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Additional Note on the 'preceding Paper. By William Thomson, M.A., F.R.S., F.R.S.E., 
Fellow of St. Peters College, Cambridge, and Professor of Natural Philosophy in the 
University of Glasgow. 
Received March 23. 
1. Synthetical Investigation of the Duty of a Perfect Thermo-Dynamic Engine 
founded on the Expansions and Condensations of a Fluid, for which the gaseous 
laws hold and the ratio (k) of the specific heat under constant pressure to the 
specific heat in constant volume is constant ; and modification of the result hy 
the assumption of Mayer’s hypothesis. 
Let the source from which the heat is supplied be at the temperature S, and let 
T denote the temperature of the coldest body that can be obtained as a refrig-erator. 
A cycle of the following four operations, being reversible in every respect, gives, 
according to Carnot’s principle, first demonstrated for the Dynamical Theory by 
Clausius, the greatest possible statical mechanical elfect that can be obtained in 
these circumstances from a quantity of heat supplied from the source. 
(1.) Let a quantity of air contained in a cylinder and piston, at the temperature 
S, be allowed to expand to any extent, and let heat be supplied to it to keep its tem- 
perature constantly S. 
(2.) Let the air expand farther, without being allowed to take heat from or to 
part with heat to surrounding matter, until its temperature sinks to T. 
( 3 .) Let the air be allowed to part with heat so as to keep its temperature con- 
stantly T, while it is compressed to such an extent that at the end of the fourth 
operation the temperature may be S. 
( 4 .) Let the air be farther compressed, and prevented from either gaining or 
parting with heat, till the piston reaches its primitive position. 
The amount of mechanical effect gained on the whole of this cycle of operations 
will be the excess of the mechanical effect obtained by the first and second above 
the work spent in the third and fourth. Now if P and V denote the primitive 
pressure and volume of the air, and if Pj and Vj, P2 and Vg, P3 and V3, P4 and V4 
denote the pressure and volume respectively, at the ends of the four successive 
operations, we have by the gaseous laws, and by Poisson’s formula and a con- 
clusion from it quoted above, the following expressions : — 
V 
Mechanical effect obtained by the first operation =PV log 
Mechanical effect obtained by the second operation 
Work spent in the third operation 
= P3V3lOg 
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