a 
Theorem for Lrreversible Cycles. 519 
negative, as this would involve a contradiction of Lord Kelvin’s 
version of the Second Law; nor can it be zero, as if so the 
process by which the system passed from the state A to the 
state B would be “reversible” (“ reversibel”’) by definition ; 
it must therefore be positive and hence the result follows. 
Clausius’ Theorem: an Enunciation. 
15. In discussions which take the course of demonstrating 
the proposition known as ‘‘ Clausius’ theorem,” in some cases 
not only is the argument obscure or fallacious, but even the 
_ statement of the theorem is wanting in clearness. At least 
two slightly different forms of the proposition have been 
discussed : for the sake of definiteness I consider the fol- 
lowing :—If a body or system of bodies undergo any irre- 
versible cycle of operations in which, in addition to interchanges 
of heat which may take place between different parts of the 
system, it receives heat from (or gives heat to) any external 
bodies, then for the cycle We is negative, where 6H denotes 
the small quantity of heat absorbed by any small portion of 
the system when at the absolute temperature 0, whether this 
heat be supplied by conduction or by radiation, and whether 
by external bodies or by any other part of the system, the 
doubled sign of integration being used to indicate that inte- 
gration is to be performed over the whole mass as well as 
throughout the cycle. 
Preliminary Objections : in some cases the statemené és 
, meaningless. 
16. The difficulty, alluded to above, of defining “ tempera- 
ture ” in extreme cases of irreversibility, of course presents 
itself here too ; and there appears to be at least one further 
difficulty. Whatis meant by the “ quantity of heat absorbed ”’ 
and how is it measurable? The justification of the phrase 
“quantity of heat” and the elucidation of its meaning is 
another of the fundamental difficulties which a treatise on 
Thermodynamics must overcome. Its measurableness, like 
that of temperature, is complicated in the present instance by 
the existence of relative motions within each small part of 
the system. Moreover, heat is or may be generated by 
“friction”’ (to use the word somewhat loosely) within ‘each 
small portion, and, as I understand the statement of the 
theorem, such heat is not to be included in the quantity 
“absorbed.” If it were so included the truth of the theorem 
as a matter of fact, leaving aside the question of proof, would 
