SCATTERING OF PLANE ELECTRIC WAVES BY SPHERES. 
199 
Let us now consider the problem of waves incident from some source (or sources) 
and reflected from the surface S. Let u denote any Cartesian component of force in 
the incident wave, and let the point P be outside S. Then (7‘l) applies, and so 
c?S = 0, 
because u has no singularity inside S. 
If v! denotes the corresponding component of force in the reflected wave, we have 
from (7*12) 
47r'?i^P, 
because u' has no singularity outside S (and in! will correspond to a divergent wave 
at infinity). 
By addition we have the result 
(7*13) Ttt'w'p = I j(?^ + '?/)|^ + 
where v is taken to be e~"‘^ /E. 
Now u + ^i' = w gives the corresponding component of force in the complete*'wave; 
and this accordingly satisfies certain known relations at the surface S (the exact 
form depending on the physical properties of S). It must, however, be clearly 
3'W 
understood that we cannot usually obtain both w and ^ by any simple methods, any 
Cp 
more than the analogous problems of electrostatics can be solved by a mere appeal 
to Green’s Theorem. 
However, we can obtain aw approximate solution, suitable to the problem of short 
wave-lengths, by assuming that near the reflecting surface, the character of u' can be 
determined from that of u by the rules of elementary geometrical optics. Thus we 
treat the reflected wave as derived from the incident by simple reflexion in the 
tangent-plane at the point of incidence. Making this hypothesis it is an easy matter 
to construct both w and when the form of u is given. 
Cv 
It will be noticed that we shall have = 0, — = 0 at all points within the 
cv 
geometrical shadow ; and so the final integral (7'13) extends only over the illuminated 
side of the surface S. 
Suppose now that we consider electric waves incident on a simple convex 
conducting surface; and take an origin O on the surface such that OP is the 
reflected ray (in the sense of geometrical optics). Take the plane of incidence as the 
plane of yz, and the normal at 0 as the axis of 2 ;. 
VOL, CCXX,—A. 
2 F 
