184 DE. T. J. I'A. BROMWICH ON THE 



and using this in equation (3 -4) we deduce that 

 (3-82) S^ W 



Combining (3 -81) and (3 -82) we have also 

 (3 83) S_, (*) + S + i (z) = 



The relations (3 -82), (3 -83) hold equally for E n (z) and C (*), as may be seen from (3 -5) and (3-6). 

 As E n (z), S B (z) are independent solutions of the equation (3 -8), it is evident that 



EW 2 f -SW^ = const. 

 Now when g is small, it is easy to verify from (3 -4) and (3'6) that 



and accordingly we have 



(3-84) (*)%-S.W^ = L 



In the discussions of 6, when ft, 2 are both large, it will be convenient to adopt 

 the following notation : 



(:r85) . | E. (z) I == R, E, (z) = He-*, so that S,, (2) = R sin ^, C n (z) = R cos ^. 

 Substituting from (.T85) in (3'84) we deduce that 



(3-86) 11*^ = 1. 



az 



Before leaving these preliminary formulae it will be convenient to quote the formula 

 for e"" in terms of our standard functions ; namely 



(3-9) 



where z r cos = >> and P n (/u) is LKGENDRE'S polynomial of order n. 



This result follows at once from the formula given in LAMB'S ' Hydrodynamics,' 

 Art. 291, on using the relation (3'4) between ^(KT) and S n (rr), already quoted. 

 It is of course evident that an expansion of the type (3 '9) might be anticipated, 

 since each side satisfies the wave-equation, is symmetrical about the axis of z, and is 

 continuous at r = ; the determination of the numerical coefficients may be then 

 carried out quickly by comparing the terms in (/rt>)" on the two sides of the equation. 



4. PLANE ELECTROMAGNETIC WAVES INCIDENT ON A SPHERICAL OBSTACLE. 



Suppose that the incident wave-train is travelling along the negative direction of 

 the axis of z (that is, from = towards 6 = *) ; and that it is polarized in the 

 plane of yz (that is, in the plane <j> = ^TT). Suppose further that the electric force in 



