — (2n — 1) ^ [ n\ 



/ul sin i = sia r, and sin 6 = fi sin 2(r—i) . j 



In this formula, t is the thickness of the film, p, the 

 refractive index, X the wave-length and 8 is the phase- 

 change occurring in total reflexion, and n is any integer 

 (counting from zero). 



The position of the dark rings in the second set is given 

 by the formula 



(fj,-l)t + ±tsmd-^==(2n + l)~. ... (2) 



The two sets of rings would be superposed on each other 

 near the centre of the halo. 



3. Comparison of Theory and Experiment for Uniform Films : 

 Case of Normal Incidence. 



It has already been indicated in the second paper of the 

 series that there is a general qualitative agreement between 

 the facts observed and the results of the theory outlined above. 

 This becomes clearer on representing the expressions for the 

 path-difference of the interfering ra}^s given in formulae 

 (I) and (2) graphically. Taking p. for egg-white to be 1*35, 

 the expressions t(l — p, sin i)(/ju cos i— \/l— p? sin 2 i) and 

 t{(p,— 1) +f sintff in formulas (1) and (2) respectively have 

 been calculated as fractions of (/jl — l)t and plotted in fig. 1 

 n gainst the corresponding values of 0, the angle of diffraction. 

 It will be seen that in one case the path-difference diminishes 

 continuously (at first rapidly and later more slowly) from 

 (fi — V)t up to a very small fraction of it for the largest angles of 

 diffraction, while in the other case the path-difference actually 

 increases with 6. We should accordingly have in the former 

 case a series of circular rings in the halo, closer together near 

 its centre and wider apart near its margin. The value of S, 

 the phase-change in total reflexion, decreases gradually 



from -- to zero as increases. The outermost ring in white 



light corresponding to a negligibly small path-difference is 

 therefore achromatic, while those following within it should 

 Phil. Mag. S. G. Vol. 42. No. 251. Nov. 1921, 2 Z 



