Method of treating Electrostatic Theorems. 



29 



converge on the gap left by the sphere, those in front will 

 correspondingly diverge. We have to find the amount of the 

 extra displacement at every point of the sphere. 



General considerations alone seem to show that the sphere 

 will be undistorted, simply translated ; for from symmetry 

 the positive charge on the one side of the sphere must be 

 numerically equal to the negative charge on the other side, 

 point by point, taking always two points the line joining 

 which is parallel to the direction of the displacement of the 

 field. The sphere is therefore uncontracted longitudinally, 

 and therefore also laterally. 



But this is proved also in the following complete solution. 



Let us examine the effect on the external medium of shifting 

 the sphere undistorted an extra distance a to the left. We 

 shall find that if we give a certain value to a, the resulting 

 strain of the external medium will cause a pressure across 

 each unit of area of the surface of the sphere equal and 

 opposite to the pressure across that area caused by the strain 

 of the medium within the sphere. Hence there will be com- 

 plete equilibrium, and we shall have found the nature of the 

 state into which the sphere and medium will naturally fall. 



Fig. 7. 



Now the state of strain into which the external medium 

 will fall in consequence of the sphere being moved a distance 

 a to the left is the same as would be produced if E 2 were 

 equal to E l5 and charges 1irr z and — 277T 8 were placed at 0' 

 and respectively. For the displacement at any point P 



*) 3 



produced by such charges would be ^-^ along P and 



27rr 3 

 4 " (VP 2 a ^ on S O'P* Since a is very small, this amounts to a 



