TRANSACTIONS OF SECTION A. 69 



two cases — (i.) wheu a ball of required specific inductive 

 capacity is placed in the field ; (ii.) when a similarly-shaped 

 ball of conducting naaterial is placed there. In the equation 



he uses, therefore, the — - divides out. 

 o 



-3 « 



The nature of the field due to a charged ellipsoidal con- 

 ductor is by this method found without much difficulty ; but, 

 as the proof differs but little, except in language, from the 

 ordinary proof, I will only state it briefly. 



It may bs shown that, if every ellipsoidal surface confocal 

 to the original one be supposed to expand siinilarLij — i.e., so 

 as to become a similar and similarly-situated ellipsoid — then 

 the strain so produced will be one of equilibrium. 



x^ 2/2 z^ 

 Let ~o + X3 + "3 = 1 ^® ^^® original ellipsoid ; and let 



Xi, Xo) '\) be the parameters of three confocals to it, Aj being 

 the ellipsoid. 



Let ^Ji, po, 2^3, be the perpendiculars from the centre on the 

 tangent planes to the three surfaces where they intersect. 



Now, if the ellipsoid Aj expand, its linear dimensions alter- 

 ing in the ratio 1 to 1-f p, we must have — 



Q being the charge. 



.•.P= ^ 



47rv/(a-HA0(6HAi)(c2 + A0 



Also the normal displacement at Aj, A.2, Ao, is easily proved to 

 bep.^Ji. 



Now, consider the tube formed by the confocals, Ao, A^ + 8A2, 

 X3, X3 + 8X3 : the area of section of this tube at Xj, X.^, X3, is 



~- X Q-^' these two factors being the lengths of the sides of 



the section. 



v/^g+XOC&M-XQF'+Xi) 

 Now, pi = — = ; 



V X.2X3 



^2 and 2^3 have similar values. 



8X., 8X3 . . ^ ^ 

 Hence, ppi X k~ x -^ is independent of Xj. 



Thus, the total displacement across this or any similar 

 tube is a constant along the tube ; and so the tubes are tubes 

 of displacement of the supposed strain. 



We have yet to show that the strain produces equilibrium. 

 But it may be easily shown that, if V be the pressure at any 



