SCATTEKING OF PLANE ELECTEIC WAVES BY SPHEKES. 
201 
Since the co-ordinates of P are (O, —mr, nr), the value of P is given by 
{y-\-mrY+ {z—nrY, 
— r^ + 2.r{'my—nz) + x^ + y^ + z^, 
where {x, y, z) is a point on the surface near to O. Thus, when r is very large in 
comparison with the dimensions of the surface (as we assumed in the previous 
investigations, §§ 4-6), we can use the approximate formula 
(7*3) R = r + my—nz. 
Thus we can write in (7*13) 
p—LK(r+my-nz) 
-y — _ _ _ Q->'K(my-nz) 
r ’ 
if Vq is the value of v at O. Then the most important term in ^ is seen to be 
Ov 
dz 
Accordingly the components of electric force in the reflected wave will be given by 
the approximation 
(7*4) ^ 
X 
Y 
Z 
^ f(2t/cnA)e=^“”^dS, 
4x J 
To. 
Aiir 
= -t- 
bTT 
„2iKnz 
„2iKHZ 
dS, 
dS. 
To evaluate the integrals in (7'4) we must write out the equation to the surface in 
the approximate form 
2 z = — (ax^ + 2/8xy + yy^). 
Then* 
lQ-^>in{a3^+2Pxy+yy^) _ 1 
J IKH y/[oLy — Yi^) 
Now ay— is the absolute (or Gaussian) curvature of the surface at the point O ; 
and since the surface is supposed convex, we represent this curvature by l/p^. 
* This is most easily found by taking the integral as lim 
e^o J 
2 F 2 
