These results, in as far as the conditions correspond, were found to agree very closely with the results obtained by Graham. Thus, through stucco at 30 inches, the comparative times of transpiration of air and hydrogen were as 2.9 to 1, Graham's results being 2.8 to 1. Through meerschaum the ratio was 3'6 to 1, Graham having found the ratio 38 to 1 through a graphite plate, which was in all probability finer than the meerschaum. The ratio of the times of transpiration of equal volumes of different gases, which Graham looked upon as varying only with the coarseness of the plates, was found, as was expected, to depend entirely on the relation between the pressures of the gases and the coarseness of the plates, the ratio of the times being the same as long as the pressures of the gases were inversely proportional to the coarseness of the plates.

Thus, at a pressure of 5 inches, the times for hydrogen and air through stucco, instead of being 2.9, as at the pressure of the atmosphere, were 3.6, or the same as through meerschaum at a pressure six times as great; the coarseness of the plates, as determined in the previous experiments, being 56. The same agreement held as long as the ratio between the pressures was maintained.

The correspondence of the results for different plates, and the complete verification of the theoretical conclusions which they afforded, is shown by comparing the logarithmic homologues of the curves in which the times of transpiration are the ordinates and the pressures the abscissæ. The fitting of the logarithmic homologues is exact, both as regards the direction of the curves and the distances between the curves, for air and hydrogen; the displacement along the abscissæ, to bring the curves into coincidence, being 819 = log. 6.5.

As this number, 6·5, is the ratio of the coarseness of the plates, it should have corresponded with the ratio obtained by thermal transpiration, which was 56, with the same plates. This discrepancy, although too small to cast a doubt upon the general agreement of the results, is too large to be attributed to experimental inaccuracy, and must have been due to some change in the plates, probably arising from the plates being hot in the one case and cold in the other.

Experiments on Impulsion with a Suspended Fibre.

A single fibre of unspun silk was suspended from one end in a vertical test-tube, closed with an india-rubber cork, and connected with an air-pump, and a microscope was arranged for observing the motion of the fibre when a hot body was brought into a certain position near the test-tube.

With air in the test-tube at the pressure of the atmosphere, it was found that the fibre was carried by the air-currents towards the hot body, and this was the case as long as the pressure was greater than 8 inches of mercury, but after the tube had been exhausted below

this point, the fibre moved away from the heat, and the motion steadily increased as the pressure became very small, such as th of an inch of mercury.

50

With hydrogen in the test-tube, the fibre moved away from the heat at all pressures below that of the atmosphere, and for small pressures the motion was somewhat larger than with air.

A spider-line was also tried, and gave results similar to the fibre of silk.

It thus appeared that both with the fibre of silk and the spider-line the phenomena of impulsion were manifest at densities many hundred times greater than the highest densities at which like results are obtained with the vanes of the radiometer which are several hundred times broader than the fibre of silk. And this verifies the law of corresponding results at corresponding densities for this class of phe

nomena.

Abstract of Part II (Theoretical Investigation).

The characteristic as well as the novelty of this investigation is that, not only is the mean of the motions of the molecules at the point under consideration taken into account, but also the manner in which the mean motion may vary from point to point, in any direction across the point under consideration. It appears that such a variation gives rise to certain stresses in the gas (tangential and normal) and it is of these stresses that the phenomena of transpiration and impulsion afford evidence.

Instead of considering only the mean condition of the molecules comprised within an elementary volume of the gas, what is chiefly considered in this investigation is the mean condition of the molecules which cross an elementary area in a plane supposed to be drawn through the point.

Q is used to indicate a quantity belonging to a molecule such as its mass, momentum, or energy, σ(Q) to express the rate at which Q is carried across a plane perpendicular to the direction x.

Two systems of axes are employed, xyz fixed axes, with respect to which u, v, w, are the component velocities of a molecule, and a system of axes parallel to xyz, but moving with the component velocities U, V, W, with respect to which §, 7, 【, have the component velocities of a molecule. U, V, W, having such values that

As preliminary to the investigation, expressions are obtained for (Q) in terms of a, U, V, and W, on the supposition that the gas is uniform. This is accomplished by the application of well-known methods.

When the condition of the gas varies from point to point, the molecules are considered as consisting of two groups, one crossing from the positive and the other from the negative side of the plane. Considered in opposite directions, the mean characteristics (the number, mass, momentum, or energy) of these two groups are not necessarily equal. They may differ in consequence of the motion of the gas, the motion of the plane through the gas, or a varying condition of the gas, and the determination of the effect of these causes, particularly the last, on the mass, momentum and energy that may be carried across by either or both groups constitutes the extension of the dynamical theory of gases.

In order to take account of the difference in the two groups it is assumed, and so far there is nothing new in the assumption itself, that the group of molecules which crosses the surface from either side will partake of the characteristics of the gas in the region from which the molecules which constitute the group have come. The first direct step in the investigation is the deduction from the foregoing assumption of two theorems (I and II), supposing that there are no external forces. Taking o'(Q) to be the approximate value of a(Q) obtained on the assumption that the gas is uniformly in the mean condition which holds at the point xyz, the theorems I and II admit of the following symbolical expression:

Where s represents a certain distance, measured from the plane of reference.

This distance, s, enters as a quantity of primary importance into all the results of the investigation.

It is proposed to calls the mean range of the quantity Q, so as to distinguish it from the mean path of a molecule.

s is a function of the mean path, but it also depends on the nature of the impacts between the molecules. It is subsequently shown that—

8= π αμ,

from which it appears that so".

P

The dynamical conditions of steady density, steady momentum, and steady energy are then considered.

Putting (Q) for the value of Q in a unit of volume, in order that (Q) may be steady, we have

whence, by giving to Q the value M, the mass of a molecule, we have

the condition of steady density, for steady momentum Q has severally the values Mu, Mv, Mw, and for steady energy Q=M(u2 + v2 + w2).

The equations of motion are then applied to the particular cases which it is the object of this investigation to explain. Two cases are considered.

The first case is that of a gas in which the temperature and pressure vary only along a particular direction, so that the isothermal surfaces and the surfaces of equal pressure are parallel planes; this is the case of transpiration.

The second case is that in which the isothermal surfaces and surfaces of equal pressure are curved (whether of single or double curvature); this is the case of impulsim and the radiometer.

Transpiration.

As regards the first case, the condition of steady energy proved to be of no importance, but from the conditions of steady momentum and steady density, an equation is obtained between the velocity of the gas, the rate at which the temperature varies, and the rate at which the pressure varies, the coefficients being functions of the absolute temperature of the gas, the diameters of the apertures, and the ratio of these diameters to the mean range, which coefficients are known for the limiting conditions of the gas, i.e., when the density is either very small or very large.

The most general form of this equation is

in which is the mean velocity of transpiration along the tube, which is taken in the direction of the axis of x. M is the mass of a molecule, the pressure of the gas, the absolute temperature, and c the semidistance across the tube.

Ρ

In which A, m, m' depend only on the shape of the section of the tube.

All these functions varying continuously between the limits here ascribed. Also

A depends on the nature of the surface of the tube, but not upon the nature of the gas, while

A and λ may depend both upon the gas and the surface.

From this equation, which is the general equation of transpiration, the experimental results, both with regard to thermal transpiration and transpiration under pressure, are deduced.

Impulsion.

In dealing with the second case, that in which the isothermal surfaces are curved, the three conditions-steady density, momentum, and energy-are all of them important.

These conditions reduce to an equation between the motions of the gas the variation in the absolute temperature and the variation in the pressure, with coefficients which involve the ratio of the mean range to the dimensions of the radii of curvature of the surfaces. The equation corresponds to the equation of transpiration, and as applied to the case in which heat is being conducted through a gas which is constrained to remain at rest, the equation becomes

p-p1 being the excess of pressure in the direction of, and due to, the variation of temperature. In the abstract of a paper read in April last, Professor Maxwell gives an equation which, transformed into my symbols is

which only differs from mine in the coefficient - 3. As Professor Maxwell indicates that he has obtained his result without taking account of the tangential stresses, this difference is not a matter of surprise.

Besides the broad lines of the investigation which have been mentioned in this abstract, there are many minor points of which it is