334 Mr Olazebrook, on the Reflexion [Feb. 9, 



the potential energy of tlie medium by M, tbe form assumed for it 

 by MacCullagh, then we can put V= M + W ; and if the vibrations 

 be transverse W does not occur in the equations either of motion 

 or of condition at the surface, but is zero if u, v, w are functions 

 of the same function of x, y, z snad t. 



MacCullagh assumes ah initio that p = p , which follows of 

 necessity from equations 5 and 8. 



I have thought it interesting to shew that these results can be 

 deduced from the correct expression for V, assuming the density of 

 the ether the same in the two media. The total intensity of the 

 reflected ray will be the same as that given by Fresnel's expres- 

 sions, that of the refracted ray will differ from his. 



The amplitude of the vibration in the plane of incidence given 

 above is the same as that for the vibration perpendicular to the 

 plane of incidence on Fresnel's theory, and vice versa. 



Moreover, we have from 9 and 13 , 



. cos ((^ - (^') 

 tan 6, = — tan 6 )^ — ^ , 



^ cos [(p + (])) 



6, 6^ being the angles which the directions of vibration make 

 with the intersection of the wave-fronts and the reflecting surface. 



If a, ^ are the angles between the plane of incidence and the 

 directions of vibration 



a = |-^, /3 = |-^„ 

 tan^ = -tan.H^^^i±|| (15). 



cos {(j) —(p) 



And if further we suppose that the directions of vibration lie 

 in the planes of polarization, this is the formula which has been 

 verified by Fresnel and Brewster. 



For the refracted wave we have 



tan 6' = tan 6 cos (^ — cf)') ; 



.: tan/3' = tx^^ 16 ; 



cos [4> — (f)) V / ' 



this also agrees with Frfesnel's formulas. 



Again, Green has shewn that the change of phase arising from 

 total reflexion can be deduced from his formulse, and the ex- 

 pressions he arrives at agree with Fresnel's and are the foundation 

 of the theory of Fresnel's rhomb. I propose to shew that the 



