Cambridge Philosophical Society. 137 



Dec. 11. — On the Formation of the Central Spot of Newton's 

 Rings beyond the Critical Angle. By G. G. Stokes, M.A., Fellow 

 of Pembroke College, Cambridge. 



It has long been known that when Newton's rings are formed be- 

 tween the under surface of a prism and the upper surface of a lens, 

 or of a second prism, so as to allow of increasing the angle of inci- 

 dence at pleasure, the rings disappear when the critical angle is 

 passed, but the central spot remains. The existence of the spot 

 under these circumstances has even been attributed to the disturbance 

 in the second medium, which, when the angle of incidence exceeds 

 the critical angle, takes the place of that disturbance which at a 

 smaller incidence constitutes the refracted light ; but the expression 

 for the intensity has not hitherto been given, so far as the author is 

 aware. The object of the author in the present paper is to supply 

 this deficiency. 



The author has not adopted any particular dynamical theory, but 

 has deduced his results from Fresnel's expressions for the intensities 

 of reflected and refracted polarized light. When the angle of inci- 

 dence becomes greater than the critical angle these expressions be- 

 come imaginary. "When the imaginary expressions are interpreted 

 in the way in which physical considerations show that they must be 

 interpreted, it becomes easy to obtain the expression for the intensity 

 of the light, whether reflected or transmitted, in the neighbourhood 

 of the spot. When the first and third media are of the same nature, 

 the following expression is obtained for the intensity (I) of the re- 

 flected light, the incident light being polarized in the plane of inci- 

 dence, and its intensity being taken for unity. 



In this expression jw, is the refractive index of the fii*st medium, i the 

 angle of incidence on the surface of the second medium, or inter- 

 posed plate of air, D the thickness of that plate at the point con- 

 sidered, A the length of a wave in air, 2 9 the acceleration of phase 

 due to total internal reflexion. When the light is polarized perpen- 

 dicularly to the plane of incidence, it is only necessary to replace 2 fl 

 by 2 f , the angles 9, ^ being those so denoted in Airy's Tract. The 

 intensity of the transmitted light is obtained by subtracting that of 

 the reflected light from unity. 



From the expression for the intensity, the author has deduced the 

 following results, all of which he has verified by observation. 



The spot is comparatively large near the critical angle, and becomes 

 smaller and smaller as the angle of incidence increases. Near the 

 critical angle the fainter portion, or ragged edge, of the bright spot 

 seen by transmission is broad ; at considerable angles of incidence 

 the light decreases with comparative abruptness. Towards the edge 

 of the spot there is a predominance of the colours at the red end of 

 the spectrum, causing the ragged edge to appear brown. Near the 

 critical angle the spot is larger for light polarized perpendicularly to 

 the plane of incidence than for light polarized in that plane : at con- 



