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Oscillatory Theory of Light. 411 



composed of the incident wave and those synchronous with it, 

 and for the set of waves retarded by one quarter of an undulation. 



The results of these conditions have been investigated in 

 detail for singly refracting substances. 



The indices of refraction of such substances are proportional 

 to the square roots of the moments of inertia of the loaded lumi- 

 niferous atoms in a given space. Thus, if the coefficients M', 

 M" are proportional to these moments in two given substances 

 respectively, then the index of refraction of the second substance 

 relatively to the first is 



w 

 w 



In the case of light incident on a plane surface between two 

 such media, the axes of coordinates may be assumed respectively 

 perpendicular to the reflecting surface, perpendicular to the 

 plane of reflexion, and along the intersection of those two planes ; 

 and oscillations round axes normal and parallel to the plane of 

 reflexion may be considered separately. 



When the axes of oscillation are normal to the plane of re- 

 flexion, that is to say, when the light is polarized in that plane, 

 the formulae for the intensities of the reflected and refracted light 

 agree exactly with those of Fresnel. When the reflexion takes 

 place in the rarer medium, the reflected light is retarded by half 

 an undulation ; when in the denser, there is no change of phase 

 unless the reflexion is total, when there is a certain acceleration 

 of phase depending on the angle of incidence. In the last case, 

 the disturbance in the second medium is an evanescent wave, 

 analogous to those introduced into the vibratory theory by 

 M. Cauchy and Mr. Green ; that is to say, a wave in which the 

 amplitude of oscillation diminishes in proportion to an exponen- 

 tial function of the distance from the bounding surface (called 

 by M. Cauchy the modulus), and which travels along that surface 

 w r ith a velocity less than the velocity of an ordinary wave ; the 

 square of the negative exponent of the modulus being propor- 

 tional to the difference of the squares of those velocities, divided 

 by the square of the velocity of an ordinary wave. 



This is an evanescent wave of oscillation round transverse axes. 



How large soever the coefficient of polarity for oscillations 

 round longitudinal axes may be, an evanescent wave of such 

 oscillations may travel along the bounding surface of a medium 

 with any velocity, however slow, provided the negative exponent 

 of the modulus is made large enough. Consequently, in framing 

 the formulae to represent oscillations round axes parallel to the 

 plane of incidence, we must introduce in each medium two such 

 evanescent waves of suitable exponents and indeterminate ampli- 

 tudes ; one travelling along the surface with the incident wave, 



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