Forced Vibrations of Electromagnetic Systems. 139 

 given by 



Knowing thus the effects due to initial elements of E and 

 H , we have only to integrate with respect to z to find the 

 solutions clue to any arbitrary initial distributions. I forbear 

 from giving a detailed demonstration, leaving the satisfaction 

 of the proper conditions to be the proof of (42) to (48) ; 

 since, although they were very laboriously worked out by 

 myself, yet, as mathematical solutions, are more likely to have 

 been given before in some other physical problem than to be 

 new. 



Another way of viewing the matter is to start with s = 0, 

 and then examine the effect of introducing s, either + or — . 

 Let an isolated plane disturbance of small depth be travelling 

 along in the positive direction undistorted at speed v. We 

 have E = yttvH in it. Now suddenly increase k, making s 

 positive. The disturbance still keeps moving on at the same 

 speed, but is attenuated with greater rapidity. At the 

 same time it leaves a tail behind it, the tip of which 

 travels out the other way at speed v, so that at time t, 

 after commencement of the tailing, the whole disturbance 

 extends through the distance 2vt. In this tail H is of the 

 same sign as in the head, and its integral amount is such 

 that it exactly accounts for the extra-attenuation suffered 

 by H in the head. On the other hand, E in the tail is 

 of the opposite sign to E in the head ; so that the integral 

 amount of E in head and tail decreases faster. As a special 

 case, let, in the first place, there be no conductivity, k = and 

 g = 0. Then, keeping g still zero, the effect of introducing k 

 is to cause the above-described effect, except that as there was 

 no attenuation at first, the attenuation later is entirely due 

 to k, whilst the line-integral of H along the tail, or 



including H in the head, remains constant. This is the per- 

 sistence of momentum. 



If, on the other hand, we introduce g, the statements made 

 regarding H are now true as regards E, and conversely. The 

 tail is of a different nature, E being of same sign in the tail as 

 in the head, and H of the opposite sign. Hence, of course, 

 when we have both k and g of the right amounts, there is no 



