478 PROFESSOR WILLIAM THOMSON ON THE 
will, when any set of data are proposed, make it manifest whether or not they are 
sufficient, and will point out the methods, whether of summation or of analytical 
integration, according to the forms in which the data are furnished, to be followed 
for determining the value of ¢ for every value of v. Or the data may be such that, 
while the thermal capacities would be derived from them by differentiation, values 
of ¢ may be obtained from them without integration. Thus, if the fluid mass consist 
of water and vapour of water at the temperature ¢, weighing in all one pound, and 
occupying the volume »,* and if we regard the zero or “ standard” state of the mass 
as being liquid water at the temperature 0°; the mechanical energy of the mass, 
in the given state, will be the mechanical value.of the heat required to raise the 
temperature of a pound of water from 0° tot, and to convert = of it into vapour, 
diminished by the work done in the expansion from the volume A, to the volume 
»; that is, we have 
mts (c¢+L 22) —p 0-0) centimetre). 
The variables, c, L, and p (which depend on ¢ alone) in this expression have been 
experimentally determined by REGNAULT, for all temperatures from 0° to 230°, and 
when 7¥ is also determined, by experiments on the density of saturated steam, the 
elements for the determination of ¢ in this case will be complete. The expressions 
investigated formerly for M and N in this case (§ 54) may be readily obtained by 
means of (4) and (5 of § 84), by the differentiation of (8). 
89. If Carnor’s function has once been determined by means of observations 
of any kind, whether on a single fluid, or on different fluids, for a certain range of 
dp 
temperatures, then, according to (6) of § 85, the value of S for any substance 
whatever, is known for all temperatures within that range. It follows that when 
the values of M for different states of a fiuid have been determined experimentally, 
the law of pressures for all temperatures and volumes (with an arbitrary function 
of v to be determined by experiments on the pressure of the fluid at one particular 
temperature) may be deduced, by means of equation (6); or conversely, which is 
more likely to be the case for any particular fluid, if the law of pressures is com- 
pletely known, M may be deduced without farther experimenting. Hence the 
second member of (4) becomes completely known, the equation assuming the fol- 
lowing form when, for M, its value according to (6) is substituted :— 
* The same notation is used here, as formerly in § 54, viz. pis the pressure of saturated vapour 
at the temperature ¢, vy the volume, and L the latent heat of a pound of the vapour, A the volume of 
a pound of liquid water, and c the mean thermal capacity of a pound of water between the tempera- 
tures 0 and ¢. A mass weighing a pound, and occupying the volume v, when at the temperature ¢, 
lS vapour, and 7" of water. 
—i y-* 


must consist of a weight 
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