80 
PROFESSOR THOMSON ON THE SPECIFIC HEATS OF AIR. 
which involves no hypothesis, the expression 
J 
for Carnot’s function, which Mr. Joule had suggested to me in a letter dated 
December 9, 1848, as the expression of Mayer’s hypothesis, in terms of the nota- 
tion of my “ Account of Carnot’s Theory.* * * § ” Mr. Rankine-I" has arrived at a formula 
agreeing with it (with the exception of a constant term in the denominator, which, 
as its value is unknown, but probably small, he neglects in the actual use of the 
formula), as a consequence of the fundamental principles assumed in his Theory of 
Molecular Vortices, when applied to any fluid whatever, experiencing a cycle of 
four operations satisfying Carnot’s criterion of reversibility (being in fact precisely 
analogous to those described above, and originally invented by Carnot) ; and he 
thus establishes Carnot’s law as a consequence of the equations of the mutual con- 
version of heat and expansive power, which had been given in the first section of his 
paper on the Mechanical Action of Heat:|:. 
2. Note on the Specific Heats of Air. 
Let N be the specific heat of unity of weight of a fluid at the temperature C kept 
within constant volume, v ; and let k'N be the specific heat of the same fluid mass, 
under constant pressure, p. Without any other assumption than that of Carnot’s 
principle, the following equation is demonstrated in my paper§ on the Dynamical 
Theory of Heat, § 48, 
^N-N= 
where ^ denotes the value of Carnot’s function, for the temperature t, and the dif- 
ferentiations indicated are with reference to v and t considered as independent 
variables, of which p is a function. If the fluid be subject to Boyle’s and Mariotte’s 
law of compression, we have 
dp p , 
dv V 
and if it be subject also to Gay-Lussac’s law of expansion, 
dp Ep 
dt 1 -j- 
* Royal Society of Edinburgh, January 2, 1849, Transactions, vol. xvi. part 5. 
f On the Economy of Heat in Expansive Engines. Royal Society of Edinburgh, April 21, 1851, Trans- 
actions, vol. XX. part 2. 
J Royal Society of Edinburgh, February 4, 1850, Transactions, vol. xx. part 1. 
§ Royal Society of Edinburgh, March 17, 1851, Transactions, vol. xx. part 2. 
