DISPERSION. 49 



(61) The conclusion of analysis to which we have just 

 adverted, namely, that the velocity of wave-propagation 

 is constant in the same homogeneous medium, is deduced 

 on the particular supposition, that the sphere of action of 

 the molecules of a vibrating medium is indefinitely small 

 compared with the length of a wave. If this restriction 

 be removed, we have no longer any ground for concluding 

 that the waves of different lengths will be propagated with 

 the same velocity ; and the conclusion hitherto acquiesced 

 in must be regarded but as an approximate result. It was 

 in this point of view that the question presented itself to 

 M. Cauchy . Eesuming the . problem of wave-propagation 

 with the more general equations, he has proved that there 

 exists, generally, a relation between the velocity of propagation 

 (or the refractive index in vacuo) and the length of the wave; 

 and, therefore, that the rays of different colours will be dif- 

 ferently refracted. 



(62) Let us make, for abridgment, 



z 2* 27T 



/4 ' = 1T> s = 



A T 



in which X and r denote, as before, the wave-length and time 

 of vibration. M. Cauchy has proved that k and s are con- 

 nected by an equation of the form 



s* = atf + ajt + a^ + &c., 



in which the coefficients a^ a z , a 3) &c., vary with the medium. 

 Now the velocity of wave-propagation is 



F =-=v 



r k 

 consequently, 



&c. 



