DUTY OF A PERFECT THERMO-DYNAMIC ENGINE. 
79 
Work spent in the fourth operation 
Now, according to the gaseous laws, we have 
P,V.=PV; P,V,=P,V,|^; 
P3V3=P2V2; and (since V4=V), P4=P. 
Also by Poisson’s formula, 
(vj “(vy “l + ET* 
By means of these we perceive that the work spent in the fourth operation is equal 
to the mechanical effect gained in the second ; and we find, for the whole gain of 
mechanical effect (denoted by M), the expressions 
M= (PV - P,V.) log: ^=PV log ^ 
All the preceding formulee are founded on the assumption of the gaseous laws 
and the constancy of the ratio {k) of the specific heat under constant pressure to 
the specific heat in constant volume, for the air contained in the cylinder and piston, 
and involve no other hypothesis*. If now we add the assumption of Mayer’s hypo- 
V 
thesis, which for the actual circumstance is PVlogY=JH, where FI denotes the 
heat abstracted by the air from the surrounding matter in the first operation, and J 
the mechanical equivalent of a thermal unit, we have 
M=JH 
E(S-T) 
l + ES ■ 
The investigation of this formula given in my paper on the Dynamical Theory 
of Heat, shows that it would be true for every perfect thermo-dynamic engine, if 
Mayer’s hypothesis were true for a fluid subject to the gaseous laws of pressure and 
density, whether, for such a fluid (did it exist), k were constant or not. 
It was first obtained by using, in the formula 
* From the sole hypothesis that k is constant for a single fluid fulfilling the gaseous laws, and having E for 
its coefficient of expansion, I find it follows, as a necessary consequence, that Carnot’s function would have 
JE 
the form ; where C denotes an unknown absolute constant, and t the temperature measured by a 
1 d-E^ -j- C 
thermometer founded on the equable expansions of that gas. From this it follows, that for such a gas sub- 
V 1-1- ES 
jected to the four operations described in the text, we must have PV log JH ^ and consequently, 
•p/C 'P'v 
M = JH- — which is Mr. Rankine's general formula. 
1 +Eo + O 
