of a pump that makes it possible easily and reliably to obtain a sufficient preliminary rarefaction, after which Langmuir’s pump can give the final evacuation. Experiments carried out at the Physical Institute of the Moscow Scientific Institute by N. Ya. Selyakov, with the advice of N. D. Papaleksi, make it possible to resolve this last question as well. From the experiments it turns out that, in order to obtain a vorvacuum, one may use an ordinary glass water-jet pump, provided only that a sufficient drying of the water vapor is produced, which can be done with phosphoric anhydride. Under these conditions, a small Langmuir pump connected in series makes it possible to obtain a rarefaction down to 0.01 mm of mercury, so that two Langmuir pumps connected in series, together with the vorvacuum achieved by a water pump, can readily give the limiting rarefaction.

These investigations are of very substantial importance not only for laboratory practice, but also for technology in the broad sense of the word. In life itself, at the present time, in view of the shortage of fuel, vessels of the “Thermos” type, which make it possible to keep water and food hot for a long time, must come more and more into use. These vessels require as perfect an evacuation as possible, and here pumps of the type described will find wide application. Furthermore, the evacuation of electric lamps and Roentgen tubes requires a large number of pumps, and in this field the experiments described above may have a very substantial significance.

I. Lazarev.

FROM CURRENT LITERATURE

On the principle of similitude.

(Richard C. Tolman. The Principle of Similitude. Phys. Review, 3, p. 344. 1914.)

The author establishes a new physical principle of the following kind: to every real physical world there is similar an imaginary world in which there exist the same general physical laws as in the real one. A real observer is supplied with a meter, a clock, and other measuring instruments. An imagined observer of the micro-world is supplied with a shorter unit of length than the meter, a clock with a changed period, etc. If the first, measuring a physical length, finds the value $l$, then in the second world it is $l'$, with

$l' = xl$.

Likewise, for the interval of time between two events,

$t' = xt$.

Having established these relations, the author shows that from them there follows a series of others, for example: for pressure,

$p' = p/x^4$,

for volume,

$V' = x^3 V$,

for absolute temperature,

$T' = T/x$.

The principle of similitude formulated above makes it possible to find relations between various physical quantities. For example, an ideal gas is given. It is asked how the product of pressure and volume depends on the absolute temperature. Suppose that the general law is

$pV = F(T)$,

where $F$ is an unknown function. By the principle of similitude, for the micro-world it must be

$p'V' = F(T')$,

whence

$pV = x F(T/x)$.

Since $x$ is any number, the only solution of this functional equation compatible with

$pV = F(T)$

is

$pV = kT$,

i.e. the law of Mariotte—Gay-Lussac is obtained.

By the same method, Tolman establishes the independence of the heat capacity of an ideal gas from temperature, the Stefan–Boltzmann law, the dependence of the density of electrostatic energy on voltage, the dependence of the electromagnetic mass of a spherical electron on its radius, and the proportionality between the magnitude of the electron’s radiation and the square of its acceleration. Wien’s law, though it cannot be derived from the principle of similitude, does not contradict it. Maxwell’s electromagnetic equations likewise agree with the principle. Newton’s law of gravitation in the formulation \(f = km_1m_2/l^2\) contradicts the principle of similitude, whence the author concludes that the formulation of the law is not sufficiently general.

K. Shaposhnikov.

(Richard C. Tolman. The Specific Heat of Solids and the Principle of Similitude. Phys. Review, 4, p. 145, 1914).

The author, from the principle of similitude he has established, derives Debye’s well-known formula for the heat capacity of a solid or of a liquid elastic body, which Debye obtains from quantum theory. He defines the energy of a body as the energy of standing longitudinal and transverse waves that arise in the body at its equilibrium; he indicates the functional dependence of the energy on two arguments: the frequency of oscillations and temperature; then the various frequencies of the waves are expressed by known formulae of the theory of elasticity through constants: the coefficient of compression \(\chi\), Poisson’s coefficient \(\sigma\), the volume \(V\), and the density \(\rho\). One obtains one functional dependence of the energy on the temperature and these constants. The principle of similitude gives another. From a comparison of the arguments of the one and the other function, a conclusion is drawn as to the form of the function itself. The proof of Debye’s formula proposed by the author is wholly independent of quantum theory.

K. Shaposhnikov.

(Richard C. Tolman. The Principle of Similitude and the Principle of Dimensional Homogeneity. Phys. Review 4, p. 219, 1915).

In this article the author shows that his principle of similitude should essentially be distinguished from the principle of dimensionality. Thus, the equation of state of an ideal gas and the Stefan–Boltzmann law may be obtained from the principle of similitude, but cannot be derived from the principle of dimensionality. The former assumes complete relativity in the measurement of physical quantities (the Relativity of Size), whereas the latter is not connected with this. The author further points out that the application of the principle of similitude has certain limits.

K. Shaposhnikov.

(T. Ehrenfest-Afanassjewa. On Mr. R. C. Tolman’s “Principle of Similitude”. Phys. Review, 8, p. 1, 1916; Richard C. Tolman. Note on the Homogeneity of Physical Equations. Phys. Review 8, p. 8, 1916).

Both authors analyze in detail the conditions that Tolman’s system of units of the micro-world must satisfy in order that, in the latter, the physical equations of the actual world be homogeneous in the sense of dimensionality.

K. Shaposhnikov.