OF THE REFLECTION AND REFRACTION OF LIGHT. 
699 
reflection. Hence these equations cannot explain metallic reflection. Indeed, this 
question of change of phase seems to be one of a higher order than I am here dealing 
with, and requires a discussion of the nature of the transition from one medium to 
another, which, of course, cannot be abrupt, as our equations suppose, nor indeed 
probably, in these cases, even very small compared with the vibrations. In order that 
these terms involving the time should disappear from the equations when %—0 we 
must have 
\ 0 : \' 0 : X l : \\ : : sin i 0 : sin i' 0 : sin i x : sin i\ :: 1 : 1 : s : s' 
which involves that the angle of incidence should be equal to the angle of reflection ; 
and if the second medium were an ordinary one, so that s=- = s' we should have that 
the ratio of the sines of the angles of incidence and refraction was constant. Putting 
in these values, our equations reduce to 
T 0 cos a 0 +T' 0 cos a' 0 =v T\ cos a 1 -|-'y / T / 1 cos a\ 
T 0 cos /3 0 +T' 0 cos /3' 0 = vl 1 cos ^ 1 -{-vT' l cos /3\ 
together with the condition that when z=0 the superficial displacements should be the 
same to whichever medium they belong, namely, 
£=£', C=Z' 
As each of these is a resolved part of the vibrations rj 0 , t/ 0 and r) 1} r/\ we get three 
additional equations, the last of which, however, is the same as the second of the 
former ones and there result, consequently, but four equations from which the four 
quantities, namely, the three intensities T 0 , T 1? T' L , and the azimuth of T 0 are to be 
determined. It is remarkable that whether we assumed or no that £=£' it is here 
introduced. That is, however, no proof that it is wrong to omit it, as in Fresnel’s 
method of obtaining the intensities of the reflected and refracted rays, for the fact of 
its turning up independently shows that there is something at least debateable about 
it, and as I shall have cause to omit this equation as leading to inconvenient results in 
a subsequent part of my paper, I thought it well to mention that there is something 
curious about it even here. These equations are those long ago given by M’Cullagh 
in the ‘Transactions of the Royal Irish Academy,’ vol. xxi., to solve the problem of 
crystalline reflection and refraction, and from which he deduces his beautiful theorem of 
the polar plane and thus marvellously simplifies an extremely complicated problem. 
Indeed, as the forms into which I have thrown T and W are identical with his 
expressions for what are practically the same quantities, my whole investigation so fai¬ 
ls but a modification of his. In the simple case of the second medium being also 
isotropic we have that v=v'=s=s, and if we in the first place suppose r) Q to be in the 
plane of incidence, we have at once a () = 90°, and consequently a 0 =a'=90°, and T'j 
