510 Prof. W. McF. Orr on Clausius? 
state to the other by a “reversible ’’ (“reversibel”) process 
without leaving changes in other bodies*. This is proved 
substantially as follows :—Let the initial state of the 
system be given by the temperatures 0, 62,...0n, and 
the entropies ¢), $:,... x, and the final state by the 
corresponding dashed letters. Bring the first gas and the 
nth to a common temperature by a reversible adiabatic 
change (the necessary work in this and following ex- 
pansions or compressions being done by or on _ bodies 
external to the system +) ; then place them in thermal com- 
munication, and by indefinitely slow compression or expansion 
compel heat to pass between them until the entropy of the 
first gas is changed from ¢;, to ¢,’; let the second, third, and 
remaining gases by similar interchanges if ae with the nth 
change their entropies from do, 3, . . . &C., 0 Go, cag one 
&e. Then, since in each interchange Sra sum of the entropies 
of the two gases concerned is unaltered, es since by sup- 
position d+ 2+ ..- br=,/ Hho’ + ... dn’ it follows that 
the entropy of the nth gas is now @¢,’. Finally, let each gas 
be expanded or compr essed adiabatically until they attain the 
temperatures 0,', 0,',. . . On’, respectiv ely, and they are then 
in the given final state since each has the assigned tempera- 
ture and entropy. 
3. Itis assumed as a result of experience that the expansion 
of a perfect gas without performing any external work is 
“irreversible ” (“‘irreversibel”) and also, as usual, that in 
such a case when the gas after expansion comes into a state 
of equilibrium its final temperature is the same as the initial 
provided it does not interchange heat with any external 
bodies, so that the entropy, from its definition, is increased 
by such an expansion. From these postulates it follows that 
it is impossible to diminish the entropy of a perfect gas 
without leaving changes in other bodies f. 
The last deduction is then extended from the case of a 
single gas to that of a system of gases: it is proved that it is 
impossible to diminish the total entropy of such a system 
without leaving changes in other bodies§. The demonstra- 
tion is as follows : Suppose, if possible, that the total entropy 
could be diminished without leaving changes in other bodies ; 
let the initial entropies be dq), ¢5,... $i and the final 
P's e's - . bi’, where 3’ <=d. Then itis possible, accord- 
ing to the proposition of Art. 2 above, to bring the system, 
without leaving changes in any other bodies, from the final 
* Loe. cit. Art. 128. 
7 This point is commented on below. See § 5, 
t Loc. cit. Art. 124. § Loc. cit, Art. 125. 
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