free path; for a velocity is not, like a path, measured only by a length, but for its complete specification it requires a time unit also to be laid down. It is therefore well not to express the numerical values of the friction in the way usual for values of pressure, but, after deduction of the formula, which on replacement of Nm by the density p we bring into the form

to refer them to the units of density, length, and time. We thereby obtain also this advantage, that the numerical values become independent of the value of gravity, which alters with the latitude of the place of observation. I shall therefore give the numerical values of the viscosity which follow in such units that they contain the density of water, the centimetre or square centimetre, and the second as fundamental units.

76. Theoretical Laws of Gaseous Friction

The final formula of the kinetic theory of viscosity, which was found by Maxwell,' leads to a very surprising result if we introduce into it the value of the free path.

Since we have worked out the above considerations as if all the molecules possessed equal speeds and attained equal free paths-a mode of calculation which is sufficient only for an approximation-we must use the Clausius expression for the free path, and not that of Maxwell. Referring to the Mathematical Appendix (§§ 46*-48*) for the stricter calculation of viscosity, we put here

with Clausius, and find for the viscosity-coefficient the value

n = Nλ3mG/4πs2.

In this formula N is the number of molecules per unit volume, and λ3 the volume occupied by a single molecule; consequently Nx3 = 1, as we have already (§63) several

'Phil. Mag. 1860 [4] xix. p. 31; Scientific Papers, 1890, i. p. 390.

times seen, so that the coefficient of viscosity takes the simpler form

n = mG/4πs2.

In this form the expression for the viscosity-coefficient contains no factor which at all depends on the pressure of the gas, but only magnitudes which depend on the mass, speed, and sphere of action of the molecules, and thus generally on their state. The formula therefore gives the proof of the well-known law of Maxwell that the viscosity of a gas is independent of its density.

At first sight this law must seem but little probable. According to it the friction should retain the same intensity when the gas increases in rarity. This seems to lead to a conclusion which, although apparently admissible by the last formula, contains a contradiction in itself, viz. that a gas rarefied to density 0, and thus rarefied out of existence, exerts the same friction as one that actually exists. We see the fallacy of this conclusion when we consider how the formula was obtained; it is a transformation of the formula

n = pLG

1

given in § 75, according to which the viscosity ʼn vanishes with the density p, so long as neither the mean free path L nor the mean speed G becomes infinitely great. But this limiting case is obviously excluded in the deduction of the formula given in § 75, and therefore the theoretical formula no longer holds for the coefficient of friction in the limiting case for which p = 0.

With the exception of this limiting case, Maxwell's theoretically deduced formula seems still surprising enough o justify our more closely describing the causes of its >eing obtained which are hidden in the mathematical reasonng. For such an explanation in words the last formula, whose meaning is easily perceived, offers itself suitably. The friction n is the quantity of momentum which is carried over from layer to layer under the before-mentioned circumstances. The transfer occurs by means of the

heat-motions of the molecules; consequently, as the formula shows, it must be proportional to the speed G. It, further, is carried out by the molecules themselves, and therefore will be the greater the more there are of them; hence the formula for the friction contains the density of the gas as a second factor. The transfer can only take place between layers whose distance apart can be traversed by a molecule; the friction must therefore be the greater the wider the range of effective layers, and it must therefore, in the third place, be proportional to the molecular free path. The fourth factor, the coefficient, is explained in the same way as in the exactly similar formula for the pressure, namely, by the circumstance that only a third part of the molecules which are moving in all directions, and therefore symmetrically with respect to the three dimensions of space, come into account in regard to transference in the direction of one of these three dimensions.

This explanation of the formula we have found contains at the same time a reason for this remarkable law. Of the factors in the formula there are only two, p and L, which are variable with the compression or rarefaction of the gas, and they vary so that, if the density p increases, the free path L of the molecules, which are hindered in their motion by the constriction of the space containing them, becomes smaller and vice versa. In this way it is possible that the product of these two quantities, of which one increases while the other diminishes, may always keep the same value; and therefore after this consideration the paradoxical law of Maxwell will have lost much of its improbability.

The coefficient of viscosity is not, however, independent of the temperature, as it is of the pressure. Of its fetors only G and L can be variable with the temperature. With respect to the former we know from experiments on the pressure of gases that it is proportional to the square root of the absolute temperature, or that it increases with the temperature 9 measured on the usual scale in the rati √(1+a9): 1, where a denotes the thermal coefficient of expar sion of the gas. As to the free path L, the theory leaves it ur decided whether it alters with the temperature or not. T

simplest assumption would be that it does not depend upon temperature; but, as we have seen in § 71, other possibilities are not excluded. From this, then, we cannot take it as certain, but only as a probable consequence of the theory, that the viscosity of gases increases with the temperature. Gases would therefore in this respect behave oppositely to liquids, whose viscosity is the less at the higher temperatures.

77. Observations on the Friction of Gases at Different Pressures

The remarkable laws of the viscosity of gases deduced by Maxwell in 1860 from the kinetic theory of these media challenge experimental proof, not only on account of their apparently innate improbability, but also especially because an experimental proof of the laws of viscosity might in general give at the same time a decision as to the truth and admissibility of the kinetic theory. For if we do not verify by experiment the laws that are consequences of the theory, the theory which requires them must be rejected as erroneous. The importance of this question prompted both Maxwell and myself almost simultaneously to carry out experimental investigations, which were published in the next following years, and were founded on exactly similar methods.

Of the methods employed up to that time for the determination of the viscosity of liquids, that invented by Coulomb1 presented itself first of all as the most suitable, because with it the same pressure is exerted everywhere throughout the gas investigated. If a circular disc is suspended horizontally by a wire fastened to its centre, it may, by means of the torsion of the wire, be put into oscillation in its own plane about that centre. If the disc is in a fluid, the amplitude of the oscillations will gradually decrease by reason of the friction which is exerted on each other by the layers of the fluid that are set in motion, and, indeed, the decrease of successive amplitudes follows the law of a geometrical progression. If we measure the amount of decrease, and determine therefore the so-called logarithmic decrement of that progression, we can from the observed 1 Mém de l'Inst. National, an IX, iii. p. 246.

magnitudes calculate the amount of friction that has taken place and the value of the coefficient of viscosity. The external friction that also comes into play in the experiment, that is, the friction that occurs between the fluid and the surface of the disc, may be eliminated by repetition of the experiments with more discs of various sizes.

To make this method more suitable for the determination of the friction of the air, I altered it in my own experiments by employing three oscillating discs with a common axis, instead of one, and arranging them either apart from each other so as to put into oscillation by their six faces the air above and below each of them, or all close together, like a single disc, so as to move the air by two faces only. The combination of the results of the two experiments allowed the coefficient of viscosity to be calculated in a simple way from the difference of the two observed logarithmic decrements. Determinations of the coefficient of viscosity of air carried out at different pressures by this method did not, indeed, show a complete agreement with each other as Maxwell's theory required, but the differences between the values found were small enough to prove the law at least within certain limits of the pressure. As an instance I cite the numbers 2 which I obtained by experimenting with an apparatus provided with three glass discs. Since the method does not lead directly to a knowledge of the viscosity-coefficient itself, but rather, first of all, to the square root of the coefficient, I put the latter, referred to centimetres, in the following table, which contains also the temperature and the pressure, the former in Centigrade degrees, the latter in centimetres of mercury.

1 Amtl. Ber. der Naturf.-Vers. in Stettin, 1863, p. 141; Pogg. Ann. 1865, CXXV. p. 177.

2 Calculated on an improved theory (Sitzungsber. d. Münchener Akad. 1887, xvii. p. 343; Wied. Ann. 1887, xxxii. p. 642), account being taken of a correction suggested by W. König (ibid. 1887, xxxii. p. 193).