60 On the " Loss of half an undulation " in Physical Optics. 



according to the law of the inverse square. Taking account of 

 this circumstance, and the extremely small value of r, it may be 

 presumed that any effect of the reflected velocity on the fluid at 

 a position P, is sensibly due only to reflexions from atoms situ- 

 ated within a very small distance from P. That is, if a be the 

 cUstance of P from a fixed plane, the effect at P of the reflected 

 velocities may be supposed to be due only to atoms situated at 

 distances from the plane between a — a. and a + u, u being very 

 small. Hence, if V be the velocity of the fluid at any distance 

 z, and it be assumed that through the small interval 2«, the 



value of —j— is very nearly constant at each instant, the re- 

 sultant {v) of the reflected velocities is at the distance ^ in a 

 given ratio to V, and the actual velocity at that distance is com- 

 pounded of the velocities V + v and —v. Thus, by the retarda- 

 tion due to the atoms, the effective accelerative force of the fluid is 

 diminished in a constant ratio. Hence the fluid situated at the 

 plane boundary of the medium is subject to precisely the same 

 disturbance as that communicated by the diaphragm in the case 

 above supposed, of aerial waves in a cylindrical tube ; that is, on 

 the fluid in this position an accelerative force is impressed, which 

 has a given ratio to that which would be due to the condensation 

 of the incident waves if there were no disturbance. At the 

 instant of intromittence into the medium, the disturbance acts 

 as a sudden retardation, and corresponds to a value of q less than 

 unity. The reflected wave is, therefore, in the same phase as 

 the incident. There is no reflexion after intromittence, because 

 the retarding force then acts continuously, and has only the 

 effect of altering permanently the relation of the condensation to 

 the velocity, and the rate of propagation. On the passage of the 

 waves out of the medium into vacuum, the retarding force sud- 

 denly ceases, the rate of propagation and the relation of the con- 

 densation to the velocity return to what they were before intro- 

 mittence, and the effect on the fluid situated at the plane 

 boundary of the medium, is the same as if an accelerative force 

 were there impressed, having a given ratio q, greater than unity, 

 to the accelerative force due to the original waves in vacuum. 

 Consequently, the waves reflected back into the medium have 

 the contrary phase to that of the incident waves. In this 

 manner, as I conceive, the facts of observation are shown to be 

 consistent with the undulatory theory of light, and the apparent 

 loss of half an undulation is accounted for. 



Another instance of the supposed loss of half an undulation 

 is mentioned in Art. 966 of Sir John Ilerschel's ' Treatise on 

 Light.' When a plane-polarized ray is bifurcated by transmis- 

 sion through a thin plate of a iiniaxal crystal, and the two por- 



