60 ELECTRICAL EQUILIBRIUM. 



it is equal to the radius. This example shows that the electrostatic 

 capacity of a conductor is a linear quantity. 



74. ELLIPSOID. If a conductor bounded by the surface of an 

 ellipsoid is covered by a homogeneous electrical layer, bounded itself 

 by a second ellipsoidal surface concentrical and similarly placed to 

 the former, the action of the layer on an internal point P is null. 



Let us suppose, in fact, that this layer is very thin, and let us 

 draw through the point P (Fig. 15) an infinitely slender cone */w, 

 which cuts an element of surface ^S at M at the distance u, and in 

 the layer, a volume element the height of which along the radius 



Fig. 15- 



vector is du. The action at P of this volume element is in the 

 direction of the radius vector, and calling p the density, its value is 



The action of the opposite element at M' is also pdudu'. As the 

 heights du and du' are equal and the forces are directly opposed, 

 their resultant is zero ; this is also the case for all the elements 

 of surface two by two, and the action of the entire layer on the 

 point P is null. An electrical layer distributed on an ellipsoid 

 according to this law will be then in equilibrium and will have a 

 constant potential in the interior. 



Let (i +a) be the ratio of similitude of the two surfaces supposed 

 to be very close. The thickness of the layer at a point N is pro- 

 portional to the distance of the tangent planes from the two 

 homologous points N and N', and is equal to /a, p denoting the 

 perpendicular OQ let fall from the common centre on the tangent 

 plane in N ; the value of the surface density cr is 



