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XCI. Some Further Problems connected with the Motion of 

 Charged Spheres. By G. H. Livens, B.A., Fellow of 

 Jesus College, Cambridge ; Lecturer in Mathematics, Sheffield 

 University *. 



I. The Vibration of a Charged Conducting Sphere. 



4 GLANCE at the method previously employed to solve 

 ±\. the case of the uniformly accelerated motion o£ the 

 charged sphere, will suffice to show that it is merely the 

 simplest of the soluble cases. In fact, the solution of 

 the problem for any acceleration can be at once obtained, 

 provided only that it is such that the displacement of the 



centre of the sphere in a time comparable with - is small 



compared with a. The solution for uniform motion was 

 obtained by integrating the surface condition 



a 2 f r/ (ct) + af(ci) +/(<*) -*f =0, 



where f was taken to be \st 2 . The solution consisted of a 

 vibratory part not depending on f and the particular integral. 

 In the general case 



£ = [ sdtdt, 



Jo Jo 



and the success of the solution merely depends on being able 

 to carry out these integrations and then in obtaining the 

 particular integral of the above differential equation with 

 the form of f thus obtained. As a case let us assume that 



£ = —.cos nt. 

 rr 



The particular integral is then easily seen to be 



i 1 -^)' 



an . 

 I cos nt -\ sin nt 



Act) = ' 



/ (fn*\ a~ if 



I 1 " -IF) + T 



The complementary integral representing the vibratory 

 part of the solution will be exactly the same as before. If 

 s is small enough the solution thus obtained will apply for 

 any period of time, because the displacement of the sphere is 

 always small. Thus the vibratory terms represented bv the 

 complementary function will ultimately decay and become 



Communicated bv the Author. 



