66 



EEPORT — 1891. 



displacement — a sphere of dielectric different from that of the 

 rest of the field. 



Let El be the elastic constant of the field, E2 of the sphere, 



and suppose Eg < Ei . 



It is obvious that, since 

 the medium of which the 

 sphere is composed is weaker 

 than that surrounding the 

 sphere, the sphere will give 

 as a whole, and be displaced 

 further in the direction of 

 the existing displacement of 

 the field. The lines of dis- 

 placement behind the sphere 

 will converge on the gap left 

 by the sphere ; those in front 

 will correspondingly diverge. 

 General considerations show that the sphere will not be 

 distorted, but simply translated ; for, from symmetry, the 

 positive charge at A must be numerically equal to the negative 

 charge ac B, and so for other corresponding points on the 

 sphere. Hence, as the sphere is not distorted longitudinally, 

 and is incompressible, it is not distorted laterally. 



Let us suppose then that the sphere is displaced bodily to 

 the left through a distance a, in addition to the displacement 

 it has in common with all the rest of the field — viz., b. We 

 shall find that by this supposition we do indeed obtain complete 

 equilibrium, and that we have therefore solved the problem ; 

 and we can find the value of a. In the figure, 



AA' = BB' = 0' = rt. 



The effect on the surrounding dielectric of moving the sphere 

 bodily a distance a to the left is the same as would be produced 

 if (the dielectric being for the moment supposed uniform) there 

 were placed at a charge — 27rr\ and at O' a charge + 27rr^ 



For the normal displacement at P due to these charges 



, , , 1 2777-3 I 2Trr' 

 wouldbe^.^^--.^, 



_ r'iJ: L1 



OP^i 



"~ 2(0'P-' 

 _ r' 2.0 0'cos (^ 

 ^ 2* OP^ ' 

 = ft cos <^. 



since OP nearly = O'P. 



— the thickness of the gap left by the sphere. 



Bo the state of the field is simply this : The sphere is moved 

 to the left a distance a -\- b, and the surrounding dielectric 



