204 DR. T. J. FA. 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 nearly equal to TT ; 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 = -e'", 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" f 



where (treating the tangent-plane as the reflecting surface) 



f = z+p (alxmynz) 

 and p is determined by 



( lpf+( mpf + (\-npf = 1. 

 Thus 



p = 2n. 



Further, the resultant of (X, Y, Z) and (X', Y', Z') at the point of incidence must 

 be along the normal ; and so 



A'-l = B' = (7. 

 I m n 



Also (A', B', C') is perpendicular to the reflected ray ; and so 



A'-l ^B'_C'_ 2nl _ 2l 



I m n n2n 



A' - 1-2P, B' = -2lm, C' = -2ln. 



The components X + X', Y + Y', Z + Z' at the point of incidence are accordingly 

 equal to 



-2(l 2 > lm,ln)e lfan . 



We have still to evaluate the normal differential coefficients, which are found to be 

 (7'64) n(-l-A', -B', -C')e" an = 2n(P-l, Im, ln)e" an . 



The value of B is now seen to be given by 



+ (z-an) a = r i -2a(lx+my+nz) + a 3 . 



