520 Dr. J. W. Nicholson on the Bending of 



and leads after some reduction to 



w = - | 2; 2mr* I (-)»J_ m - t J„,(fo-) }>- r IJ» (fo-j) (1 -ft **- , 



and is- obviously continuous with (o) at r=?* b except perhaps 

 in derivates. 



But by separating off the portion of yp which is infinite 

 at the pole, a mode of determination can be found which 

 leads necessarily to continuous series. The oscillator, 

 having an external magnetic force 



behaves, in its own neighbourhood, like a simple doublet, 

 since we may then write k = 0, a time factor being present 

 in the strength of the doublet. The expansion of this 

 simplified function in an harmonic series'is, on reduction, 



so that in the actual case, with (5), 



where the series vanishes with k, and is finite everywhere, 

 including the neighbourhood of the oscillator. 



The disturbance y 2 p produced by the sphere must, near 

 the oscillator, be that due to its electrostatic image in the 

 sphere, which consists of (1) a doublet of strength —a 3 /^ 3 

 (that of the oscillator being unity), and (2) a residual charge 

 a/r, 2 , both situated at a distance a 2 /n from the centre of the 

 sphere. These are referred to the original doublet as a unit. 

 The magnetic effect of the image, the strengths having a 

 time factor, becomes on reduction, if B/ denote distance from 

 the image, 



a?p B _1 _^_ rfi—rj 



and its harmonic expression for external points r>a 2 /r 1 is 



— sin^ 6 2, n n . 2 —5— 5 



1 n r n r{ l +* dfi 



and the complete disturbance due to the obstacle is, being 



(6) 



