THEOREMS RELATING TO CLOSED SURFACES. 45 







that of ^ m. .A layer + M, distributed in the same way on the 

 surface S, will have everywhere ' dn the outside a potential equal, 

 and of the same sign, to that of the masses in question, ^ m. 



The layer M will not in general be in equilibrium of itself; that 

 is to say, that it will not have the distribution which would result 

 from the form of the surface, and from the action of external 

 masses; the lines of force will not cut it perpendicularly. 



60. The layer +M. will be in equilibrium of itself, if the surface 

 S is an equipotential surface of the primitive system, taking into account 

 (he external masses. 



Let us denote by ^ m the external masses. The lines of 

 force of the field are the same at the exterior of S for the system 

 ^m and ^m, and for the system M and ] m' ; these lines 

 being, by supposition, normal to S, the layer M which covers it 

 is in equilibrium, and has a constant potential V. 



We may then, for external points, replace the system ^m by 

 an equal mass in equilibrium on an equipotential surface which 

 surrounds it, the density at each point being determined by the 

 condition 



_ F _ i_JV 

 4/r 4?r dn 



This substitution always modifies the field in the interior of the 

 surface S ; in the present case the internal potential has become 

 constant and equal to V, for it is constant on the surface, and the 

 cavity no longer contains electricity. 



If the potential is constant, the force is zero ; the external system 

 m', and the layer M, exert equal and contrary actions at each 

 point of the interior of S ; a layer - M would exert actions equal 

 to, and of the same sign as, that of m'. Thus for all the internal 

 points of the equipotential surface S, we may replace the action of 

 external masses by that of a layer in equilibrium, equal to the internal 

 masses, and of the opposite sign. 



61. The following theorems follow from this: If we consider 

 an equipotential surface S in any electrical system, we may 



i st. For all external points replace the internal masses by a mass 

 M, equal and of the same sign, in equilibrium on this surface ; 



2nd. For points in the interior, we may replace the external masses 

 2? ?n by the same mass M with changed sign; that is to say, by a 

 mass equal and opposite to the internal masses, this layer being still 

 in equilibrium. 



