Herapath was initially interested in developing an explanation of gravity in terms of the impacts of particles of an ethereal fluid, somewhat along the lines of the "kinetic theory of gravity" proposed by G. S. LeSage and many others. Herapath's version was somewhat different: he proposed to take account of the effect on the gravific particles of the high temperatures in the space near the Sun. In this way he came to consider the relation between temperature and particle velocity.
Herapath was puzzled by the old paradox of collisions between atoms: if they meet head-on they must stop at least for an instant before rebounding. Is the collision elastic or inelastic? It can't be elastic, since an atom by definition has no smaller parts and cannot change its size, so how can it store its kinetic energy during that instant? But if it were inelastic, both atoms would have to stop; then what happens to their energy?
To avoid the paradox, Herapath decided to adopt momentum instead of energy as the fundamental measure of motion, since momentum is always conserved whether collisions are elastic or inelastic. Herapath simply assumed that the scalar momentum mv of a particle is a measure of its absolute temperature and that the total momentum of a system is conserved in collisions while individual momenta tend to be equalized. (One consequence of this unorthodox definition was that temperature of a mixture of hot and cold fluids should be somewhat lower than one would expect using the ordinary definition of temperature.)
Herapath arrived at the same relation between pressure, volume, and particle velocity that Bernoulli had derived (he apparently did not know of Bernoulli's work). But he expressed it somewhat differently: PV is proportional to T_H^2, since T_H, which he called "true temperature," is proportional to mv rather than to mv^2. (I have added the subscript "H" to avoid confusion with the absolute temperature T used elsewhere in this chapter.)
Herapath submitted his first paper on kinetic theory to the Royal Society of London in 1820, hoping to get it published in the Philosophical Transactions. That would have given his views wide circulation in the international scientific community; moreover, as Herapath himself frankly admitted, it would have enhanced his personal reputation so that he could embark on a career of teaching and scientific research.
Humphry Davy (1778-1829), a well-known chemist, became president of the Royal Society shortly after the submission of Herapath's paper and was mainly responsible for its fate. Davy had earlier supported the general idea that heat is molecular motion rather than a substance, and thus he might have been expected to be receptive to a theory that gave this idea a precise mathematical formulation. But his reaction was negative, for several reasons. The only one he made explicit to Herapath was his reluctance to consider heat as a simple quantity that could be completely extracted from a body by annihilating the motion of its molecules, thus implying the existence of a lowest temperature ("absolute zero"). It is also evident that he found Herapath's derivations difficult to follow and would have preferred a less abstract, less mathematical approach emphasizing instead the correspondence between concepts and observations at each step: that seems to be a persistent difference in the attitudes of chemists and physicists.
Finally, we may conjecture that Davy had a metaphysical repugnance for the basic assumption of kinetic theory -- that particles move through empty space with no interactions except when they collide. By this time Davy had adopted some of the sentiments of Romantic nature philosophy; for example, seeing the world as an interconnected system dominated by all-pervading forces rather than by push-pull mechanisms.
Herapath was told that his paper would not be published in the Philosophical Transactions and he was advised to withdraw it, since according to the custom of the Royal Society once a paper was "read" (formally presented, if only by title or abstract, at a meeting) it became the property of the Society and could not be returned to its author. Herapath complied with this advice and sent his paper to an independent scientific journal, the Annals of Philosophy, where it was published in 1821. Later he attacked Davy and the Royal Society in a series of letters in the Times of London.
Denied recognition by the scientific establishment, Herapath nevertheless did have an opportunity to present his ideas to a larger public. The Annals of Philosophy, though now forgotten, was not by any means an obscure journal in the early 19th century. Michael Faraday and other reputable scientists published there. It was similar in content and circulation to the major physical science journals Philosophical Magazine (with which it merged in 1826), the Annalen der Physik und der Chemie, or the American Journal of Science. So if Herapath's theory was ignored by most scientists in the 182Os, we cannot simply blame Davy and the Roval Society but must recognize that the theory was out of harmony with prevailing ideas about the nature of gases and heat, and failed to convince its readers that those ideas should be revised.
Herapath later became editor of the Railway Magazine, a position that give him an opportunity to publish his scientific work though to a limited and perhaps unappreciative audience. In 1836 he presented a calculation of the speed of sound, which he had completed four years earlier. This was in fact the first calculation of the average speed of a molecule from the kinetic theory of gases. (J.P. Joule, was is usually credited with this accomplishment, was simply following Herapath's method.) Herapath found that the speed of sound in air at 32 F should be about 1090 feet per second, in good agreement with the experimental results available at that time.
In the 1840s, stimulated by the publications of Thomas Graham on gas diffusion and of Regnault on compressibility, Herapath revised and elaborated his kinetic theory and published the two-volume treatise Mathematical Physics (1847). He claimed that he had calculated from his theory in 1844 a formula for the time required for a given volume of gas to pass through a small hole into a vacuum, and that this formula was subsequently confirmed by the experimental results of Graham.
Herapath also claimed to have predicted in advance Regnault's result (1846) that the pressure of very dense gases is greater than that given by Boyle's law. This is indeed what the model of atoms as impenetrable spheres would lead one to expect if one ignores short-range attractive forces. However, earlier experiments had indicated deviations in the opposite direction, suggesting that attractive forces are more important than repulsive. (Later kinetic theories managed to account for the fact that attractive forces dominate at low temperatures, repulsive at high.)
The British physicist James Prescott Joule (1818-1889), one of the founders of the law of energy conservation, is the only scientist known to have read Herapath's Mathematical Physics during the first few years after its publication. Joule presented a short paper on kinetic theory, based on Herapath's work, at scientific meetings in 1848, but very few scientists learned about it.
Herapath lived long enough to see the kinetic theory revived by others. In 1860, after Maxwell's first paper was reported in a British magazine, Herapath published a letter calling attention to his own earlier work, and thus helped to ensure that he would be remembered as one of the pioneers of kinetic theory -- though by this time he could not be credited with the first kinetic theory, since Daniel Bernoulli's chapter in Hydrodynamica had also been rediscovered.
As I mentioned at the beginning of this section, Herapath's version of kinetic theory is inconsistent with Avogadro's hypothesis, a fact that passed unnoticed in the 1820s but would have called for revision if anyone had tried to integrate it with chemical atomic theory after 1860.
Like Herapath, Waterston was interested in the problem of explaining gravity by impacts of particles, and his efforts on this problem led him to develop a kinetic theory of gases. Waterston was employed as a Naval Instructor to the East India Company's cadets at Bombay, India. In 1843 he published a book that included some of his early results on the kinetic theory of gases. His most significant conclusion was that "equilibrium of temperature depends on molecules, however different in size" having the same kinetic energy. This was a special case of what later became known as the "equipartition theorem." There is no evidence that any physical scientist read the book; perhaps it was overlooked because of its misleading title, Thoughts on the Mental Functions.
Two years later Waterston submitted a long manuscript, presenting a detailed account of the kinetic theory of gases, to the Royal Society of London. Two members of the Society, asked to review the paper, recommended that it should not be published -- primarily because they disagreed with its fundamental premises. But no one had told Waterston, still far away in India, that once his paper had been officially "read" to the Society (i.e. presented by title or abstract, not read word for word) it would not be returned to him; and Waterston had not retained a copy for himself. Thus he could not easily follow Herapath's course of publishing the original paper in an independent journal, although he did try to call attention to his theory by circulating shorter versions, and by mentioning it when he published papers on related subjects.
In 1851 Waterston presented a short paper on his kinetic theory at the annual meeting of the British Association for the Advancement of Science. The published abstract of that paper clearly states that in gas mixtures, the average kinetic energy of each kind of molecule is the same; thus he established his priority for the first statement of the equipartition theorem. He also indicated in this abstract that Avogadro's hypothesis follows from the kinetic theory.
The German chemist and physicist August Karl Kroenig (1822-1879), who published a short paper proposing a kinetic theory of gases in 1856, was probably familiar with the published abstract of Waterston's 1851 paper and may have been influenced by it (see Daub, 1971).
Waterston also attempted to calculate the ratio of the two specific heats (c_p, at constant pressure, and c_v, at constant volume). Because of a numerical mistake he obtained the value c_p/c_v = 4/3 for a monatomic gas instead of the correct theoretical value 5/3. Since his value was fairly close to the observed ratios for air and other gases he didn't realize his error.
In 1858 Waterston published a paper arguing that Laplace's calculation of the speed of sound from the caloric theory of adiabatic compression and expansion could be based equally well on Waterston's own kinetic theory of gases and thus did not provide evidence for the caloric theory of heat.
In 1891 the British physicist Lord Rayleigh, surveying the literature on acoustics for his comprehensive treatise on that subject, came across Waterston's 1858 paper on the theory of sound, which referred to his unpublished manuscript lying in the Royal Society Archives. Rayleigh was Secretary of the Royal Society at that time, and had no difficulty in retrieving the manuscript and arranging for its belated publication in the Philosophical Transactions. In his introduction to the paper, Rayleigh remarked: "the history of this paper suggests that highly speculative investigations, especially by an unknown author, are best brought before the world through some other channel than a scientific society, which naturally hesitates to admit into its printed record matter of uncertain value. Perhaps one can go further and say that a young author who believes himself capable of great things would usually do well to secure the favourable recognition of the scientific world by work whose scope is limited, and whose value is easily judged, before embarking on greater flights" (see Haldane, 1928, pp. 209-210).
These remarks of Lord Rayleigh do not justify the original refusal of the Royal Society to publish Waterston's brilliant paper, but they hint at one of its failures as an organization for advancing scientific knowledge. By rejecting work by authors without established reputations, or theories that contradict established doctrines, a scientific society shirks one of its most important functions. In the case of the kinetic theory of gases, the net result of the Royal Society's refusal to publish the works of Herapath and Waterston was to retard the progress of molecular physics by a decade or two, this permitting the German scientists August Kronig and Rudolf Clausius to gain the major share of credit as founders of the theory and damaging the Society's own reputation.