300 
PROF. H. M. MACDONALD ON THE EFFECT PRODUCED BY 
and the components of the electric force at the point x, y, z are given by 
0^ a 
a' /3 + 
0' 
r_ 
1 
0^ 
a 
_dx^ r 
dxdy r 
dxdz 
r 
de 
T _ 
0^ a 
+ ^ + 
0^ 
1 
1 
0^ 
_dxdy r 
dy'^ r 
dydz 
r 
de 
0^ a 
+ a* 
1- 
1 
02 
r1 
dxdz r 
dydz r 
02^ 
r 
V2 
0^2 
dS + 
477 dt 
■_^Z 
_cy r 
~_^X 
_02 r 
'IX 
dx r 
IX’ 
02 r _ 
ll” 
037 r_ 
AX' 
dyr_ 
c/S, 
dS, 
dS, 
where 
a = fny\ — 7i^\ 
X = mZ'i-nY' 
yS = na\ — ly\ 
Y = nX.\-lZ', 
y = 
z =: lY\-mX': 
and the integrations are taken over the surface of the obstacle. The quantities 
a, y8, y are the components of the electric current distribution on the surface which 
would produce the tangential magnetic force on the surface due to the distribution of 
sources inside it, and the quantities X, Y, Z are the components of the magnetic 
current distribution on the surface which would produce the tangential electric force 
on the surface due to the distribution of sources inside it. The waves incident on the 
surface can be represented as the effect of a distribution of Hertzian oscillators, and 
it will therefore be sufl&cient to consider the effect of a Hertzian oscillator situated at 
a point, 0, outside the surface and emitting waves of a definite wave-length, 
Now, the conditions to be satisfied at the surface of the obstacle are linear relations 
involving the components of the electric and magnetic forces, and therefore the 
integrands in the above expressions for X, Y, Z, a, yS, y will each contain the factor 
g-iK(i-+ 2 er') distance of the point {x, y, z) from the point (^, y, Q on the 
surface, and the quantities r' are other distances. The remaining factors of the 
integrands will l^e non-oscillatory unless the surface has corrugations on it, for which 
the interval between successive corrugations is comparable with the wave-length 2tt/k 
of the oscillations. The principal parts of the expressions for X, Y, Z, a, /3, y are 
therefore contributed by the portion or portions of the surface in the neighbourhood 
of the point or points for which r-l-Se/ is stationary. If the wave-length of the 
oscillations is small compared with both principal radii of curvature of the surface at 
a point at which r + ter' is stationary, the corresponding principal parts of X, Y, Z, 
a, y8, y are the same for a point, P, very near to this point Q on the surface as if the 
point Q were situated on a plane surface. Hence the principal parts of X, Y, Z, 
a, /3, y at points on the surface are related to the principal parts of the components 
of the electric and magnetic forces in the incident waves in the same way as if the 
surface were plane, but it should be observed that the incident waves to be taken 
