= and Eu2 97.66; the former of these is 1.84 times the latter. In this case we found the ratio of Σ to Zu2v to be 1.87. § 54. In conclusion, I wish to refer, in connexion with Class 11., § 28, to a very interesting and important application of the doctrine, made by Maxwell himself, to the equilibrium of a tall column of gas under the influence of gravity. Take, first, our one-dimensional gas of § 50, consisting of a straight row of a vast number of equal and similar atoms. Let now the line of the row be vertical, and let the atoms be under the influence of terrestrial gravity, and suppose, first, the atoms to resist mutual approach, sufficiently to prevent any one from passing through another with the greatest relative velocity of approach that the total energy given to the assemblage can allow. The Boltzmann-Maxwell doctrine (§ 18 above), asserting as it does that the time-integral of the kinetic energy is the same for all the atoms, makes the time-average of the kinetic energy the same for the highest as for the lowest in the row. This, if true, would be an exceedingly interesting theorem. But now, suppose two approaching atoms not to repel one another with infinite force at any distance between their centres, and suppose energy to be given to the multitude sufficient to cause frequent instances of two atoms passing through one another. Still the doctrine can assert nothing but that the timeintegral of the kinetic energy of any one atom is equal to that of any other atom, which is now a self-evident proposition, because the atoms are of equal masses, and each one of them in turn will be in every position of the column, high or low. (If in the row there are atoms of different masses, the Waterston-Maxwell doctrine of equal average energies would, of course, be important and interesting.) § 55. But now, instead of our ideal one-dimensional gas, consider a real homogeneous gas, in an infinitely hard vertical tube, with an infinitely hard floor and roof, so that the gas is under no influence from without, except gravity. First, let there be only two or three atoms, each given with sufficient velocity to fly against gravity from floor to roof. They will strike one another occasionally, and they will strike the sides and floor and roof of the tube much more frequently than one another. The time-averages of their kinetic energies will be equal. So will they be if there are twenty atoms, or a thousand atoms, or a million, million, million, million, million atoms. Now each atom will strike another atom much more frequently than the sides or floor or roof of the tube. In the long run each atom will be in every part of the tube as often as is every other atom. The time-integral of the kinetic energy of any one atom will be equal to the time-integral of the kinetic energy of any other atom. This truism is simply and solely all that the Boltzmann-Maxwell doctrine asserts for a vertical column of a homogeneous monatomic gas. It is, I believe, a general impression that the Boltzmann-Maxwell doctrine, asserting a law of partition of the kinetic part of the whole energy, includes obviously a theorem that the average kinetic energy of the atoms in the upper parts of a vertical column of gas, is equal to that of the atoms in the lower parts of the column. Indeed, with the wording of Maxwell's statement, § 18, before us, we might suppose it to assert that two parts of our vertical column of gas, if they contain the same number of atoms, must have the same kinetic energy, though they be situated, one of them near the bottom of the column, and the other near the top. Maxwell himself, in his 1866 paper ("The Dynamical Theory of Gases")*, gave an independent synthetical demonstration of this proposition, and did not subsequently, so far as I know, regard it as immediately deducible from the partitional doctrine generalized by Boltzmann and himself several years after the date of that paper. § 56. Both Boltzmann and Maxwell recognized the experimental contradiction of their doctrine presented by the kinetic theory of gases, and felt that an explanation of this incompatibility was imperatively called for. For instance, Maxwell, in a lecture on the dynamical evidence of the molecular constitution of bodies, given to the Chemical Society, Feb. 18, 1875, said: "I have put before you what "I consider to be the greatest difficulty yet encountered by "the molecular theory. Boltzmann has suggested that we are to look for the explanation in the mutual action between "the molecules and the ethereal medium which surrounds them. I am afraid, however, that if we call in the help of "this medium we shall only increase the calculated specific "heat, which is already too great." Rayleigh, who has for the last twenty years been an unwavering supporter of the Boltzmann-Maxwell doctrine, concludes a paper " On the Law of Partition of Energy," published a year ago in the Phil. Mag., Jan. 1900, with the following words: "The difficulties "connected with the application of the law of equal partition "of energy to actual gases have long been felt. In the case of "argon and helium and mercury vapour, the ratio of specific "heats (167) limits the degrees of freedoms of each molecule Addition, of date December 17, 1866. Collected works, vol. ii. "to the three required for translatory motion. The value “(1·4) applicable to the principal diatomic gases, gives room "for the three kinds of translation and for two kinds of "rotation. Nothing is left for rotation round the line joining "the atoms, nor for relative motion of the atoms in this line. 'Even if we regard the atoms as mere points, whose rotation means nothing, there must still exist energy of the last"mentioned kind, and its amount (according to law) should "not be inferior. 66 "We are here brought face to face with a fundamental "difficulty, relating not to the theory of gases merely, but "rather to general dynamics. In most questions of dynamics, a condition whose violation involves a large amount of "potential energy may be treated as a constraint. It is on "this principle that solids are regarded as rigid, strings as "inextensible, and so on. And it is upon the recognition "of such constraints that Lagrange's method is founded. "But the law of equal partition disregards potential energy. "However great may be the energy required to alter the "distance of the two atoms in a diatomic molecule, practical "rigidity is never secured, and the kinetic energy of the relative motion in the line of junction is the same as if the "tie were of the feeblest. The two atoms, however related, "remain two atoms, and the degrees of freedom remain six "in number. แ "What would appear to be wanted is some escape from "the destructive simplicity of the general conclusion." The simplest way of arriving at this desired result is to deny the conclusion; and so, in the beginning of the twentieth century, to lose sight of a cloud which has obscured the brilliance of the molecular theory of heat and light during the last quarter of the nineteenth century. 1. II. The Absorption of the Ionized* Phosphorus OR reasons of both theoretical and practical import it the under which the phosphorus nucleus vanishes on passing Whoever writes on subjects relating, like the present, to certain features of ionization is obliged to make free use of the admirable work (Thomson, C. T. R. Wilson, Townsend, Rutherford, Zeleny, and others), which has been sent out by the Cavendish Laboratory under the direction of Prof. J. J. Thomson. These researches, like those of Chattock, Elster and Geitel, and others (cf. H. Becquerel in Nature,' Feb. 21st, p. 396, 1901), are so recent and well known that detailed reference would be cumbersome; but I desire to make my acknowledgments here. + Communicated by the Author. through tubes at a definite velocity; or, in general, on being retained in any vessel, or put in contact with any barrier in a definite way, for a definite time. The experiments of the present paper thus relate to the absorption of condensationproducing atmospheric nuclei by surfaces or by suspended particles. They show, I think, that such absorption takes place as though each nucleus of a nearly saturated region Sectional Elevation of the Colour-tube C, the Absorption-tube t and appurtenances. Scale. travelled in the entire absence of an electric field, with a velocity of about 3 millims. per second; or, if it be put roughly that of the total number travel in a given cardinal direction, as though each nucleus had a velocity of about a centimetre per second. It is curious to note how close this lies to the velocity of the ion in a unit electric field. 2. The method of experimentation has been indicated in my first paper, and is based on colour criteria obtainable with the steam-jet. Here I need only recall that a current of moderately dry air is furnished by a gasometer-train (Mariotte flask, volume-flask, pressure-gauge, desiccator), terminating in the fine screw-stopcock F. On opening the latter, this passes through the phosphorus-tube P (containing pellets of phosphorus between strips of wire-gauze), where it is highly charged with the ionized emanation. This saturated air is conveyed into the colour-tube CC (old pattern of which j is the simple jet), through the absorption-tube t, of the length, diameter, and material to be examined. The tube t is sealed into P, while the other end dips slightly into the lateral influxpipe (" of the colour-tube. The arrangement of C (length, &c.) has very little, if any, effect on the results, as was pointed out in my last paper. The draft due to j is sufficient to capture all the nuclei from the open end of the absorptiontube t, and the whole of it is impressed on the jet. 3. To illustrate the method of work, an example of the data for a single tube is given in the following table (Ì.). The other results are briefly summarized (Table II.) or expressed in the chart. Length x and radius r refer to the absorption-tubing employed. The volume (V litres per minute) of charged air passing through P is the amount needed to produce a definite colour (jull blue) in the field of the colour-tube C. The velocity of the air-current through t is given under v in centimetres per second. The constant k in the final column is the absorption velocity, computed from the equation k=2·65(V/ræ) log (V/V.), where Vo is the volume in litres per minute giving the identical blue colour when the absorption-tube t is removed, and the phosphorus-tube conveys its contents directly into C. The other data (p, the pressure of the steam issuing from the jet, always low, and the temperature of the air at influx measured by the thermometer T) are of little immediate interest. In most cases many observations (often four or five) were made for each tube-length æ in each series given, the difficulty being to select the same standard blue. The table contains only the mean values. * Phil. Mag. [6] vol. i. p. 572 (1901). ; preliminary results in 'Science,' vol. xi. p. 201 (1900); Amer. Journal of Science, March, 1901. |