POINTS AND LINES OF EQUILIBRIUM. 37 



it occupies, is a centre of the emission of force in all directions ; 

 if it is negative, it is a centre Of absorption for all directions. In 

 both cases the adjacent equipotential surfaces are closed surfaces. 

 In the limit the surfaces are spherical even; for at a small 

 distance r from the point A, the potential of the external masses 



may be neglected as compared with the potential - of the mass m. 



The equipotential surfaces being closed about A, the potential 

 has a maximum or minimum value ; maximum, if the mass m is 

 positive; minimum, if it is negative. 



Conversely, there is electricity wherever the potential is 

 maximum or minimum. 



For, starting from a point A of maximum, the potential decreases 

 in all directions, the adjacent equipotential surfaces are necessarily 

 closed, and the flow of force which traverses, for example, a very 

 small spherical surface comprising the point A, has a finite value 

 Q. In the interior of this surface there is therefore a quantity 



of positive electricity equal to . 



4?r 



In like manner, a minimum of potential is a centre of absorption 

 of force, where there is a corresponding mass of negative electricity. 



Hence, in any given electrical system, there can neither be absolute 

 maximum nor absolute minimum df potential outside the acting masses. 



47. POINTS AND LINES OF EQUILIBRIUM. Let V be the 

 potential at a fixed point P . The potential V at an adjacent point 

 P, whose co-ordinates referred to axes passing through the former 

 are x, y, z, may be expressed as a function of increasing powers of 

 the co-ordinates, 



H 15 H 2 ---- H n being homogeneous functions of the first, of the 

 second ---- of the n th degree of the co-ordinates. 



As the equation of Laplace should be satisfied separately by 

 each of these functions, we shall have in general 



If the function H } is identical with zero, that is to say, if the 

 three partial differentials of the potential 



p 



- 



