694 
MR. G. F. FITZGERALD OR THE ELECTROMAGNETIC THEORY 
I shall now proceed to integrate this by parts relatively to x, y, z, and in order to 
express the result conveniently I shall assume ds to be an element of the surface of 
the medium and 91, with components l. m, n, to be a unit normal to this element of 
surface, when we evidently obtain 
E.f 
S(91.<£Fv 01.891)cfe-f111 j/xS9lS91+^S(V Vcf>Vv 01.89 i)^dxdydz 
dt =0 
or 
r 
+*f 
“hi 
dy dh dz dg) \dz df dx dh) ^ \ffcc dg dy df / 
dxdydz 
d U dVX^i.dV dV\s ,/ dV ,dU\- 
iK n ^- m M) s H l M- n vr + r j 
ds 
dt =0 
It is evident that the superficial and general integrals must vanish separately, and 
as 801 or Si, Sy and S£, is arbitrary in the general integrals, we must evidently have 
47r/x01+ V v <f> V v 01 =0 
or 
M+i 
' d 
dU 
d . 
dV\ 
dy 
’ dh " 
dz 
dg) 
' d 
fiU 
d , 
fiU\ 
1 rU 
1 ^ 
df~ 
dx 
dh) 
d , 
dU 
d 
dV\ 
dx 
dg 
dy 
' df ) 
As it will not take long I will deduce the ordinary equations for the transmission of 
plane-waves from these before proceeding to discuss the superficial condition. In the 
first place, as our axes of coordinates are perfectly arbitrary I shall assume 2 to be 
normal to the plane of the wave, and consequently i, g, £ to be functions of 2 and t 
only, so that V and — = —= 0, and consequently £=0 and h — 0 and 
Fv8 
n y v rJ n 
so that if 
and consequently 
dz 
<f)p = \S ip +/xS \jp fi- v&lcp 
<f>Vv 
dz dz 
TT7 \ d* v 
