distance apart of the molecules, and therefore on the same elements which regulate the molecular free path. This explanation makes the connection of the two magnitudes to appear no longer surprising.

Clausius has developed the theory of these relations after the method of Maxwell and Helmholtz. His theory, with the assumption that the molecules are spherical in shape and are perfect conductors of electricity, gives the dielectric capacity K in the form

K = (1+2g)/(1 — g),

where g denotes the fraction of the volume containing the gas which its molecules actually occupy. Pyra¬sformation then we obtain the value of g expressed in tc dielectric capacity K, viz.

་

of the

We see at once that this magnitude g is closely allied in its meaning to the coefficient of condensation » introduced by Loschmidt; for both ratios represent exactly the same thing if the molecules come into actual contact in their utmost state of compression. But it is possible, and even probable, that the spherical surfaces on which the electric charges of the molecules reside, do not come into actual contact with each other, even when the molecules are on the point of entering within the range of their spheres of action. The fraction denoted by g may therefore be less than that denoted by v, and can at most be equal to it.

If, therefore, we replace y in Loschmidt's formula

s = 6/2 vL

by g, we shall probably obtain a smaller value for the molecular diameter s than is given by either Loschmidt's or van der Waals's formula. Dorn2 is the first who has carried out this calculation of s by the formula

s = 6√/2 gL = 6√√2 L(K − 1)/ (K + 2),

and he combined the values of the dielectric capacity K 1 Mechanische Wärmetheorie, 1879, 2. Aufl. ii. p. 94.

2 Wied. Ann. 1881, xiii. p. 378.

determined by Boltzmann, as well as one observation by Ayrton and Perry, with the values of the molecular free paths L calculated from Graham's experiments on transpiration. The numbers found by him are really much smaller than those obtained by the other modes of calculation, as we see from the following comparison of his numbers with those calculated by the methods of Loschmidt (§ 116) and van der Waals (§ 117).

We might therefore consider it possible that Dorn's numbers represent too small values of the molecular diameters, and that they even form an inferior limit, as Loschmidt's give a superior limit, for the magnitudes, whose true value lies somewhere between the two. I will not contest the admissibility of this idea, but I must consider it very uncertain, as so many kinds of unproved and even improbable assumptions underlie all these calculations. The hypothesis that the molecules are shaped like spheres, which occurs in the discussion of their dielectric behaviour, seems to me to be especially doubtful; for the dielectric polarisation will, if the molecules are of a flat shape (§ 112), be entirely different from what it will be if they are spheres.

Franz Exner1 and Ph. Guye have devoted attention to these relations, in addition to Dorn. The former gives extensive tables of values, among which are some that have been calculated from the indices of refraction observed by Dulong. For the fraction g can be calculated

1 Wien. Sitzungsber. 1885, xci. Abth. 2, p. 850; Exner's Repert. 1885, xxi. p. 446.

2 Arch. d. Sc. Phys. et Nat. 1890 [3] xxiii. p. 197.

not only from the dielectric constant K by means of the relation

but also from the index of refraction n by means of the analogous formula

g= (n2 - 1)/(n2 + 2),

which on Maxwell's theory is identical with it. To avoid repetitions I will cite these figures only from Exner's

These numbers, too, like those calculated by Dorn from the dielectric capacity, are considerably less than those which were deduced from the coefficients of condensation, and from the deviations from Boyle's law. They agree in magnitude with Dorn's values with striking accuracy, since, almost without exception, they are equal to 0.2 millionth of a millimetre. Sulphurous acid is the only exception to this rule in Dorn's table, and in his opinion this is due to inaccuracy in the value of the dielectric capacity used in the calculation; and since Exner's value is much smaller, we may fall in with Dorn's conjecture.

We must therefore consider it established that, if electrical or optical measurements are employed in the calculation of the diameter of the molecular sphere of a gaseous molecule, the value

s = 0.2 millionth of a millimetre

is found on the average. On the contrary we find s = 1 millionth of a millimetre

at least, if we rely upon observations within the domain of mechanics. We should not pretend to see any agreement in these numbers, which vary in the ratio 1 to 5, if we

had to do with magnitudes which are capable of direct measurement. But since these magnitudes are much smaller than the smallest that is microscopically visible, and since a knowledge of them is attainable only in roundabout ways by the use of many kinds of measures and uncertain conclusions, we must rejoice and, at least provisionally, be content that we have found values which differ only so little from each other that they, in all cases, are of the same order of magnitude.

We shall be able to form a judgment with greater certainty as to the trustworthiness of these figures when we shall have compared them with the values deduced from different speculations (§ 122). But we may now attempt to decide the question, which method of determining the absolute magnitude of the molecules deserves preference over the other. The answer can scarcely be doubtful if we remember that the determination by means of the mechanical measurements cannot give too small, but only too large values of the molecular diameter. The smaller of the values found is therefore to be looked upon as the more credible, and I therefore use the value

s = 0.2 × 10-7 cm.

in some further conclusions as to the state of gaseous molecules.

But there is still a further reason which we may give for preferring the values calculated from the dielectric capacities and the refractive indices. The equality of the path-volume (§ 69), which the former values failed to give (§ 116), comes into view when the latter are employed in the calculation, as is proved with sufficient accuracy by the following values of the product sL calculated by Dorn :

These numbers have for unit of length the millionth part of a millimetre.

119. Section and Volume of Molecules

Although we may not pretend to see an exact evaluation of the size of the molecular diameter in the value 0.2, which we have assumed as a mean, yet it seems justifiable to suppose that this number may serve as an approximately correct estimate. It is therefore not lost trouble, and it is more than a play with figures, if we calculate the sectional area and the volume of a molecule from this estimate of its diameter. We shall be conscious that the calculation can give us only approximate values, since we must once more introduce the assumption of a spherical figure, which is not strictly correct.

If, then, we put s = 2 x 10-8 cm. as the average diameter of a gaseous molecule, its sectional area will be πς? πs2 = 3 x 10-16 sq. cm., and its volume s3 = 4 × 10-24 ccm. Referred to the millionth part of a millimetre as unit of length, these numbers are 0.2, 0.03, 0.004 respectively, the units of area and volume then being the face and volume of a cube of which the edge is a millionth of a millimetre.

120. Number and Distance apart of Molecules

Now that we have attained to a knowledge of the size of the molecules, there opens out the possibility of taking a further step towards the knowledge and measurement of an invisible world by determining first the value of N, or the number of molecules contained in unit volume. This is at once obtained if we compare the value of the sectional area just computed with the sum of the sections Q discussed in § 109, which we likewise know in absolute measure, having calculated it numerically in § 110. By this magnitude Q we understand the area covered when we range close together on a plane all the molecules contained in one cubic centimetre of a gas under atmospheric pressure. Since we now know, at least approximately, the size of the