42 GENERAL THEOREMS. 



centre is the mean of the values of the potential at the different points 

 of the surface. 



For let R be the radius of the sphere, m one of the masses at 

 a point A, at a distance d from the centre, and r the distance of 

 a surface element ^/S from the point A; the mean value, on the 

 surface, of the potential due to the mass m, is 



the sum I is the value at A, of the potential of a homogeneous 

 layer of density m, which would cover the sphere ; it is equal (43) to 



d ' 



The mean value of the potential on the sphere is then 



V -- 



v nt j) 



a 



that is to say, equal to the value of the potential at the centre. 



This reasoning manifestly extends to any number of masses. 



That being admitted, let us suppose that in a portion of the 

 dielectric bounded by the surface S, the potential has a constant 

 value V; if the value of the potential were different from V on 

 the outside, it would always be possible to draw a sphere having 

 its centre in the inside of S, and on the outside only meeting 

 points where the potential would be always either greater or less 

 than V ; but, in that case, the mean value of the potential on the 

 sphere would be different from its value at the centre. The 

 potential on the exterior of the surface S cannot, therefore, be different 

 from V. 



55. When a closed surface S surrounds all the acting masses, 

 and the potential on this surface has a constant value V, at each of 

 the external points there is a value comprised between V and zero. 



Let us suppose V positive. As the potential is zero at an 

 infinite distance, it cannot have a higher value than V, at a point 

 P external to the surface, without there is somewhere a maximum 

 potential, and therefore electrical masses, which is contrary to the 

 hypothesis. In like manner the potential at P cannot be less than 

 V 3 for otherwise there would be a minimum. The potential on 



