MOTION OF SMALL BODIES IN THE ELECTRICAL FIELD. 167 



Consider now a sphere so small that, placed at any point of a 

 variable field, it becomes electrified as it would be in a uniform field 

 where the force was the same ; the electrical moment of this sphere 

 will be 



or = UO-Q = uK<f>, 



and the axis of electrification being parallel to the force <, will be 

 perpendicular to the equipotential surfaces which include it. 



Suppose that we fix on the surface the two electrical layers, which 

 are equivalent to the system of two solid spheres, or of two equal 

 and opposite masses m at an infinitely small distance 8, such that 

 7 = m8. If Vj and V 2 are the potentials at the points occupied by 

 the masses - m and + m, the energy of this sphere in the field is then 



W 



= f*V = - 



At a point P' where the force is equal to <f> + d$> the energy of 

 the sphere will be 



W'= - 



The work necessary for bringing the sphere from the point P to 

 the point P' is then 



W'-W= - 



In reality, if the layers are not fixed, the electrification changes 

 with the displacement of the sphere, and the work in question is 

 between -CT^ and -(cr + ^fer)^; this work only differs therefore 

 from the value found by an infinitely small expression of the second 

 degree. 



The energy dW expended in effecting the displacement is then 



(33) u 



dW= 



Thus, when an infinitely small sphere passes from a point in the 



