Forced Vibrations of Electromagnetic Systems. 377 



subsequent history depends upon the constitution of the media 

 beyond the boundaries, which can be summarized in two 

 boundary conditions. The expression for E/H is, in general, 



by (120), extended, the extension being the introduction of ?/, 

 which is a differential operator of unstated form, depending 

 upon the boundary conditions. Let y and y x be the tfs on 

 the inner and outer side of the surface of/. The differential 

 equations of H a , the magnetic force there, is then 



>={(E/H) (0Ut) -(E/H) (i ^H„, . . (285) 



as in § 19. Applying (284) and the conjugate property 

 (114) of the functions u and ic, (since there is no change of 

 medium at the surface of/), this becomes 



H _ 4nrk + cp (uq-yoWaXug-yiiCq) . ,_ 



from which the differential equation of H at any point 

 between a and a is obtained by changing u a —y w a to 

 (a/r)(u —yoiv) ; and at any point between a and a 1 by changing 

 Ua—yiWa to (a/r)(u—y 1 iv'). 



Unless, therefore, there are singularities causing failure, 

 the determinantal equation is 



yi-yo=o 3 (287) 



and the complete solution between a and a 1 due to /constant 

 may be written down at once. Thus at a point outside the 

 surface of/ we have 



(out) H= 4?7/j4 C P a - (P-y^^-y^fi^-ifi (288) 



and therefore, if /starts when t = 0, 



H = L + &y <*-?$"-*"> . *&±3L+ % . (289) 



(f> r p(d/d P )(yi-yo) q 



p being now algebraic, given by (287) ; (p the steady <£, from 

 (218) ; and y the common value of the (now) equal y's ; which 

 identity makes (289) applicable on both sides of the surface 

 of/. 



45. Construction of y x and y . — In order that y x and y 

 should be determinable in such a way as to render (286) true, 

 the media beyond the boundaries must be made up of any 



