
DYNAMICAL THEORY OF HEAT. 481 
data from which it is deduced. It is not improbable that such a Table or Empi- 
rical Function, and a similar representation of the pressure, may be found to be 
the most convenient expression for results of complete observations on the com-- 
pressibility, the law of expansion by heat, and the thermal capacities of a vapour 
or gas. 
94. The principles brought forward in a former communication “Ona Means 
of discovering experimentally, &c.” (which is now referred to as Part IV. of aseries 
of papers on the Dynamical Theory of Heat), may be expressed in a more con- 
venient, and in a somewhat more comprehensive manner than in the formule 
contained in that paper, by introducing the notations and principles which form 
the subject of the present communication. Thus, let ¢ be the temperature, and u 
the volume of a pound, of air flowing gently in a pipe (under very high pressure it 
may be) towards a very narrow passage (a nearly closed stopcock, for instance), and 
let p be its pressure. Let /’, uw’, and p’ be the corresponding qualities of the air, 
flowing gently through a continuation of the pipe, after having passed the “rapids” 
in and near the narrow passage. Let Q be the quantity of heat (which, according 
to circumstances, may be positive, zero, or negative) emitted by a pound of air 
during its whole passage from the former locality through the narrow passage, to 
the latter; and let S denote the mechanical value of the sound emitted from the 
“yapids.” The only other external mechanical effect, besides these two, produced 
by the air, is the excess (which, according to circumstances, may be negative, zero, 
or positive) of the work done by the air in pressing out through the second part 
of the pipe above that spent in pressing it in through the first; the amount of 
which, for each pound of air that passes, is of course p’ u’—pwu. Hence, the 
whole mechanical value of the effects produced externally by each pound of the 
air, from its own mechanical energy, is 
JQ+Stp'w—pu, . : ; , : : ; 3 (15). 
Hence, if ¢ (v, ¢) denote the value of ¢ expressed as a function of the independent 
_ variables, » and ¢; so that ¢ (wu, t) may express the mechanical energy of a pound 
of air before, and ¢ (w’, 7’) the mechanical energy of a pound of air after, passing 
the rapids; we have 
b (uv, t)=h (,f)-{JQ+St+puw—puy . . . « (16). 
95. If the circumstances be arranged (as is always possible), so as to prevent 
the air from experiencing either gain or loss of heat by conduction through the 
pipe and stopcock,we shall have Q=0; and if (as is perhaps also possible) only 
a mechanically inappreciable amount of sound be allowed to escape, we may take 
S=0. Then the preceding equation becomes’ 
p (vw, t)=p (u, t)—(p' w—p u) : : . : : (17). 
