THE MOTIONS OF THE ATMOSPHERE AND ESPECIALLY ITS WAVES.* Address delivered before the Meteorological Section of the Association of German Naturalists and Physicians at the Annual Meeting held in Vienna, September 25, 1894. BY DR. E. HERRMANN. The inadmissibility of those views according to which the motions of the atmosphere consist in the development of independent cyclones and anti-cyclones is, of late years, more and more plainly recognized. This conclusion has been arrived at, not so much through a severe criticism of the fundamental basis upon which these erroneous views had been established, as by the power of the facts that resisted introduction into this artificial system. It is now nearly ten years since the theory of cyclones began to totter, and especially under the influence of a memoir published by Hann.† Following soon upon this the idea was developed by von Helmholtz (Berlin Sitzungsb., 1888), "that in the atmosphere, by continuously acting forces, the formation of surfaces of discontinuity is possible, and that the anticyclonic motion of the lower strata and the extensive, gradually increasing cyclone of the upper strata that are to be expected around the north and south poles, resolve themselves into a great number of irregularly progressing cyclones and anticyclones with an excess of the former." Von Helmholtz further says§ (Berlin Sitzungsb., 1889): "if as in the case of the earth, the lower stratum is the denser, then it can be shown that the disturbances must at first proceed as do the waves of water raised by the wind." In this way the origin of cyclones and anticyclones is traced back to the general atmospheric circulation which is itself dependent on the difference in temperature between the equator and the poles; but the cyclones and anticyclones when once formed can still be considered as independent phenomena complete within themselves. * Translated from Verhandlungen der Gesellschaft Deutscher Naturforscher und Aerzte, pp. 42-50 and 323-324, by Professor CLEVELAND ABBE. † Comp. von Bezold, Berliner Sitzungsberichte, math.-nat. Kl., 1890, p. 831. Translated at pages 92, 93 of the Mechanics of the Earth's Atmosphere. Translated at page 94 of the Mechanics of the Earth's Atmosphere. The proof of this proposition was obtained by von Helmholtz by the application of the so-called theorem of the conservation of areas to a specific portion of the atmosphere, i. e., a ring of air. But the application of this theorem in this manner is incorrect; it can only be applied to a free system of bodies (See Met. Zeit., 1894, p. 114). Von Helmholtz applies this theorem to a ring of air cut out of the atmosphere; now such a ring of air does not form a free system but is subject to the conditions of its connection with the rest of the atmosphere, and the theorem of the conservation of angular momentum, or of areas, is not applicable to it. The application of this theorem to a ring-shaped portion of the atmosphere is only allowable in case the conditions within the atmosphere are of such a nature that they correspond entirely to the conditions at the boundary surface of a fluid and any other body. The conditions for the boundary surface of a fluid are: equal pressure on both sides of the boundary surface and equal values for the normal component of the velocities in the two masses that come in contact at the surface. (See Kirchhoff, Mechanik, 2d ed. 1877, p. 165; 3d ed. 1883, p. 164). These conditions are not fulfilled in the earth's atmosphere, which is in motion and not in static equilibrium. Therefore, that theory of atmospheric motions that depends upon their analysis into cyclones and anticyclones is deceptive and not so well established as it was believed to have been by the investigations of von Helmholtz. Now, in order to find the point at which the prevailing views as to the motions of the atmosphere diverge from the reality it will be necessary to follow up a train of thought based upon indubitable well-established mechanical theorems. In doing this we will, as usual, begin with the consideration of an atmosphere that, without friction, surrounds the ideal figure of the earth, i. e., a homogeneous ellipsoid of rotation, and whose particles are attracted by the mass of this ellipsoid, according to the general law of attraction of masses. It will be further assumed that, for a certain initial temperature, uniform throughout all its strata, the atmosphere has arrived at a uniform rate of revolution about the axis of rotation of the ellipsoid and with a velocity such that the ellipsoid represents the figure of equilibrium of an incompressible fluid. Let this initial equilibrium of the atmosphere be now disturbed by introducing a different distribution of temperature, viz., one depending only on altitude above the ideal surface of the earth and on latitude, but uniform along any circle of latitude and every circle concentric therewith.. The first question now is whether a new state of equilibrium can develop, perhaps by means of a new rate of rotation that shall be different in the various strata of air vertically above each zone of latitude but be uniform in each circle concentric with and lying in the same plane as the circle of latitude. The conditions for the equilibrium of a rotating fluid where the axis of z is the axis of rotation; is the density of the element of mass; r is the distance of the element of mass from the axis of z; w is the angular velocity; R and Z are the components of the forces parallel to r and z; p is the elastic pressure. Let V be the potential of the attraction of the mass of the terrestrial ellipsoid; T the absolute temperature of the element of fluid mass. Let the characteristic equation of elasticity for gases be: By differentiating this with respect to r, substituting the value of А др thus obtained in the first equation and exe cuting a partial integration we obtain: F(r) and F' (r) are determined by the surface conditions, in the present case by the circumstances existing in the highest layer of the atmosphere in which a uniform temperature and a velocity of rotation equal to that |