These equations also replace the equations of Gulberg and Mohn by integration of the fundamental equations. Exner obtained the trajectory of air masses; at the same time it turned out that oscillatory motion must occur even with rectilinear isobars; the case of a stationary flow is improbable.

In agreement with this are the results of Dines’s observations1 on the internal structure of the wind, made with a Sang anemometer adapted for recording. Using his self-recording instrument during kite ascents, Dines showed, in addition, the decrease of the stormy character of the wind with height; thus, if the frequency of wind gusts from 0 to 500 English feet is taken as 100, then at a height of 1000–2000 it falls to 44%, and at 2000–3500 to 30%.

In the 6th book, Hann sets forth the views of the English meteorologists Shaw, Dines, Cave, and others on the change with height of the pressure difference between regions of high and low pressure, which has led to the establishment of the most active layer of the atmosphere, the so-called “substratosphere,” which, in their opinion, chiefly determines the displacement of baric systems and the character of the weather at the earth’s surface.

Let us denote, somewhere but at one and the same level in the regions of the maximum and the minimum, the low pressure by \(p\), the high pressure by \(p+\Delta p\), and let us seek the change of the pressure difference with increasing height (Shaw—Journal Scottish Met. Soc. vol. XVI)

then:

\[ dp=-\rho dh;\qquad d(p+\Delta p)=-(\rho+\Delta\rho)dh, \]

where \(\rho\) is the weight of a unit volume of air;

hence

\[ d(\Delta p)=-d\rho dh, \]

but

\[ \rho=\frac{1}{v}=\frac{P}{R\cdot T} \]

and

\[ d\rho=\frac{p}{R\cdot T}\left(\frac{dp}{P}-\frac{dt}{T}\right) \]

where \(dp\) and \(dt\) in the present case represent the changes of \(p\) and \(t\) between the regions of the maximum and the minimum, i.e. they are taken in the horizontal direction.

Finally we have:

\[ \frac{d(\Delta p)}{dh}=\frac{p}{R\cdot T}\left(\frac{dt}{T}-\frac{dp}{P}\right) =0.0311\,\frac{P}{t}\left(\frac{dt}{T}-\frac{dp}{P}\right). \]

This equation gives the change of the pressure difference between the verticals to the two regions per unit change of height.

To the expression in parentheses in the last formula Shaw attaches special

importance for the dynamics of the atmosphere; the variation of this difference with height proceeds in a truly remarkable manner.

In the lower and middle parts of the troposphere the regions of maxima are usually warmer than the regions of minima; thus, \(dp\) and \(dt\) have the same signs, and the whole expression \(\frac{dt}{T}\) and \(\frac{dp}{P}\) will have a positive sign. Above 9 klm. this relation, as Teisserenc de Bort showed, changes: anticyclones become colder, and the whole expression, having passed through zero somewhere at a height up to 9 klm., becomes negative and begins rapidly to increase in absolute value; however, over the regions of minima the stratosphere begins at a lower level than over the maxima, as a result of which the gradient over cyclones that have already passed into the stratosphere becomes extremely small and even changes its sign, while at the same height over anticyclones there is still troposphere and the temperature continues to decrease. This leads to the fact that \(dt\) is rapidly evened out, and the expression under investigation again slows its growth.

Thus, in the greater part of the troposphere the effect of the vertical change \(\frac{d(\Delta p)}{dh}\) is small, excluding the lower layers of regions of high pressure with cold near the earth’s surface, and gradually turns to zero; with height, approximately around 9 klm. and up to 11, its greatest growth occurs, which then again slows.

The observations of Cave and Peppler, who found a maximum of wind force each time at the beginning of the stratosphere, agree with this scheme; at higher levels its decrease sets in more rapidly; this is also confirmed by the investigations of Köppen and Wedemeyer.

In connection with what has been set forth, it is appropriate to point out the fundamental change, arising from this, in views on the causes of meteorological processes: whereas until recently all atmospheric disturbances were reduced to differences of temperature at the earth’s surface, now their origin is being transferred to a height of 9—11 klm., and the preeminent role is ascribed to convection currents of air of different origin, which Shaw formulates as follows: “the dynamics of the atmosphere is conditioned by the substratosphere; physics belongs to the lower layers.”

In Chapter VI of his book, concerning the general circulation of the atmosphere, Hann, giving a survey of the most recent observations in tropical regions, comes to the conclusion that at the present time we are sufficiently well informed about horizontal and vertical currents between the tropics; however, the conditions have proved more complex than was previously thought; the latter is clearly seen, for example, from the distribution of winds over Java, according to the observations of van Bemmelen1.

From October to November — SE up to \(5^{1}/_{2}\) klm. (trade wind).
            above NE up to 17 klm. (antitrade wind).

From December to February — W up to 5 klm. (monsoon)
            above SE up to 9 klm. (trade wind)
            above NE up to 17 klm. (antitrade wind)
            above SE up to 22 klm. (“upper trade wind” according to van Bemmelen).

From March to April — NW and SW up to 6 klm. (monsoon)
above SE up to 10 klm. (trade wind)
above NE (anti-trade wind).

From May to September — SE up to 3 klm. (trade wind)
above NE up to 16 klm. (anti-trade wind)
above, “upper trade wind” with powerful intermediate layers of western winds, also found by Berson in equatorial Africa.

For the non-tropical regions, Perpler’s tables are of interest—the pressure fall in mm. between the equator and the pole at various latitudes:

Thus, the existing pressure fall, which the trade wind follows, has its greatest magnitude at the surface, then diminishes and at a height of 4 klm. changes into a fall directed toward the pole; the latter reaches a maximum at a height of 12 klm. and, perhaps, again changes sign at a height of 20 klm.; the pressure fall between the subtropical anticyclone and the high latitudes, i.e. the fall for the western wind of the high latitudes, increases up to 8 klm., remains almost constant up to 11 klm., then decreases and disappears completely at a height of 20 klm. This height is the level at which, in summer, the general circulation between the pole and the equator ceases.

As regards observations in the Arctic countries, they do not confirm the existence of a western wind in the high layers, which is a contradiction of the usual scheme of the general circulation in the regions of the great polar vortex. For example, from the pilot-balloon observations of the Swiss expedition of 1912/13 under the direction of de Quervain it follows that in the high layers a southerly wind prevails; the explanation of this in our former notions is not easy and, in Hann’s opinion, must in the present case be reduced to the action of a minimum over Davis Strait and a maximum over Greenland; aerological investigations on Spitsbergen, undertaken by Hergesell with the aid of pilot balloons and kites, gave the following distribution of winds:

Thus, the expected dominance of western winds was not found,

In the fifth book, “On Disturbances in the Atmosphere,” Hann gives both the newest general ideas on the origin and development of baric systems, expressed during this time by Ekholm, Köppen, Meinardus, Sreznevsky, Hellmann, Ficker, and others, and also a large number of data and tables concerning various questions: the distribution of winds in different parts of cyclones and their velocities, the speed of motion of maxima and minima, precipitation in depressions, the paths of cyclones, etc.

We shall note only the experimental works cited there by W. Schmidt on the penetration of cold masses of air into warmer ones, intended to explain the origin and properties of squalls. These works are of interest also because they belong to a still small, but very promising, field of experiment in meteorology and geophysics.

In carrying out his investigations, W. Schmidt found that cold masses move beneath warm ones not wedge-like, as had earlier been supposed, but that their penetration takes place in the form of a crest, appearing already at a temperature difference of \(7^\circ\); the form of the crest changes with the change of temperature; the air situated before the squall wave is lifted upward, which explains the powerful cloudiness and precipitation of the squall; maximum velocities are obtained at some distance behind the crest.

The velocity of motion of cold masses depends on their thickness and the temperature difference. The height of the crest, for the same temperature difference, is proportional to the square of the velocity, as follows from Helmholtz’s law.

Therefore:

\[ h = cv^2 \]

or

\[ v = c'\sqrt{h} \]

Schmidt found from experiments the constant \(c'\):

\[ \begin{array}{c|ccc} \text{for } dt = & 2^\circ & 7^\circ & 14^\circ \\ \hline c' = & 0.74 & 1.5 & 2.36 \end{array} \]

where \(v\) is in km/hour, \(h\) in meters.

In nature \(v\) is usually observed, while \(h\) is unknown; however, it can be calculated from the change of pressure during the passage of the squall and from the temperature difference. Schmidt found the following relation:

\[ h = \frac{db}{dt}\cdot T \cdot z, \]

where \(T\) is the mean absolute temperature of the air column, and \(z\) is the baric step. Ficker found complete agreement with observations when applying this formula to the squall of May 11, 1911, in Bavaria (observed velocity—55 km/hour; calculated—54).

Turning to the second area of the intensive development of meteorology—investigations of the upper layers—we shall dwell on some of the data collected in the Course: in Chapter Five of the first book, “On the Temperature Conditions of the Upper Layers of the Atmosphere,” Süring gives tables of mean tempera-

“surface” there are interesting tables from Emden’s book1 concerning the values, for the earth’s surface, of the thermal radiation of the atmosphere itself; from these it is found that, for latitudes from \(0^\circ\) to \(30^\circ\), the annual radiation of the atmosphere onto the earth’s surface is somewhat less than the annual influx of heat directly from the sun at the boundary of the atmosphere; for higher latitudes the former begins to predominate—thus, for example, in January Central Europe receives, from the radiation of the atmosphere, 2–3 times more heat than from the sun. The investigations of Angström point to the same circumstance; he established that the diffuse radiation of the sky toward the earth is always greater than the radiation of the earth toward the sky.

In calculating the mean pressure for the entire surface, Hann, using its values reduced to sea level and to normal gravity, obtains the value 758 mm; however, it does not correspond to the actual mass of air situated above the earth’s surface. Taking into account the mean elevation of the solid shell above sea level, the following values are obtained:

Thus, over the southern hemisphere the layer of air is almost 7 mm of mercury more powerful; the mean pressure over the whole earth’s surface is taken to be 737 mm.

In the first half of Chapter 4 of the third book, investigations are given of the velocity of fall of raindrops. Lenard and Schmidt gave experimental values of the velocity of fall as a function of their radius from \(r = 0.001\) cm, for which \(v = 0.10\) cm/sec, to \(r = 0.175\) cm, when \(v = 0.74\) cm/sec; with a further increase of the radius of the drop, as is known, upon falling it undergoes deformation, and its velocity of fall decreases; moreover, the drop attains its maximum velocity, 8 cm/sec, at a radius equal to 0.225 cm; hence it follows, incidentally, that an ascending current in the atmosphere with a velocity of 8 cm/sec can keep the largest drops from falling.

Stokes gave the formula for the fall of a water sphere in air of mean density:

\[ v = 1.26 r^2 10^6, \]

where \(v\) and \(r\) are expressed in cm,

which was verified experimentally within the limits from \(r = 0.001\) cm to \(r = 0.00016\). Subsequently this formula was improved by Cunningham and Mc. Kennan2, whereby the slip of the drop was taken into account. Mc. Kennan gave the formula, for very small values of \(r\), where slip plays the primary role:

\[ v = c\sqrt{\delta : d} \]

where \(c = \mathrm{Constans}\), \(\delta\) is the specific weight of the body, \(d\) the density of the air.

Taking, for the earth’s surface, \(C = 1344\) cm; \(\delta = 0.86\) (for ice),

obtain approximately, for water drops, \(v = 1344\sqrt{r}\), for hail grains \(v = 1246\sqrt{r}\); for great heights a factor \(\sqrt{B/l}\) is introduced.

Finally, Schmidt’s experimental formula is given, having significance for all \(r\) and, for small values of it, passing into Stokes’s formula:

\[ v = 1 : (0.00787 : r^2 + 0.150 : \sqrt{r}). \]

We conclude our survey with a brief account of the third chapter of Stüring’s book, “Clouds, Their Forms and Origin.”

At the beginning of the chapter Stüring pays tribute to Howard’s classification of clouds and explains its durability by the extraordinarily successful choice of the distinction of classes according to external appearance, into which it later proved possible to invest physical and genetic meaning.

For example, the forms Cumulus, distinguished by Howard by their dark and dense covering, are classified at the present time also into the separate class of “clouds of ascending currents.” Layered clouds, in their genetic meaning, form the class of “clouds of horizontal currents,” and so on.

Pointing to the tendency, observed in recent times, toward a more detailed classification of clouds, Stüring gives the following established names of additional forms: 1) “Wavy clouds” (Undulatus) of various types, appearing before depressions and, by their radiants, indicating the direction toward its center; 2) “Clouds of descending currents” (Ci Cu, ACu, AS—lenticularis, or mammatoides (Föhn clouds, negative Ci Cu according to Stüring)); 3) “Cloud caps” and Falsi Cirri; 4) “Dynamic Cumuli” (ACu—castellati), characterized as Cumuli not connected with ascending currents rising directly from the ground; 5) Mammato-Cumuli (Festoon Cloud, pocket cloud). Of the methods for determining the height of clouds, Stüring considers photogrammetric measurement so much more accurate than the previous determinations that, in his opinion, the latter have only historical significance.

As regards contemporary theories of cloud formation, the greatest difficulties are encountered in explaining the origin of Cirrus. Stüring points out that the formation of Cirrus and Cirro-Stratus fibers is most clearly seen over limited areas of low pressure, for example, over thunderclouds; in this case Cirri represent paths of outflows in the upper layers from the center of air masses; the outflow may, apparently, in some cases occur at lower altitudes, which leads to the formation of Falsi-Cirri.

However, Cirri are not always products of a nearby or distant depression; sometimes forming under a clear sky, before the eyes of the observer, they represent independent cloud formations and, moreover, very frequent ones. In explaining such types, it should be borne in mind that the newest ascents have shown that in the high layers there is almost constantly a thin layer of ice needles, imperceptible to the terrestrial observer and only slightly changing the visible coloration of the vault of heaven. Thus, material for the formation of Cirrus is always present; the immediate cause of formation may be mixing at the boundaries of differently moving air masses, which will lead to the formation of CiS or “strips of fall,” i.e. Cirrus of the types “litis,” “mare’s mane,” “strip”

and the like. Shaw indicates that cirri of the ACu type, in the form of waves, may be formed as a result of a lowering of pressure in the substratosphere.

Privat-Docent S. Bastamov.

Rozhdestveno.
August 1918.

Calculation and Measurement of Self-Induction and Capacitance

(W. H. Nottage. The calculation and measurement of inductance and capacity. London 1917).

The enormous theoretical and experimental material on the calculation and measurement of self-inductions and capacitances, accumulated over many years in physical and technical journals and very heterogeneous in quality, has long been in need of critical treatment and systematic compilation. Nottage’s book satisfies this need to a considerable degree, being a fairly complete collection of the formulas and schemes proposed up to the most recent time (1916).

Unfortunately, the author has limited himself only to a juxtaposition of final results, without giving derivations and, what is especially unpleasant, without always indicating the degree of accuracy of the formulas. There is likewise no critical comparison of the methods of measurement. In this respect the book is considerably inferior to the corresponding summaries in Fleming’s and Eccles’s books. The tables and graphs necessary for practical use of the formulas are not always given; instead, references are made to the corresponding journal literature. This circumstance to a considerable extent deprives the book of practical independence.

Quite a few pages are devoted purely to radio-telegraphic measurements and calculations: for example, a series of remarkable formulas for calculating the capacitance of radio-telegraphic networks proposed recently by Howe is given. In the last chapter alone are described various standards and instruments necessary for accurate measurements: Duddell’s alternator, the alternating-current “chronometer” of Fleming and Dyle, wave-meters, electrostatic telephones, and so forth.

There are some regrettable omissions; for example, one of the most general and exact formulas for calculating the self-induction of a concentric coil, Rayleigh’s, is absent.

In any case, however, the book fills a substantial gap in the series of reference books necessary for every physical and electrical-engineering laboratory.

S. Vavilov.

1. Emden—Über Strahlungsgleichgewicht und atmosphärische Strahlung, München, 1913. ↩↩↩

2. Proc. R. Soc. 83, A, 1910, 357 and Phys. Zeitschr. XIII, 106, 1912. ↩