156 Dr. T. J. I'a. Bromwich on 



It will be recognized at once that the calculation o£ (4*7) 

 from the equivalent Cartesian formulae (4*5) and (4*6) 

 would be much longer, although it must lead to the same 

 formula finally. It is also more troublesome to form a 

 mental picture of the field from the Cartesian formulae 

 than from those given in (4*3) in terms of spherical polar 

 coordinates. 



§ 5. Special case of simple harmonic leaves. 



If we assume that the waves are simple harmonic with a 

 wave-length equal to 27r Jk in free space, we can suppose that 

 t occurs only in a time-factor e tKCt ; as usual we suppose that 

 in the final results the real (or the imaginary) parts of the 

 formulae are retained. 



Then the functions /, g in (3*49) above will be exponentials 

 of the types 



/"_ gucifat— »*i) 3 g — 0«i(*i*+»") | 



- (5*1) 

 where k y ci = kc, or k^ = A\/(yu,K). ( 



Thus, if we now suppress the time-factor for brevity, we 

 see that the functions F w are expressed by the formulae 



1 O \ n (e~ lKVi 



i divergent waves -, 



' (5-21 



(1 ~0 \ n /e~ bK i T \ 

 ^- ) ( ) divergent waves , 

 r or \ t J 



or ? ^+i/ — d\ f sm [Ktf ) I i nterna i problems, j 



In place of using the general lormulae {D'2) it is often 

 more convenient to write 



F w = E^r) or $ n ( Kl r), . . . (,r21) 



where the functions E n , S« are defined by * 



Some of the properties of these Junctions will be given 

 below, as well as their connexion with tie functions used in 

 other solutions of these problems. 



* Tables of the functions S„(s), C n (z), E„(z)=G n (z)rS„(z) lave been 

 calculated in.m s=l to 2=10 by Mr. A. T. Doodson, and will be found 

 in the British Association Report for 1914. 



(5-22; 



