280 PROFESSOR WILLIAM THOMSON ON THE 
that the true value of 4 might be “ inversely at the temperatures from zero;” * 
and values for various temperatures calculated by means of the formula, 
E 
14+E¢ . . . . . . (11), 
were given for comparison with those which I had calculated from data regarding 
steam. ‘This formula is also adopted by CLausius, who uses it fundamentally in 
his mathematical investigations. If 4 were correctly expressed by it, we should 
have 
S 1+ES 
Jp Htt= Flos tans 
and therefore equations (1) and (2) would become 
B=J 

S-—T 
Weta «| dp pom, Ly eee 
Ets 
aa 
Bee (13). 
SS 
E 
43. The reasons upon which Mr Jouue’s opinion is founded, that the preceding 
equation (11) may be the correct expression for Carnov’s function, although the 
values calculated by means of it differ considerably from those shewn in Table I. 
of my former paper, form the subject of a communication, which I hope to have 
an opportunity of laying before the Royal Society previously to the close of 
the present session. 
Part [1J].—AppiicaTIoNs OF THE DYNAMICAL THEORY TO ESTABLISH RELATIONS 
BETWEEN THE PHYSICAL PROPERTIES OF ALL SUBSTANCES. 
44. The two fundamental equations of the dynamical theory of heat, inves- 
tigated above, express relations between quantities of heat required to pro- 
duce changes of volume and temperature in any material medium whatever, 

* If we take u=k i _ where i may be any constant, we find 
k 
w=J (—) J. 
ge 
i 1" 
which is the formula I gave when this paper was communicated. I have since remarked, that Mr 
Joute’s hypothesis implies essentially, that the coefficient & must be as it is taken in the text, the 
mechanical equivalent of a thermal unit. Mr Rankine, in a letter dated March 27, 1851, informs 
me that he has deduced, from the principles laid down in his paper communicated last year to this 
Society, an approximate formula for the ratio of the maximum quantity of heat converted into 
mechanical effect to the whole quantity expended, in an expansive engine of any substance, which, 
on comparison, I find agrees exactly with the expression (12) given in the text as a consequence of 
the hypothesis suggested by Mr Jouxe regarding the value of at any temperature.—[April 4, 1851.] 

