nl (L J)l.i/jince In JJitjlectrU'S. (>5 



every p )int on 8. Vii may tlioreforo in-ike U = V — Vq, wliich 

 .siitisfie.s that coii(iitioii, siiico V = Vo at ovcry [)oiiit on kS. In 

 tliat ca.-30 



dx ~ dx dy dij dz dz 

 an 1 (2) heconios 



Now K cannot be negative, and cannot be assumed always 

 zero. The al)ove equation, then (which, as we have seen, holds 

 universally if V be constant over the surface fS), cannot be 

 satisfied unless 



dx ^' dy ^' dz ^' 



. . dY . 



and V = Vo at every point within S. It follows that -.-, in 



equation (B), wdiich is the rate of increase of V with the normal 

 on the inside of 8), mast bo zero. Applying, therefore?, equa- 

 tion (B) to our surface, cr being the surface-density, and making 



d\r . 



,-> zero, we nave 

 dv' ./y 



K-V-4-47ra- = 0. 

 dv 



dV . 

 Here, as will be remembered, -,-- is the rate of increase of V 

 ^ dv - 



per unit of length of the normal irom S outwards. 



. dY . 



IV. Hence as K and crare known at everv ijoint of S, —- is 



" ^ ' dv 



also known at every point of 8, and we can therefore describe 



round 8 the equi})otential surface 8i corresponding to A'^q — 8V, 



5V bein^ any iniinitelv small increment of ijotential. It will 



. . ' K 



be at a distance from 8 equal to -, f)V. 



^ 47ro- 



Further, having now obtained two consecutive equipotential 



surfaces belonging to our distribution, the property of our 



d\ . " 

 surfaces investigated above, namely that aK ~ is constant 



throughout every tube of force, enables us to construct all the 



others. For if we take as base of our tube any element of 



dN 

 area on 8, and call it a, then aK — - is known at the base of 

 ' ^ dv 



the tube ; also if normals forming the tube be drawn from all 



points on the boundary of a, and if they cut out from the 



Flvd, Mag. S. 5. Yol. 4. No. 22. July 1877. F 



