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



Such a distribution may be called a double or sliding dis- 

 tribution, and a body possessing such a distribution is said to be 

 polarized in the direction h. The conception of the sliding 

 distribution is due to Poisson. The moment of the couple 

 experienced by the body when placed in a uniform field of 

 strength unity whose direction is perpendicular to the direction 

 of polarization, is called its moment of polarization, and I, the 

 moment of unit volume, the intensity of polarization. J is a 

 vector quantity, for the whole couple may evidently be sup- 

 posed to arise from three bodies occupying the same space, 

 and polarized in three mutually perpendicular directions, the in- 

 tensities of polarization being respectively 



4=Icos(/0), B = Icos(Iy), <7 = 

 For if the components of the field are 



X=FcoB(Fai), Y=Fcos(Fy), Z = 



the polarization B produces the couple BZ about the X-axis, and 

 the polarization G the couple CY about the same. In like 

 manner the couples about the other axes are obtained, 



(5) L = BZ-CY, M=CX-AZ, N = AY-BX. 



We accordingly find that the resultant couple is the vector product 

 of the intensity of polarization and the field-strength, whose 



magnitude is 



FI sin (FT), agreeing with (4). 



Suppose now that we have a body of such a nature that every 

 element of its volume has a double distribution, although the 

 direction and magnitude of the polarization / may vary from 

 element to element. Such a body is polarized in the most general 

 manner, and the volume-density will not vanish throughout. Let 

 us seek its value in terms of the polarization at each point. Con- 

 sider a rectangular element of volume dr, whose sides are dx, dy, dz, 

 and in which the values of the component polarizations are A,B,C. 

 Then the face dydz on the side next to the origin has the charge 

 Adydz, while the opposite face has the equal and opposite 

 charge Adydz. In the next element of volume on the right, 

 whose center is at a distance dx from the center of the first, the 

 ^-component of polarization will be 



dA . 

 A+^-dx, 



ox 



