482 PROFESSOR W. THOMSON ON THE DYNAMICAL THEORY OF HEAT. 
If by experimenting in such circumstances it be found that / does not differ sen- 
sibly from ¢, MayEr’s hypothesis is verified for air at the temperature ¢, and, as 
yw would then be equal to pw, according to BoyLE and Mariorre’s law, we 
should have 
— w= (ud) 
which is in fact the expression of MayEr’s hypothesis, in terms of the notation 
for mechanical energy introduced in this paper. If, on the other hand, ¢’ be found 
to differ from ¢;* let values of p, p’, ¢, and i’ be observed in various experiments 
of this kind, and, from the known laws of density of air, let ~ and w’ be calculated. 
We then have, by an application of (13), to the results of each experiment, an 
equation shewing the difference between the mechanical energies of a pound of 
air in two particular specified states as to temperature and density. All the par- 
ticular equations thus obtained, may be used towards forming, or for correcting, 
a table of the values of the mechanical energy of a mass of air, at various tempera- 
tures and densities. 
96. If, according to the plan proposed in my former communication (§ 72), the 
air, on leaving the narrow passage, be made to pass through a spiral pipe immersed 
in water in a calorimetrical apparatus, and be so brought back exactly to the pri- 
mitive temperature ¢, we should have, according to Boyie’s and Mariorre’s law, 
p w—pu=0; and if H denote the value of Q, in this particular case (or the 
quantity of heat measured by means of the calorimetric apparatus), the general 
equation (16) takes the form, 
@ (Ww, )=>(u)-JH+8) . . . Ma ak) 
If in this we neglect S, as probably insensible, and if we pase for (uw, ¢) 
and ¢# (w’, t) expressions deduced from (9), we find, 
1 E eh uw 
He (3-7ceED} palogee oe , } OS ee 
which agrees exactly with the expression obtained by a synthetical process, 
founded on the same principles, in my former communication (§ 76). 
* If the values of ~ I have used formerly be correct, ¢ would be less than ¢, for all cases in 
which ¢ is lower than about 30° cent.; but on the contrary, if t be considerably above 30° cent., ¢ 
would be found to exceed t. (See Account of Carnor’s Theory, Appendix II.) It may be shewn, 
that if they are correct, air at the temperature 0° forced up with a pressure of ten atmospheres towards 
a small orifice, and expanding through it to the atmospheric pressure, would go down in temperature 
by about 4°-4; but that if it had the temperature of 100° in approaching the orifice, it would leave 
at a temperature about 5°2 higher; provided that in each case there is no appreciable expenditure 
of mechanical energy on sound. 

