122 
MR. MACQUORN RANKINE ON THERMO-DYNAMICS. 
Section II.— PROPOSITIONS RELATIVE TO HOMOGENEOUS SUBSTANCES. 
(7.) Proposition II. — Theorem. In Jig. 5, let AiA 2 M, B^N be any two curves of 
no transmission , indefinitely extended in the direction of X, intersected in the points , 
Fig. 5. 
Aj,B 15 A 2 , B 2 , by two isothermal curves, QjAjBjQj, Q 2 A 2 B 2 Q 2 , which are indefinitely 
near to each other ; that is to say, which correspond to two quantities of actual heat, 
Q, and Q 2 , differing by an indefinitely small quantity Q x — Q 2 =^Q. 
Then the elementary quadrilateral area, AjB^Aa, bears to the whole indefinitely- 
prolonged area MA,BiN, the same proportion which the indefinitely small difference of 
actual heat bears to the whole actual heat Q x ; or 
area AjBjB^Ag 8CJ 
area MAjBjN Qj* 
(Demonstration.) Draw the ordinates A X V A1 , A 2 \ A2 , BjVm, B 2 V B2 . Suppose, in the 
first place, that 5Q is an aliquot part of Q 15 obtained by dividing the latter quantity 
by a very large integer n, which we are at liberty to increase without limit. 
The entire indefinitely-prolonged area MAjBjN represents a quantity of heat which 
is converted into potential energy during the expansion of the substance from V A1 to 
V B1 , in consequence of the continued presence of the total actual heat Q x ; for if no 
heat were present no such conversion would take place. Mutatis mutandis, a similar 
statement may be made respecting the area MA 2 B 2 N. By increasing without limit 
the number n and diminishing £Q, we may make the expansion from V A2 to V B2 as 
nearly as we please an identical phenomenon with the expansion from V A1 to V B1 . 
The quadrilateral AjB^A., represents the diminution of conversion of heat to poten- 
tial energy, which results from the abstraction of any one whatsoever of the n small 
equal parts c$Q into which the actual heat Q x is supposed to be divided, and it there- 
fore represents the effect, in conversion of heat to potential energy, of the presence of 
any one of those small portions of actual heat. And as all those portions are similar 
and similarly circumstanced, the effect of the presence of the whole actual heat Q, in 
causing conversion of heat to potential energy, will be simply the sum of the effects 
of all its small portions, and will bear the same ratio to the effect of one of those 
small portions, which the whole actual heat bears to the small portion. Thus, by 
