OF THE REFLECTION AND REFRACTION OF LIGHT. 
695 
Now if we had originally assumed 
U=A/ 3 + Bcf +CA 3 + 2F.gh +2G. hf+ 2H fg 
we should have 
so that we obtain 
\=Af+H/+G£, ii=H.i+Bj+m 
4 Mit+h+K)+i (B.g-H.0) +j (a*-H 5 
= 0 
so that if we assume i and j parallel to the axes of the section of S p$p= 1 by the wave- 
plane we evidently have got H = 0, and consequently £=0 and 
47r/x£=B.£f, 4,7Tixr] = A 6 ~^ 
and as A and B are inversely proportional to the squares of the axes of the section of 
$p<f>p=l or U=1 by the wave-plane, we get M’Cullagh’s result that each component 
in these directions is propagated independently and with a velocity inversely propor¬ 
tional to the perpendicular axis of the section of Spcf)p=l by the wave-plane. 
This comes out at once from the Cartesian equations, for as In— 0, U reduces to 
U=A/+2H/y+B/ 
and by choosing x and y parallel to the axes of this section, we have H = 0, so that 
dg 
— 2B .g 
and 47 r/== — y- and 4 tu/= ^ 
and the result of putting these in is evidently the same as before. 
Returning now to the superficial conditions, I shall follow M’Cullagh, and assume 
that at each point of the surface of separation of two media the values of the elements 
of the integrals must be equal for the two media, and, indeed, I think it is pretty 
evident that if the original integrals are to express the whole state of affairs in the 
case of a motion propagated from one medium into another, the superficial integrals 
must vanish when the limits introduced in them are the functions corresponding to 
the two contiguous media. 
Using the suffix 0 for one medium and t for the other, and assuming the normal to 
the surface as k, we get, as SR is evidently arbitrary and the same for both media at 
the surface of separation, 
.’.Yk<f> 0 Vv 3t=YJc(f) 1 Vv di 
4 u 2 
