348 
MR. J. P. JOULE AND PROFESSOR THOMSON ON THE 
v, V, V' denote, with reference to air at the temperature of the bath, respectively, the 
volumes occupied by a pound under any pressure /?, under a pressure, P, equal to that 
with which the air enters the plug-, and under a pressure, P', with which the air escapes 
from the plug - ; and JK^ is the mechanical equivalent of the amount of heat per 
pound of air passing- that would be required to compensate the observed cooling 
effect S. The direct use of this equation for determining - requires, besides our own 
results, information as to compressibility and expansion which is as yet but very 
insufficiently afforded by direct experiments, and is consequently very unsatisfactory, 
so much so that we shall only give an outline, without details, of two plans we have 
followed, and mention the results. First, it may be remarked that, approximately, 
w=(l +E<)H log P , and ^=EH log p , 
H being the “ height of the homogeneous atmosphere,” or the product of the pressure 
into the volume of a pound of air, at 0° Cent. ; of which the value is 26224 feet. 
Hence, if IS denote a certain mean coefficient of expansion suitable to the circum- 
yj 
stances of each individual experiment, it is easily seen that -j— may be put under the 
dt 
form ji+C and thus we have 
J 
v- 
JKS-(PV'-PV) 
EH log | 
since the numerator of the fraction constituting the last term is so small, that the ap- 
proximate value may be used for the denominator. The first term of the second 
member may easily be determined analytically in general terms ; but as it has refer- 
ence to the rate of expansion at the particular temperature of the experiment, and 
not to the mean expansion from 0° to 100°, which alone has been investigated by 
Regnault and others who have made sufficiently accurate experiments, we have not 
data for determining its values for the particular cases of the experiments. We may, 
however, failing more precise data, consider the expansion of air as uniform from 
0° to 100°, for any pressure within the limits of the experiments (four or five atmo- 
spheres) ; because it is so for air at the atmospheric density by the hypothesis of the 
air-thermometer, and Regnault’s comparisons of air-thermometers in different con- 
ditions show for all, whether on the constant- volume or constant-pressure principle, 
with density or pressure from one-half to double the standard density or pressure, 
a very close agreement with the standard air-thermometer. On this assumption 
then, when we take into account Regnault’s observations regarding the effect of 
variations of density on the coefficient of increase of pressure, we find that a suitable 
J 
mean coefficient E for the circumstances of the preceding formula for - is expressed, 
V* 
