If a be the index of refraction when light passes from vacuum to a medium (A), and a' when it passes from vacuum to a medium (A'), then is the index when the ray is transmitted directly from the former to the latter.
For if we look at a star through a medium bounded by parallel planes, as a plate of glass, its position will not be affected, and therefore the emergent light is parallel to the incident ; but since the second angle of incidence is equal to the first angle of refraction by the parallelism of the planes, and the second angle of refraction is equal to the first of incidence by the parallelism of the rays, therefore the index of refraction out of a medium into vacuum is the reciprocal of that from vacuum into the medium. Again, if we place in optical contact two plates of different refracting media n, A', as for example a horizontal plate of glass covered with water, the emergent light is still parallel to the incident. Now the second angle of incidence or first of refraction is given by the 1 equation sin = . sin I ; and the second angle of refraction or third of incidence, by sin I = sin : whence sin t = sin Hence, generally, if the emergent ray be supposed to become incident, the latter will take the place of the emergent.
This fact shows that the velocity of light which traverses several media is the same as if transmitted directly from vacuum to the last medium, which is consonant to both the theories of light. In the wave theory, the velocity of the wages in a medium is independent of the mode of their propagation, and in that of emission the incre ment of the square of the velocity generated at one surface of a medium is destroyed by like forces on its emergence at the second, so that the only increment it finally receives is that generated by the surface of the last medium it enters, and which it would receive if it entered this medium directly from vacuum.
The index of refraction is greater than unity from a rarer to a denser medium, and less than unity from a denser to a rarer. Hence in the latter case there is a limit to the angle of incidence, beyond which it is impossible for the ray to emerge into the rarer medium, for since sin u = 1 sin a, it follows that R is a right angle when sin 1:=/.4, ' A or the emergent ray is then parallel to the surface ; but if sin i > then sin R > 1, which is impossible. Observation shows that the light
is then totally reflected.
Let us now trace the progress of a ray passing through a medium terminated by planes inclined at a given angle a, as in the case of light refracted by a glass prism. Let a, IL' be the indices of refraction into the medium through its first bounding plane, and out of it through the second, and let a, n, be the first angles of incidence and refraction, it', the second, and D the total deviation, and suppose the plane of incidence to be perpendicular to both planes, so that there may be no deviation in planes ; the following equations fully describe the progress of the ray : sin sin ; sin sin ; a χ ; D = χ : thus a being given, the first equation determines R, the third a', the second the fourth D.
When the deviation is a minimum we have 717 = 1, and generally de du ; and by differentiating the other equations we have dR do cos a =1.t cos it . cosi'. =IA' oos u'; therefore cos I cos 1!=a cos it cos by squaring a) a) (1 le); Or if m = then a= COO 1 2 1 and (1 it = When the ray after the second refraction moves in the same medium as before the first incidence, we find the miniumni deviation when iv: and ; the internal part of the ray is then equally a D+ a inclined to both planer. We have then it f(n+s) which affords a simple method of determining the index sine.
of refraction of media capable of being formed into prisms.
NYheri light emanating from one point is refracted accurately to another, if r, represent the Incident and refracted rays, and an arc of the curve by the revolution of which the refracting surface is dr generated, j, 4t are the sines of the angle. of incidence and re dr fraction (abstracting from their algebraical signs), therefore ± ;Az Lad rt coma, : this equation belongs generally to a curve of the fourth order, but If r be infinite, or the Incident light parallel to the axis, it gives a conic section, and if the arbitrary constant vanishes the equation represents a circle. If one surface be given, it is easy to find second by which homogeneous light may he refracted accurately to a given point..