;;08 Lloyd's Fringes for Internal Reflexion. 



IT 



Both these tangents vanish at (/> = ~ y and therefore the 



change of phase at this incidence— grazing incidence — is 

 or 7T. As cf) decreases each tangent increases in magnitude, 

 changes sign on passing through infinity, and becomes zero 

 again at cf> = a. Each change of phase therefore changes by 

 the value ir corresponding to half an undulation. 



From the 'critical angle as cf> increases, the formulae for 

 reflexion involve no further change of phase except when we 

 reach the incidence corresponding to the polarizing angle for 

 external reflexion, when the light polarized perpendicularly 

 to the plane of incidence gives, just as in the case of external 

 reflexion, a zero amplitude for the reflected light — there is 

 then a change of phase again of 7r, in other words the ampli- 

 tude of the reflected light changes sign. From this incidence 

 there is no change up to normal incidence. 



Now from the character of the fringes it appears that at 

 grazing incidence the values of 7* 'and y p are both 7r, as there 

 is the loss of half an undulation in both cases. Therefore at 

 the critical incidence the values of y t and y p are 2ir or equiva- 

 lent to 0. Now at this incidence the Stokes investigation 

 applies, and therefore the corresponding external reflexions 

 must involve a loss of phase 7r, and these reflexions are of 

 course those at grazing incidence. This result is in agree- 

 ment with the Lloyd experiment, and therefore our results 

 are consistent. For incidences between this critical incidence 

 and normal incidence the Stokes law still holds, and the 

 results are again in complete agreement. 



We have not taken account of the modifications necessary, 

 if we assume the transition layer at the surface and not a 

 sudden discontinuity, but these only affect the question of the 

 light polarized perpendicularly to the plane of incidence 

 at incidence very near the principal incidence. The 

 modification makes in each case — internal and external 

 reflexion — a continuous change in the change of phase and 

 not a sudden alteration of it. But this does not affect the 

 general argument, which is to connect up the complete system 

 of changes and to show that the experimental verification of 

 the reflexion formula? for the case of grazing internal 

 incidence enables us to consider all the phenomena as one 

 connected whole. 



