EARNSHAW'S THEOREM. 49 



_ X. ----_... - - - r 



Applying Green's equation to the surface of this sphere, we 

 shall have 



pw 



As the integral dS becomes null, it follows that the 



aw J * r 



differential is negative for certain directions, and positive for 



others ; hence, the body A is not in equilibrium, and tends to move 

 towards places for which the energy W diminishes. 



aw 



There is equilibrium if ^ is always zero ; that is to say, if the 



energy is constant, or passes through an absolute maximum or 

 minimum. We have then 



and in like manner, 



The components X, Y, Z of the force of the field, produced by 

 the external masses, may be developed as a function of increasing 

 powers of the co-ordinates a, b, c, and of the components X , Y , Z , 

 relative to the point P , which gives 



HP H 2 , ---- H w .. being functions of the i, 2 ---- n.. degrees of 

 the co-ordinates a, b y and c. 

 We get then 



..... +H n ) = 0. 



To satisfy this equation, all the coefficients of the development 

 must be zero. That necessitates, in the first place, that all the 

 differentials of X be null ; then, that the position of the point P 

 be arbitrary; and, lastly, that we have 



Hence, either the total mass ^m of the body A must be zero, or 

 the components X , Y , and Z of the force must themselves be zero. 



E 



