198 PR. T. J. I'A. BROMWICH ON THE 



These results in-(6'9) are precisely the same as would be found by writing = 

 in the approximations (67), and accordingly the formula (67) do remain valid right 

 up to the axis 9 = 0. 



When 6 is nearly equal to TT, 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 KO, = 10, the formulae (67) do give the forces 

 with ' a fair degree of accuracy up to an angle Q = f TT ; 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 2 + At = 0, A 2 v + K 2 v = 0, 

 at points within a closed simple surface S, then 



(7-1) 



where t'he 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 v is a solution which behaves like e~ llcK /E, 

 near a particular point P (R denoting the distance measured from P), we see that 



in't i\ f / dv ou 



(711) \(u -v 



J \ ov ov 



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 



/.-, \ f / 3v ou\ TO 



(712) \(u~ -- v )dS = + 4-irUp ; 



J \ ov ov] 



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 1/r). 



* Compare MACDONALD, loc. tit., pp. 120-122. The cause of the difficulty 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 equal to z, and in all such cases more complicated analysis is inevitable. In fact the approximations 

 to S n (z) and E n (z) require to be modified by different formulae corresponding to the cases n > z, n < z 

 and to the cases in which |m-| is of order V or of lower order. 



