MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
155 
action of heat, when either their specific heats, or their volumes for unity of weight 
at a given pressure and temperature, or both these classes of quantities, are different. 
Hence a portion of a liquid, and a portion of its vapour, enclosed in the same 
vessel, though chemically identical and mutually transformable, are heterogeneous, 
and are to be treated as an aggregate, with respect to the expansive action of heat. 
M. Clausius and Professor William Thomson have applied their formulae to the 
aggregate composed of a liquid and its vapour, and have pointed out certain relations 
which must exist between the pressure and density of a liquid and its vapour, and 
the latent heat of evaporation. 
I shall now apply the geometrical method of this paper to the theory of the ex- 
pansive action of heat in an aggregate, especially that consisting of a liquid and its 
vapour. The total volumes are, for the present, supposed not to be large enough to 
exhibit any appreciable differences of pressure due to gravitation. 
(35.) Proposition XIII. — Theorem. In an aggregate in equilibria , the pressure of 
each ingredient must be the same ; and the quantity of heat in unity of weight of each 
ingredient must be inversely proportional to its real specific heat ; that is to say , the 
temperature must be equal. 
The following is the symbolical expression of this theorem, with certain conclusions 
to which it leads. 
Let 7 — k be the common temperature of the ingredients, as measured from the 
point of absolute cold ; 
P their common pressure , 
Wi, n 2 , n 3 , &c. their proportions by weight, in unity of weight of the aggregate ; 
t) u v 2 , v 3 , &c. the respective volumes of unity of weight of the several ingredients ; 
V the volume of unity of weight of the aggregate ; 
q^,q 2 , q z , &c. the respective quantities of actual heat in unity of weight of the 
several ingredients ; 
fci, k 2 , k 3 , &c. their respective real specific heats ; 
Q the quantity of heat in unity of weight of the aggregate ; 
a thermo-dynamic function for the aggregate. 
Then these quantities are connected by the following equations : — 
2.^=1. (50.) 
V=2.nv. (51.) 
(52 -> 
Q —'2.nq — (r—z).'2.nk (53.) 
, r2.n(fe+/ , .r) frfP 
°=j - -•*+) < 54 -) 
It is evident that all these equations hold whether the proportions of the ingre- 
