70 EEPOKT — 1891. 



point, the rate of variation of Y in any direction is equal to 

 — Ex displacement in that direction. 



In this case the displacement at Ai, X^, A3, is p .2h ', and the 

 normal distance between the surface Ai and the surface Ai + 8A1 

 . 8 Ax 



IS 



Q 



/^(a2+Ai)(62+Ai)(c2+Ai) 



Hence, if V be constant over one confocal ellipsoid, it is 

 constant over all. But V is zero at infinity, and one of the 

 confocal ellipsoids is an infinite circle. 



Hence the confocal ellipsoids are surfaces of equal pressure. 

 This proves the proposition. 



I should like here to point out the curious fact that in an 

 ordinary proof of part of the above theorem — viz., that part 

 which asserts the distribution of the charge on the ellipsoid to 

 be represented by the expanding of the ellipsoid similarly — the 

 reasoning has to do with attractions inside the ellipsoid, and 

 takes no account of actions outside. 



It should be noticed that this ordinary form of proof is 

 very incomplete, for it is upset entirely if the dielectric within 

 the ellipsoid be not uniform, as in that case calculation of the 

 attractions would become impossible. The proof here given is, 

 of course, independent of the nature of the dielectric within the 

 elhpsoid. Moreover, the ordinary proof would apparently be 

 unaffected by any want of uniformity in the external dielectric : 

 a student might therefore be easily led astray. 



In connection with this subject it is worth noticing that the 

 method of this paper leads to a most simple proof of the law 

 that a closed conductor screens its interior from the action of 

 external charges. 



For the only way in which the pressure of the ether in the 

 conducting shell could affect the dielectric — uniform or not — in 

 the interio]", would be by changing the shape of it, pushing 

 it in in some places, and therefore allowing it to bulge in 

 others. But the pressure is uniform over the conductor, and 

 the interior of the shell is unalterable in volume, and so no 

 work can be done by any such deformation. Hence no energy 

 can be imparted to the interior of the conductor, and conse- 

 quently no chaiiges can be forced to take place in it. 



I hope that I have made it clear that this method of treat- 

 ing electrostatic theorems is of value. It seems to me to 



