which will produce the same pressure as arises from the actual impacts. This quantity v² is known as the mean square of the molecular velocity, and is so used by Waterston.
In a paper in the Philosophical Magazine for 1858 Waterston gives an account of his own paper of 1846 in the following terms:—“Mr. Herapath unfortunately assumed heat or temperature to be represented by the simple ratio of the velocity instead of the square of the velocity, being in this apparently led astray by the definition of motion generally received, and thus was baffled in his attempts to reconcile his theory with observation. If we make this change in Mr. Herapath’s definition of heat or temperature—viz., that it is proportional to the vis-viva or square velocity of the moving particle, not to the momentum or simple ratio of the velocity—we can without much difficulty deduce not only the primary laws of elastic fluids, but also the other physical properties of gases enumerated above in the third objection to Newton’s hypothesis. [The paper from which the quotation is taken is on ‘The Theory of Sound.’] In the Archives of the Royal Society for 1845–46 there is a paper on ‘The Physics of Media that consist of perfectly “Elastic Molecules in a State of Motion,”’ which contains the synthetical reasoning on which the demonstration of these matters rests.... This theory does not take account of the size of the molecules. It assumes that no time is lost at the impact, and that if the impacts produce rotatory motion, the vis viva thus invested bears a constant ratio to the rectilineal vis viva, so as not to require separate consideration. It does, also, not take account of the probable internal motion of composite molecules; yet the results so closely accord with observation in every part of the subject as to leave no doubt that Mr. Herapath’s idea of the physical constitution of gases approximates closely to the truth.”
In his introduction to Waterston’s paper (Phil. Trans., 1892) Lord Rayleigh writes:—“Impressed with the above passage, and with the general ingenuity and soundness of Waterston’s views, I took the first opportunity of consulting the Archives, and saw at once that the memoir justified the large claims made for it, and that it marks an immense advance in the direction of the now generally received theory.”
In the first section of the paper Waterston’s great advance consisted in the statement that the mean square of the kinetic energy of each molecule measures the temperature.
According to this we are thus to put in the pressure equation—½ m v² = T, the temperature, and we have at once—p V = ⅔ N · T.
Now this equation expresses, as we know, the laws of Boyle and Gay Lussac.
The second section discusses the properties of media, consisting of two or more gases, and arrives at the result that “in mixed media the mean square molecular velocity is inversely proportional to the specific weights of the molecules.” This was the great law rediscovered by Maxwell fifteen years later. With modern notation it may be put thus:—If m₁, m₂ be the masses of each molecule of two different sets of molecules mixed together, then, when a steady state has been reached, since the temperature is the same throughout, m₁ v₁² is equal to m₂ v₂². The average kinetic energy of each molecule is the same.
From this Avogadros’ law follows at once—for if p₁, p₂ be the pressures, N₁, N₂ the numbers of molecules per unit volume—
p₁ = ⅓ N₁ m₁ v₁², p₂ = ⅓ N₂ m₂ v₂².
Hence, if p₁, is equal to p₂, since m₁ v₁² is equal to m₂ v₂², we must have N₁ equal to N₂, or the number of molecules in equal volumes of two gases at the same pressure and temperature is the same. The proof of this proposition given by Waterston is not satisfactory. On this point, however, we shall have more to say. The third section of the paper deals with adiabatic expansion, and in it there is an error in calculation which prevented correct results from being attained.