198 
DR. T. J. I’A. BROMWICH ON THE 
I’hese results in (6’9) are precisely the same as would be found by writing 0 = 0 
in the approximations (6‘7), and accordingly the formulce (67) do remain valid right 
wp to the axis 0 = 0. 
When 0 is nearly equal to x, the calculation on the present lines becomes more 
difficult,* and we shall accordingly obtain the corresponding approximation by a 
different process in the next section (§ 7). 
It appears that for the special value Ka = 10, the formulae (67) do give the forces 
with a fair degree of accuracy up to an angle 0 = f x; the approximation in fact 
appears to be better than might have been expected. [See p. 176 above.] 
§ 7. Alternative Method, applicable to any Conductor whose Dimensions 
ARE Large compared with the Wave-length. 
It follows at once from Green’s theorem that if u, v are solutions of the equations 
A^u + k^u = 0, A^v + k^v = 0, 
at points within a closed simple surface S, then 
where the integral is taken over the surface S {dv being the element of outward 
normal), and it is supposed that u, v are both free from singularities in the interior 
of S. 
Similarly if u has no singularities and p is a solution which behaves like e ‘''“/R 
near a particular point P (R denoting the distance measured from P), we see that 
(7'11) ^ —p c^S = — 4xWp, 
provided that P is inside the surface S, 
Equations of similar forms apply when the space considered is outside the surface 
S ; but then the sign of the last equation (7‘11) is reversed, giving 
(7‘12) 1^ — p dS =-f-4xMp; 
it is then necessary to assume also that at infinity, u, v both correspond to 
divergent waves (unless it is known that u tends to zero more rapidly than ijr). 
* Compare Macdonald, Iog. cit., pp. 120-122. The cause of the difl&culty is to be found in the fact 
that now the stationary value may be expected to arise from values of n for which a is small. Then n is 
nearly ecpaal to z, and in all such cases more complicated analysis is inevitable. In fact the approximations 
to S„ (z) and En (z) require to be modified by different formulae corresponding to the cases n > z, n < z 
and to the cases in which \n-z\ is of order ^1^ or of lower order. 
