182 The Aether as an Elastic Solid. 



to become imaginary for certain kinds of light, in order to 

 explain satisfactorily both the surface colours of the aniline 

 dyes and the strong reflecting powers of the metals. 



Dispersion was the subject of several memoirs by the 

 founders of the elastic-solid theory. So early as 1830 Cauchy's 

 attention was directed* to the possibility of constructing a 

 mathematical theory of this phenomenon on the basis of 

 Fresnel's " Hypothesis of Finite Impacts "f i.e. the assumption 

 that the radius of action of one particle of the luminiferous 

 medium on its neighbours is so large as to be comparable with 

 the wave-length of light. Cauchy supposed the medium to 

 be formed, as in Navier's theory of elastic solids, of a system 

 of point-centres of force : the force between two of these 

 point-centres, m at (x, y, z), and //, at (x + A#, y + Ay, z + Az), 

 may be denoted by m^/(r), where r denotes the distance between 

 m and p. When this medium is disturbed by light- waves pro- 

 pagated parallel to the z-axis, the displacement being parallel 

 to the #>axis, the equation of motion of m is evidently 



- 



+ p ) - rT , 



where denotes the displacement of m, ( -i A If) the displace- 

 ment of p, and (r + p) the new value of r. Substituting for p its 

 value, and retaining only terms of the first degree in A?, this 

 equation becomes 



DT r dr 



Now, by Taylor's theorem, since depends only on z, we have 



Substituting, and remembering that summations which 

 involve odd powers of Az must vanish when taken over all 



* Bull, des Sc. Math, xiv (1830), p. 9 : " Sur la dispersion de la lumiere," 

 . Exevcwe* de Math., 1836. t Cf. p. 132. 



