289 



LIGHT. 



LIGHT. 



270 



into two rays, of which neither lies in the plane of incidence. The 

 same phenomena take place when light passes from a dense to a rarer 

 medium, except that in this case the whole of the light may under a 

 certain incidence be totally reflected. 



This alteration of the path of light passing from one medium to 

 another, which is familiarly observed in the apparently bent form of 

 a straight stick partially immersed in water in an oblique direction, is 

 called refraction. As double refraction is of comparatively rare occur- 

 rence among the instances of refraction which ordinarily fall under 

 our notice, we shall first attend to the laws of single refraction. 



Let N P c represent a solar beam in vacuo and incident at e on a 



transparent medium (as water), to the surface of which D c K is normal. 

 When the medium is fluid, place a graduated circle D 8 E in the plane 

 of incidence with its centre at c ; a portion of the light will be reflected 

 in the direction L, and another entering the medium will be refracted 

 in the direction c R. If uninfluenced by the medium, its direction 

 would have been c s. The angle K c F. is the angle of refraction, DON 

 or K c s of incidence, and s c K of deviation. The arcs D P, D I,, art- 

 equal by the law of reflection, and if we compare the arcs D p, E R, 

 their sinea will be found in a constant ratio, depending on the nature 

 of the medium, but independent of the angle of incidence. Thus if I 

 be the angle of incidence, and R that of refraction, the two are con- 

 nected by the simple relation sin I = n sin n. The constant ^ peculiar 

 to the medium is called its index of refraction. When the medium is 

 solid, we can easily compare the tangents of the angles, and thence 

 their sines. The above law will be found rigorously exact. 



This law may be accounted for on the theory of emission. Let v 

 be the velocity of the ray before incidence, which is decomposable 

 into a horizontal velocity, v sin I, and a normal one, v cos I. The 

 former will not be affected by the medium ; the square of the latter 

 will be increased at the confines of the medium by a quantity ro', which 

 is the sum of the products of twice the force into the element of the 

 normal throughout that inappreciable space in which the forces of the 

 medium do not destroy each other, in consequence of proximity to the 

 urface. Therefore the normal velocity of the refracted ray is 

 V ! v 2 cos 3 I + * j and its actual velocity V { y2 + " 2 1 > so that the 

 horizontal velocity in the medium is V (v- + n : ) ein R, which being 

 equated with v sin I, its value before incidence gives sill I=ju sin R, 



Vtv-'+n') 

 where n= 



How are we to account for the reflected ray c L / Why is not the 

 whole incident light refracted ? Even when the incident light is per- 

 pendicular to the refracting surface, a portion of the light is reflected ; 

 and when the ray has but a very Email inclination to the surface, a 

 portion will yet be intromitted. Hence we may consider generally 

 that the incident light consists of portions which are differently dis- 

 posed to be subject to the repulsive and attractive forces of the medium, 

 or, in Newton's language, are in Jiti of eay nflexi'in or traiamialon. 

 When the angle of incidence increases, the normal velocity of the ray 

 diminishes, the effect of the repulsive forces is therefore augmented, 

 or the reflexion is more copious. 



If r, / be any portions of the incident and refracted rays measured 

 to fixed points in their directions, and v, V the corresponding 



dr 

 velocities, and we make A c B the axis o x, wa have sin I = -T-, 



dr 1 d 



sin n = -7-, and since v sin i = v' sin n, therefore -7- (v r + v* )') = ; 



ax x 



and vr + v'/ = minimum, which result is agreeable to the dynamical 



'le of least action. 



On the undulatory theory each portion of an incident wave, as it 

 cirrives at the reflecting and refracting surface, is conceived to be the 

 of a disturbance which spreads out 'in each medium with the 

 velocity of propagation appropriate to that medium. Thus, setting 

 aside the incident waves, the disturbance in the two media is conceived 

 of as the aggregate of the disturbances due to these various elementary 

 or secondary waves, a mode of conception the legitimacy of which 

 rests directly on the general dynamical principle of the superposition 



all motions. In this way the laws of reflection and refraction 

 are simply deduced [UNDULATORY THEORY OF LIGHT], and with 

 respect to refraction, it is shown that the ratio of the sine of incidence 



to the sine of refraction is that of the velocity of light in the first 

 medium to its velocity in the second. , Hence, according to this 

 theory, light must travel more swiftly in vacuo than in a refracting 

 medium in the ratio of /u to 1, while, according to the theory ,pf 

 emission, it must travel more slowly in the inverse ratio. This leads 

 to a crucial experiment for deciding between the two theories, which 

 is decisively against the theory of emission. [UNDULATORY THEORY.] 



The fact that the differently coloured rays have different refractive 

 indices has been thought by some to offer a great difficulty to the 

 undulatory theory, inasmuch as the velocity of propagation within 

 refracting media must depend on the periodic time of the disturbance, 

 which is contrary to the laws of elastic fluids. The circumstances 

 are, however, different, as the fluid in this case envelops the material 

 particles of the medium, instead of constituting one uninterrupted 

 mass. Besides, the fact of dispersion has been accounted for on this 

 theory, and even its law (in the case of substances of low dispersive 

 power) deduced, by adopting certain dynamical hypotheses not im- 

 probable in themselves. 



If n be the index of refraction when light passes from vacuum to a 

 medium (A), and /u' when it passes from vacuum to a medium (A'), 



u! 

 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 A, 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 



equation sin i' = . sin I; and the second angle of refraction or third 



n' 



of incidence, by the equation sin I = /i' sin H' : whence sin i' = sin n'. 



Hence, generally, if the emergent ray be svipposed 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 waves 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 R = sin I, it follows that R is a right angle when sin i = ,u, 



or the emergent ray is then parallel to the surface ; but if sin i > p., 

 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 cuse of light 

 refracted by a glass prism. Let /i, /u' be the indices of refraction into 

 the medium through its first bounding plane, and out of it through the 

 second, and let I, R, be the first angles of incidence and refraction, l', 

 R', 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 l=jusinR; sin i'=/t' sin R'; a=R + f; D = R +1 a: 

 thus i being given, the first equation determines H, the third i', the 

 second R', the fourth D. 



rfR' 



When the deviation is a minimum we have -r- = 1, and generally 

 -7- 



-J- -7- and by differentiating the other equations we have 



R R 

 cos i =n cos B . ^j-; cost'. -^ =M' cos R'; 



therefore cos I cos i' = p /*' cos n cos R' 

 by squaring (1 sin- I) (1 /u' 2 sin' 2 n') = p' a (p- sin 3 1) (1 sin' R'); 



or if m = then(m 2 1) cos* I = 0x s 1) cos 2 n' 



and ( 1 ; 



= ( 1- 1 cos' i'. 



When the ray after the second refraction moves in th same 



