314 CONSTITUTION OF MAGNETS. 



the same conditions, is the surface cut on a sphere of radius equal to 

 unity, whose centre is the point P, by a cone described with vertex 

 at P on the surface of this element as base ; it is the apparent surface 

 of this element. The angle to, or the apparent surface of the 

 whole shell, is therefore defined by a limited cone on the contour 

 of this shell; it is positive or negative according as the surface 

 of the shell which the point P views throughout the contour is itself 

 positive or negative. The potential of the shell is therefore inde- 

 pendent of its form, and only depends on its magnetic power and on 

 its contour. From this follows the important theorem of Gauss : 



The potential of a simple magnetic shell at an external point is 

 equal to the magnetic power of the shell by its apparent surface seen 

 from this point. 



In order that the potential shall be zero at this point, the appa- 

 rent surface of the shell must be zero. 



The apparent surface of 'the shell is null if, the contour being a 

 plane, the point in question is situate in this plane. 



It is zero, whatever may be the shell, when it consists of parts 

 of opposite signs whose algebraical sum is zero. 



As a particular case, if the shell forms a closed surface, the 

 potential for any external point is zero. For a point inside the 

 shell, the angle o> is equal to 477, the potential is therefore constant 

 and equal to 4^ ; it is of the same sign as the internal surface. 

 The value of this potential being constant both inside and outside, 

 the action of the closed shell on any given point is zero. 



330. If two equally strong magnetic shells S and S' (Fig. 77) 

 have the same contour, and if their surfaces which face each other 

 are of opposite signs, their potentials are equal for all points outside 

 the space which they comprise ; these potentials differ, on the con- 

 trary, by 47r ( i > for all points between the two surfaces. For the 

 potential of one of the shells S is positive and equal to 3>w, that 

 of the other shell S' is -<a>'; the difference is therefore 



In like manner, for two infinitely near points situate on each 

 side of a magnetic shell at a finite distance from the contour, the 

 difference of potentials is equal to 4^, for it is 3>(o for the one 

 and < (4?r co) for the other. Hence, when the point in question 

 traverses a shell in the direction of the magnetisation that is to 

 say, from the negative to the positive face the potential suddenly 

 increases by 



