148 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



so small that the force of gravity can be regarded as constant in 

 direction and in intensity, we may use Y to denote the potential of 

 the force of gravity, and express the variation of the part of the 

 energy which is due to gravity in the form 



-/Y 8 Dm -/Y A Dm. (238) 



We shall then have, for the general condition of equilibrium, 



fSDe +/AZ>e -/Y SDm -/Y ADm ^ ; (239) 



and the equations of condition will be 



(240) 



(241) 



We may obtain a condition of equilibrium independent of these 

 equations of condition, by subtracting these equations, multiplied each 

 by an indeterminate constant, from condition (239). If we denote 

 these indeterminate constants by T y M l} ...M n , we shall obtain after 

 arranging the terms 



JSDe-Y3Dm-TSDr]-M l SDm l ...-M n SDm n 



> 0. (242) 



The variations, both infinitesimal and finite, in this condition are 

 independent of the equations of condition (240) and (241), and are 

 only subject to the condition that the varied values of De, Zty, 

 Dm v ...Dm n for each element are determined by a certain change 

 of phase. But as we do not suppose the same element to experience 

 both a finite and an infinitesimal change of phase, we must have 



SDe - Y SDm -T8Dt]-M l 8Dm 1 ...-M n SDm n ^ 0, (243) 

 and &De-'YADm-T&Dr]-M l &Dm 1 ...-M n ADm n ^(). (244) 



By equation (12), and in virtue of the necessary relation (222), the 

 first of these conditions reduces to 



n ^(); (245) 



for which it is necessary and sufficient that 



(246) 



(247) 



* The gravitation potential is here supposed to be defined in the usual way. But if 

 it were defined so as to decrease when a body falls, we should have the sign + instead 

 of - in these equations ; i.e., for each component, the sum of the gravitation and 

 intrinsic potentials would be constant throughout the whole mass. 



