Mr. stokes on THE FORMATION OF THE CENTRAL SPOT, ETC. 



fracted polarized light, which in the case considered became imaginary, interpreting these imaginary 

 expressions, as has been done by Professor O'Brien*, in the way in which general dynamical con- 

 siderations show that they ought to be interpreted. 



By means of these expressions, it is easy to calculate the intensity of the central spot. I have 

 only considered the case in which the first and third media are of the same nature : the more general 

 case does not seem to be of any particular interest. Some conclusions follow from the expression 

 for the intensity, relative to a slight tinge of colour about the edge of the spot, and to a difference in 

 the size of the spot ascending as it is seen by light polarized in, or by light polarized perpendicularly 

 to the plane of incidence, which agree with experiment. 



1. Let a plane wave of light be incident, either externally or internally, on the surface of an 

 ordinary refracting medium, suppose glass. Regard the surface as plane, and take it for the plane 

 xy ; and refer the media to the i-ectangular axes of w, y, z, the positive part of the latter being 

 situated in the second medium, or that into which the refraction takes place. Let /, m, n be the 

 cosines of the angles at which the normal to the incident wave, measured in the direction of 

 propagation, is inclined to the axes; so that ra = if we take, as we are at liberty to do, the axis 

 of y parallel to the trace of the incident wave on the reflecting surface. Let V, F, V denote the 

 incident, reflected, and refracted vibrations, estimated either by displacements or by velocities, it 

 does not signify which ; and let a, a , a denote the coefficients of vibration. Then we have the 

 following possible system of vibrations : 



V = a cos — (vt — Ix — nz). 



F= a^cos — {vt - Iw + nz), 

 V = a cos — (v t — I X - n z). 



{A). 



In these expressions u, u' are the velocities of propagation, and X, X' the lengths of wave, in the 

 first and second media; so that «, v , and the velocity of propagation in vacuum, are proportional 

 to X, X', and the length of wave in vacuum : H is the sine, and n the cosine of the angle of incidence, 

 I' the sine, and ri the cosine of the angle of refraction, these quantities being connected by 

 the equations 



- = -, n = '\/l-f, n = y/l - I'-. (1). 



V V 



2. The system of vibrations (A) is supposed to satisfy certain linear differential equations of 

 motion belonging to the two media, and likewise certain linear equations of condition at the surface of 

 separation, for which z = o. These equations lead to certain relations between a, a., and a', by virtue 

 of which the ratios of a^ and a' to a are certain functions of /, v, and v, and it might be also 

 of X. The equations, being .satisfied identically, will continue to be satisfied when I' becomes 

 greater than 1, and con.sequently n imaginary, which may happen, provided v > v ; but the 

 interpretation before given to the equations (A) and (l) fails. 



When n becomes imaginary, and equal to i/\/- i, v being equal to y/l'' - I, z, instead of 

 appearing under a circular function in the third of equations (A), appears in one of the exponentials 



«**"'-, A;' being equal to ^ • By changing the sign of \/- 1 we should get a second system 



X 

 of equations (A), .satisfying, like tlie first system, all the equations of the problem ; and we should 



Cambridge PUtoiopMcal Tramactioim , Vol. viii. p. ■JO 



