XLVIII. On the Formation of the Central Spot of Newton's Rings beyond the Critical 

 Angle. By G. G. Stokes, M.A., Fellow of PemhroJce College, Cambridge. 



[Read December 11, 1848.] 



When Newton's Rings are formed between the under surface of a prism and the upper surface 

 of a lens, or of another prism with a slightly convex face, there is no difficulty in increasing the angle 

 of incidence on the under surface of the first prism till it exceeds the critical angle. On viewing the 

 rings formed in this manner, it is found that they disappear on passing the critical angle, but that the 

 central black spot remains. The most obvious way of accounting for the formation of the spot under 

 these circumstances is, perhaps, to suppose that the forces which the material particles exert on the 

 ether extend to a small, but sensible distance from the surface of a refracting medium; so that in the 

 case under consideration the two pieces of glass are, in the immediate neighbourhood of the point of 

 contact, as good as a single uninterrupted medium, and therefore no reflection takes place at the 

 surfaces. This mode of explanation is however liable to one serious objection. So long as the angle 

 of incidence falls short of the critical angle, the central spot is perfectly explained, along with the rest 

 of the system of which it forms a part, by ordinary reflection and refraction. As the angle of inci- 

 dence gradually increases, passing through the critical angle, the appearance of the central spot changes 

 gradually, and but slightly. To account then for the existence of this spot by ordinary reflection 

 and refraction so long as the angle of incidence falls short of the critical angle, but by the finite 

 extent of the sphere of action of the molecular forces when the angle of incidence exceeds the critical 

 angle, would be to give a discontinuous expiation to a continuous phenomenon. If we adopt the 

 latter mode of explanation in the one case we must adopt it in the other, and thus separate the theory 

 of the central spot from that of the rings, which to all appearance belong to the same system; although 

 the admitted theory of the rings fully accounts likewise for the existence of the spot, and not only for 

 its existence, but also for some remarkable modifications which it undergoes in certain circumstances*. 

 Accordingly the existence of the central spot beyond the critical angle has been attributed by 

 Dr. Lloyd, without hesitation, to the disturbance in the second medium which takes the place of that 

 which, when the angle of incidence is less than the critical angle, constitutes the refracted light -f-. 

 The expression for the intensity of the light, whether reflected or transmitted, has not however been 

 hitherto given, so far as I am aware. The object of the present paper is to supply this deficiency. 



In explaining on dynamical principles the total internal reflection of light, mathematicians have 

 been led to an expression for the disturbance in the second medium involving an exponential, which 

 contains in its index the perpendicular distance of the point considered from the surface. It follows 

 from this expression that the disturbance is insensible at the distance of a small multiple of the 

 lenn-th of a wave from the surface. This circumstance is all that need be attended to, so far as the 

 refracted light is concerned, in explaining total internal reflection ; but in considering the theory of 

 the central spot in Newton's Rings, it is precisely the superficial disturbance just mentioned that must 

 be taken into account. In the present paper I have not adopted any special dynamical theory : I 

 have preferred deducing my results from Fresnel's formulte for the intensities of reflected and re- 



• I allude especially to the phenomena described by Mr. Airy 

 in a paper printed in the fourth Volume of the Cambridge Philoso- 

 phical Transactions, p. 409. 



t Report on the present state of Physical Optics. Reports of 

 the British Association, Vol. ill. p. 310. 



