TRANSACTIONS OF SECTION A. 



67 



bears in addition to its ordinary displacement {b to the left 

 everywhere) the strain due to a charge — 27rr^ at 0, and a 

 charge 27rr^ at 0'. 



Now let us see whether equilibrium is secured by sup- 

 posing the field strained in this way. 



Consider two points, P and Q, on 

 the sphere, PQ being parallel to the 

 displacement of the field. The dif- 

 ference of pressures at P and Q is, 

 since every point in PQ is displaced 

 to the left a distance a + h, 



Ea (a + b) . It cos <^. 



But, considering the strain in the 

 outer medium, the difference of pres- 

 sures at P and Q is due (i.) to a uni- 

 form strain b to the left, and (ii.) to 

 a strain from charges — 27rr^ at and 27rr^ at O'. 

 This latter produces a difference of pressures- 



Ei 



47r 



27rr^ 27rr''l 

 "OP + OTJ 



477 . Ej . r^ . a cos 4> 



47r . r^ 



= — El . r a cos <^. 



Hence, for equilibrium we must have — - 



E.2 (a + b) 2r cos ^ = Ej 6 . 2r cos ^ — Ej ?• a cos <^. 



a] __ 2Z>(Ei-E.,) 



of."" 



B.,{a + b) = EJb 



or a = 



2E, + El 

 the left at B 



{i.e., the 



Thus the total displacement to 

 density of the charge) = a -\- b 



= b.-^... 



2E, + Ei 



If Eg =0 — i.e., if the sphere is a conductor — the total dis 

 placement to the left at B =Sb. 



The loss of energy occasioned by the pressure in the field 

 of this sphere can also be easily found. 



To do this, let us calculate the work that must be done first 

 to restore the surrounding dielectric to its uniform state, and 

 then to replace the sphere of elasticity E^, displaced a distance 

 a-{-b by a sphere of elasticity Ei, displaced a distance b. 



Since the difference of pressures at P and Q is, when the 

 sphere is displaced a distance x to the left, 



Ei& . 2rcos^ — Ei7' . x . cos(f> (see above) 

 then, if fZs be the sectional area of such a tube as that run- 

 ning from P to Q, the work done in restoring the external 



