Theorem for Irreversible Cycles. 525 
be deduced from the principles of mechanics. For the force 
which the steam exerts on the moving piston is not necessarily 
always less in the former expansion than that at the corre- 
sponding stage in the latter. In the first expansion, the laws 
of mechanics tell us indeed that when the velocity of the 
steam is being accelerated, the force which the steam exerts 
on the piston is less than that which it exerts on the fixed end 
of the cylinder, and the fact that the sides of the cylinder exert 
friction on the steam shows that the difference is greater than 
the mass of the steam multiplied by its average acceleration ; 
the same laws tell us also, however, that towards the end of the 
expansion, when the velocity is diminishing, the force on 
the piston may exceed that on the fixed end, and would do so 
by the mass of the steam multiplied by its average retardation 
were it not that the friction between the sides of the cylinder 
and the steam now tends to equalize these forces. Mechanical 
principles may thus enable us at any stage in the former 
expansion to compare the force which the steam exerts on the 
moving piston with the force which it exerts on the fixed end of 
the cylinder ; but they do not enable us to compare either of 
these forces with the force exerted on the piston in the other 
expansion. It is thus quite possible that towards the end of 
the former expansion the driving-force on the piston’may 
exceed that in the latter, especially since, as a matter of fact, 
the steam in the final stages is on the average hotter in the 
former. Mere mechanical considerations thus do not enable 
us to say in which case more work is done on the piston, even 
if we do take account of the friction of the cylinder-sides. 
To settle the question we must invoke the principles of 
Thermodynamics. Let us suppose that the steam firstly 
undergoes the sudden expansion, secondly, settles into 
equilibrium at the end of it, thirdly, is then brought by con- 
tinuous heating or cooling at constant volume into the state 
which it would have had at the end of the indefinitely slow 
adiabatic expansion, and, finally, is restored to its original 
state by an indefinitely slow adiabatic compression, the reverse 
of the slow expansion. ‘The steam has now undergone a cycle 
and must therefore, by the principle of irreversibility *, have 
actually at some stage given out heat ; that is to say, the third 
process is actually one of cooling: hence, from the First Law, 
work must have been done on the steam in the cycle asa whole ; 
that is, the work which it does in the sudden expansion is 
less than that done on it in the indefinitely slow compression, 
* T,e, the principle stated in the preceding section, 
