234 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. 



From this, the lines of force due to the doublet, or the lines 

 ^ = const., are drawn by a simple geometrical construction, Fig. 53. 

 (Nature, Vol. XXL, p. 371.) The combination of this field with a 

 straight field in the direction of the axis of the doublet, drawn by 

 the method of 106, is shown in Figs. 74 and 75, according as the 

 axis of the doublet is in the direction of, or opposite to that of 

 the field. The lines are drawn for equal increments of M*. 



123. Potential due to uniform Polarization. We may 



easily find a convenient expression for the potential at any point 

 due to uniform polarization, / = constant. Let H represent the 

 potential at a point P whose coordinates are x, y, z of a body occupy- 

 ing the space r and filled with a single or ordinary distribution 

 of the uniform density unity. Then after the body A of the 

 double distribution has been displaced the distance h in the direc- 

 tion /, the potential at P is the same as if the body had remained 

 fixed while P had been displaced the distance h in the opposite 

 direction, that is, 



The potential at P of the negative body B is 



and the potential of the double distribution is the sum of these 

 two, or 



and inserting the value of the polarization, 



'. an _ , an 



Consequently if we know the value of the potential due to a 

 single distribution of constant density, we may obtain by differen- 

 tiation the potential for a body of the same form uniformly 

 polarized. The expression holds both for inside and outside points. 

 The potential due to a doublet illustrates this, for the potential 

 due to a point m is 



a-s 



