204 
DE. T. J. I’A. BROMWICH ON THE 
IS small. That is, near the edge of the shadow, in the ordinary phrase of geometrical 
optics. 
In the application to the sphere, the region excluded by this condition corresponds 
to values of 6 nearly equal to x; and in this region the specification of the scattered 
wave by means of Cartesian co-ordinates seems simplest. 
We consider then the incident wave as specified in Cartesian form by 
(7-6) X = Y = 0, Z = 0, 
and consider the approximation to the reflected wave incident at the point {al, am, an) 
on the sphere for which the direction-cosines of the normal are I, m, n. The expressions 
will be of the form 
(7-61) (X', Y', Z') = (A', B', C')e‘“^ 
where (treating the tangent-plane as the reflecting surface) 
I — z+p {a—lx—my—nz) 
{ — lpY-\-{—mpY + {\—npY = 1 . 
p = 2n. 
and p is determined by 
Thus 
Further, the resultant of (X, Y, Z) and (X', Y', Z') at the point of incidence must 
be along the normal; and so 
A^-1 ^ B' ^ c;. 
I m n 
Also (A', B', C') is perpendicular to the reflected ray ; and so 
A'(—2n/)-f-B'( —2nm) + C'{l—2n^) = 0. 
A'-l B' C' 2nl 
Hence 
(7*62) <! 
or 
7 - - = - 2 /, 
L m n n—2n 
A' = 1 - 2Z^ B' = - 2lm, C' = - 2ln. 
The components X + X', Yq-Y', Z-f-Z' at the point of Incidence are accordingly 
equal to 
(7’63) —2 Im, Zn)e““”. 
We have still to evaluate the normal diflerential coefficients, which are found to be 
(7*64) i/cn(-l-A', -B', -C')e‘'“" = 2<icn(Z'-l, Im, 
The value of E, is now seen to be given by 
= {x—alY-^{y—amy-\-{z—anY = 7^—2a{lx+my + nz) + a^. 
