48 GENERAL THEOREMS. 



Let m denote the electrical mass of a volume element of A at 

 a point P, where the potential of the external masses is V. The 

 energy of the body in the field is 



(i) W = mV. 



For stable equilibrium, the differential must be null or 

 positive in any given direction r. 



Let x, y, z be the co-ordinates of the point P ; f , 77, f those of a 

 point P taken in the interior of the body A, and a, b, c that of 

 the point P with reference to three new axes parallel to the first, 

 and passing through the point P ; we shall have 



The potential V may be considered as a function of a, b, c, and 

 f > ^ f If the body A is constrained to move parallel to itself, 

 we shall have 



dz=d(, 

 and therefore, 



3 2 V W 3 2 V W D 2 V W 



(2) AV = VT + VT + ^ = ^ + ^- + ^79- 



cte 2 ty/ 2 c)^ 2 Sf 8 <V ^i 2 



As this sum is zero for each of the terms nN of the second 

 member of the equation (i), we shall also have 



^) 2 W t) 2 W _ 



~ ++ ~ 



The energy W is therefore a function of the co-ordinates of the 

 point P , and this function satisfies Laplace's equation, as long as 

 equation (2) itself is satisfied. We may assume that the point P is 

 comprised within a sphere of radius r, so small that the body in 

 question, A, does not meet any external masses. The variation of 

 energy for any displacement parallel to one of the radii of this 

 sphere, will be 



