90 APPLICATION OF THE PRELIMINARY RESULTS 



Hence we deduce successively 

 titf d 



d do- dE 



If now we make the radius a vanish, X t must become equal to 

 X'j the value of X at the pointy', and there will result 



_ _ 



dx' dx'~ ' 'dx~dx' 



But ~-T expresses the value of the force acting in the 



direction of x positive, on a point p within the infinitely small 

 sphere dv, arising from the whole of A exterior to dv; sub- 



~cW 

 stituting now for -p its value just found, the expression of this 



force becomes 



**-% ;: 



Supposing V to represent the value of the potential function at 

 p, arising from the exterior bodies which induce the magnetic 

 state of A, the force due to them acting in the same direc- 



tion, is 



dT 

 dx" 



and therefore the total force in the direction of x' positive, 

 tending to induce a magnetic state in the spherical element 

 dv, is 



4 dp dV T 



7rX -r-i -j-r=X. 

 dx dx 



In the same way, the total forces in the directions of y' arid z' 

 positive, acting upon dv', are shown to be 



dV , ^, dty' dV 



"~ = 



