Forced Vibrations of Electromagnetic Systems. 437 



Besides these two sets of solutions, we sometimes require 

 to use a third set. A pair of solutions of the J equation is 



U = r-M'(R+S), W = r-*e-« r (R-S),l 

 where R + S=l + * + ™. ± «» + . . . > (™> 



Sqr + \2{8qr) 2 ± |3(tyr) 8 + * * * j 



The last also defines the Rand Se in (308). R is real whether 

 q 2 be -f or—, whilst S is unreal when q 2 is — , or Si is then 

 real, s 2 being +. 



When qr is a + numeric, the solution U is meaningless, 

 as its value is infinity. But in our investigations q 2 is a 

 differential operator, so that the objection to U on that score 

 is groundless. We shall use it to calculate the shape of an 

 inward progressing wave, whilst W goes to find an outward 

 wave. The results are fully convergent within certain limits 

 of r and t. From this alone we see that a comprehensive 

 theory of ordinary linear differential equations is sometimes 

 impossible. They must be generalized into partial differ- 

 ential equations before they can be understood. 



The conjugate property of U and W is 



UW'-U'W=-2^, .... (310) 



if the ' = d/dr. An important transformation sometimes re- 

 quired is 



J 0r -iGor=2iW(27rq)-* ; .... (311) 



or, which means the same, 



W=-(J)Vorlog<?r + L 0r ]. . . (312) 



When we have obtained the differential equation in any 

 problem, the assumption s <2 = a + constant converts it into 

 the solution due to impressed force sinusoidal with re- 

 spect to t and z ; this requires d?/dz 2 =. — ra 2 , and d^/dt 2 

 = — n 2 , where m and n are positive constants, being 2tt times 

 the wave-shortness along z and 2tt times the frequency of 

 vibration respectively. 



After (309) we became less exclusively mathematical. To 

 go further in this direction, and come to electromagnetic 

 waves, observe that we need not concern ourselves at all 

 with F the radial component, in seeking for the proper 

 differential equation connected with a surface of curl of im- 

 pressed force ; it is E and H only that we need consider, as 

 the boundary conditions concern them. The second of (302) 

 derives F from H. 



