704 
MR, G. F. FITZGERALD ON THE ELECTROMAGNETIC THEORY 
Returning to the superficial equations, and using / as a sign of substitution, we get 
j^: V(9l . V v SH) + 4ttC( VSfc^jj-eYv5ft)=0 
As I intend only to work out the results in Cartesian coordinates, I shall confine 
myself to them, and for simplification assume the axes to be 2 normal to and x and y 
in the surface, and consequently l—m— 0, n— 1, and e=y. C may also be supposed 
to vanish for one of the media for which K x is the dielectric inductive capacity. I 
shall also assume that S£=0 as the vanishing of its coefficient leads to inconvenient 
results, and the assumption may be to some extent justified by considering that as 
these S£, Sy, SC are superficial values, no virtual displacement out of the surface, as 
this would be, is admissible. From the other two, S£ and Sy, we evidently get 
K\dz 
JL® 
KjU/ 
A remarkable point about these equations is that they admit of being integrated 
with regard to the time as far as their right-hand members are concerned, so that in 
addition to the values which satisfy them as they stand, and which are the only ones 
of much interest, there are other periodic values of £, y, C independently of y x> Ci 
which satisfy these superficial conditions, and which may consequently be looked upon 
as a sort of free vibration of the surface of the medium. But even if this could be 
propagated into the rest of the ether it is improbable that the resultant vibrations 
would be of such a period as to be visible, though some energy might be expended 
on them. 
I shall now further assume that the axis of x is the intersection of the surface with 
the plane of incidence of a plane-wave, and that consequently none of my quantities 
are functions of y, which reduces these equations to 
In order still further to simplify the problem I shall now consider separately the 
cases in which the magnetisation of the medium is normal and that in which it is 
in the surface. In the first place, it is evident that if it were all in y we should 
have a—y— 0, f3=l OZ, and the coefficient of C would vanish in both equations because 
