144 J. W. Glhbs —Equilibriuin of Heterogeneous Substances. 



the subscript nuraertils refer, (but not excluding tlie case in which 

 // z= 1 and the composition of tlie body is invariable,) there is a rela- 

 tion between the quantities enumerated in any one of the above sets, 

 from which, if known, with the aid only of general principles and 

 relations, we may deduce all the relations subsisting for such a mass 

 between the quantities e, i/-, x, I', '/, v, rjt ^, m.^, . . , ///„, t, p, /.ij, ju.^, 

 . . . //„. It will be observed that, besides the equations which 

 define i/:, x, and 'C, there is one finite equation, (93), which subsists 

 between these qiiantities independently of the form of the fundamental 

 equation. 



Other sets of quantities might of course be added which possess 

 the same property. The sets (100), (101), (102) are mentioned on 

 account of the important properties of the quantities i/-, j, 'Q, and 

 because the equations (88), (90), (92), like (86), ufiTord convenient 

 definitions of the potentials, viz., 



;,,=(*) ={'PL) =(m =(^) (104) 



etc., where the subscript letters denote the quantities which remain 

 constant in the differentiation, m being written for brevity for all the 

 letters m^, mg, . . . in„ except the one occurring in the denominator. 

 It will be observed that the quantities in (103) are all independent 

 of the quantity of the mass considered, and are those which must, in 

 general, have the same value in contiguous masses in equilibrium. 



0)1 the quantities i/\ j, t. 

 The quantity //' has been defined for any homogeneous mass by the 



equation 



if' -€ ~ tt]. (105) 



between 



the second equation may evidently be derived from the first. But the first equation 

 cannot be derived from the second; for an equation between 



^' \dn) - ^. "in w?) • • • w„ 



Kd?! 



is equivalent to one between 



drj 

 de 



f, V, m,, m.^, . . . m„. 



which is evidently not sufficient to determine the value of ?/ in terms of the other 

 variables. 



