

 DISTRIBUTION OF ELECTRICITY ON CONDUCTORS. 21 



Representing by AV the sum 'of the second partial differentials 

 of the potential in reference to the co-ordinates, we may write : 



(9) AV=-47i7>. 



If the element of volume is not electrified, p = and 



(10) AV = 0. 



Thus, the sum at a point of the three second partial differentials 

 of the potential in reference to three rectangular axes is equal, and 

 of opposite sign, to the product of 4?r by the density of the mass acting 

 at this point. 



This sum is zero when there is no electricity near the point. 



This theorem of the second differentials was first enunciated by 

 Laplace in the form (10). The more general equation is due to 

 Poisson. 



32. If the equipotential surfaces are concentric spheres, the 

 force F is inversely as the square of the distance r from the 

 common centre, and we have 



F __^X_A c) 2 V 2 A 2 F 



~^7~^' ^*-z-- 



Taking the z axis along the perpendicular to the surface, the 

 two others will be in the tangent plane ; if we measure the distance 

 in the direction of the force, we get 



2F 



and therefore, by Laplace's theorem, 



If the equipotential surface is of any given form, it is readily 

 seen that the second differential of the potential along the tangent to 

 a principal section is the same as for the osculating circle. The z axis 

 being always perpendicular, and the two others along the tangents to 

 the principal sections, whose radii of curvature are R x and R 2 , we 



shall have 



W _ F W _ F 



^2 = R/ "Sy^R^' 



and, therefore, 



