156 
MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
clients &c. are constant, as in an aggregate of chemically distinct substances, or 
variable, as in the aggregate of a liquid and its vapour. 
Let be the beat which disappears in consequence of a small expansion of the 
aggregate at constant temperature represented by 
&V=2.&k, (55.) 
lu representing any one of the parts arising from the changes undergone by the dif- 
ferent ingredients, of which the whole expansion of the aggregate, W, is made up. 
Then 
m = (56.) 
but the pressure P is the same for every ingredient, as well as the temperature ; there- 
fore the factor (r— x) -jg is the same for every ingredient, and consequently for the 
whole aggregate ; that is to say, 
m = (r-*)^.iV=(r-*)M> (57.) 
This equation shows, that the relation of temperature to the mutual transformation 
of heat and expansive power is the same in an aggregate as in a homogeneous 
substance. 
Consequently, if we define Isothermal Curves for an Aggregate to be Curves of 
Constant Temperature, we arrive at the following conclusion : — 
Proposition XIV. — Theorem. Isothermal curves on the diagram of energy of an 
Aggregate, have the same properties, with reference to the mutual transformation of 
Heat and Expansive Power, with those on the diagram of energy of a homogeneous 
substance. 
It is unnecessary to enunciate separately a similar proposition for curves of no 
transmission ; for the demonstration of Proposition I., on which all their properties 
depend, is evidently applicable to an aggregate constituted in any manner. 
Hence it appears, that if the isothermal curves for an aggregate be drawn accord- 
ing to the above definition, all the propositions proved in this paper respecting homo- 
geneous substances become true of the aggregate. 
(36.) Proposition XV. — Theorem. Every Isothermal line for an aggregate of a 
liquid and, its vapour, is a straight line of equal pressure, from the volume corresponding 
to complete liquefaction to the volume corresponding to complete evaporation. 
This is a fact known by experiment. The Theorem is equivalent to a statement, 
that the pressure of a liquid and its vapour in contact with each other, is a function 
of the temperature only. 
Corollary . — Theorem. At any given temperature, the volume of an aggregate of 
liquid and vapour is arbitrary between and up to the limits of total liquefaction and 
total evaporation. 
To express this symbolically, let P be the pressure of an aggregate of liquid and 
