FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OF MOTION. 13 
velocity and specific weight are variable, these variables are bound by the relation 
expressed by v 3 = 1/w, which signifies that if two media are in equilibrium of pressure 
and have in the same volume of each the same number of molecules, the squares of 
their molecular velocities must be inversely as their specific molecular weights. 
Hence, we deduce that if any number of separate media have equal density and 
tension, the molecidar velocity of each must be 'proportional to the inverse square root 
of their specific molecidar weight, or to the inverse square root of the specific gravity 
of the media respect ively * .V. 
§ 8. But media may be in equilibrium of pressure without being of equal density, 
for a deficiency of density may be compensated by an excess in the molecular velocity. 
It is plain that if e and u 3 are constant in any two media they may still be in 
equilibrium of pressure if w is proportional to 1/A 3 , or if the molecular weight of each 
medium is inversely as its density. If the specific molecular weights are in a constant 
ratio to each other and the tension and velocity also constant, the media must be kept 
in equilibrium of pressure if the density of each is reciprocally proportional to the 
specific molecular weight of the other. 
We have supposed hitherto that the media are separate while their respective 
elasticities are compared. Let us now enquire into the effects of allowing them to have 
access to each other. The united media immediately obtain a heterogeneous character, 
for it requires no demonstration to convince us that the molecules of each will 
permeate through the volume occupied by the other, the vacuities in the space 
occupied by each presenting no more obstacle to the motion of one set of molecules 
than it does to the other; and as collision must take place amongst them in every 
possible manner and direction, the common space of the united media are free alike to 
each individual molecule of both to range through in its zig-zag course. Consequently, 
media in contact with each other become gradually equally diffused through their 
common volume .VI. 
The internal condition of the mixture must after a time become settled so that 
in any infinitesimal portion the same mean velocity will be found proper to the 
molecules of each medium respectively. 
But as each of the two sets of molecules, although completely mixed together, 
preserve their specific weights, so must they have corresponding specific velocities 
that remain intact, notwithstanding that they as often impinge on molecules of the 
other set as on the molecules of their own kind. It is of consequence to settle what 
the ratio of these specific velocities is, for upon this point depends the nature of the 
vis viva equilibrium of different media, and we have to determine the relative 
condition of two media when they are in equilibrium both of pressure and of vis viva. 
* [The deduction of V. appears to be correct, though much embarrassed by the irrelevant g. In his 
first memoir on the Theory of Gases (‘ Pogg. Ann.,’ vol. 100, 1857), Clausius arrives at the same con¬ 
clusion. His assumption that the density (in Waterston’s sense) of various gases is the same, appears 
to have been made upon chemical grounds.—R.] 
