THERMODYNAMIC THEORY. 227 



mined by the relation between pressure and density as exhibited at various 

 depths within the earth at any single epoch. For since the compression has 

 been treated as adiabatic, this identification of the two thermodynamic 

 paths means that as long as the effect of the conduction is insensible the 

 nucleus would be in a condition of isentropic or convective equilibrium 

 such as described by Ritter and Kelvin. The generaUzation of equation (21) 

 consists in supposing that it would give the form of the pressure-density 

 curve corresponding to convective equilibrium for any value of the entropy 

 by proper choice of the constants h and p^. It is, then, required to deter- 

 mine the functions /i, /a, (p, so as to satisfy the given conditions. 

 Equation (134) shows that 



de=^dd + ^^dv (143) 



o o 



in which the coefficient of dv must therefore be a function of v only, which 

 for convenience may be written 



p+y'w ,„ . m (144) 



d (p{v) 



and then the integral of (143) is 



Oil,T=(^=e^ (e = Naperian base) ^ ' 



in which the constant of integration is considered to be absorbed in the 

 undetermined function ^. 



Elimination of 6 from (144) and (145) gives 



P=<^^-/ (146) 



which must be identical with (142), and thus gives 



Hence 



/,(£)=aAc + a /,{£) = -oBc-h (147) 



and 



^ = (A^Bv + Cy' (148) 



<p=--{-bv (149) 



The constant of integration in (p is omitted, since only differences in the 

 intrinsic energy are in question, so that there remain five undetermined 

 constants. In terms of these auxiliary functions, then, the properties of the 

 substance are described by equations (141), (144), and (145), from which 

 may be deduced the following: 

 15 



from which come 



