410 Mr. \V. J. M. Rankine's General View of an 



ing the ratio of tiese quantities as finite, there is obtained an 

 equation of the sixth order, representing a wave-surface of three 

 sheets, differing somewhat from that of the propagation of vibra- 

 tions in an elastic crystalline solid ; inasmuch as the former has 

 always three circular sections, while the latter has none, unless 

 it is symmetrical all round one axis at least. By increasing the 



A 



ratio tt without limit, this equation is made to approximate 



indefinitely to the product of the equation of Fresnel's wave- 

 surface by the following, 



which represents a very large ellipsoidal wave of oscillations 

 round axes parallel to the direction of propagation. 



5. Of Reflexion and Refraction. 



According to the proposed hypothesis of oscillations, the laws 

 of the phase and intensity of light reflected and refracted at the 

 bounding surface of two transparent substances are to be deter- 

 mined by conditions analogous to those employed in the hypo- 

 thesis of vibrations by M. Cauchy and Mr. Green. They are 

 the consequences of the principle, that if we have two sets of 

 formulae expressing the nature and magnitude of the oscillations 

 in the two substances respectively, then either of those formulae, 

 being applied to a particle at the bounding surface, ought to give 

 the same results. 



According to this principle, the following six quantities for a 

 particle at the bounding surface must be the same at every 

 instant, when computed by either of the two sets of formulae : — 



The three angular displacements round the three axes of 

 coordinates, 



The three rotative forces round the same three axes. 



There is, generally speaking, a change of phase when light 

 undergoes refraction or reflexion. It is known that we may 

 express this change of phase by subdividing each reflected or 

 refracted disturbance into two, of suitable intensities and signs ; 

 one synchronous in phase with the corresponding incident dis- 

 turbance, and the other retarded by a quarter of an undulation. 

 There are thus twelve quantities to be found, viz. the amplitudes 

 of the six components of the reflected disturbance, and those of 

 the six components of the refracted disturbance. To determine 

 these quantities there are twelve conditions, viz. the equality at 

 every instant, according to the formulae for either medium, of 

 the total angular displacements, and of the total rotative forces, 

 round each of the three axes of coordinates, for the set of waves 



