INFLUENCE OF THE CASE. 175 



The principal term in each of these expressions represents the 

 first approximation given by the equations (4). It may be observed 

 in particular that, for the expression of the force as a function of the 

 electrical masses, the simple formula used by Coulomb does not 

 involve a relative error of 0*02 when c= 6 that is, when the distance 

 of the centres is equal to three times the diameter of the spheres. 



y 803. INFLUENCE OF THE CASE. When the case is of glass, no 

 correction is possible, for we do not know the electrical condition 

 which it acquires for a given condition of the balls. If, moreover, 

 some parts of this surface are accidentally electrified in a few points, 

 very serious errors may arise. Hence the internal surface must be a 

 conductor, and in connection with the earth. The charge of the 

 case is then equal, and opposite to the algebraic sum of the charges 

 of the two spheres, and its potential is zero. 



The presence of electricity induced on the conducting case may 

 materially affect the reciprocal action of the spheres, and therefore 

 the calculation of the charges as a function of the repulsive force ; 

 but it more particularly changes the value of the potentials, and this 

 influence is greater the smaller is the case, for the quantity of induced 

 electricity is the same in all cases. 



The calculation of this correction presents in general the greatest 

 analytical difficulties. In order to give an idea of its importance, we 

 will consider the case of a spherical shade. We shall first assume 

 that, the axis of suspension of the needle being eccentric, the two 

 balls are near the centre. The induced electricity forms then an 

 almost uniform layer at a constant internal potential ; the reciprocal 

 action of the two spheres will be expressed then as a function of the 

 charges, just as if the case were not there. 



This is no longer the case with the potentials. If, as before, V is 

 the potential due to the charges m and m', U the true potential, and 

 R is the radius of the cage, we have, for a point near the centre, 



m + m' 



When the charges are equal, and we take the first approximation 

 given by equation (4), we have, for the potential of each ball, 



w - ' ~* v ' ^T^~ Y i ' ~ * ^ :rr^ I 



