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Rh in the heat received from the sun; for instance, what variation in the earth’s atmosphere corresponds to the periodic variations of the solar spots. The general current of Helmholtz’s investigations shows that no periodic change in the earth’s atmosphere can be maintained for any length of time by a given periodic influence outside of the atmosphere. On the other hand, it is barely possible that wave and vortex phenomena on the sun’s surface may have the same periodicities as regular phenomena in the earth’s atmosphere, so that there may be a parallelism without any direct connexion between the two.

An important paper on the application of hydrodynamics to the atmosphere is that by Professor V. Bjerknes, of Stockholm, Sweden, which was read in September 1899 at Munich, and is now published in an English translation in the U.S. Monthly Weather Review, Oct. 1900 (“On the Dynamic Principle of Circulatory Movements in the Atmosphere”). In this memoir Bjerknes applies certain fundamental theorems in fluid motion by Helmholtz, Kelvin and Silberstein, and others of his own discovery to the atmospheric circulation. He simplifies the hydrodynamic conceptions by dealing with density directly instead of temperature and pressure, and uses charts of “isosteres,” or lines of equal density, very much as was proposed by Abbe in 1889 in his Preparatory Studies, where he utilized lines of equal buoyancy or “isostaths,” and such as Elkholm published in 1891 as “isodenses” and which were called “isopyks” by Müller-Hauenfels. Bjerknes has thus made it practicable to apply hydrodynamic principles in a simple manner without the necessity of analytically integrating the equations, at least for many ordinary cases. He also gives an important criterion by which we may judge in any given case between the physical theory, according to which cyclones are perpetually renewed, and the mechanical theory, according to which they are simply carried along in the general atmospheric current. Bjerknes’s paper is illustrated by another one due to Mr Såndström, of Stockholm, who has applied these methods to a storm of September 1898 in the United States The further development of Bjerknes’s methods promises a decided advance in theoretical and practical meteorology. His profound lectures at Columbia University in New York and in Washington in December 1905 aroused such an interest that the Carnegie Institution at once assigned the funds needed to enable him to complete and publish the applications to meteorology of the methods of analysis given in detail in Bjerknes’s Vorlesungen (Leipzig, i. 1900, ii. 1902), and in his Recherche sur les champs de force hydrodynamiques (Stockholm), Acta Matematica (Oct. 1905). In his lectures of 1905 at Columbia University Bjerknes treated the atmosphere as a continuous hydrodynamic field of aerial solenoids and forces acting on them, to which vector analysis can be applied, as was done by Heaviside for electric and magnetic problems. Every material point is a small spherical mass of air free to extend or contract with pressure, temperature or moisture; free to rotate about each of three movable axes passing through its centre and to move along and revolve about three fixed axes through the centre of the earth. These numerous degrees of freedom are easily expressed in Bjerknes’s notation and by his typical equations of motion. The density at any point is recognized as the fundamental “dimension” controlling inertia and movement. The observed atmospheric condition at any moment is shown by a series of isodense surfaces intersecting potential surfaces of equal gravity and thus forming a continuous mass of unit solenoids. This field becomes either an electric, magnetic or hydrodynamic field according to the interpretation assigned to the notations—in either case the analytical processes are identical. The analogies or homologies of these three sets of phenomena are complete throughout, and those of one field elucidate or illustrate those of the two other fields. This is the outcome of the study of such analogies begun by Euler, Helmholtz, Hoppe, and extensively furthered by Maxwell and Kelvin, but especially by C. A. Bjerknes. The homologies or analogies by V. Bjerknes are given at p. 122 of his Recherche (1905), and include the following six triads:—

which have been slightly rectified by Dr G. H. Ling, ''Am. Jour.'' Math. (1908). In the application of Bjerknes’s methods of study to the daily weather map Såndström draws special maps to represent the solenoids and the forces. Barometric pressures are reduced from the observing stations not only down to sea-level but up to other level surfaces of gravity. The differences between these level surfaces represent the work done in raising a unit mass from one level to the next (see Bjerknes and Såndström, A Treatise on Dynamic Meteorology and Hydrography, Washington, 1908).

The Diurnal and Semi-diurnal Periodicities in Barometic Pressure.—For a long time attempts were made to explain the periodic variations of the barometer by a consideration of static conditions, but it is now evident that this problem, like that of the circulation of the atmosphere, is a question of aerodynamics. A most extensive series of researches into the character of the phenomena from an observational point of view has been made by Hann, who gave a summary of our knowledge of the subject in the ''Met. Zeit.'' for 1898, translated by R. H. Scott in the ''Quart. Jour. Roy. Met. Soc. (Jan. 1899) (see also an important addition by Hann and Trabert in the Met. Zeit., Nov. 1899, and the summary of his results as given in his Lehrbuch'', 1906). Hann has shown that at the earth’s surface three regular periodic variations are established by observation, viz. the diurnal, semi-diurnal and ter-diurnal. On the higher mountains these variations change their character with altitude. (1) At the equator the diurnal variation is represented by the formula 0·30 mm. sin (5°+x), where x is the local hour angle of the sun. In higher latitudes either north or south the coefficient A1＝0·30 mm. diminishes, but the phase angle, 5°, varies greatly, generally growing larger. It is therefore evident that this diurnal oscillation depends directly on the hour angle of the sun, and probably, therefore, principally on the amount of heat and vapour received by the atmosphere from the ocean and the ground at any locality and season of the year. It is apparently but little affected by the wind, but somewhat by altitude above sea; the amplitude diminishes to zero at a certain elevation, and then reappears and increases with the opposite sign; the phase angle does not change. (2) Superimposed upon this diurnal oscillation is a larger semi-diurnal one, which goes through its maximum and minimum phases twice in the course of a civil day. The amplitude of this variation is largest in equatorial regions, and is expressed by the formula A2＝(0·988 mm.–0·573 mm. sin2) cos2 as given by Hann, or A2＝(0·92 mm.–0·495 sin2) cos2 as revised by Trabert. This amplitude also may be considered as variable along each zone of latitude having a maximum value on certain central local meridians. The times at which the semi-diurnal phases of maximum and minimum occur are subject to laws different from those for the diurnal period. Within the tropics the phase angle is 160° and at 50° N. it is 147°, and between these limits it seems to be the same over the whole globe, so that the phase does not depend clearly upon the hour angle of the sun or on the local time. The amplitudes appear to depend on the excess of land in the northern hemisphere as compared with the water and cloud of the southern hemisphere. The amplitude also varies during the year, being greatest at perihelion and least at aphelion. Hann suggests that this is an indirect effect of the sun’s heat on the earth, as the northern hemisphere is hotter when the earth is in aphelion than is the southern hemisphere when the earth is in perihelion, owing to the preponderance of land in the north and water in the south. (3) The ter-diurnal oscillation has the approximate value shown by the formula 0·04 mm. sin (355°+3x). The phase angle is sensibly the same everywhere, and the amplitude varies slightly with the latitude. Both phase and amplitude have a pronounced annual period which is as remarkable as that of the semi-diurnal oscillation; the maximum amplitude occurs in January in the northern hemisphere, and in July in the southern.

The physics of the atmosphere has not yet been explored so exhaustively as to explain fully these three systematic barometric variations, but neither have we as yet any necessity for appealing to some unknown cosmic action as a possible cause of their existence. The action of the solar heat upon the illuminated hemisphere, and the many consequences that result therefrom, may be expected to explain the barometric periods. The variations of sunshine and cloud must inevitably produce periodic variations of temperature, moisture, pressure and motion, whose exact laws we have not as yet fathomed. Among the many methods of action that have been studied or suggested in connexion with the barometric variations the most important of all is the so-called tidal wave of pressure due to temperature. Laplace applied his investigations on the tides to the gravitational tide of the ocean, and when he passed to the corresponding solar and lunar gravitational tides of the atmosphere he was able to show that they must be inappreciable, unless, indeed, certain remarkable relations existed between the circumference of the earth and the depth of the atmosphere. As these relations do not exist, it is generally conceded as certain that the gravitational tides, both diurnal and semi-diurnal, cannot exceed a few thousandths of an inch of barometric pressure. On the other hand, the same process of mathematical reasoning enables us to investigate the action of the sun’s heat in producing a wave of pressure that has been called a pressural tide, due to the expansion of the lower layer of air on the illuminated half of the globe. The laws that must govern these pressural tides have been investigated by Kelvin, Rayleigh (Phil.