384 Mr. 0. Heaviside on Electromagnetic Waves, and the 



The conjugate property of U and W is 



UW'-U'W=-2 ? , (112) 



which is continually useful. 



We have next to combine U and W so as to produce 

 functions suitable for use inside spheres, right up to the 

 centre, and finite there. Let 



w =i(U + W), u»«i(U-W). . . . (113) 



It will be found that when m is even, w/r is zero and u/r 

 infinite at the origin ; but that when m is odd, it is u/r that 

 is zero at the origin and w infinite. 



The conjugate property of u and w is 



uw' — u'w = q, (114) 



corresponding to (112). 



19. Construction of the Differential Equations connected with 

 a Spherical Sheet of Vorticity of Impressed Force. — Now let 

 there be two media — one extending from r = Q to r=a, in 

 which we must therefore use the u function or w function, 

 according as m is odd or even, and an outer medium, or at 

 least one in which q has a different form in general. Then, 

 within the sphere of radius a, we have 



H = Ar- 1 M , (115) 



-k l K = Ar- l u', (116) 



where &, = 4 , 7r&-j-cp, and we suppose m odd. It follows that 



l-li- ■■■■■ -mi 



In the outer medium use W, if the medium extends to 

 infinity, or both U and W if there be barriers or change 

 of medium. First, let it be an infinitely extended medium. 

 Then, in it, 



K = Br-\u-ic), (118) 



-k 2 E = Br-\u'-u^), (119) 



where k 2 = 4:7rk + cp in the outer medium. From these 



E 1 u' — w' 



H ~~ k 2 u — w' ^ ' 



(117) and (120) show the forms of the resistance-operators on 

 the two sides *. 



* Some rather important considerations are presented here. On what 

 principles should we settle which functions to use, internally and exter- 

 nally, seeing that these functions U and W are not quantities, but diffe- 

 rential operators ? First, as regards the space outside the surface of origin 



