Forced Vibrations of Electromagnetic Systems. 385 



Now, at the surface of separation, r = a, H is continuous 

 (unless we choose to make it a sheet of electric current, which 

 we do not) ; so that the H in (117) and in (120) are the same. 

 We only require a relation between the E's to complete the 

 differential equation. 



Let there be vorticity of impressed force on the surface 

 r=a, and nowhere else (the latter being already assumed) . 



curl e = curl E (121) 



is the surface-condition which follows; or, if/ be the measure 



of the curl of e, /= E 2 -E 1 , (122) 



E 2 meaning the outer and Ej the inner E. Therefore 



/= H °(t-D> < 123 > 



H a denoting the surface H. So, by (117) and (120), used 

 in (123), 



the required differential equation. Observe that u x only 

 differs from w 2 and w l from w 2 in the different values of q 

 inside and outside (when different), and that r = a in all. 



of disturbances. The operator ei r turns f(t) into f(t+r/v), and can there- 

 fore only be possible with a negative wave, coming to the origin. But 

 there cannot be such a wave without a barrier or change of medium to 

 produce it. Hence the operator e-w alone can be involved in the external 

 solution when the medium is unbounded, and we must use W. Next, 

 go inside the sphere r=a. It is clear that both U and W are now needed, 

 because disturbances come to any point from the further as well as from 

 the nearer side of the surface, thus coming from and going to the centre. 

 Two questions remain : Why take U and W in equal ratio ? ; and why 

 their sum or their difference, according as m is odd or even ? The first is 

 answered by stating the facts that, although it is convenient to assume 

 the origin to be a place of reflexion, yet it is really only a place where 

 disturbances cross, and that the H produced at any point of the surface 

 is (initially) equal on both sides of it. The second question is answered 

 by stating the property of the Q V OT function, that it is an even function of 

 fx when m is odd, and conversely ; so that when m is odd the H dis- 

 turbances arriving at any point on a diameter from its two ends are of 

 the same sign, requiring U+W ; and when m is even, of opposite signs, 

 requiring U — W. 



Similar reasoning applies to the operators concerned in other than 

 spherical waves. Cases of simple diffusion are brought under the same 

 rules by generalizing the problem so as to produce wave-propagation with 

 finite speed. On the other hand, when there are barriers, or changes of 

 media, there is no difficulty, because the boundary conditions tell us 

 in what ratio U and W must be taken. 



