THERMAL EFFECTS OF FLUIDS IN MOTION'. 
357 
we adopt the form to which Mr. Rankine was led by his theory of molecular vortices, 
and which he has used with so much success for the expression of the pressure of 
saturated steam and the mechanical properties of gases ; with this difference, that the 
series we assume proceeds in descending powers of the absolute thermo-dynamic 
temperature, while Mr. Rankine’s involves similarly the temperature according to 
what he calls “ the scale of the perfect gas-thermometer.” 
Now any variable part of <p 0 (v), and the whole series of terms following it, must 
correspond to deviations from the gaseous laws, since the general expression of these 
laws would be simply jw = A£+B, if A and B be constant. Hence for atmospheric 
air any variable part that <p 0 (v) can have, and all the terms following it in the series, 
must be very small fractions of pv. We shall see immediately that the various devia- 
tions from the gaseous laws which have been established by experiment, as well as 
the cooling effects which we have observed, are all such as to be represented by ex- 
pressions derived from the preceding formula, if the variable part of <p 0 (v), and the 
whole functions <p x {v), <p 2 (v) } &c. be taken each of them simply proportional to the 
density directly, or to the volume (v) of a pound inversely. We may then, to avoid 
unnecessary complications, at once assume 
pv— A£+b+(c+ 
where A, B, C, D and G are all constants to be determined by the comparison with 
experimental results, and denotes a particular volume corresponding to a standard 
state of density, which it will be convenient to take as 12 , 387 cubic feet, the volume of 
a pound when under the atmospheric pressure n (=2117 lbs. per square foot) of 
29 - 9218 inches of mercury in latitude 45°. The series is stopped at the fifth term, 
because we have not at present experimental data for determining the coefficients for 
more. The experimental data which we have, and find available, are (1) the results 
of Regnault’s observations on the coefficients of expansion at different constant den- 
sities, (2) the results of his observations on the compressibility, at a temperature of 
4°*75 Cent., and (3) our own experimental results now communicated to the Royal 
Society. These are expressed within their limits of accuracy (at least for pressures of 
from one to five or six atmospheres, such as our experiments have as yet been con- 
fined to), by the following equations : — 
. E=.00366 5+ ^(i-l), 
or E=-00365343+-000011575 j (18) 
PV — P , V , ='008163 — PV, at temperature 4 0- 75 Cent., . . . (19) 
P— P' 
and d = ’26 n , at temperature 17° Cent (20) 
D G\ 4> 
/ “I J 
( 17 ) 
