“At best you would be entering into a discussion (perhaps not in
    bad company) in which words would play a greater part than precise
    ideas.

    “This is the way I feel about it.

    “I remain,
    “Yours faithfully,
    “J.W.  GIBBS.”

Professor Gibbs’s criticism of the illustration of water-mixture is evidently just.  Another might well have been used where the things mixed are not material—­for instance, the value of money deposited in a bank.  If A and B each deposits $100 to C’s credit and C then draws $10, there is evidently no way of determining what part of it came from A and what from B. The structure of “value”, in other words, is perfectly continuous.  Professor Gibbs’s objections to an “atomic” theory of the structure of energy are most interesting.  The difficulties that it involves are not overstated.  In 1897 they made it unnecessary, but since that time considerations have been brought forward, and generally recognized, which may make it necessary to brave those difficulties.

Planck’s theory was suggested by the apparent necessity of modifying the generally accepted theory of statistical equilibrium involving the so called “law of equipartition,” enunciated first for gases and extended to liquids and solids.

In the first place the kinetic theory fixes the number of degrees of freedom of each gaseous molecule, which would be three for argon, for instance, and five for oxygen.  But what prevents either from having the six degrees to which ordinary mechanical theory entitles it?  Furthermore, the oxygen spectrum has more than five lines, and the molecule must therefore vibrate in more than five modes.  “Why,” asks Poincare, “do certain degrees of freedom appear to play no part here; why are they, so to speak, ’ankylosed’?” Again, suppose a system in statistical equilibrium, each part gaining on an average, in a short time, exactly as much as it loses.  If the system consists of molecules and ether, as the former have a finite number of degrees of freedom and the latter an infinite number, the unmodified law of equipartition would require that the ether should finally appropriate all energy, leaving none of it to the matter.  To escape this conclusion we have Rayleigh’s law that the radiated energy, for a given wave length, is proportional to the absolute temperature, and for a given temperature is in inverse ratio to the fourth power of the wave-length.  This is found by Planck to be experimentally unverifiable, the radiation being less for small wave-lengths and low temperatures, than the law requires.

Still again, the specific heats of solids, instead of being sensibly constant at all temperatures, are found to diminish rapidly in the low temperatures now available in liquid air or hydrogen and apparently tend to disappear at absolute zero.  “All takes place,” says Poincare, “as if these molecules lost some of their degrees of freedom in cooling—­as if some of their articulations froze at the limit.”