Theorem for Irreversible Cycles. 521 
perfectly reversible Carnot engine which continually brings 
it from a source whose temperature is uniform and constantly 
equal to 8,. In a complete cycle the total quantity of heat 
° oH . 
taken from the source is 0, 7a Hence, since a system 
e 
which undergoes a complete cycle cannot take heat from a 
body whose temperature is uniform and constant unless 
some other body of different temperature be also present, 
6 
frictional ” cycle, and that for all other cycles it is negative. 
That the integral cannot be positive for any cycle, and hence 
that it is zero for a reversible cycle, is a legitimate inference 
from the premisses ; but the grounds of the deduction that it 
is negative for a frictional cycle are not, I maintain, stated 
with sufficient definiteness, especially as the precise meaning 
which the author attaches to the word “ non-frictional” is 
not clear. He has previously * stated that “in much that 
follows the term ‘friction’ is used in a general way to 
include every kind of irreversibility as well as friction 
proper ;”’ but, on the other hand, that a certain process + (the 
slow heating of a saturated solution of sulphate of magnesium 
along with an excess of salt) is “‘not reversible, even though 
it is a non-frictional process.” 
The author, indeed, is evidently himself dissatisfied with 
this discussion, as in a later one ¢ he states that ‘‘the case of 
an irreversible cycle cannot be discussed in its generality ; 
but on taking some typical cases of irreversibility, as in 
OH . 
it is inferred that the integral iE is zero for a “‘ non- 
/ // 
Art. 37, it is found that 35 2 rf +... is negative. 
Jee i ie 
We therefore infer that we may put s a + ot Beal 
according as the cycle is reversible or irreversible.” 
Proofs by Poincaré, Kirchhoff’, Voigt, and Buckingham. 
; vie 6H 
19. Poincaré proves § that in any cycle thas cannot be 
positive, not that in an irreversible cycle it must be negative. 
* Loc, cit. footnote, p. 118. 
+ Loe. cit. p. 108. a avira 
t ‘Thermodynamics treated with Elementary Mathematics,’ 1894, 
02 rae 
§ Thermodynamique, ‘Chap. xii. 
