Forced Vibrations of Electromagnetic Systems. 387 



/ concerned. This happens, when / is a sinusoidal function 

 of the time, at definite frequencies. Also, inside the sphere, 



f H = ^^u(u a -w a )f m , . . (134) 



(in) << W r 



As for the radial component F, it is not often wanted. It is 

 got thus from H : — 



-c P F=~(vR):, (136) 



where for cp write 4:7rk+cp in the general case. Thus, the 

 internal F corresponding to (135) is 



(in) qpF^S^i^^-^/.Q.. (137) 



21. Practical Problem. Uniform Impressed Force in the 

 Sphere. — If there be a uniform field of impressed force in the 

 sphere, parallel to the axis, of intensity f 1} its vorticity is 

 represented by /j sin on the surface of the sphere. It is 

 therefore the case m = l in the above. Let this impressed 

 force be suddenly started. Find the effect produced. We 

 have, by (132) *, 



(out) H = M a (u-«0— ; (138) 



fj> vr 



* It will be observed that the operator connecting f and H is of such 

 a nature that the process of expansion of H in a series of normal functions 

 fails. I have examined several cases of this kind. The invariable rule 

 seems to be that when there is a surface of vorticity of e, leading to an 

 equation of the form /=(/>H, and there is a change of medium somewhere, 

 or else barriers, causing reflected waves, the form of <£ is such that we 

 can, when /is constant, starting at £=0, solve thus 



H - f+K fi# 



extending over all the (algebraical) p roots of 0=0, which is the determi- 

 ne ntal equation. But should there be no change of medium, the conjugate 

 property of the functions concerned comes into play. It causes a great 

 simplification in the form of <p, and makes the last method fail completely, 

 all traces of the roots having disappeared. But if we pass continuously 

 from one case to the other, then the last formula becomes a definite inte- 

 gral. On the other hand, we can immediately integrate /=$H in its 

 simplified form, and obtain an interpretable equivalent for the definite 

 integral, which latter is more ornamental than useful. In the simplified 

 form, $ may be either rational or irrational. The integration of the irra- 

 tional forms will be given in some later problems. 



