126 PROFESSOR W. THOMSON ON THE 
a joint communication to be made by Mr Jove and myself to the Royal Society 
of London before the close of the present session. 
101. A corollary from the second General Law of the Dynamical Theory stated 
above in § 98, equivalent to the law itself in generality, is, that if a material 
system experience a continuous action, or a complete cycle of operations, of a 
perfectly reversible kind, the quantities of heat which it takes in at different 
temperatures are subject to a linear equation, of which the coefficients are the 
corresponding values of an absolute function of the temperature. The thermo- 
metric assumption which has been adopted is equivalent to assuming that this 
absolute function is the reciprocal of the temperature; and the equation conse- 
quently takes the form 
t Uv t 
if ¢, ¢, &c., denote the temperatures of the different localities where there is 
either emission or absorption of heat, and =H, == H,, == H,, &c., the quantities 
of heat taken in or given out in those localities respectively. To prove this, con- 
ceive an engine emitting a ‘aad H, of heat at the temperature ¢, and taking 
in the corresponding quantity > “H, at the temperature 7’; then an engine emitting 
the quantity ; rH, +H, at?¢’, and taking in the corresponding quantity 7” (= es & = 
at the temperature ¢’; another emitting 7’ ee =) + H, at ¢’, and taking in the 
corresponding quantity 7” (= +e +t) at ?/’”; and so on. Considering n—2 
such engines as forming one system, we have a material system causing, by reversible 
operations, an emission of heat amounting to H, at the temperature 7, H, at the 
temperature /’,-...and Hy» att; and taking in ¢"~” (= ae 5 pees ae) 
at the temperature ¢~”. Now this system, along with the given one, constitutes a 
complex system, causing on the whole neither absorption nor emission of heat at 
the temperatures #, ¢’, &c., or at any other temperatures than ¢”~”, &; but giv- 
ing rise to an absorption or emission equal to =| «> (= — +. | 
- Hon | at ¢—, and an emission or absorption equal to = H,» at ¢. This com- 
plete system fulfils the criterion of reversibility, and, having only two tempera- 
tures at localities where heat is taken in or given out, is therefore subject to 
Law II.; that is, we must have 
Hy = a _ [a (A+ 3+ BOON sae) +Ho-» | 
which is the same as 
H, Hy He-y Hye 
ete o+ joy + ro) 
=0 ie a ern Tee e's 
