hitherto exclusively considered, in which the molecules slide or roll over each other without, on the average, coming nearer to or going further from each other. But the different mathematical theories do not agree together in respect to the numerical ratio of these two coefficients of friction.

The value which the kinetic theory of gases requires for the second coefficient of friction can be calculated from the same formulæ as before, if only the single alteration is made, that instead of the velocity-component v, which is parallel to the surface of friction dy dz, we consider a velocity u perpendicular to it. We have, consequently, in the formula given in § 47* for the momentum carried across the element dy dz, no further change to make than to employ the exponent q with the value

and to exchange the factor m sin s cos

for m cos s.

Thus the momentum normal to the surface of friction which is carried over

which corresponds to a former assumption, and that

eq = e -km2 (1 + 2kmwu' cos s)

with sufficient approximation, we find the values

is the pressure, and y' and ŋ' are constants whose meaning is

The latter is the second coefficient of viscosity for which we are seeking; y disappears from the difference Q2

Q between the

momenta carried the one way and the other, which has the value

Our theory agrees, therefore, with all the older theories in giving the value of the friction that accompanies alterations of density as larger than that of the ordinary friction. The ratio we have found of 3:1 is the same as that which, on Poisson's theory, should hold between the elastic constants of extension and torsion. In a theory of friction which I formerly developed from other hypotheses I found the same value for this ratio. Its determination has, however, but slight practical value, since, as indeed the last formula shows, this kind of friction gives rise to forces which are not distinguished from the pressure, and may therefore be reckoned in the value of the pressure.

50*. External Friction

The considerations and formulæ of § 47* at once supply the means of determining the external friction which a gas experiences at the surface of a solid body.

Consider a gas which flows along the surface of a body at rest and has everywhere the same velocity v; then in each unit of time a number of particles, which have the momentum

Q=&v=jmNov

in the direction of flow, strike unit of surface and rebound from it. Each particle rebounds from the solid wall with the same speed with which it struck it, but not always in a direction inclined to the wall at the same angle as that of the impact; for the solid wall, which is made up of molecules grouped together, is, in respect of a striking molecule, an exceedingly rough surface. Therefore the colliding molecules lose a part of their momentum in the direction parallel to the wall, and this becomes transformed into heat-motion. This loss appears as external friction, whose intensity, therefore, is given by the expression

BQ = 4BmNQv,

where ẞ is a numerical coefficient.

'Crelle's Journal für Mathematik, 1873, lxxviii. p. 130; with Addition lxxx. p. 315, with improvements by Stefan and Boltzmann. Other theories of internal friction have been given by Navier (Mém. de l'Acad. de Paris, 1823, vi. p. 389), Poisson (Journ. de l'Éc. Poly. 1831, xiii. cah. 20, p. 139), Stokes (Camb. Phil. Trans. 1849, viii. p. 287), Cauchy (Exerc. de Math. 1828, 3rd year, p. 183), Barré de St. Venant (Comptes Rendus, 1843, xvii. p. 1240), and Stefan (Wiener Sitzungsber. 1862, xlvi. Abth. 2, p. 8).

If the wall is very

This factor 3 need not be a pure fraction. rough, many of the particles meeting it will be jerked back by its unevennesses with an oppositely directed motion; they undergo, then, a diminution of their velocity v by an amount which may rise to 2v. It may consequently be very possible that the particles which strike upon the wall receive a mean motion which is oppositely directed to their initial motion, as certainly in like manner occurs at the edge of flowing water; and in this case we should have to assume ẞ > 1. According to Kundt and Warburg's observations, which were discussed in § 83, it appears that

β B = 4,

whence the coefficient of external friction would have to be put

The assumption herein contained, that the striking molecules lose their whole velocity of translation and gain an opposite one, is not in contradiction with the fact that the gases slide along solid surfaces with a sensible speed; for not all the molecules of the surface layers of gas lose their velocity, but only those that strike against the solid surface.

APPENDIX V

DIFFUSION

51*. General Theory of Diffusion

It is not my intention to investigate the theory of diffusion of gases with the same mathematical rigour as the simpler theory of viscosity. I limit myself here to supplying the mathematical explanations desirable for those going more deeply into the theory of diffusion developed in the text, and these I shall found upon Maxwell's law.

As to the distribution of the two gases, I make only the assumption that the whole pressure of the mixture

possesses everywhere the same constant value P, and therefore keeps this same value always; and also that, corresponding to it, there are always at every point the same number

N1 + N2

of molecules of the two kinds in unit volume.

As in the investigation given in the text, we determine for one of the two kinds of gas the number of molecules which in unit time pass in the direction of increasing through a surfaceelement dS of a section of the diffusion tube at a distance x from the beginning of the tube. We form this sum with the assumption of the validity of Maxwell's law of distribution of speeds. This assumption is not strictly admissible, since the deduction of this law presupposes the state of motion of the whole gas to be everywhere the same. But the application of this law to our problem is allowable as a good approximation, if we can look upon the ratio of mixture of the two gases in the space filled by them, not simply as a continuous function of the position, but also as one that varies very slowly. For, with this hypothesis, that ratio and the whole state of the mixture can be assumed to be constant

within a tolerably large region, throughout which, therefore, Maxwell's law may be considered to hold.

According to a formula which we have developed before, and used several times, the element dS is reached by a number

N(km/)Be-Br/we-km2 w2 dw dS cos s sin s ds dr do

of particles, which proceed in unit time with speed from the volume-element r2 dr sin s ds do expressed in polar coordinates with dS as origin and the normal to dS as axis. N, m, and B are here magnitudes which have different values for the two kinds of gas, and must therefore be distinguished by subscripts 1 and 2. N and B are also functions of the position; but it will be sufficient in the case of B, the collision-frequency, to assume a mean constant value, and consequently to take into account only with respect to N that we must employ that value of it which is proper for the position of the volume-element r2dr sin s ds do, and which should be indicated by the argument x

↑ cos s.

r is small, the function N with this argument may be put

Since

where the letter N without any argument denotes the value at the position x.

We are not concerned with the whole number of particles that pass through dS, but only with the difference between the numbers which pass from the right and from the left; this difference does not depend on the absolute value of N, but is conditioned only by its variation. Hence, on introducing into the above formula the expression we have developed for N, we neglect the first term and investigate only the second

dN

dx

14

(km/π) * Be-Br/we

-km2 w2dw dS cos2 s sin s ds rdr dø,

which we have to integrate between 0 and ∞ in respect to w and r, over unit area as regards dS, from 0 to 1 in respect to s, and from 0 to 2 in respect to p. We thus obtain as expression for the number which pass through unit area in the direction of increasing a, in consequence of the unequal distribution,

For the number passing in the opposite direction the same expression holds, but with changed sign.