20 ON POTENTIAL. 



so that the preceding theorem may be expressed analytically by either 

 of the equations 



Vv 



- | cos / dS = 4?rM, 

 (8) 



which are due to Green. 



31. EQUATIONS OF LAPLACE AND OF POISSON. Let X, Y and Z 

 be the components of the force at a point P, whose co-ordinates are 

 x, y, z, and let us consider the element of volume dx t dy, dz. 



If the medium contains the acting masses distributed in a con- 

 tinuous manner, and if ^M is the total mass contained in the element, 

 then, denoting the density by p, we have 



dM = pdxdydz. 



The flow of force which enters by the surface dydz, passing 

 through the point P, is 



Jidydz; 



the flow of force which emerges from the opposite face is 



/ 3>X \ 



( X + dx \dydz. 



\ ^ x /' 

 The excess of the flow which emerges is equal to 



dX <> 2 V 



- dxdydz= --dxdydz. 



^x ^x 2 



Repeating the same reasoning for the other co-ordinates, it will 

 be seen that the total flow of force which proceeds from the element 

 of volume is 



/W c) 2 V 3 2 V\ 



~ ( ^-T + ^TV + TV \dxdydz. 

 \J)x 2 <Vy 2 c) 2 I 



In virtue of the preceding theorem, this excess is equal to the 

 product of 4?r by the total mass of electricity comprised in the 

 volume, which gives the equation 



