SMITHSONIAN MISCELLANEOUS COLLECTIONS. 843 THE MECHANICS OF THE EARTH'S ATMOSPHERE. A COLLECTION OF TRANSLATIONS j;y CLEVELAND ABBE. CITY OF WASHINGTON: PUBLISHED BY THE SMITHSONIAN INSTITUTION. 1891. CONTENTS. Page. Introduction 5 I. Hagen, 1874. The measurement of the resistances experienced hy plane plates when they are moved through the air in a direction normal to their planes 7 II. Helmholtz, 1858. On the integrals of the hydro-dynamic equations that represent vortex motions 31 III. Helmholtz, 1868. On discontinuous motions in liquids 58 IV. Helmholtz, 1873. On a theorem relative to movements that are geo- metrically similar, together with an application to the problem of steering balloons 67 V. Helmholtz, 1888. On atmospheric motions; first paper 78 VI. Helmholtz, 1889. On atmospheric motions ; second paper. On the theory of wind and waves « 94 VII. Helmholtz, 1890. The energy of the billows and the wind 112 VIII. Kirchhoff, 1869. The theory of free liquid jets 130 IX. Oberbeck, 1877. On discontinuous motions in liquids 139 X. Oberbeck, 1882. The movements of the atmosphere on the earth's surface 151 XI. Oberbeck, 1882. On the Guldberg-Mohn theory of horizontal atmos- pheric currents 171 XII. Oberbeck, 1888. On the phenomena of motion in the atmosphere ; first paper 176 XIII. Oberbeck, 1888. On the phenomena of motion in the atmosphere ; second paper 188 XIV. Hertz, 1884. A graphic method of determining the adiabatic changes in the condition of moist air 198 XV. Bezold, 1888. On the thermo-dynamics of the atmosphere; first paper. 212 XVI. Bezold, 1888. On the thermo-dynamics of the atmosphere ; second paper 243 XVII. Bezold, 1889. On the thermo-dynamics of the atmosphere ; third paper 257 XVIII. Rayleigh, 1890. On the vibrations of an atmosphere 289 XIX. Margules, 1890. On the vibrations of an atmosphere periodically heated """ XX. Ferrel, 1890. Laplace's solution of the tidal equations 319 o THE MECHANICS OF THE EARTH'S ATMOSPHERE: A COLLECTION OF TRANSLATIONS. By Cleveland Abbe. INTRODUCTION. The complexity of the phenomena of the atmosphere has rendered it necessary to delay their mathematical treatment until our knowledge of hydro-dynamics and thermodynamics could attain the perfection which it began to acquire about the middle of this present century at the hands of Helmholtz, Clausius, Sir William Thomson, and their disciples During the past few years some of the fundamental prob- lems of meteorology have been treated analytically and graphically with great success. The present collection of translations presents some of the best memoirs that have lately been published on the re- spective subjects by European investigators ; a few earlier memoirs of great excellence are included in the collection because of the references subsequently made to them. Other mathematical memoirs by Guldberg and Mohu, Marchi and Diro Kitao have been omitted because their length would have made this collection too large for the present mode of publication. There is a crying need for more profound researches into the me- chanics of the atmosphere, and believing as I do that meteorology can only be advanced beyond its present stage by the devotion to it of the highest talent in mathematical and experimental physics, I earnestly commend these memoirs to such students in our universities as are seeking new fields of applied science. I have taken a very few liberties in translating the language and notation -of the distinguished authors whose works are here collected. I have frequently used the word liquid instead of " Wasser," " Tropf bar- Elussigkeit,'" -'Iukoinpressible Flussigkeit," and the word gas or vapor as equivalent to compressible or elastic fluid, and have used the word fluid when the more general term including liquids, vapors, and gases is needed. As the ideal or " perfect " liquid is absolutely incompressi- ble and devoid of all resistance to mere change of shape, having neither elasticity nor viscosity, namely, internal friction, it seems more proper 6 THE MECHANICS OF THE EARTH'S ATMOSPHERE. to use the general terms liquid, gas, and fluid when neglecting the re- sistance, compressibility, elasticity, and viscosity as in dealing with these ideal substances, and to reserve the terms air, water, etc., for use when dealing with actual natural fluid phenomeua where slight com- pressions and expansions and resistances occur. The relation between elastic pressure, volume, and temperature, as deduced by Boyle, Mariotte, Gay-Lussac, and Charles, that characterizes a gag, and the equation for which the Germans call the " Zustands- Gleichung" in common with other equations of condition, I have pre- ferred to speak of as the equation of elasticity or the characteristic equa- tion of a perfect gas. In view of the remarkable want of uniformity existing in English and American works in respect to the notation for total and partial differentials I have decided to make such alterations in the original notations of these papers as shall make the whole series consistent with the elegant and classical notation that is rapidly being adopted in Ger- many, and that will, 1 hope, eventually be accepted by ail English and French writers. In accordance with this I shall always express the total differential by d, as first introduced into geometry by Leibnitz for the infinitesimal difference; the small increment or variation by 6, as introduced by Lagrange; the large finite difference by J, first used by Euler; the partial differential by J, (" the round d,") as used by Jacobi. Occasionally the dotted variable x will indicate the rate of variation with regard to the time, or the fluxion as first introduced into mathe- matical physics by Sir Isaac Newton, a notation which has lately been extensively revived in Euglaud by those devoted to classic authority. Evidently the problems here treated by elegant mathematical meth- ods are not always precisely the problems of nature. The differences between the conclusions of Kayleigh, Margules, and Feirel as to the diurnal and semidiurnal tides due to heat, or the differences between Ferrel, Oberbeck, and Siemens on the one hand and nature on the other as to the general circulation, show that by the omission of apparently minor local and periodical irregularities we have constructed for our- selves problems that still differ from the case of the earth's atmosphere, although they may more closely represent the conditions of such a planet as Jupiter. I have to acknowledge the assistance of my friend, Mr. G. E. Curtis, in copying a portion of the formulae for these translations, and renew the expression of my hope that a coming generation of American meteor- ologists may prosecute to further conquests the mathematical studies begun by Ferrel and perfected by our European colleagues. Cleveland Abbe. February, 189 1. THE MEASUREMENT OF THE RESISTANCES EXPERIENCED BY PLANE PLATES WHEN THEY ARE MOVED THROUGH THE AIR IN A DIRECTION NORMAL TO THEIR PLANES.* By Professor G. H. L. Hagen. Sometime since I submitted to the Academy the results of a series of observations that I had instituted upon the motions of air and of water when the uniform flow of these fluids is interrupted by means of interposed planes.t By means of small bits of paper or tin foil floating from the tips of needles the direction of the motion could be perceived at every point. The velocities were indeed too feeble to be capable of direct measurement, but the disposition of particles of pulverized am- ber that were strewn over the water showed the limits of the strongest current, and when the coarser particles came to rest before the finer ones it was to be inferred that there was a gradual diminution of ve- locity at such points. In general it was concluded that air and water alike swerve in curved paths in front of such obstacles and flow towards the free openings. In the latter and directly adjoining the outer ends of the obstacle the strongest current is formed which here retains its direction unaltered, therefore free from all variations. The deviation in front of the obstacle does not take place at any definite distance from it, but rather extends up to the- obstacle itself and even when the plate faces the current it is seen that a feeble motion still exists immediately adjoining it. Behind the obstacle the fluid by no means remains at rest, but rather there is always formed here a counter current whose length is equal to four or five times the distance of the head of the obstacle from the neigh- boring side wall of the channel, which counter current, however, is not only fed at its rear enc, but principally also at two intermediate points by the steadily broadening main current. The latter immediately be- hind the head of the cross-wall meets the outcomiug counter-current *Read before the Academy of Sciences, Berlin, January 22. February 16, and April 20, 1874. (Translated from The Mathematical Memoirs [ Abhandlungen] of the Royal Academy of Sciences at Berlin for the year 1874, pp. 1 to 31.) tSee tbe Monats-Berichte for 1872, p. 861. 7 8 THE MECHANICS OF THE EARTH'S ATMOSPHERE. and here, as also at the two intermediate places just mentioned, whirls are formed which set in rotation the little vanes placed thei c The phe- nomena agree with those that one observes in streams and rivers in front of and behind sharp protruding rocks or piers. It must still be mentioned that neither water nor air rebounds like elastic spheres from the obstacle against which it strikes, as is fre- quently assumed. Even strong streams of water that I allowed to play against the plates did not rebound, but continued their onward path close to the obstacle, producing a strong current there. I had instituted these experiments in order to see in what manner the resistances originate that the liquid experiences in such deviations and which cause the pressure against the opposing plate. However, I thought it was allowable to assume that when the plate is itself moved through stationary water or air the ratios remain nearly the same and that similar currents of the fluid occur in its neighborhood. The pres- sure that the plate experiences in this latter case is the object of the following investigation which is moreover confined to plane disks moved through the air in a direction perpendicular to their planes. Already 40 years ago I had occupied myself with the same problem,* but the apparatus used at that time was too imperfect to give useful results. In essential points I have retained the earlier arrangements, but many changes have been made in order to remove the defects. The accompanying plate shows the apparatus now used by front and side elevations (see figs. 1 and 2), as also by a horizontal sec- tion (see tig. 3) through the line A B, of fig. 2. Two thin arms of straight- grained pine wood which are JYff. 7 *Some of the series of observations made at that time are communicated as exam- ples of the application of the method of least squares in the first edition of the Grundzuge der Wahrscheinlichkeits-Rechnuug." PAPER BY PROF. HAGEN. 9 bevelled 011 the sides that cut through the air, rest upon a vertical metal axis which communicates the rotary motion to them. Each of these arms is 8 feet or 96 Rhenish inches long and on its end the disk is fas- tened whose resistance is to be measured. In order to prevent the bending of the arms they are held not far from their ends by small ttg.2. wires which pass over a sup- port 18 inches high vertically above the vertical axis. The drawing presents only the connection of the two arms between themselves and with the axis. The latter is in its upper portion turned slightly conical and carries the corres- ponding hollow hub which is screwed to the brass plate under the arms. The rotation is brought about by the tension of two small threads which are wound in the same direction around the ivory spindle that is fas- tened to the axis, and are then drawn in opposite directions over two rollers and drawn taut by light scale pans with weights therein. These rollers I had formerly fastened at the greatest possible distance on the opposite walls of the room in order that when winding up the weights the threads might lie uniformly alongside of and not over each other, but this design was by no means certainly attained and the far-stretched threads ma- terially increased the labor of the observation, especially since the arms and the disks fastened to them occasionally came in contact with these threads. When in the past summer I again undertook the observations I placed the rollers, as the drawing shows, close to the axis, but did not let the latter stand upon a fixed point, but rather provided it with a screw thread on its lower part whose motber is cut into a thick plate of brass. By rotation the axis therefore rose or sank uniformly, MM fi JO B 1A 11 3oW Fiff. 3. HorUonttU Section onAB- ,z Zoll 10 THE MECHANICS OF THE EARTH'S ATMOSPHERE. and the threads simultaneously arranged themselves alongside of each other on the spindle from which they were drawn always in a horizontal direction. Underneath the roller I connected both threads by means of a light rod, and on this hung the scale-pans for weights ; I also fastened thereon a pointer which slid close to the graduated scale and served to measure the velocity. Notwithstanding the great convenience of this change it introduced the troublesome consideration that the friction became disproportion- ately great and varied so much during the observation that its magni tude and its influence on the measured velocity could not be determined with the necessary accuracy. This great evil I removed in that I al- lowed a steel point to work in the conical depression already formed by the turning lathe at the lower end of the axis, which point exerted an upward pressure equal to the weight of the arms, the discs, and the axis. The axis is therefore completely supported by the steel point, and the screw serves only as a guide in order to raise and lower the spindle corresponding to the windings of the thread. This steel point forms the upper end of a stout wire 12 inches high, whose lower end, ground to a wedge shape, stands in a metallic groove that is fastened at the end of a lever whose equal arms are 19 inches long. This lever, whose center of gravity lies in its axis of rotation, was so formed that its axis lay in a straight line with the metal groove and the point of suspension of the scale-pan, and was equally distant from both. This pan, with the counterpoise, corresponded exactly to the pressure of the axis on the wire when no resisting discs were placed upon the arms, but as soon as the latter occurred the counterpoise was always increased in a corresponding degree by an appended light cup with shot. Before attaching the discs these were laid upon a balance and the cup was partly filled with shot until brought into equilibrium with it. Since the lever changes its position during the rotation of the axis the steel wire deviates somewhat from the vertical position, but, as will be shown in the following, so slightly that this may be overlooked. The result of these changes in the apparatus proved to be very favora- ble, for whereas before at least 3 Prussian loths had to be placed in each scale in order to set the arms in permanent motion, now, the weight of the rod and the scale-pan, which together weighed 3.3 loths, sufficed without any additional weight to produce a uniform motion. At the ends of the arms pieces of perforated cork are glued, and in these the stems of the various discs find their support. The discs were always pushed so far on that they closely touched the ends of the arms. The disiance of the disc from the axis of rotation is found from the known lengths of the arras ; the stems of the discs did not extend through the corks, therefore the resistance of the air against the arms was only increased by that which the discs themselves experienced. Therefore, after the resistance which the arms experienced at each velocity had been determined by observation of the rotation of the PAPER BY PROF. HAGEN. 11 arms under various loads, this could be subtracted each time from the resistance observed with the disc in place and thereby the resistance of the various discs for various velocities be determined. The ivory spindle around which the threads wind was, like the axis, very carefully turned cylindrical, and is 1.1 inches high and 1.6 inches in diameter. The portion of the axis extending above the spindle is also turned cylindrical so that for any position of the upper perforated brass plate it is securely held with very little play. Under its slightly coni- cal flat head are found, as the figure shows, two openings perpendicular to each other, one square and the other circular. The first serves for the introduction of a small crank handle by means of which the axis is turned backwards when the weights are being raised. Before taking off the arms a wire is put through the circular opening which pre- vents the axis from turning forward while the observer is taking off tue crank aud putting ou the arms and discs. Moreover, at a distance of 12 inches from the axis there is placed a beut lever, one arm of which stands upright and hinders the turning of the arms that carry the discs when the weight fastened to the other arm of the bent lever hangs freely. While the arm carrying the discs is thus held by the bent lever the stout wire is withdrawn aud the air is allowed to come to rest, it the weight be placed on the neighboring table then the bent-lever arm falls and the apparatus starts in motion. The pitch of the screw of the axis below the spindle is 0.05 inch, and this distance corresponds to tue width of both threads so that the latter lie regularly close to each other on the surface of the spindle. This always occurred very regularly even when the axis was turned very rapidly by means of the crank handle. The threads, the so-called " iron twine," were so strong that each with safety carried 4 pounds, which weight, however, was never even distantly approached in practice. The threads were so light that 40 feet weighed only 0.1 loth, so that the fall of the index by 6 feet increased the driving power by only 0.03 loth. Nevertheless, for very feeble loads in the scale-pans a slight increase in the velocity was apparent during the descent, and in order to prevent this the small increase iu the weight was annulled by means of two equal threads suspended from the scale pans to the floor. Since the two former or driving threads were fastened to the rod they were thereby prevented from turning aud unwinding, which I had been able to avoid in my earlier work only by guiding the scale-pans by means of taut wires. Even if, however, the threads by this method of fastening did not materially change, still it remained to be proved whether perhaps they lengthened sensibly with greater tension, in which case the relation between the path of the index and the rotation of the arms could not remain constant. Such an extension could not be mistaken when I laid a weight of 1 pound on the empty scale pans when they were at their lowest position. The index then sank at once 0.2 of an inch. A further extension, however, did not follow ; at least 12 THE MECHANICS OF THE EARTHS ATMOSPHERE. it was not to be observed iu the short interval occupied by each sepa- rate observation. In consequence of this extension of the threads it was incumbent to lay those weights that were to be used to set the axis iu motion during the next observation upon the scale-pan while the latter was at its lowest position. The threads were therefore always wound up under the same tension with which they were to do the work. The question now arose whether with stronger tensions the spiral windings of the threads perhaps lay flatter on the spindle than with weaker tensions, and whether therefore the length of a winding or the path that the index described for one turn of the arms became shorter. This point was decided in that with various loads in the scale-pan 1 measured the path that the index described during a certain number of revolutions. The above-mentioned bent lever offered the opportu- nity of always stopping the arms at the same point, but it was necessary to bring them to rest by gentle pressure, because with a strong blow against the upright standing arm the horizontal arms carrying the discs could easily turn somewhat on the conical head of the axis. After the position of the index was read off I allowed the arms to make five com- plete turns and again read off the position of the index on the scale, estimating only to the hundredth part of an inch. The lengths of the paths for the corresponding weights in each scale- pan are as follows : Weight.* Path.t 0 25.69 4 .67 8 .68 16 .66 21 .67 28 .05 * Prussian loths. t Rhenish inches. A very slight shortening of the path appears from this to occur for the heavier loads, but if it actually exists it is so small that it is far less than the accuracy of the measurement of the path of the index on the divided scale. It may therefore be assumed that the velocity of the index stands in a constant ratio to that of the arms or disks. The lengths of the individual windings of the thread around the spindle as resulting from the above measures do not correspond in all accuracy to the circumference of a circle that is normal to the axis of the spindle, and at a distance therefrom equal to that of the central axis of the threads, inasmuch as the threads lie spirally around the spindle. Now the pitch of the screw measures 0.05 inch ; therefore the threads on the surface of the spindle make an angle with the horizon 0° 33' 29". Since the average length of one winding of the thread is 5.134 inches, therefore the equivalent thread encircling the normal is somewhat smaller, namely, 5.1338. Hence the resulting distance of the center of the threads from the axis of rotation or the length of the lever arm by PAPER BY PROF. HAGEN. 13 which the weight acts is equal to 0.81705 inch. This figure is adopted in the following computations, where it is represented by the letter a. It remains still to investigate whether the steel wire that carries the axis may perhaps depart so far from the vertical direction by the move ment of the lever on which it rests that it occasionally may exert an appreciable side pressure and thereby in an injurious way increase the friction in the screw threads. The lever is, as was mentioned, not only perfectly balanced, but the point that carries the counterpoise is also situated in the prolongation of the straight line drawn through the supporting point of the wire and the rotation axisof the lever. Therefore for every position of the lever the foot of the wire is pressed upwards vertically with equal force, but it rises only 0.8 of an inch, while the weight that drives the disks around in the extreme case sinks 80 inches. Therefore the deviation of the foot of the steel wire from a mean position amounts only to 0.4 of an inch, or in angle 2° 24' 48", for a length of the lever arm of 9 5 inches. Therefore the deviation from the initial vertical- ly is limited to 0.0086 iuch,aud consequently the wire 12 inches long is inclined 0° 2' 38" to the vertical. Even this small inclination can be reduced by one half if we place the axis of the wire or its upper point in the vertical line that bisects the deviation of its lower end, but such accuracy in the establishment of the apparatus must not be anticipated. It is evident from this that there can be no sensible increase of the fric- tion in consequence of the movement of the lever. As regards the execution of the observations the remark must be pref- aced that the Rhenish inch, or the twelfth part of the Prussian foot according to the earlier determination of the standard, and the old Prus- sian loth, of which 32 make 1 Prussian pfuud, have been adopted as units of length and weight.* The divided scale over which the index glides is divided into tenths of inches, but this subdivision is only used for determining the length of one winding of the thread, as previously described. In all other cases only the transit of the index over the heavier division marks for each 10 inches was observed by the beatiug of the seconds clock and the corresponding whole or half seconds noted. Since at the beginning of an observation the armsdo not immediately assume that velocity for which the resistance in connection with the friction balances the acceleration, therefore the significant observations began only when the weight had fallen 20 inches or the index had passed over the twentieth inch mark. At the seventieth inch the weight- scale pan had approached the floor, and therefore here the measures must be stopped. When, however, the rotation of the arms was ob- served without disks and the weights employed were very slight, then the speed continued increasing somewhat longer and the time of transit over the twentieth inch could not be used in the calculations. [* One Rhenish inch = 1.0297217 English inch = 26.15446 millimetres. One Prus- sian loth = 0.032226 pounds avoirdupois = 14.616 grammes. (See Barnard's Weights and Measures, C. A.] 11 THE MECHANICS OF THE EARTHS ATMOSPHERE. In order to determine with tbe greatest aceuraey the resistance of the air against each separate pair of disks it certainly would have been advantageous to employ very different weights and thereby attain very different velocities. This intention, however, could not be carried out by reason of the moderate length of tbe arms, which was limited by the dimensions of the room. If I loaded each scale pan with more than 1 pfund then the whole mass of air in the room, especially when using larger disks, assumed a rotatory motion, in which case the resistance dur- ing the individual observation is always less or the velocity is always greater. Even with a load of 1 pfund the light paper vanes that floated at the tips of the needles already showed a feeble continuous rotation, although the flame of a caudle did not allow of its recognition. In all the following observations therefore in the extreme cases only 28 loth was placed in each scale pan. To this it is to be added also that the measurements for very large velocities lose in accuracy on account of the relative maguitude of the unavoidable error. According to this the index should not move faster than an inch in 1.8 seconds. On the other hand, however, on account of the excessive influence of the very variable friction, the movement became highly irregular, when more than 8 secouds elapsed while the index described 1 inch. Within these limits the times in which 10 inches were described did not easily devi- ate more than half a second from the average value. The velocities of the disks were therefore not greater than 66 and not less than 17 inches per second.* In order to attain a uniform tension with reference to the axis the weights placed in the two scale pans were always equal and since on each occasion the disks attached to the arms were also always of equal magnitude, therefore each of these weights corresponded to the resist- ance of one disk. To this indeed should still be added one-half of the weight of the rod and the two scale pans but this may be disregarded since for each individual observation the value of the constant term which indicates the friction has to be especially computed. This con- stant term will then be the sum total of these weights less the friction, and presented itself always with the negative sign because the friction remained less than the weight of the rod and the scales. In order to simplify the computation I have at first referred not to the velocity of the disks, but only to that of the index, whence as above mentioned the velocity of the rotation can be easily deduced. In this way the opportunity was offered at each observation with disks to take into consideration that resistance which the arms alone experienced for the corresponding velocity of rotation. Before and after each series of observations, which generally oc- cupied 3 or 4 hours, the barometer and thermometer were read oft, the latter being at the same altitude above the floor as that at which the arms revolved. The computed coefficients of resistance were re- * Between 3.6 miles and O.'J mile per hour. PAPER BY PROF. HAGEN If, duced to the barometric pressure of 28 Paris iuches and the tempera- ture 12° Keaumur or 15° C. Assuming that the resistance of the air is proportional to its density I formed a table of the logarithms of this correction whereby the separate reduction is very easy. In case the temperature sensibly changes during the time of observation it must be assumed that this change occurred gradually and therefore for each individual observation the correction corresponding to the time is adopted. When especially large variations occurred readings were also made in the intervals; still, in such cases very large deviations were sometimes apparent, and it was repeatedly remarked that then the movement of the arms steadily increased or that the times in which the index sank 10 inches became smaller the lower its position was, which never occurred with uniform temperature. The reason of this is cer- tainly nothing else but this, that the equilibrium of the warmer and colder air in the room gave rise to special currents that were combined with the movement of the disks. When the temperature during a se- ries of observations changed by two degrees or more, the results deduced became so discrepant that they had to be rejected as entirely useless. For this reason the room before and during the observation could not be heated warm. On the contrary, the oven used for heating the room must be cooled down completely. Even when the suu shone on the window whose shutters could not hinder the warming, nothing remained but to stop the observations. Almost equally troublesome was the friction in the various parts of the apparatus. This varied perpetually, wherefore its value lor each individual observation had to be especially determined. Of course it diminished when fresh oil was introduced between the rubbing surfaces, but then the variations became of such magnitude and were often so sudden that the observations were again useless. Only after many days and after the arms had remained for a long time continuously in motion there was established a greater regularity. When this, how- ever, became evident from the measures immediately following each other, then again on the next day the conditions would be remarkably changed. It was therefore necessary that the whole of any series of observations that were to be compared among themselves should be made in immediate succession. In order to render this possible it was necessary to reduce the number of measures as much as was any way allowable, namely, to the number of the desired constants. Such a course is defensible also because the individual readings, in a long series of observations, accord much more closely with the law deduced therefrom than with the similar measures repeated at other times. These preliminary remarks are the result of the great number of ob- servations that I have executed during a half year. These were, es- pecially at the first, extremely unreliable, and only gradually were all the circumstances perceived that come into consideration. The follow- ing observations, which are the only ones serving as a basis for the sub- 16 THE MECHANICS OF THE EARTH'S ATMOSPHERE. sequent computations, were made at recent dates with the greatest possible care and under quite favorable external conditions. The resistance that the arms alone experience for different velocities must first be determined because this must be subtracted every time from the total resistance of the disc and the arms. The following table contains the measures made on this point. G is the weight [in lothsj that is placed in each scale-pan, and t the number of seconds oc- cupied by the index in passing over 1 inch. The velocity of the index is therefore equal to -- according to the adopted unit of measure. The v observations were made twice tor each load in the scale-pan, and in the second column of the table the two values tr and t2 thus found are given separately, while the third column contains the mean value (t) adopted in the succeeding computation. G. «,. t2. t. 5. 725 A. 0.040 Diff. B. Diff. 0.0 5.725 5.725 + 0.040 —0. 009 —0. 009 0.5 4.238 4.225 4.2315 0.514 + 0.014 +0. 498 —0. 002 1.0 3.488 3.500 3. 404 1.001 + 0. 001 1.007 + 0.007 2.0 2.725 2.735 2.730 1.979 —0. 021 2. 006 + C 3.0 2.300 2.312 2.306 2.986 -0. 014 3.018 + 18 4.0 2.038 2.038 2.038 3 972 —0. 028 4.001 + 1 6.0 1.700 1.700 1.700 5.941 —0. 059 5.946 — 054 8.0 1. 475 1.675 1. 475 8.066 +0.066 8,029 + .029 Earlier observations had shown that the resistances could be ex- pressed by the simple formula G + ?8 On attempting to introduce a third term containing as factor the first power of the velocity the constant coefficient corresponding had a very slight value and even sometimes a negative one. Therefore I now first chose the preceding expression, and by the method of least squares found z= - 0.531 S = + 18.703 By the introduction of these constants I obtained the values for G, which are given in the column headed A. The next following column shows the error or the differences {A—G) for each of the weights acta ally used. We notice that these errors progress very regularly in that both for the smallest and largest values of G they attain the largest positive values while between these they become negative. From this circumstance it may be inferred that the form of the formula has not been appropriately chosen, and I therefore repeated the computation using the expression G = z + 1p + 1s 1 tl PAPER BY PROF. HAGEN. 17 This then gave, z=- 0.724 p= + 1.034 s = +15.518 According to this last we obtain for G the values given in the column headed B, whose errors B — G are shown in toe last column. We remark that these latter do not occur regularly, owiug to the change of the signs for the heavier weights, and therefore can be looked upon as accidental errors of observation. The sum of the squares of the errors amounts in the last case to 0.004252, whereas in the first case it was 0.011055, therefore more than twice as great. There is still another reason that favors the introduction of the first power of the velocity. So long as I neglected this term there occurred without exception the inexplicable phenomenon that for observations with disks the numerical value of the constant rafter the negative sign was always greater, therefore the friction was always smaller, the larger and heavier the disks were. This anomaly disappeared upon the in- troduction of such a second term. There is, moreover, as the observations show, a peculiar condition in connection with the second term. The coefficient p assumes a very small value or entirely disappears when the screw on the axis is freshly oiled. From this we may conclude something as to its significance, i. e., it indicates the resistance that arises from the viscosity of the oil and which is proportional to the velocity. When disks are attached, the resistance peculiar to them is found when we subtract from the observed resistance that which the arms experience for equal velocities. This latter, however, is so variable that we must measure it anew every time, and since it assumes various values within even short intervals, therefore there remains only one method to determine the value of the three constants *, p, and s, namely, to allow the arms to revolve alone with three different velocities both before and after each observation. When, however, as usually hap- pened, a second measure again gave somewhat different values, then the appropriate mean value corresponding to the intervening time should be used in the computation. In the resistances of the disks found in this manner the second term proportional to the velocity is no longer contained, because the influ- ence of the viscosity of the oil has already been allowed for in the resistances of the arms. The constant z is, on the other hand, so variable that it must be specially deduced from each series of observa- tions. 80 -a 2 18 nu: MECHANICS OF THE EARTH'S ATMOSPHERE. The following observations were made with two square disks of 6 inches on each side,* 6 indicates the weight placed in each scale pan. and this changes to Q when we subtract the weight required to over- come the resistance of the arms for equal velocities. The second col- umn contains as before the times during which the index sinks by 1 inch, as found from the two measurements respectively. tot*. 1 - 4 i •. .'- -- t lot*. A • ■■ lot*. loth. ! 9.4-: - 1.117 1.9vv5 1.064 1.963 -0.053 - .003 " - .- : - . B7S - >. 54 739 - .MS i - - Hi - . . - ' . 1 - U4 '•- 1 Ml i " 10 " - 061 - . - 14 235 - 1 ' . - - . 17.625 - . ' 1 :■ m . ^23 ..' . - u - . DM Adopting the expression. Q = : -'- I find as most probable values - 0 - 124.24 From this the values of G given in the column marked A are deduced for the respective times. The errors of these, as contained in the fol- lowing column, vary so much in sign that we can consider them as accidental ami there is no reason in introduce still another term in the above expression. In this connection it must still be mentioned that when in the computation of theeariierobservations I have assumed the coefficient ;> equal to sero, a satisfactory agreement of the resistances appears for larger disks as soon as 1 set the resistance proportional to the square of the velocity. This is explained by the fact that the value of the term " is very small in comparison with the stronger resistances which the disks experience. [*In»lltl jit is 1 rsl . s for the • - a ofth< - - - - - .i been made with tlu a - - terminal combined t :. for arms Pll> • ;e due to the arms luis been puted. - _ Mne the fiietkm pins the iwBBtauioe of tbe ~ a tnm ~ _r required to o- : ion plus resistai r of the air to the motiou of the disks : 3 is the weight required to overcome the ,-ht required to overcome the resistance! to the disks. C. >.] PAPER BY PROF. HAGEN. H> The resistance of the air against the disks is therefore pi oportional lo the square of the velocity, and a single observation would suffice to give the coefficient r if the value of z were known, but since this is so very variable, therefore at least two observations at two different veloci- ties arc necessary. The further extension of the measures is unneces- sary, as already before mentioned, because the greater accuracy at- , tained surpasses the other inevitable errors; but for greater security and especially to avoid possible mistakes I have always repeated these two measures, and in such a way that beginning with the less velocity I then execute the two measures with the greater velocity and finally re- turn again to the less. From the values of r found in this manner the pressure that the disk experiences for various velocities is directly given. Let a be the known distance of the axis of rotation from the center of the threads wound round the spindle and R the distance of the same axis from the center of pressure of the air against the disk, then this pressure becomes D= " (G-z)= a r R PR But - is the velocity of the thread, hence the velocity of the center of pressure of the disk is at and D= a r &. R? if we introduce the pressure on a unit of surface, since F is the whole surface of the disk, we have D «' r , F R? F In order to reduce the constant r to the barometric pressure of 28 inches or 336 Paris lines, and to reduce the temperature to 15° C, we have for an observed pressure, A, in Paris lines, and an observed tem- perature r in centigrade degrees during the observations the reduced r =^ (0.9480+0.003477) r. A The distances R, on account of the great lengths of the arms in com- parison with the width of the disks, agree quite nearly with the dis- tances of their centers of gravity from the axis of rotation, but they are always somewhat larger and there is no reason to omit this correction, which is easily executed. We consider first a rectangular disk whose height is h and width b. As the origin of abscissas we may take its center of gravity whose dis- tance from the axis of rotation is A, and consider the disk divided into 20 THE MECHANICS OF THE EARTH'S ATMOSPHERE. elementary portions, the area of any one of which is hdx, and the pres- sure that it experiences is dD=—(A+xfdx aH2 consequently the pressure against the whole disk, found by taking the integral from x — — h, b to x = + £ b, is D= — (A2+ A, V) aH2 or the average normal pressure on a unit of surface is -=— (A*+ i ¥). F a2t2 If uow I seek that value of x which belongs to the elementary area that experiences a pressure the same as this average, then it represents the center of pressure for the whole disk. The result is, A+x=R=\J 'a*+&V For circular disks we again take A as the distance of the center from the axis of rotation, while the radius of the disk is p. In the division of the disk into elementary vertical sections I indicate the limits of these by the angle cp which is measured from the horizontal diameter. The area of such a section is then 2 cp sin cp2 d cp and the pressure that it experiences is t~ \ (( J By expanding the binomial and converting the cos2

)>r • and Hit' section tbat experiences this average pressure is that whose cp satisfies the equation — A+p cos (p=R=\A2+jp2 It follows that in both kinds of disks the difference between A and the desired /.' remains very small when, as in my apparatus, A is very large compared with b and p. Next, a series of observations will be communicated, made with five pairs of circular disks, whose diameters were 2.5, 3.5, 4.5, 5.5, 6.5 inches. Each time only two different weights were laid in the scale-pan; with PAPER BY PROF. HAGEN. 21 these, however, as above mentioned, the measures were executed twice. The resulting values of z and r are given in the last columns. The other letters correspond to those above given : p G> U u t G z r 1.25 0. 75 5.42 5.42 5.42 —0. 041 9.0 2.00 1.988 1.994 +4. 058 —0. 683 18.850 1.75 1.5 5. 31 5.30 5.305 0.690 14 2.00 1.98 1.990 9.054 —0. C79 38. 545 2.25 o 5. 76 5.68 5. 720 1.302 20 2.04 2.03 2.035 15. 270 -0. 722 66. 243 2.75 3 5.89 5.86 5.875 2.345 21 2.24 2.24 2.240 20. 079 —0. 671 104. 117 3.25 3 6.97 6.89 6.930 2.521 28 2.43 2.44 2.435 24. 670 . —0. 599 149. 827 In order from this to find the pressure on thu unit of surface, or 1;, we have to assume the lever arm «=0.8i705 inch, as already shown above. The following table contains the values of R, as well as the reduced r, and the surfaces of the disks F, as to which latter it is to be noticed that after more careful measuremeuts the radii of the second and third disks resulted 1.745 and 2.245: Table I. p 1.25 1.745 2.245 2. 75 3.25 99. 260 ! R 97. 252 97. 754 98. 256 9G. 700 * Reduced /• 18. 791 38, 463 60 105 : 04. 095 149. 942 F 4.909 9.566 15.834 2 i. 758 33. 182 k 2. 2700 2. 3476 2. 4028 2 4810 2. 5199 r i. e., Reduced to standard density of air. In order to avoid too small numbers these values of Jc are given too large, and must be divided by one million in order to present the desired constant factors, which, multiplied by the squares of the velocities in inches, will give the pressures in loths for each square inch of the disk. This same multiplication of A: is also continued in the following para- graphs. Many days later I repeated these observations with the same disks. The results were — p G1 'i t2 t G z r 1.25 1 5.00 5.02 5.01 0.168 —0. 592 19.091 10 1.91 1.91 1.91 4.641 1.75 1.5 16 5.21 1.87 5.22 1.87 5. 215 1.87 0.721 10. 405 -0. 708 38. 861 2.25 2 20 5.67 2.05 5.70 2.04 5.685 2.045 1.329 15.291 —0. 746 i;-,.ot;t; 2.75 3 24 5.74 2.23 5.79 2. 24 5.765 2.235 2.338 20. 025 —0. 790 103. 983 3.25 4 28 6.06 2.45 6.09 2.43 6.075 2.44 3.391 24. 633 —0. 694 150. 786 00 the mechanics of the earth s atmosphere. The values of R and F are the same as in the first series. The follow- ing values of Tc are computed from the reduced r : Table II. p 1.25 1.745 2.245 2.75 3.25 Reduced /• 18.052 38. 576 66. 575 103. 221 149.683 2. 2894 2. 3549 2. 4176 2. 4602 2. 5154 It evidently results that k becomes larger as soon as the surface of the disk increases, as also that the differences are proportional, not to the increase of the surfaces, but to the increase of the radii. Measures were also made with square disks whose sides measured b= L', 3, 4, 5, 6 inches, respectively. These gave — b G1 (> <_• 1 5.83d G z r 2 0.5 5.80 5.86 —0. 188 10 1.84 1. S3 1.835 +4. 104 —0. 660 16.012 o 1 6.00 5.95 5.975 +0. 346 14 1.97 1.96 1.965 8.840 —0. 684 36. 774 4 2 6.06 6.03 6.045 1.364 20 2.08 2.08 2.080 15. 383 -0.519 08. 798 5 3 5.99 6.06 6.025 2.364 24 2.30 2.28 2.290 20. 168 —0. 643 109. 135 o 4 6.50 6.43 6.465 3.443 24 2.55 2.54 2.545 24. 874 -0.488 1G4.270 The closer investigation showed again that the surfaces of the disks in part needed some small corrections, as in the following Table III : Table III. b 2 3 4 5 6 It 97. 002 97. 504 98. 008 98.512 99. 015 Reduced r 15.C07 35.810 67. 053 106. 455 160. 522 F 4.000 8.977 16. 000 24. 958 36.(00 k 2. 3317 2. 3472 2. 4281 2. 4338 2. 5055 i The following results were given by a subsequent repetition of the same observations : b 2 G] ('. 5 5. 76 5.79 t O z r ! 5.775 —0. 149 —0. 630 16. 020 111 1.84 1.83 1.835 +4. 128 3 1 5.96 5. 94 5.950 +0. 397 —0. 641 36. 744 14 1.96 1.97 1.965 8.876 4 •> 5. 74 5.78 5.760 1.371 —0. 608 68, 976 •_o 2.07 2.07 2.070 15.387 5 ;i 5.92 5. 93 5.925 2.415 —0. 714 109. 855 24 2.29 2.29 2.290 20. 233 6 4 6.26 0.26 6.260 3.485 —0. 700 164. 000 28 2.53 2.53 2.530 24. 922 PAPER BY PROF. HAGEN. According to this, the values of A- are: Table IV. 23 . - 3 4 5 6 Reduced r . . k .. 15. 704 2. 3461 35.998 67.524 107.493 2.3595 2.4452 , 2.4574 160. 378 2. 5032 By connecting among themselves the two first, as also the two last series of observations, the law according to which the value of k de- pends on the size of the disk may be approximately recognized, but the relation between the two forms of disks does not appear clearly. In order to discover this I tried allowing circular and square disks to run one immediately after the other, the radius of the first being 0.5 greater than the side of the latter. From this, however, it could only be in- ferred that for equal areas the resistance of the square disk is the greater. In order to recognize the influence of the shape, I tried also disks which formed equilateral triangles of 7.6 inches on each side, which were fastened in such a way that one of the sides stood vertically at the end of an arm. The area of each disk measured 25 square inches, agreeing, therefore, to within a very small quantity, which subsequent accurate measures showed, with that of the square disk of 5 iuches on a side. As I observed these two pair of disks one immediately after the other under the same load, it appeared that the square disk re- volved somewhat more rapidly. This result, however, was not decisive, in that the distances of the centers of pressure from the axis of rota- tion, or R, did not remain the same. In this respect it may be men- tioned that when the side of the equilateral triangle =& and its altitude =h=b cos 30° and the distance of the center of the surface from the axis of rotation is A, we then find R=V(A* + ^ti>) A complete series of observations, together with the preliminary and the concluding determinations of the value of p and s, gave the following: t , equal to three different transverse lines drawn through the center of the disk. First, the smallest transversals, for which of course the sides of the square and the diameters of the circles were directly introduced. This gave a = 2.204 ft = 0.0487 [xx] = 0.01282 For the greatest transversals, namely, the diagonals of the squares and diameters of the circles, I obtained a = 2.230 ft = 0.0354 [xx\ = 0.02221 Finally, for the average transversals which I drew [centrally] across the disks at distances apart of every 3 degrees, aod took the arithmeti- cal mean of all, I found a = 2.200 ft = 0.04675 [xx] = 0.00966 It is evident that this latter method must lead to very nearly the same result as the introduction of the square root of the surface since ft diminishes in the same ratio as the coefficient of ft increases. Judging by the sums of the squares of the errors it would, according to this, be advisable to introduce the square roots of the surfaces as factors, but this is impossible, even although the results of the observa- tions made with the long disk should be included under this same law. There only remains to introduce the circumference as a factor, even although in this case notable departures still remain. These are in no- wise however errors of observation, but result principally from the inevitable variations in friction. An error of 1 per cent, in the time could scarcely have been made, but still such discrepancies and even larger ones show themselves very frequently since the friction induced now faster and now slower motion. Nevertheless, from the following collection of all the observations it results that these have led to a quite trustworthy result. PAPER BY PROF. HAGEN. Radii and sides. k P A Diff. Squares. Circle p = 1.25 2.270 7.854 2.338 +0. 068 0. 004624 1.75 2.348 10. 996 2.368 4-0. 020 0400 2.25 2.403 14. 137 2.397 -0.006 0036 2.75 2.481 17. 279 2.427 -0. 054 2916 3.25 2.520 20. 420 2.456 -0. 064 4096 Circle p — 1.25 2.289 7.854 2.338 +0. 049 2401 1.75 2.355 10. 996 2.368 4- 0. 013 0169 2.25 2.418 14. 137 2.397 -0.021 0441 2.75 2. 460 17. 279 2.427 -0.033 1089 3.25 2.515 20. 420 2. 456 -0.059 3481 Square b — 2. 2.332 8.0 2.339 + 0.007 0049 3. 2.347 12.0 2.377 4-0. 030 0900 5. 2.428 16.0 2.415 -0. 013 0169 5. 2.434 20.0 2.452 +0. 018 0324 6. 2.505 24.0 2.490 -0.015 0225 Square b = 2. 2.346 8.0 2.339 -0.007 0049 3. 2.360 12.0 2.377 4-0.017 0289 4. 2.445 16.0 2.415 -0.030 0900 5. 2.457 20.0 2.452 -0.007 0049 6. 2 503 24.0 2.490 -0. 013 0169 Triangle 2.503 22. 795 2.479 2.452 -0.024 4-0. 003 0576 0009 Square b = 5. 2.449 20.0 Parallelogram 2.529 34.0 2.584 +0. 055 3025 Parallelogram 2.518 34.0 2.584 4-0. 066 4356 0.030742 ; From this table there results as the most probable values a = 2.2639 /i = 0.009410 The values of 1c computed from this are giveu in the column A; from the differences in the next column, with reference to the observed values of k, there results the probable error 0.0252, and we find the probable error of a equal to 0.01338, or about i per cent., and of ft equal to 0.000719, or about lh per cent. Although the reliability of these results, especially in their applica- tion to still larger surfaces and greater velocities, leaves much to be desired, still scarcely any important higher degree of accuracy is to be attained with apparatus that is similar to that above described. On the other hand the concluded law of resistance would be in an impor- tant degree confirmed or corrected, if on a firm rod in front of a loco- motive, disks are fastened, whose pressure could be measured by the tension of a spring, while the milestones on the roadside would serve very conveniently for the determination of the velocity.* * [This experiment has been carried on recently by Wild and others, but the resulting value of k is not so reliable as that deduced from observations with large whirling machines.— C. A.] 28 THE MECHANICS OF THE EARTH'S ATMOSPHERE. From the preceding it results that the pressure of the air against a plane disk turned normally towards it is _ 2.264 + 0 00942 x p F 2 D = ~~ 1,000,000 C Where D is expressed in old Prussian loths and p, F, and c in [Rhenish] inches. According to the above, the pressure against a square disk of 1 square foot area, moving with a velocity of 50 feet per second, would for example be 140.8 lotbs, or nearly 4.4 pfund. For reduction to metric measures and weights 1 take not the metre itself but the decimetre as the unit of measure for lengths and surfaces, in order to remain within the limits of the observations. Therefore the resistance of the air for a temperature of 15° C. and a barometric pressure of 28 Paris inches,* expressed in grammes, amounts to (0.00707 + 0.0001125^) Fc\ Where p represents the circumference of the disk, F the sectional area, and c the velocity expressed in decimetres. The pressure that very small disks experience when struck normally by a current of air is also given by another simple consideration, whose correctness has in general been confirmed by many experiments. These experiments indeed are limited, so far as known, to streams of water; but the expansibility of the air is certainly in this case without influ- ence, since the observations mentioned in the preface, upon the direc- tion and strength of currents deviated in front of opposing disks, showed identical results with water and with air. Imagine a vessel filled to the height h with a fluid of which .one unit of volume or 1 cubic inch weighs y loths. The bottom of the vessel therefore experiences on each square inch a pressure equal to yh, when no side pressure exists. If there is suddenly made therein an opening of 1 square inch, the outflow of the fluid through it begins with the velocity c = 2v/ ghf, and if we catch the stream by an equally large sur- face directed normally against it, then the pressure D upon this is again equally as great as before upon the bottom of the vessel, namely, yh. From this we have J)=yh = fgC For the density of the air above adopted its specific weight is 0.001223; therefore a cubic inch weighs 0.001495 loth, and g is equal to 187.0, if the semi-acceleration due to gravity is expressed in inches. From this we have these results: D =0.000001992 = 1.992 millionths of a loth. " The density is that of air at 15° C. aucl 28 Paris inches or 757.96inm under gravity at Berlin (52° 30), but strictly speaking the pressure should be stated in standard measure as 758.47""" under gravity at 45° and sea level. t g is the height fallen through in 1 second, or one-half the acceleration due to gravity. PAPER BY PROF. HAGEN. 29 As the first term of the above value of Jc comes out 2.264 or larger than this by nearly 14 per cent., the stronger resistance deduced from the observations is explained by the rarefaction of the air occurring at the rear of the disk, which rarefaction in the case of an assumed out- flow into empty space does not take place. Although the present investigation is confined only to those posi- tions in which the disks are turned normal to the direction of their motion, still it was important to be convinced that slight and unavoid- able deviations from this normal position had no important influence. The pins by means of which the disks were fastened to the arms were directed radial' y towards the axis of rotation. Thus the disks could be given any desired inclination to the direction of their motion. One such experience however showed this arrangement to be en- tirely unallowable in the observations, in that the simple relation between the resistance and the velocity of the disk completely disap- peared. The reason for this irregularity is apparent. According as the two disks were inclined downwards or upwards they were pressed up or down by the impinging air, and by so much the more the greater their velocity was. The arms with the inclined disks and with the axis of rotation therefore pressed variably upon the steel point on which the axis rested, and accordingly the screw threads on the axis were varia- bly pressed up or down, whereby the friction each time experienced an important change. When however I inclined one disk upwards and the opposite disk downwards, the axis was pressed to one side, and by so much the more, the greater the velocity was. In order not to change the simple arrangements for fastening the disks, I provided the two 5-inch square disks with roof shaped piece, in addition, so that in front of the lower half of the disk the inclined plane was turned upwards, and in front of the lower half an equal plane with the same inclination was turned downwards. Each of the two disks thus changed was thus both raised and depressed by equal forces for all velocities, so that the injurious effect upon the axis of rotation dis- appeared. • A complete series of observations (wherein both at the beginning and at the end the arms were set in motion without disks in order to deter- mine the resistance) gave — (a) When the roof surface was inclined 40° to the vertical or to the plane disk, r = 83.92. (b) For an inclination of 20° to the vertical, r = 101.16. (c) And for the plane disk itself, therefore, after removing the addi- tions r = 110.93. 30 THE MECHANICS OF THE EARTH'S ATMOSPHERE. If we divide these values by the cosines of 40, 20, and 0 degrees, respectively, there results 109.55, 107.65, and 110.93. The resistances are therefore in accordance with the ordinary as- sumption, proportional to the cosine of the inclination. In case the plane of the plane disk does not include the axis of rota- tion, we should also have to consider the diminution of the surface opposed to the impinging air in consequence of the projection upon the direction of motion, and for both reasons the resistance diminishes in the ratio of the square of the cosine of the deviation. Since the disks were always adjusted by the plumb line, therefore an error of 2 degrees, by which the resistance would only be diminished by its thousandth part, could not easily remain unnoticed. Finally, it still remains to be investigated whether the nature of the surface of the disks, according as they were smooth or rough, had any influence on the resistance. To this end I took two disks, each of which was covered on one side with very smooth paper but on the other with very coarse sandpaper. I allowed these to run with various velocities, exposing each time first the smooth and then the rough side to the impinging air. In both cases the times in which the index described 10 inches remained very nearly the same. The differences were very irregular, and not larger than occurred in repeated experiments with equal pairs of disks. Hence the nature of the surface of a plane disk has no iufluence on the resistance of the air when the surfaces are normal to the direction of motion. II. ON THE INTEGRALS OF THE HYDRO-DYNAMIC EQUATIONS THAT REPRESENT VORTEX-MOTIONS,* By Prof. Hermann von Hki,mholtz. Hitherto the integrals of the hydro-dynamic equations have been sought almost exclusively under the assumption that the rectangular components of the velocity of every particle of liquid can be put equal to the differential quotients in the correspoudiug directions of a certain definite function that we will call the velocity potential. On the one hand Lagranget had proven that this assumption is al- lowable whenever the movement of the mass of water has arisen and is maintained under the influence of forces that can be expressed as the differential quotients of a force potential, and even that the influ- ence of moving solid bodies that come in contact with the liquid do not affect the applicability of the assumption. Since now most of the forces of nature that are easily expressed mathematically can be presented as the differential quotients of a force potential, therefore also by far the majority of the cases of fluid motion that are treated mathematically fall into the category of those for which a velocity potential exists. On the other hand, even Euler J had called attention to the fact that there are cases of fluid motion where no velocity potential exists; e. g., the rotation of a fluid with equal angular velocities in all its parts about an axis. The magnetic forces that act upon a fluid permeated by electric currents, and especially the friction of fluid particles on each other and on solid bodies, belong to the forces that can give rise to such forms of motion. The influence of friction on fluids could not hitherto be mathe- matically defined, and yet it is very large in all cases where we are not treating of infinitely small vibrations, and causes the most important deviations between theory and nature. The difficulty of defining this influence and of finding methods for its measurement certainly lay * Crelle's Journal fur die reive und angewandte Mathematik, 1858, vol. lv, p. 25- 85. Helmholtz, Wissenschaftliche Abhandlungen, 1882, vol. I, pp. 101-134. London, Edinburgh, and Duhlin Philosophical Magazine, June, 1867 (4), xxin, pp. 485-510 t Me'canique Analytique, Paris, 1815, vol. II, p. 304. X Histoire de VAcade'mie des Sciences de Berlin, auuo 1755, p. 292. 31 32 THE MECHANICS OF THE EARTH'S ATMOSPHERE. mostly in the fact that we had no idea of the forms of motion that fric- tion produces in the fluid. Therefore in this respect an investigation of those forms of motion in which no velocity potential exists seems to me to be of importance. The following investigation will now show that in those cases in which a velocity potential does exist the smallest particles of liquid have no motion of rotation, but that when no velocity potential exists then a part at least of the liquid particles are in the act of rotation. By vortex lines (Wirbellinien) I designate lines that are so drawn through the mass of liquid that their directions everywhere coincide with the direction of the instantaneous axis of rotation of the liquid particles at that point of the line. By vortex filaments (WirbelfMen) I designate the portion of the mass of liquid that is cut out when we construct the corresponding vortex lines passing through every point of the circumference of an infinitely small element of the surface. The following investigation shows that when a force potential exists for all the forces that act upon the fluid then : (1) No particle of liquid acquires rotation that was not in rotation from the beginning. (2) The particles of liquid that at any moment belong to the same vortex line remain belonging to the same vortex line, even although they have a motion of translation. (3) The product of the sectional area by the velocity of rotation of an infinitely slender vortex filament is constant along the whole length of the filament and also retains the same value during the translatory motion of the filament. Therefore the vortex filaments must return i nto themselves within the liquid or can only have their ends at the bounda- ries of the fluid. This last proposition makes it possible to determine the velocities of rotation when the form of a particular vortex filament is given at dif- ferent moments of time. Further we solve the problem to determine the velocity of the particles of liquid for a given moment of time when the velocities of rotation are given for this moment, but in the solution there remains undetermined one arbitrary function that must be util- ized to satisfy the boundary conditions. This last problem leads to a remarkable analogy between the vortex motions of liquids and the electromagnetic actions of electric currents. When in a simply connected space* filled with moving liquid a ve- locity potential exists, the velocities of the liquid particles are equal to and in the same direction as the forces that a certain distribution of * I use this expression (einfach zusammenhiingenden Raume) in the same sense in which Riemann (Journal fiir die reine und angewandte Mathematik, 1857, Liv, p. 108) speaks of simple and multiple-connected surfaces. A space that is Ji-times connected is therefore one such that n — 1 but not more intersecting surfaces can pass through it without cutting the space into two completely separate portions. A ring is there- tore in this Bense a doubly-connected space. The intersecting surfaces must be com- pletely surrounded by the lines in which they cut the surface of the space. PAPER BY PROF. HELMHOLTZ. 33 magnetic masses on the surface of the space would exert upon a mag- netic particle in the interior. On the other hand, when vortex threads exist in any such space the velocities of the liquid particles are equal to the forces exerted upon a magnetic particle by a closed electric current that flows partly through the vortex filaments in the interior of the mass and partly in the bound- ary surface, and whose intensity is proportional to the product of the sectional area of the vortex filament by its velocity of rotation. I shall therefore in the following lines often allow myself to hypoth- ecate the presence of magnetic masses or of electric currents, simply in order thereby to obtain shorter and more perspicuous expressions for the nature of functions that are just the same functions of the co- ordinates as the potential functions, or the attractive forces for a mag- netic particle, are of the magnetic masses or electric currents. By these propositions the forms of motion concealed in that class of integrals of the hydro-dynamic equations not hitherto treated of be- come accessible at least to the imagination even although it be possible to execute the complete integration only in a few of the simplest cases where only oue or two rectilinear or circular vortex filaments are pres- ent in masses of liquid that are either unlimited or partially bounded by one infinite plane. It can be demonstrated that rectilinear parallel vortex filaments in a mass of water that is bounded only by planes perpendicular to such filaments, rotate about their common center of gravity, when in the determination of this center we consider the velocity of rotation as equivalent to the density of a mass. In this rotation the location of the center of gravity remains unchanged. On the other hand, for cir- cular vortex filaments, all stauding perpendicular to a common axis, the center of gravity of their cross section advances parallel to the axis. I. DEFINITION OF ROTATION. At a point within a liquid whose position is defined by the rectangular coordinates x, y, z, and at the time f, let the pressure be p, the three com- ponents of the velocity u. v, w, the three components of the external forces acting on the unit mass of the liquid X, Y, Z, and h be the den- sity whose changes can be considered as negligible; the established equations of motion for an interior point of the fluid are : _Upjm su du J* 1 A hdx Jt^ dx M dz YJL®^ dv iv dv x hdy dt dx W dz I z hdz tf & W & • • (1) 80A- 34 THE MECHANICS OF THE EARTH'S ATMOSPHERE. Hitherto, almost exclusively, only those cases have been treated where not only the forces X, Y, Z, have a potential V so that they can be expressed in the form, x=jyY=jyz=dY (ia) dx' dy' w bat also where a velocity potential q> can be found so that U=WV = W,W = W . ' (16) The problem is thereby greatly simplified since the first three of equations (1) give a common integral equation from which to find p after we have determined

that satisfies this last differential equation within a simply connected space,* can be expressed as the potential of a definite dis- tribution of magnetic masses on the boundary surface of the space as I have stated already in the introduction. In order that we may be able to make the substitution required in the equation (16) we must have ^-^=0, * *?=0, f-f=0, (1a) dy dx dz dy dx as ' In order to understand the mechanical significance of these last three conditions, we may imagine the change that any infinitely small volume of water experiences in the elementary time dt to be com- pounded of three different motions : (1) a motion of transference of the whole through space: (2) an expansion or contraction of the particle along the axis of dilatation, whereby every rectangular parallelopipe- don of water whose sides are parallel to the principal axis of dilata- tion remains rectangular while its sides change their lengths but re- main parallel to their original directions : (3) a rotation about some temporary axis of rotation having any given direction, which rotation can by a well-known proposition be always considered as theresultaut of three rotations about the three coordinate axes. *In manifold-connected-spaces cj> can have several values, but for such many val- ued functions as satisfy the above differential equations the fundamental proposition of Green's theory of electricity no longer holds good (see Crelle Journal, xliv, p. 360,or "The Mathematical Papers of the late George Green"), and therefore fail also a greater part of the propositions resulting from this which Gauss aud Green have demonstrated for the magnetic potential functions, which functions are in their very nature always uni-valued. PAPER BY PROF. HELMHOLTZ. 35 If the conditions (lc) are satisfied at the point whose coordinates are j, l), 5, and if we designate the values, of w, v, w, and their dif- ferential quotients as follows: dx dy dz dy dz dx ' We obtain for the point whose coordinates x, y, z, differ differentially from 5, \), 5 : u=A+a («-5)+r (y-W+/(*-a), v=B+y (*-$)+& (y-D) + a («-a)i «?=£+/? (#-£)+ a (y-t;)+c (s-j), or when we put : +J a(*-s)*+i&(y-W*+4o(*-a)a +a(y-l)) («-d) + /? (»-5) (»-i) +y-(*-5j (y-D)i there results : u= — —j u= — -, tc = — — dx dy dz It is well known that by a proper selection of another system of rec- tangular coordinates Xi, yu 2X, whose origin is at the point 5, 1), 5, the ex- pression for 95 can be brought into the form : 93=^1 Xx+Bi 2/1+C1 Zx+% ax a?i2+^6i yi2+£ cx ^2 where the component velocities Wi, »i, w?i, along these new coordinate axes have the values : Mi=A1+aia?i, »i=jBi+&i-yii «*i=Cri+Ci«1. The velocity Mj parallel to the axis of ^ is therefore alike for all liquid particles that have the same value of a?1? therefore particles that at the beginning of the elementary time dt lie in a plane parallel to that of yi Zi are also still in tbat plane at the end of the elementary time dt. This same proposition is true for the planes X\ yx and xx Z\. There- fore when we imagine a parallelopipedon bounded by three planes parallel to the last named coordinate planes and infinitely near to them, the liquid particles inclosed therein still form at the end of the time dt a rectangular parallelopipedon whose surfaces are parallel to the same coordinate planes. Therefore the whole motion of such an indefinitely small parallelopipedon is, under the assumption expressed 36 THE MECHANICS OF THE EARTH'S ATMOSPHERE. in (lc) compounded only of a motion of translation in space and an ex- pansion and contraction of its edges and it has no rotation. We return now to the first system of coordinates, that of %,y, 0, and imagine added to the hitherto existing motion of the infinitely small mass of liquid surrounding the point 5, 1), j, a system of rotatory motions about axes that are parallel to those of a?, y, 0, aud that pass through the point 1;, 1), 5, and whose angular velocities of rotation may be £, 77, C, thus then the component velocities parallel to the coordinate axes of x, y, 0, as resulting from such rotations are respectively : Parallel to x\ o. -(s-5) 7> Parallel to y : o, -(#-£) C, Parallel to 0: o. Therefore the velocities of the particles whose coordinates are x, y, z, become : u=A + ct(x - E)+( r + 0 (2/ - i)) + (/5 - 7) (*-i)» m,= C+ {0+t?) (a?- J) + («-£) (y-t))+c(0-j), whence by differentiation there results : J<5« 3«J ^M? dz'W ?y ?x fc,_J (2) Therefore the quantities on the left baud side, which according to equation (lc) must be equal to zero in order that a velocity potential may exist, are equal to double the velocity of rotation about the three coordinate axes of the liquid particles uuder consideration. The exist- ence of a velocity potential excludes the existence of a rotary motion of the particles of liquid. As a further characteristic peculiarity of fluid motions that have a velocity potential, it may be further stated that in a simply-connected space 8, entirely inclosed within rigid walls and wholly filled with fluid, no such motion can occur ; for when we indicate by n the nor- mal directed inwards to the surface of such space then the component velocity l directed perpendicular to the wall must be everywhere PAPER BY PROF. HELMHOLTZ. 37 equal to zero. Therefore, according to the well-known Green's theo- rem,* where, on the left hand, the integration is to be extended over the whole of the volume 8, but on the right hand over the whole surface 8 whose elementary surface is designated by dco. If, nowf ^ is to be equal to zero for the whole surface, then the integral on the left hand must also be zero, which can only be true when for the whole volume 8 dcp _ d _0 dx ?y d*~ ' that is to say, when there exists no motion whatever of the liquid. Every motion within a simply connected space of a limited mass of fluid that has a velocity potential is therefore necessarily connected with a motion of the surface of the fluid. If this motion of the surface, i. , and fcp ?2qp _ in the interior of the space 8. and also the condition for the surface of 8, where ip indicates the value of —-- deduced from the assumed motion of the surface, then would the function ((Pz — cp,,) also satisfy the first condition for the interior of the space S, but for the surface this function would give 3{ d2P + $p + dz* tfP = 0, (5) 3 dx? t dy2 T dz2 The method of integrating these last equations is well known, L, M, N are the potential functions of imaginary maguetic masses distributed through the space Si with the densities ~- ^5 Zji • P is the poten- 2 n' 2 7t 2 7t7 tial function for masses that lie outside of the region 8. If we indicate by r the distance from the point x, y, z to the point whose coordinates are a, 6, c; and by %a , tja , Ca the values of £, ?;, C at the point a, b, c, then L = M = N = - (5a) C, aa di tfc, 5 where the integration is extended over the space Si and - da db dc, P = where Jc is an arbitrary function of a, b, c and the integration is to be extended over the exterior space Si, that includes the region 8. The PAPER BY PROF, HELMHOLTZ. 43 arbitrary function A- must be so determined that the boundary con- ditions are satisfied, a problem whose difficulty is similar to those [difficulties that are met with in problems] on the distribution of elec- tricity and magnetism. That the values of u, v, and w, given in equation (4), satisfy the condition (14), is seen at once by differentiation and by considering the fourth of equations (5). Further, we fiud by differentiation of equations (4), and considering the first three of equations (5) that : dz " " dy ' " dx L dx dy dz J dx dz v dy L dx "*" dy dz J dy dz dz L dx + dy dz J The equations (2) are also equally satisfied when it can be shown that throughout the whole region Si we have ^ + £^4.^ = 0 dx ~*~ dy 'dz That this is the case results from the equations (5a) (56) d_L dx 2tz [Za(x-a)dadbd^ t.' «.' t' or after partial integration dh _ 1 dx M dy dz' ~2tT M 2tt -a dbdc Z7T r 3s /==0 ■■••••■ (2a) (26) 44 THE MECHANICS OF THE EARTH'S ATMOSPHERE. therefore both integrals are equal to zero and the equation (5b) as well as the equations (2) are satisfied. The equations (4) and (5) or (5a) are thus true integrals of the equations (14) and (2). The analogy mentioned in the introduction between the action at a distance of vortex filaments and the electromagnetic action at a dis- tance of conducting wires, which analogy affords a very good means of making visible the form of the vortex motiou, results from this proposi- tion. When we substitute in the equation (4) the values of L, M, N, from the equation (5a) and designate by An, Av, Aw the infinitely small portions of the velocities u, v and w in the integral which depend on the material elements da, db, dc and designate their resultant by Ap, we obtain 1 (y-b)ta-{z-c)Vada 2n r3 ± («-o)g.-(*-fi)C. da db d = 1 (x-a)th-(y-b)Sada db ^ 2n rs From these equations it follows that, Au(x— a) + Av(y— b) + Aw(z— c)=0, that is to say, Ap, the resultant of An, Av, Aw, is at right angles to r. Further, ZaAu+yaAv+CaAw=0, that is to say, this same resultant, Ap, also makes a right angle with the resulting axis of rotation at the point a, b, c. Finally, da db do Ap= y/(Auf-\-{Avf + (Awf=-^ — 2~ °"sm y» where a is the resultant of [the elementary velocities of rotation] B,a, ?/„, Ca, and ?' is the angle between this resultant and r, as determined by the equation, ffr cos v=(x-a)£a+(y-b)i/a+(s-c)Za Therefore every rotating particle of liquid a causes in every other particle b of the same mass of liquid a velocity that is directed perpen- dicularly to the plane passing through the axis of rotation of the particles a and b. The magnitude of this velocity is directly proportional to the volume of 'A, to its velocity of rotation, and to the sine of the angle between the line ab and the axis of rotation, and inversely proportional to the square of the distance of the two particles. The force that an electric current, moviug parallel to the axis of rota- tion at the point a, would exert upon a magnetic particle at b, follows exactly the same law as above. The mathematical relationship of both classes of natural phenomena PAPER BY PROF. HELMHOLTZ. 45 consists in the fact that in the case of liquid vortices there exists in those parts of the liquid that have no rotation a velocity potential qj, which satisfies the equation : tf

_n cic2"1" dy2 d*2 ' which equation fails to hold good only within the vortex filaments them- selves. But when we imagine the vortex filaments as closed, either within or without the mass of liquid, then the region in which the above differencial equation for cp holds good is a manifold-connected space, for it remains still connected when we imagine intersecting planes passing through it, each of which is completely bounded by a vortex filament. In such manifold-connected spaces a function cp that satisfies the above differential equation becomes many-valued, and it must be many- valued if it is to represent re-entering currents : for since the velocities [u, v, w,] of the liquid particles outside of the vortex fila- ments are proportional to the [partial] differential coefficients of cp [with reference to x, y, z], therefore, following the liquid particle in its motion one would find the values of cp steadily increasing. Therefore, if the current returns into itself, and if one by following it comes finally back to the place where he before was, he will find for this place a second value of cp larger than before. Since we can repeat this process in- definitely therefore for every point of such a manifold connected space, there must be an infinite number of different values of cp, which differ from each other by equal differences, like the different values of tang {y which is such a many-valued function as satisfies the above differ- ential equation. The electro-magnetic effects of a closed electric current have relations similar to the preceding. The current acts at a distance as would a certain distribution of magnetic masses over a surface bounded by the conductor. Therefore, outside of such a current the forces that it ex- erts upon a magnetic particle can be considered as the differential quotients of a potential function V which satisfies the equation -|2T7 V2T" ff „ }x* T dy2 T dz2 Here also the space that surrounds the closed conductor and through- out which this equation holds good, is manifold-connected, and V is many-valued. Therefore in the vortex motions of liquids, as in the electro-magnetic actions, velocities or forces respectively external to the space occupied by the vortex filaments or the electric currents depend upon many- valued potential functions which moreover satisfy the general differ- ential equations of the magnetic potential function, while on the other hand within the space occupied by the vortex filaments or electric cur- 46 THE MECHANICS OF THE EARTH'S ATMOSPHERE. rents, instead of potential functions which can not exist here, there occur other common functions such as are expressed in the equations (4), (5), and {5a). On the other hand, for simple progressive movements of liquids and for the magnetic forces, just as for gravitation, for electric attractions and for the steady flow of electricity and heat, we have to do with single-valued potential functions. The integrals of the hydro-dynamic equations, for which a single-val- ued velocity potential exists, we can call integrals of the first class. Those on the other hand for which there are rotations in one portion of the liquid particles, and correspondingly a many-valued velocity potential for the non-rotating particles we call integrals of the second class. It can happen that in the latter case only such portions of the space are to be considered in the problem as contain no rotatory particles of liquid, e. g., in the case of the movements of liquid in a ring-shaped vessel, where a vortex filament can be imagined traversing the axis of the vessel, and where notwithstanding this the problem belongs to those that can be resolved by means of the assumption of a velocity potential. In the hydro-dynamic integrals of the first class the velocities of the liquid particles have the same direction as, and are proportional to the forces that would be produced by a certain distribution of the magnetic masses outside of the liquid acting on a magnetic particle at the loca- tion of the particle of liquid. In the hydro dynamic integrals of the second class the velocities of the liquid particles have the same direction as, and are proportional to forces acting on the magnetic particle such as would be produced by a closed electric current flowing through the vortex filament and having a density proportional to the velocity of rotation of this fila- ment, combined with the action of magnetic masses entirely outside the liquid. The electric currents within the liquid would How forward with the respective vortex filaments, and must retain a constant intensity. The adopted distribution of magnetic masses outside of the liquid or on its surface must be so defined that the boundary conditions are satisfied. Every magnetic mass can also, as is well known, be replaced by electric currents. Therefore instead of introducing into the values u, v, and w, the potential function P of an exterior mass 1c, we can obtain an equally general solution if we give to the quantities £, ?/, and £ external to the fluid or even only on its surface, such arbitrary values that only closed current filaments arise, and then extend the integration of the equa- tions (oa) over the whole region for which £, //, and C differ from zero. IV. VORTEX SHEETS AND THE ENERGY OF THE VORTEX FILAMENTS. In the hydro-dynamic integrals of the first class it suffices, as I have already shown, to kuow the movements of the surface; the movement in the interior is then entirely determined. For the integrals of the second class, on the other hand, the movements of the vortex filaments PAPER BY PROF. HELMHOLTZ. 47 located within the fluid are to be determined, taking account of their mutual influences and of the boundary conditions whereby the problem becomes much more complicated. However, for certain simple cases even this problem can be solved, especially in those cases where the rotations of the liquid particles take place only on certain surfaces or lines and the forms of these surfaces and lines remain unchanged dur- ing the translatory motions. The properties of surfaces that adjoin an indefinitely thin layer of rotating fluid particles are easily seen from the equations (5a). When {p -MJ+ h liq2) qcosO (66) dt When the liquid mass is entirely inclosed within rigid walls then at all points of the surface q cos 6 must be zero, therefore then will d I\ — = 0, or K become constant. dt If we imagine this rigid wall to be at an infinite distance from the origin of coordinates and all vortex filaments that may be present to be at an infinite distance from this origin, then will the potential functions L, M, N [of imaginary maguetic matter], whose masses $, rj, C, or densities ~—, — rJ-. ^^ |, each and all are equal to zero, diminish 2/T 2n 2,71 \ at the infinite distance 91 as — and the velocities [which are the par- tial differential coefficients of L, M, N], will vary as — , but the element- al3 ary surface doo, if it is always to correspond to the same solid angle at the origin of the coordinates, will increase as W. The first integral in the expression for E, equation (6«), which is extended over the surface of the liquid mass, will diminish as and therefore will be zero for 9t 1 ' >H3 equal to infinity. The value of K then reduces to the expression, K= -lifff(LZ + Mv+NQdxdydz (6c) and this quantity is unchanged during the movement. V. RECTILINEAR PARALLEL VORTEX FILAMENTS. We will first investigate the case where only rectilinear vortex threads exist parallel to the axis of g, either within a liquid mass of infinite extent or which comes to the same thing, in one that is bounded by two infinite planes perpendicular to the vortex filaments. In this case 80 A 4 50 THE MECHANICS OF THE EARTH'S ATMOSPHERE. all motions take place in planes that are perpendicular to the axis of z and are precisely the same in all such planes. Therefore we put _ yu __ dv_= ?p = ?y = 0t W ~~ dz dz ' dz ~ A- Then equations (2) reduce to c n o or - <>u dv £=0,77=0,2:-—-^, the equations (3) become ^ = 0 6t Therefore the vortex threads, in so far as they have constant sectional areas, have also constant velocities of rotation. The equations (4) reduce to, In this I have put P = 0 in accord with the remark in Sect. ill. Therefore the equation of the streamline is N = constant. In this case N is the potential function of infinitely long lines ; this function itself is infinitely large, but its differential coefficients are finite. Let a and b be the coordinates of a vortex filament the area of whose cross-section is da db, then is <)iV Qdadb x—a '^x~ n r2 tW Cdadb y—b v ~~ d% 7t r2 Hence it follows that the resultant velocity q is perpendicular to the r drawn perpendicular to the vortex filament and its value is Cdadb If within a liquid mass of indefinite extent in the direction x aud y we have many vortex filaments whose coordinates are respectively xx, yx ; o?2, y2, etc., while the products of rotatory velocity by the sectional area are for each distinguished by m:, m2, etc., and if we form the sums, U = mi Ux + m2 u2 -f m3 1*3, etc., V = mx Vi + m2v2 + m3 v3, etc., then these sums are each equal to zero, because that part of each sum that is due to the action of the second vortex filament on the first is counterbalanced by the action of the first vortex filament on the sec- ond. That is to say, the two effects are, respectively, w?2 Xi—x2 . wii x2—Xi mx • • — =— and m2 • — • — 5— > 1 71 r2 ' n r2 PAPER BY PROF. HELMHOLTZ. 51 and so on through all the other pairs of sums. Now U is the velocity in the direction of x, of the center of gravity of the masses mh m2, etc., multiplied by the sum of these masses; similarly V is the velocity taken in the directiou of y. Both velocities are therefore zero, unless the sum of the masses is zero, in which case there is no center of grav- ity at all. Therefore the center of gravity of the vortex filaments remains unchanged during their motion, aud since this proposition holds good for every distribution of the vortex filaments, therefore we may also apply it to the individual filaments of infinitely small cross section. Hence result the following consequences : (1) If we have but one individual rectilinear vortex filament of infi- nitely small cross-section within a liquid mass of infinite extent in all directions perpendicular to the filament, then the movement of the par- ticles of water at a finite distance from the filament depends only on the product £ da db = m, or the velocity of rotation multiplied by the area of the cross-section, and not on the form of the cross-section. The liquid particles rotate about the filament with the tangential velocity 7)1 — where r denotes the distance from the center of gravity of the vor- tex filament. The location of the center of gravity, the velocity of rotation, the area of the cross section, aud therefore also the quantity m remains unchanged although the form of the infinitely small cross- section may change. (2) If we have two rectilinear vortex filaments of infinitely small cross- sections and an indefinitely large liquid mass, each will drive the other in a direction that is perpendicular to the line joining them together. The length of this connecting line will not be changed thereby; there- fore both will revolve about their common center of gravity, remain- ing at equal constant distances therefrom. If the rotatory velocity is in the same direction in the two filaments and therefore has the same sign, then their center of gravity must lie between them. If the rota- tions are mutually opposed to each other and therefore of opposite signs, then their center of gravity lies in the prolongation of the line connect- ing the filaments. If the products of the rotatory velocity by the cross section are numerically equal for the two but of opposite signs, thereby causing the center of gravity to be at an infinite distance, then both filaments advance with equal velocity and in the same direction per- pendicular to their connecting line. The case where a vortex filament of infinitely small section lies close to an infinitely extended plane surface parallel to it can be reduced to this last case. The boundary condition for the movement of the liquid along a plane (i. e., that the motion must be parallel to this plane) is satisfied when we imagine a second vortex filament, which is as the re- flected image of the first, introduced on the other side of the plane. Hence it follows that the vortex filament within the liquid mass ad- 52 THE MECHANICS OF THE EARTH'S ATMOSPHERE. vances parallel to tbe plane in the direction in which the liquid parti- cles, between it and the plane, themselves move, and with one-fourth of the velocity possessed by the particles that are at the foot of the per- pendicular drawn from the filament to the plane. The assumption of the infinitely small cross-section leads to no inad- missible results, because each individual filament exerts no force upon itself affecting its own progression, but is driven forwards only by the influence of the other filaments that may be present for by the action at the boundary]. But it is otherwise in the case of curved filaments. VI. CIRCULAR VORTEX FILAMENTS. In a liquid mass of indefinite extent let there be present only circu- lar filaments whose planes are perpendicular to the axis of z, and whose centers lie in this axis, so that all are symmetrical about this axis. Transform the coordinates by putting x = x c°s £ , a = g cos e, y = x sin €, b = g sin e, z = z, c = c. Agreeably to the assumption just made, the velocity of rotation a is only a function of x aua* 2, or of g and c, and the axis of rotation is every- where perpendicular to x (or g) and to the axis of z. Therefore the rec- tangular components of the rotation at this point whose coordinates are g, e, and c become B,= — 6 sin e, ?/=a cos e, C=0. In the equation (5a) we now have, r2=(z— c)2+x2+g2— 2xg cos (e-e) L=2n J J J ~r — g d9 de dc From the equations for L and M by multiplying by cos s and sin e and adding we obtain L sin e-M cos e = - 2~J J J ~ _L_ _J g dg d{e_e) f7c? L cos e+M sin *= hfjj'^f^ 9 dg d(e-e) dc, In both these integrals the angles e and e occur only in the connec- tion (e-e) and this quantity can therefore be considered as the variable under the sign of integration. In the second integral the terms that contain (e-e)=e balance those that contain/ («-.«) =2tc- e; therefore this integral is equal to zero. PAPER BY PROF. HELMHOLTZ. 53 Therefore if we put 1 f f f G cos e 9 d& de d° ZxJJJ V{z-c)*+x'+(f-2gXcose W tben will M cos e — L sin e = ip M sin a + L cos 6 = 0, or L = — ?/,' sin s, M = ip cos £. (7a) Let r denote the velocity in the direction of the radius j, and con- sider the fact that on account of the symmetrical position of the vortex ring in reference to the axis z the velocity must be zero in the direction of the circumference of the circle, we must have u=r cos f, v=r sin e and according to equations (4) Hence it follows that dip jtp i// dz* ?X^ x or r* = -if> «XJ-<§£. <7») 0" O A. Therefore the equation of the stream line is ip X — const. When we execute the integrations indicated in the value pf y, first for a vortex filament of infinitely small cross-sectiou, putting therein mx = a dg dc and indicating by ipm the part of ip depending thereon, we have ±9X wherein F and E indicate the complete elliptic integrals of the first and second order respectively for the modulus x. For brevity we put U=2(F-E)-hF, It where U is therefore a function of «, then is Ml lay ?U z~c If now a second vortex filament m exist at the point determined by X and 2, and if we let tx be the velocity in the direction of g that m communicates to the filament mu we then obtain the value of this ve- 54 THE MECHANICS OF THE EARTHS ATMOSPHERE. locity if iu the expression for r we substitute n, g, Xi c> z> mi in Place of r, x, 9, z, ci wi- , a In this process n and U remain unchanged and we obtain, mrx-\-/)»irig=0 (8) If now we determine the value of the velocity w parallel to the axis, caused by the vortex filament »*i whose coordinates are g and e, we find : n o o ,m, lg jj.mi l~dU * (z-c)2+g2-X2 w*=l^x +^vgxU Sx (g+W+(*-rcf If now we call w, the velocity at the locality of m, parallel to the axis of 2, which is caused by the vortex ring m whose coordinates are z and j, then in order to determine this, we only need to execute the interchange of appropriate coordinates and masses as above shown. Thus we find : 2mwx2-^miUhg2—,mTX!S—mlT1gc=—-^^JgxU. . . (8a) Sums similar to (8) and (8a) can be found for any number of vortex rings. For the nth of these rings I designate the product a dg dc by mn ; the components of the velocity that is communicated to this ring by all the other rings are rn and u\, in which however I provisionally omit the velocities that every vortex ring can communicate to itself. Fur- ther I call the radius of this ring pn and its distance from a surface perpendicular to the axis A, which two latter quantities agree with x and z as to direction, but, as belonging to this particular ring, they are functions of the time and not independent variables as are x aQd %• Finally let the value of ?/<, in so far as it depends on the other vortex rings, be tpn. By forming and adding the equations (8) and (8a) corre- sponding to each pair of vortex rings, there results 2 [mn pn rnJ=0. 2 [2 mH wn f?n—mn tn pn ln} = 2 [mn pv ipn]. So long as we have in these sums only a finite number of separate and infinitely slender vortex rings, we must understand by w, r, and if) only those parts of these quantities that are due to the presence of the other rings. But when we imagine an infinite number of such rings keeping the space continuously filled, then y> becomes the poten- tial fuuction of a continuous mass, w and r become partial differential coefficients of this potential function, and it is known* that both for such functions and for their differential coefficients, the portions of the function that depend upon the presence of matter within an infinitely small space surrounding a point for which the function is determined are infinitely small with respect to those portions that depend on finite masses at finite distances. * See Gauss, Allgemeine Theorie des Erdmagnetismiis in the Resaltate des magnetischen Vereins im Jahre, 1839, page 7, or the translation iu Taylor's Scientific Memoirs, vol. II. PAPER BY PROF. HELMHOLTZ. 55 Therefore if we change the sums into integrals we can understand by to, r, and ip the total values of these quantities that exist at the point in question, and can put dX dp dt ' dt To this end we replace the quantity m by the product adpdX, and the summations thus become couverted into the following integrals : f/GdpdX, and let B2 be the mean value of p2 for all the elementary masses, then /J'ffp.pdpdX=MB2, and, since this integral and the value of 2Ji do not vary with the time, it follows that B also remains unchanged during the motion of transla- tion. Therefore if there exists in the unlimited mass of liquid only one circular vortex filament of infinitely small sectional area, then its radius remains unchanged. According to equation (6c), the total living force in our case is K=-lifff{L$+Mrj)da db dc. =i — Urj,J'q6,p dp d_X de. = —InlifJ'^o-pdpdX. This also does not change with time. Furthermore, because a dp dX does not vary with time, therefore, %ffaf?X dp dX = 2/fapX % dp dX+ffap2 1 dX dp ; therefore if we indicate by I the value of X for the center of gravity of the vortex filament treated of in equation (9a), and multiply (9) by this I, and add the result to (9a), and substitute therein the equation last given, we obtain 7 T\ ^Lf/^2XdP<1X+r°ff0f){l-X)TtdpdX = '^h ' ' m 56 THE MECHANICS OF THE EARTH'S ATMOSPHERE. When the section of the vortex thread is infinitely small and € is an infinitely small quantity of the same order as (I— A) and the remain- ing linear dimensions of the section, but a dp d\ is finite, then ip and also E are ot the same order of infinitely large quantities as log s. For very small values of the distance v from the vortex ring we have K2 = l- V2 =Wl loo- A. 7T °8g In the value of K, ip is multiplied by p or g. If g is finite, and v of the same order as s, then K is of the same order as log s . Only when g is infinitely large of the order - will K be infinitely large of the order — log £. But in this case the circle becomes a straight line. On the other hand, if p, which is equal to-', is of the order -, then the sec- dt dz * ond integral will be finite, and for a finite value of p will be infinitely small with respect to K. In this case we can, in the first integral, substi- tute the constant I in place of X and obtain 2d(mRH)_ K dt 2rch or 2mm=c-J^t 2,7th Since 8ft and R are constant, I can only vary proportionally to the time. When §Jt is positive the motion of the liquid particles on the outer side of the ring is directed toward the side of positive £, but on the inner side of the ring toward the negative z. K, h, and R are by their nature always positive. Hence it follows that for a circular vortex filament of very small cross-section in an infinitely extended mass of liquid the center of grav- ity of a cross-section has a motion parallel to the axis of the vortex ring, which is of approximately constant and very large velocity, aud which is directed toward the same side as that toward which the liquid flows through the ring. Infiuitely slender vortex filaments of a finite radius will have infinitely large velocities of propagation. But if the radius of the vortex ring is infiuitely large of the order _ , then will (£) R2 be infinitely large with respect to Zl, aud I will be constant. The vortex filament which has thus transformed itself into a straight line will be stationary, as we had already previously found for rectilinear vortex filaments. PAPER BY PROF. HELMHOLTZ. 57 We can now in general see how two circular vortex threads having a common axis will behave with respect to each other, since each one independent of its own translatory motion also follows the movement of the liquid particles caused by the other filament. If they have the same direction of rotation, then they both advance in the same direc- tion, and at first the preceding one enlarges, then it advances more slowly while the following one diminishes and advances more rapidly; finally, if the progressive velocities are not too different, the second catches up with the first and passes through it. Then the same perform- ance is repeated by the one that is now in the rear so that the rings alternately pass through each other. If the vortex filaments have the same radii, but equal and opposite rotatory velocities, then they will approach each other and simultane- ously enlarge, so that finally when they have come very close together their movement towards each other grows continually feebler, while on the other baud the enlargement goes on with increasing rapidity. If the two vortex threads are perfectly symmetrical, then midway be- tween them the velocity of the liquid particles in the direction parallel to the axis is equal to z ero. Therefore one can imagine a rigid wall located here without disturbing the motion and thus obtain the case of a vortex ring that encounters a rigid wall. I remark further that we can easily study these movements of circular vortex rings in nature if we draw a half-immersed circular disk or the approximately semicircular end of a spoon rapidly for a short distance along the surface of a liquid and then quickly draw it out. There then remain in the liquid semi- vortex riugs whose axes lie in the free upper surface of the liquid. The free upper surface thus forms, for the liquid mass, a boundary plane that passes thr ough the axis whereby no im- portant change is made in the motions. The vortex rings advance, broaden when they encounter a screen, and are enlarged or diminished by the action of other vortex rings precisely as we have deduced from the theory. III. ON DISCONTINUOUS MOTIONS IN LIQUIDS.* By Prof. H. VON Helmholtz. It is well known that the hydro-dynamic equations give precisely the same partial differential equations for the interior of an incompressible fluid that is not subject to friction and whose particles have no mo- tion of rotation, as obtain for stationary currents of electricity or heat in conductors of uniform conductivity. One might therefore expect that for the same external form of the space traversed by the cur- rent and for the same boundary conditions the form of the current (ex- cept for differences depending on small incidental conditions), would be the same for liquids, for electricity, and for heat. In reality however in many cases there exist easily recognizable and very fundamental differences between the currents in a liquid aud the above mentioned imponderables. Such differences are especially notable when the currents flowing through an opening with sharp edges enter into a wider space. In such cases the stream lines of electricity radiate from the openiug outwards immediately towards all directions, while a flowing fluid, water as well as air, moves from the opening at first forward iu a compact stream which at a less or greater distance then ordinarily resolves itself into a whirl. The portions of the fluid iu the larger receiving vessel lying near the opening but outside the stream can, on the other hand, remain almost at perfect rest. Everyone is familiar with this mode of motion, especially as a current of air impregnated with smoke shows it very plainly. In fact the compressibility of the air does not come much into consideration in these processes, and with slight variations air sho'ws the same forms of motion as does water. On account of the great differences between the facts as observed aud the results of theoretical analysis as hitherto achieved the hydro- dynamic equations must necessarily appear to the physicist as a prac- * From the MounisJxrichic of the Royal Academy of Science, Berlin. 1868, April 2:?, pp. 215-228. Helmholtz JVhsenschaftliche Abhandlungen, vol. I, pp. 146-157. Ber- lin, 1832. 58 PAPER BY PROF. HELMHOLTZ. 59 tically very imperfect approximation to the reality. The cause of this might be suspected to lieiu the internal friction or viscosity of the fluid, although all forms of infreqent and sudden irregularities (with which certainly everyone has to contend who has instituted observations on the movements of fluids) can evidently never be explained as the eflect of the steadily and uniformly acting friction. The investigation of cases where periodical movements are excited by a continuous current of air, as, for example, in organ pipes, showed me that such an effect could only be produced by a discontinuous motion of the air, or at least by a hind of motion coming very near to it, and this has lead me to the discovery of a condition that must betaken into consideration in the integration of the hydro-dynamic equations, aud that, so far as I know, has been overlooked hitherto, whose considera- tion on the other hand, in those cases where the computation can be carried out, really gives, in fact, forms of motions such as those that are actually observed. This condition is due to the following circumstance : In the hydro-dynamic equations the velocity and the pressure of the flowing particles are treated as continuous functions of the coordinates. On the other hand, there is no reason in the nature of a liquid, if we consider it as perfectly fluid, therefore not subject to viscosity, why two contiguous layers of liquid should not glide past each other with defi- nite velocities. At least those properties of fluids that are considered in the hydro-dynamic equations, namely, the constancy of the mass in each element of space and the uniformity of pressure in all directions, evidently furnish no reasons why tangential velocities of finite differ- ence in magnitude should not exist on both sides of a surface located in the interior. On the other hand, the components of velocity and of pres- sure perpendicular to the surface must of course be equal on both sides of such a surface. I have already in my memoir on vortex motions called attention to the fact that such a case must occur when two moving masses of liquid previously separate and having different motions come to have their surfaces in contact. In that memoir I was led to the idea of such a surface of separation,* or vortex surface as I there called itf through the fact that I imagined a system of parallel vortex filaments arranged continuously over the surface whose mass was indefinitely small without losing their moment of rotation. Now, in a liquid at first quiet or in continuous motion a definite dif- ference in the movement of immediately adjoining particles of liquid can only be brought about through moving forces acting discontinu- ously. Among the external forces the only oue that can here come into consideration is impact. But in the interior of liquids there is also a cause present that can ['Ordinarily called surface of discontinuity or " a discontinuous surface " by English [t That is, an infinitely thin layer of parallel vortex filaments, the " vortex sheet " ol English writers.] 60 THE MECHANICS OF THE EARTH'S ATMOSPHERE. bring about discontinuity of motion— namely, the pressure, which can assume any positive value whatever while the density of the liquid will continuously vary therewith ; but as soon as the pressure passes the zero value and becomes negative, a discontinuous variation of the density occurs; the liquid is torn asunder. Now, the magnitude of the pressure (at any point) in a moving fluid depends on the velocity (at that point), and in incompressible fluids the diminution of pressure under otherwise similar circumstances is directly proportional to the living force of the moving particles of liquid. Therefore if the latter exceeds a certain limit the pressure must, in fact, become negative, and the liquid tears asunder. At such a place the accelerating force, which is proportional to the differential quotient of the pressure, is evidently discontinuous, and thus the con- dition is fulfilled which is necessary in order to bring about a discon- tinuous motion of the liquid. The movement of the liquid past auy such place can now take place only by the formation from that point onward of a surface of discontinuity. The velocity that will cause the tearing asunder of the liquid is that which the liquid would assume when it flows into empty space under the pressure that the liquid would have at rest at the point iu ques- tion. This is indeed a relatively considerable velocity; but it is to be remarked that if liquids flow continuously like electricity the velocity at every sharp edge around which the current bends must be infinitely great.* Thence it follows that at every geometrically perfect sharp edge past which liquids flow, even for the most moderate velocity of the rest of the liquid, it must be torn asunder and form a surface of discontinuity. On the other hand, for imperfectly somewhat rounded edges such phe- nomena first occur for certain larger velocities. Pointed protuber- ances on the surface of a canal through which a current flows will have similar effects. As concerns gases, the same circumstance occurs as with liquids, only with this difference, — that the living force of the motion of a particle is not directly proportional to the diminution of the pressure (p); but taking into consideration the cooling of the air by its expansion the living force is proportional to the diminution of pm, where m=l— — and y is the ratio of the specific heat at constant pressure to that for constant volume. For atmospheric air the exponent m has the value 0.291. Since this is positive and real, therefore pm, like p, for high values of the velocity can only diminish to zero and not become negative. It would be otherwise if gases simply followed the law of Mariotte and experienced no change of temperature. Then instead of pm the quan- tity log p would occur, which can become negative and infinite without *At the very small distance p from a sharp edge -whose surfaces meet each other 7T — (X at the angle a the velocities will be infinite, or as p — m, where m- 2n-a PAPER BY PROF. HELMHOLTZ. Q\ p being negative. Under this condition the tearing asunder of the mass of air would not be necessary. It is possible to convince one's self of the actual existence of such discontinuities when we allow a stream of air impregnated with smoke to issue from a round opening or a cylindrical tube with moderate velocity so'that no hissing occurs. Under favorable circumstances one obtains thin rays or jets of this kind of a few lines diameter and a length of many feet. Within the cylindrical surface the air is iii mo- tion with constant velocity, but outside it, on the other hand, in the immediate neighborhood of the jet it moves not at all or very slightly. One sees this very sharp separation clearly when we conduct a steadily flowiug cylindrical jet of air through the point of a flame, out of which it cuts a sharply defined piece, while the rest of the flame remains en- tirely undisturbed, and at most a very thin stratum of flame, which corresponds to the bouudary layer of the jet influenced by friction, is carried along a little way. As concerns the mathematical theory of this motion I have already given the boundary conditions for the existence of an interior surface of separation within the liquid. They consist in this that the pressures on both sides the surface must be equal and equally so the components of the velocity normal to the discontinuous surface. Since now the movement throughout the entire interior of a liquid whose particles have no motion of rotation is wholly determined when the motion of its entire exterior surface and its interior discontinuities are given, therefore in general for a liquid whose exterior boundary is fixed, it is only necessary to know the movement of the surfaces of separation and the variations of the discontinuity. Now such a discontinuous surface can be treated mathematically pre- cisely as if it were a vortex sheet, that is to say, as if it were continu- ously enveloped by vortex filaments of indefinitely small mass but finite moments of rotation. For each element of such a vortex sheet there is a direction for which the components of the tangential veloci- ties are equal. This gives at once the direction of the vortex filaments at the corresponding place. The moment of this filament is to be put proportional to the difference existiug between the components, taken perpendicular to it, of the tangential velocity on both sides of the surface. The existeuce of such vortex filaments in an ideal frictiouless liquid is a mathematical fiction that facilitates the integration. In a real liquid subject to friction, this fiction becomes at once a reality inasmuch as by the friction the boundary particles are set in rotation, and thus vortex filaments originate there having finite gradually increasing masses, while the discontinuity of the motion is thereby at the same time compensated. The motion of a vortex sheet and the vortex filaments lying in it is to be determined by the rules established in my Memoir on Vortex 62 THE MECHANICS OF THE EARTH'S ATMOSPHERE. Motions. The mathematical difficulties of this problem however can be overcome only in a few of the simpler cases. In many other cases, however, one can from the above given method of consideration of this matter at least draw conclusions as to the general nature of the varia- tions that occur. Especially is it to be mentioned that in accordance wifth the laws established for vortex motions, the vortex filaments and with them the vortex sheets in the interior of a frictionless liquid can neither originate nor disappear, but rather each vortex filament must retain perma- nently the same constant moment of rotation ; furthermore that the vortex filaments themselves advance along the vortex sheet with a velocity that is the mean of the two velocities existing on the two sides of the discontinuous surface. Thence it follows that a surface of dis- continuity can only elongate in the direction toivards which the stronger of the two currents that meet in it is directed. I have first sought to find examples of permanent discontinuous sur- faces in steady currents, for which the integration can be executed, in order thereby to prove whether the theory gives forms of currents that correspond to experience better than when we disregard the discon- tinuity of motion. If a surface of discontinuity that separates quiet and moving water from each other is to remain stationary, then along this surface the pressure within the moving layer must be the same as in the quiet layer, whence it follows that the tangential velocity of the particles of liquid must be constant throughout the whole extent of the surface; equally so must the deusity of the fictitious vortex filament be constant. The beginning and end of such a surface can only lie on the boundary of the inclosure or at infinity. Where the former alter- native is the case they must be tangent to the wall of the inclosure, assuming that the latter is continuously curved, because the compo- nent-velocity normal to the wall of the inclosure must be zero. Moreover the stationary forms of the discontinuous surface are dis- tinguished, as experiment and theory agree in showing, by a remarkably high degree of variability under the slightest perturbations, so that to a certain extent they behave similarly to bodies in unstable equili- brium. The astonishing sensitiveness to sound waves of a cylindrical jet of air impregnated with smoke has already been described by Tyn- dall ; I have confirmed this observation. This is evidently a peculiarity of surfaces of discontinuity that is of the greatest importance in oper- ating sonorous pipes. Theory allows us to recognize that in general wherever an irregularity is formed on the surface of an otherwise stationary jet, this must lead to a progressive spiral unrolling of the corresponding portion of the sur- face, which portion, moreover, slides along the jet. This tendency to- wards spiral unrolling at every disturbance is moreover easy to see in the observed jets. According to the theory a prismatic or cylindrical jet can be indefinitely long. In fact however such an one can not be PAPER BY PROF. HELMHOLTZ. 63 formed, because in an element so easily moved as is the air small dis- turbances can never be entirely avoided. It is easy to see that such an endless cylindrical jet, issuing from a tube of corresponding section into a quiet exterior fluid and everywhere containing fluid that is moving with uniform velocity parallel to its axis, corresponds to the requirements of the "steady condition." I will here further sketch only the mathematical treatment of a case of the opposite kind, where the current from a wide space flows into a narrow canal, in order thereby also at the same time to give an example of a method by which some problems in the theory of potential func- tions can be solved that hitherto have been attended by difficulties. I confine myself to the case where the motion is steady and dependent only upon two rectaugular coordinates, x and y ; where moreover no rotating particles are present iu the frictionless fluid at the beginning, and where none such can be subsequently formed. If we indicate by u the component parallel to x of the velocity of the fluid particle at the point (xy) and by v the velocity parallel to y, then, as is well known, two functions of x and y can be found such that __ ijp _ <>£ } u~l)x- Jy I m (l = dP = d± j v * ~~ dy dx i By these equations the conditions are also directly fulfilled that in the interior of the fluid the mass shall remain constant in each element of space, viz : For a constant density, h, and when the potential of the external forces is indicated by v, the pressure in the interior is given by the equation — ^•-*K^®>»[$'+<#] • • • • a.) The curves ip = constant are the stream lines ot the fluid, and the curves cp = constant are orthogonal to them. The latter are the equi-potential curves when electricity, or the equal temperature curves when heat, flows iu steady currents in conductors of uniform conductivity. From the equation (1) it follows as an iutegral_equation that the quantity

i is a function of x + yi, where i = y/-l. The solutions hitherto found generally express

y = A>p + Ae sin tp for the value tp = ± n we have y constant and x = A

varies from -co to + a> the value of x changes at the same time from — co to — A, and then again back to — oo. The stream lines x=±7t correspond thus to a current along two straight walls, for which y=-\zA7t and x varies between — go and — A. Therefore when we consider >p as the expression of the stream curve the equation (2) corresponds to the flow out into endless space from a canal bounded by two parallel planes. On the border of the canal however where x = — A and y = ± A n and where further, cp = 0 and tp= ± 7t, we have (f;)'+ as1-* therefore fi?Y+ ( ^\2= 2 CO Electricity and heat flow in this manner, but liquids must tear asunder. If from the border of the canal there extend stationary dividing dis- continuous lines that are of course prolongations of the stream lines tp= ± it that follow along the wall and if outside of these discontinuous lines that limit the flowing fluid there is perfect quiet, then must the pressure be the same on both sides of these dividing lines. That is to say, along that portion of the line tp = ± n which corresponds to the free dividing line, in accordance with the equatiou {lb), we must have In order now, in the solution of this modified problem, to retain the fundamental idea of the motion expressed in equation (2), we will add PAPER BY PROF. HELMHOLTZ. 65 to the above expression ofx + y i still another term g + t% which is also always a function of q> + //> «, we have then x = Acp + Ae cos ip + a I 4 f • (3a) y = A ?/' + A e sin //< + r ) and must determine a + ri so that along the free portion of the discon- tinuous surface where ip= ± n we shall have This condition is fulfilled if we make - — =0 or a = Constant ....... (3&) d

log 2, then all these values become purely imaginary, there- fore 6 = 0, while 7- has the value given above in equation (3c). This portion of the lines tp=±7t therefore corresponds to the free portion of the jet. If *? dju.e) d(v.s) ftw.j) m jt dx T dy ~r dz 1 dp d« , ?U JU JU i tfu ?ll ?U * -TTx=Tt+u^+vTy+wTz-k\^+W+W> I _^i_ ) du dv dw i 3 dx \ 'dx+dy+ lz ) (la) To these are still to be added the two equations that are deduced from the latter equation (la) by interchanging x and u with y and v or with z and ic. When now for another fluid the velocities are designated by U, V, W, the pressure by P, the coordinates by X, T, Z, the time by T, the density by E, the viscosity constant by K, and if we introduce three constants q, r, and n, and put K=qk (2) E=rs (2a) U~nu X=-x n V=nv . Y= ii nJ W=niv Z=~z n P= n2rp + constant. T= %t then the quantities designated by these capital letters will also fulfill the above differential equations. If we substitute these in those equa- tions, the result, E, is as if all the terms of equation (1) were multiplied by PAPER BY PROF. HELMHOLTZ. 09 the factor — and all the terms of equation (la) by the factor— . Of the constants q, r, n, two are determined through the equations (2) aud (2a) by the nature of the fluid, but the third, », is arbitrary so far as the conditions hitherto considered come into consideration. If the fluid is incompressible, then € is to be considered as a constant and tlr==0' aiul the above equations then suffice to determine the motion in the interior. If the fluid is compressible, we can put p=a?e-c (3) 'P=A*E-C (3a) where c and G indicate constants to be added to the pressure and which have no influence on the equation la. For gases c and G are to be put equal to zero if the motion occurs under such circumstances that the temperature remains constant. For rapid variations of density in gases without equalization of temperature (namely uon-adiabatic motions), the equations (3) and (3a) would only apply for the case of slight variations in density. The equation (3a) is only satisfied by the above-given values for P and E when A2=d*ril. By this condition therefore the third constant, n, is determined. The quantities a and A in this latter equation are the velocities of sound in the respective fluids. These quantities must change in the same ratio as the other velocities. If the boundaries of the fluid are iu part iu finitely distant and in part given by moving or quiet, perfectly wetted, rigid bodies, and the coordinates aud componeut velocities of these limiting rigid bodies are transferred from one case to the other in the same manuer as has just been done for the particles of fluid, then will the boundary conditions for U, V, Wbe fulfilled when they are fulfilled for u, v, w. In this 1 assume that on completely wetted bodies the superficial layer of fluid is held perfectly adherent; that therefore the component velocities of the surfaces of the rigid bodies and those of the adherent fluid are equal. For imperfectly wetted solids it is as a rule assumed that there is a relative motion of the superficial fluid layers with respect to the solid. In this case the application of our principles would require that a cer- tain ratio be assumed between the coefficients of sliding superficial friction of the fluid on the respective rigid bodies, and the internal friction (or viscosity) of the fluid. Similarly the boundary conditions at the free surfaces of a liquid over which the surface pressure is constant, would be satisfied in case no 70 THE MECHANICS OF THE EARTH'S ATMOSPHERE. outside forces like gravity have an influence. But since this case occurs only in liquids [i.e., fluids that form drops] that can be regarded as incompressible, therefore (for these) it is not necessary to satisfy equations (3) and (3a). Therefore (for these) the constant n remains arbitrary, and when for this case this latter constant is so determined that -=1, then in equation (la) the intensity of gravity [i. e., the accel- eration, — g) can be added to the left-hand member. The boundary condition for a discontinuous surface is that the pres- sure shall be equal on both sides of such a surface, which condition will be satisfied for P when it is so for p. As regards the re action of the fluid against a solid body moving in it, the pressure against the unit of area of surface increases as n2r. In the same ratio, the frictional forces increase that are proportional to the product of Jc e, with the differential quotients such as — , auu other similar ones. But for corresponding similar portions of the surfaces of the bounding bodies of the forces of pressure and of friction increase as — ri\r = qlr. i^l" — /<<-) The work needed to be done by the immersed bodies to overcome these resistances will therefore for equal intervals of time increase as nq2r. In general therefore for compressible fluids [gases] and for heavy cohesive fluids [liquids under gravitation] with free surfaces, if the movement is to be completely and accurately transferred from the first fluid to the other, the three constauts n, q, r are completely determined by the nature of the two fluids. Only in the case of incompressible fluids without free surfaaes does one constant remain indeterminate. Now there is a large series of cases where the compressibility not only for cohesive, but also for gaseous fluids, has only an inappreciably small influence. To such cases the following considerations apply : If the constant n becomes smaller while r and q remain unchanged, this indicates that in the second fluid the velocity of sound diminishes pro- portionally with n, and similarly for the velocities of the moving mate- rial portions, whereas the linear dimensions increase proportional to the reciprocal of n. For a constant value of r, that is to say, a con- stant density of the second fluid, a diminution of the velocity of sound corresponds to an increased compressibility of the fluid. Therefore with an increased compressibility, the movements remain similar. Hence it follows that when we diminish n, while leaving the compressi- bility of the fluid unchanged, the movements of the fluid themselves change and become similar to those that a more incompressible fluid would execute in a narrower space. Therefore for smaller velocities, PAPER BY PROF. HELMHOLTZ. 71 even in extensive spaces, the compressibility loses its influence. Under such circumstances gases move like cohesive incompressible fluids [viz, liquids], as is well known practically from many examples. If the velocities of the material parts are in general very small, as in the case of exceedingly small oscillations, so that the course of the movement remains sensibly unchanged for a uniform increase in these velocities, then it will only be the velocity of sound that changes, and our proposition will take the following form : The sonorous vibrations of a compressible fluid can, in larger spaces, behave mechanically the same as more rapid oscillations of a less compressible fluid in smaller spaces. An example of the utilization of the similarity here spoken of is found in my investigations on the acoustic movement at the ends of open organ pipes.* In that study the possibility of replacing the analytical conditions of the motion of the air by the simpler ones of the motion of water depended on the principle that the dimensions of the given spaces must be very small in comparison to the wave lengths of the existing acoustic vibrations. On the other hand the viscosity also shows itself less influential in the movements of fluids in large spaces. If we let n remain unchanged while q increases we obtain the same ratio between the frictional forces and the pressure forces. That is to say, if we increase the dimensions and the friction constants in the same ratio, then the movements in the enlarged system remain similar so long as the velocities do not change. Hence it follows that in such an enlarged model, when the friction con- stant is not increased in the same ratio, but remains unchanged, the friction loses in influence for the same velocity. That which holds good for greater dimensions with unchanged velocities also obtains for increased velocities with unchanged dimensions. For one can also simultaneously let n increase proportional to q. In fact, in most practical experiments in extended fluid masses, the resistance that arises from the accelerations of the fluid,t and especially in consequence of the formation of surfaces of discontinuity is by far the most important. Its magnitude increases proportionally to the square of the velocity, whereas the resistance depending upon the fric- tion proper (internal friction or viscosity and surface-hesiou), which increases simply in proportion to the velocity, becomes appreciable only in experiments in very narrow tubes and vessels. Neglecting the friction, that is to say, if in the above equations we put the constants h=K=0 then will the constant q also become arbitrary, and we can change the dimensions and velocities in any ratio whatever. If however the force of gravity comes into consideration as in the * Borchardt's Journal fur Mathematik, 1859, vol. lvii, pp. 1-72- t [These resistances are those that I have called " collective" in my Treatise on Meteorological Methods and Apparatus.— C. A.~\ 72 THE MECHANICS OF THE EARTH'S ATMOSPHERE. case of waves on the free surface of water, then, according to the remarks already made the ratio - must remain unchanged, therefore q must be put ==rc3. Then will X— n2cc Y=nhj T-nL Z=n2z Therefore when the wave lengths increase in the ratio n2 the duration of the oscillations will increase only in the ratio n, which corresponds to the well-known law of the velocity of propagation for the surface waves of water, which velocity increases as the square root of the wave length. Thus this result is attained very simply and for all wave forms, without the necessity of knowing a single integral of wave motion. The same principle is applicable to the relative resistances that ships having n2 times the dimensions and n times the velocity, experience by reason of the waves that they excite on the surface of the water. The total resistance in this case increases as q2r, and since for the same fluid r=l therefore the resistance increases as n6 and the work needed to overcome it as w7, therefore in a rather larger ratio than the volume of the ship, while the supply of fuel and the size of the boiler that must do the work can increase only in the same ratio as the volume of the ship, namely as nG. Therefore so long as lighter machinery can not be applied (including the supply of coal) the velocity of such an enlarged ship can increase above a certain limit only by a ratio that is smaller than that of the square root of the increase of the linear dimensions. A similar computation holds good for the model of the bird in the air. When we increase the linear dimensions of a bird and would take into consideration the viscosity, we must put q and r equal to unity be- cause the medium, namely the air, remains unchanged. Let n be a vulgar fraction, then will the velocity be reduced in the same propor- tion as the volume of the bird iucreases and the pressure (of the air) against the total surface of the larger bird will only attain the same value as for the smaller bird, therefore will not be able to bear up the weight of the larger bird. If we allow ourselves to neglect the friction, which according to the above remarks we can do so much the more readily the more we increase the dimensions, or for the same dimensions increase the velocities, then q is arbitrary and the change of dimensions aud velocities must be so made that the total pressure against the surfaces shall increase as the weight of the body or we must have q2=qor q=n\ In order to ex- n3 ecute the corresponding motions, the work that will be necessary will be q2n=n7— ' q v^ m PAPER BY PROF. HELMHOLTZ 73 but the volume of the body and of the muscles that do the work in- creases only in the ratio ( i. ) Hence it follows that the size of a bird has a limit, unless the muscles can be further developed in such a manner that for the same mass as now they shall perform more work. Now it is precisely among the larger birds, that are capable of the greater performances in flying, that we find those that eat only flesh and fish ; they are animals that consume concentrated food and need no extensive system of diges- tive organs. Among the smaller birds many grain eaters like doves and the smaller singing birds are also good flyers. It therefore ap- pears probable that in the model of the great vulture, nature has al- ' ready reached the limit that can ba attained with the muscles as work- ing organs, and under the most favorable conditions of subsistence, for the magnitude of a creature that shall raise itself by its wings* and remain a long time in the air. Under these circumstances it is scarcely to be considered as probable that man even by means of the most ingenious wing-like mechanism that must be moved by his own muscles will ever possess the strength needed to raise his own weight in the air and continue there. As concerns the question as to the possibility of driving balloons for- ward relative to the surrounding air, our propositions allow us to com- pare this problem with the other one that is practically executed in many forms, namely, to drive a ship forwards in water by means of oar-like or screw-like organs of motion. In studying this we must not consider movement on the surface, but rather imagine to ourselves a ship driven along under the surface. But such a balloon which pre sents a surface above and below that is congruent with the submerged surface of an ordinary ship scarcely differs in its powers of motion from an ordinary ship. If now we let the small letters of the two above given systems of hydro-dynamic equations refer to water and the large letters to the air, then for 0° temperature and 7G0 mm. of the barometer, we have -=773 r According to the determination of O. E. Meyer and Gierke Maxwell, 4=0.8082 the velocity of sound gives for n the value w=0.2314 Hence the increase of linear dimensions is 9=3.4928 n *[That is, by the work done by its wings; this of course does not cover the case of soaring birds whose muscles do no lifting work but simply keep the wings in the best position for the wind to act on them. — ft A.~\ 74 THE MECHANICS OF THE EARTH'S ATMOSPHERE. and the increase of volume is GO'-** The work in this case is very slight, namely, The ship, including the crew and the load, must weigh as much as the volume of water displaced by it. The balloon, filled with hydro- gen, in order to carry an equal weight witli the ship, must have a vol- ume 837 times as great. If it is tilled with illuminating gas of a specific gravity 0.65 relative to that of the air, it must have a volume 2,208.5 times as great as the ship. Thus, the weight that the balloon must have for the given dimension is now determined. The weight for the *ct n -i hydrogen balloon would be -J^^tttp that of the ship; that of the illuminating gas balloon would be -o^rF = mi that of the ship. The work that is necessary under such circumstances to propel the balloon, as the above number for the value of q-nr shows, would, how- ever, for the adopted small velocity, be reduced in much greater propor- tion than that of the weight of the balloon to the weight of the ship, so that the work here required for the given weight is easy to accomplish in the balloon. Foreven when wesochoose theshipthatitsload inexcessof that of the driving machine (or in excess of the men who act as the machine) is negligible, then the weight of the illuminating-gas balloon need be only ^- part of the weight of this driving machine, but the machine thus carried by it would also have to do only the one ~ , -. of the work J 5114 of the ship's machine, it would, therefore, need to have a less weight in about this latter ratio. Especially would this latter be the case when we utilize men as the driving machine, whose work and weight both increase proportionally to the number. So far we can therefore apply the transference from ship to balloon with complete consideration of the peculiarities of air and water. As a maximum velocity for fast ships (large naval steamers), " The Engi- neer's Pocket Book," published by the society » Die Hiitte," gives 18 feet per second, or 2.7 German miles, or 21 kilometers per hour. Similarly built balloons, with relatively very feeble or small propelling machin- ery, can attain about one-fourth of this velocity. Ships of the above-given dimensions find the limit of their efficiency bounded by the limits of the power of the machinery (including the fuel) that they can carry. However, the practical experience thus far attained allows us to neglect the influence of viscosity for large, swift PAPER BY PROF. HELMHOLTZ. 75 ships, aud therefore to arbitrarily assume the constant g, as also n (when we can neglect the movements at the surface). If we assume that q increases proportionally to n, then the dimensions remain un- changed, the velocities increase as w, the resistance as n2, the work done as w3. If therefore we were able to build a marine engine of the same weight as the present ones, but of greater efficiency, we would then be able also to attain greater velocities. We must compare the balloon with such a ship, although the latter has not yet been constructed, in order to attain complete utilization of the propelling machine that goes up with it. But for this case also and for unchanged dimensions, when the velocity increases as n the work must increase as m3. Now the ratio between weight and work done by the men who are carried by a balloon can only, for balloons of very large dimensions, be perhaps more favorable than for a war ship aud its machinery. For the latter I compute from the technical data that to attain a velocity of 18 feet requires au expenditure of one horsepower to 463G.1 kilo- grams weight.* On the other hand, a man weighing 200 pounds, who under favo table circumstances can do 75 foot-pounds of work per second during eight hours daily, gives on the average for the day one horsepower per 1,920 kilograms. When therefore the balloon weighs one aud a half times as much as the laboring men whom it carries, then the ratio is the same as for the ship. Dupuy de Lome has carried out his experiments under somewhat less favorable circum- stances ; in his balloon were a crew of 14 men whose weight was one- fourth of the whole, and of whom only eight worked. Under these circumstances it is a relatively very favorable assumption when for the balloon we assume the ratio between the weight and the work to be the same as for a war steamer. We can therefore for the illuminating gas balloon increase the ratio -, ' , between work aud weight by in- 5114ftJ creasing n so that the ratio shall equal unity; that is to say, equal to the value for ships. In this case we must have n=4.G20S. Siuce now the velocity U of the balloon which wo have before com- puted under the assumption of a perfect geometrical similarity in the *'fhe special data on which the computation is based are as follows : L~ length of the ship over all = 230 Prussian feet. B = breadth of the ship over all = 54 " " H = total height of the ship = 24 feet. T= depth under water = H — \ B F = volume of water displacement = 0. 46 L. B. T. Weight of one cubic foot of sea water =63. 343 lbs. A the area of the immersed principal sectiou = 1000 sq. feet. The total work = C A V3 Where ; = 0. 46. 76 THE MECHANICS OF THE EARTH'S ATMOSPHERE. movements has ouly 0. 2314 that of the velocity u of the ship, therefore there results : JJ=0. 2314 n. m=1. 06925«. For the hydrogen balloon under the same assumptions the velocity will be somewhat larger, since in this case we have to assume 5114 Hence, %=6.390 Z7=0. 2314 .n .«=1. 4786m. which is nearly one and a half times the velocity hitherto attained in naval steamers. This last velocity for a hydrogen balloon would suffice to go slowly forwards against a fresh breeze. But it is to be remarked that these computations relate to colossal balloons whose linear dimensions are three and a half times larger than those of the immersed portion of a large man-of-war, and that the inflammable gas balloon would weigh C0220 kilograms, while that of Dupuy de Lome only weighed 3791) kilograms. In order to return to dimensions that are attainable in actual practice, one must so diminish q and n as that the ratio of the work to the weight shall re- main unchanged, therefore, so that .....*.. • (2) 80 THE MECHANICS OF THE EARTH'S ATMOSPHERE. Assume that the fluid adheres to the earth's surface where z = 0, therefore for this surface we have *>^ u = , n, and p retain the significance just given them we then have w^0 pl * J fi2 PAPER BY PROF. HELMHOLTZ. 83 and if we consider a steady mode of motion, in which £1, p, P, and s are functions of x and p only, theu the equations (1) become .£-M*-0 (3.) dX £ JX v ' -^l y.-^^u =-y f^L dp' p e'dp'p V _dP z__l dp z_ £X dp'p e'dp'p ' p^ The two last equations combine into the one following : Jt+?Tp-l? {b) Equation 1,( is satisfied by the above adopted values of u, v, u: There- fore the only equations to be satisfied are (3a) and (3b). As concerns the value of the density e, this depends upon the pressure p and the temperature 8. Since appreciable effective conduction of beat is excluded, therefore we must here retain the law of adiabatic variations between p and e ; therefore we have a> p \ I _ e wherein y again represents the ratio of the specific heats. If we indi- cate by 8 the temperature that the mass of air uuder consideration would acquire adiabatically under the pressure p0 (wherefore 8 indi- cates the constant quantity of heat contained in the air while its tem- perature is varying with the pressure), and if we put then we have e'dp\pJr'Po 'dp' or if, for further abbreviation, we put X-y=a (3c) JL^.&.p r =q ' — 1 Y P y =7t we shall have . . • (3d) 1 dP a 1* 6 dp l dp 84 THE MECHANICS OF THE EARTH'S ATMOSPHERE. wherein q indicates a constant peculiar to the gas ami independent of 0 and p. Similarly we also have and therefore within a stratum of air having a constant 6 and £1 we have, according to equations (3a) and (3b), 1 Q2 P+Z'O' *---y -? (Se) The very slight deviation of the earth from a spherical form allows us to simplify the computation on the one hand by regarding the earth's surface as a sphere, but on the other hand by giving the potential P an addition, the effect of which is that for the normal velocity of rotation oo0 of the earth, its spherical surface becomes a level surface. To this end we put r | Where 6r=normal force of gravity: r=distauce from center of gravity to point or stratum in the actual atmosphere.] This gives the component in the direction of x, of the forces acting upon the unit of mass, X= J—= -— • dx ~ r'z ' and, for the component in the direction of p, JP G.p , dp r3 P=-^- = -^'p If to the latter the centrifugal force +0 or positive for every other direction dh, in which one moves from any point of the surface towards the same side as dn. If dh is drawn toward the other side of the surface for which 7^ — ^2=0, then will l) {7tx— 7r2)<0, or negative. If now the difference is positive on that side of the surface designa- ted by the subscript index l,then in case there is an infinitely small protrusion of the boundary surface toward this side, this protrusion will be pressed back by the exterior and greater tt, ; similarly an infinitely small protrusion toward the negative side will also be pushed back, since there, on the other hand, ttx diminishes more rapidly in the interior of such protrusion. Therefore in both these cases the equilibrium is stable. On the other hand, the equilibrium is unstable when the dif- ference {ttx — 7t2) on the side of tt\ is negative. Xow we need not form the differential quotients for the direction dn. It suffices to form them for dr or dp, and to merely determine whether the positive dr or dp look toward the side whose index is 1 or that whose index i.s 2. By forming these differential quotients from the equation (3/) there results The differential quotient is positive vhen 0, > 0,. The partial dif- ferentiation with respect to r while p remains unchanged, indicates a progress in an ascending direction parallel to the earth's axis; that is to say, in the direction of a line pointiug towards the celestial pole. The equilibrium i.s stable when the strata containing the greater quantity of heat lie at higher elevations on the side towards the celestial poles. We now form the other differentia] quotients *>- >4(f-f)--KVI) ■ • • • ^ = „ ["i*.T«d_?y-'-'] (4/, If in these equations 0, indicates the greater quantity of heat, then the equilibrium is stable when everywhere along the boundary surface we have co?—co£. oof—Goi P- ,L W. PAPER BY PROF. HELMHOL7 Both these values arf- positive w ■ . bot), -here the nd prevails. The equation Ae] can also be written - --■=': ■"-:■- — ~ ]■ la order this may be positive - _ a- equality must be s tis led -.^ .- or. 1 Ordinarily this will be the cac n general 6 ses s unlta- neously with p and from a detinue value at the pok the equator. Similarly il so - nd from zero at pole to attheequa: r,s that so ine: - - from zero at the pole to a dennite 1 s :he equator. We a refbre d - ignate this s • :he uv : - - :a only occur under special conditions withiu hniit Iu the normal ease as we progress »thes 1. the wanner 3 on the side of the great that is to s . on the side towards the equator, and equally on the sule of the _ er r if we p: 9 6S toward the 1 stial t is 1 say,/ and r icereas >rd the - ..e side of the boundar - this surface si - aod that the tangent of its aieridian section intersects tin s - ere between the pole and the poiut of the hor ingimmed be- neath it. Near the equator, where the pole rises the horizon, this _ - an inclination to the boundary snrfaof sneh that it makes a very small acute ang :h the horizou. In accordance with this, equal - snsl r those cir- cnmstances 3— isneg g thebonnd rysurft - f. Therefore the r inclination of the bounding sir is in an -ending direct ml a p : sitnated beneath the « st It on the other hand exceptional lot ss exist - ■:■: g»< dr then in sneh cases rding to equal Bl be I sil : that is to say. the boundary line will ascend to higher levels s a the earth's ..\ - Since moreover equal ^ 3 isweprooi athedi tion of a line drawn re the pole, the warmer air must li g er. tin 88 THE MECHANICS OF THE EARTH'S ATMOSPHERE. tore this line can not twice intersect the boundary surface between two layers, and consequently in the abnormal case tbis line must necessarily lie between tbe boundary surface and the horizontal plane located at the pole. Therefore the tangents to the meridional section of the boundary surfaces must intersect the greater arcs on the celestial sphere somewhere between the pole and the equatorial side of the hor- izon. The smaller the difference of temperature is relative to the difference of the velocities of rotation so much the nearer does the tangent just referred to approach the pole. Moreover at different points of the bounding line of the same two layers there can occur both normal and abnormal inclinations. For since in the expression (see equation 4/<) on whose positive or negative value such occurrence depends, the £1 and 6 throughout the extent of each layer are constant, therefore for the same altitude above the earth this value can have a positive value near the equator but a negative value near the poles. Between these the boundary curve must attain a maximum altitude where the quantity under consideration passes from positive through zero to negative. At this place also, according fly to equation (4a), we have -r- = 0, therefore r is a limiting value and is here a maximum. Location of the strata in the case when the velocity of rotation varies continuously with the quantity of heat contained. — The considerations hitherto sec forth can also be extended to the case where £1 is a con- tinuous function of d, and the value of 6 in the atmospheric strata is continually changing. The individual strata are in this case to be con- sidered as indefinitely thin. Equation (4a) now becomes. dr dp G-r= r< p3 'El ,, •p — GHf p4 i. e dp P3 ^-^-^V] In order that the equilibrium may be stable the quantity of con- tained heat (see equation 4/t) must increase in the direction towards the celestial pole. But the layers of similar air are less inclined than the inclination of the polar axis at all places where the quantity D? - B. da\ < orf ff . dO but on the other hand their inclination is steeper where the left-hand side of this inequality is greater than the right. PAPER BY PROF. HELMHOLTZ. 89 IV. GRADUAL VARIATIONS OF THE EQUILIBRIUM BY FRICTION AND HEATING. It is well known how very differently the propagation of changes of temperature in the air goes on according as heat is added or withdrawn below or above. If the lower side of a. stratum of air is warmed, as occurs at the sur- face of the earth, by action of the solar rays, then the heated stratum of air seeks to rise. This is effected very soon all over the surface in small tremulous and flickering streams such as we see over any plane surface strongly heated by the sun ; but soon these smaller streams collect into larger ones when the locality affords opportunity, especially on the side of a hill. The propagation of heat goes on relatively rapidly through the whole thickness of the atmospheric layer, and when it has a uniform quantity of heat throughout its whole depth and is therefore in adiabatic equilibrium then also the newly added air seeks de nova to distribute itself through the entire depth. The same process occurs with like rapidity when the upper side of a stratum of air is cooled. On the other hand, when the upper side is warmed and the lower side cooled such convective movements do not occur. The conduction of heat operates very slowly in large dimensions, as I have already ex- plained above. Radiation can only make itself felt to any considerable extent for those classes of rays "that are strongly absorbed. On the other hand, experiments on the radiation from ice and observations of nocturnal frosts show that most rays of even such low temperatures can pass through thick layers of clear atmosphere without material absorption. Therefore a cold stratum of air can lie for a long time on the earth, or equally a warm stratum remain at an altitude, without changing its temperature otherwise than very slowly. Similar differences exist also in the case of the change of veloc- ity by friction. For the normal inclination of an atmospheric stratum its upper end is nearer to the earth's axis than its lower end. If the stratum appears at the earth's surface as a west wind, then the moment of rotation of the lowest layer is delayed [by resistance of the earth's surface], its centrifugal force is diminished, and on the polar side of the stratum this lowest portion will slide outwards, approaching the axis in order to find its position of stable equilibrium at the upper end of the stratum. This movement will ordinarily take place in small trem- ulous streams similar to the ascent of warm air and must diminish the moment of rotation of the whole layer rather uniformly, but in the upper portions a little later than in the lower. Since, however, this latter effect distributes itself throughout the whole mass of air, it will become much less apparent on the lower side of the stratum than if it were confined to the lower stratum. 90 THE MECHANICS OP THE EARTH'S ATMOSPHERE. For the east wind matters are reversed. Its moment of rotation is increased by the friction on the earth's surface. The accelerated mass of air [the ground layer] already finds itself in that position of equi- librium which it has to occupy within its stratum, and can only press forward equatorially along the earth's surface into the stratum lying in front of it. If it is also simultaneously heated then the resulting ascent takes place more slowly than would occur in a stratum of air that is at rest at the bottom. Hence it is to be concluded that in the east wind, the change due to friction is confined to the lower layer of air, and furthermore that it is relatively more effective here than in the case of a west wind of equal velocity. In general, the retarded layer of air will press forward to- ward the equator, in the Northern Hemisphere as northeast wind. In this motion it will continue to appear as an easterly wind since it is con- tinually arriving at more rapidly rotating zones on the earth. The air of the stratum lying above the retarded layer will, where the region is free from obstruction, as at the outer border of the trade wind zone, fall behind and will appear as an east wind, retaining its moment of rotation unchanged and gradually pushing toward the equator will itself in its turn experience the above described influence of friction. I would here further remark that the water so abundantly evaporated in the tropical zone also enters into the trade wind, but with the greater velocity of rotation of the revolving earth and must diminish the retardation of the latter with respect to the earth. The lower layers of the trade wind can press in under the equa- torial calm zone itself only when any difference between their velocity of rotation and that of the earth's surface is entirely destroyed. They then blend with the zone of calms and increase its mass so that the lat- ter broadens with its inclined boundary surface always higher above the layer of diminishing east wind beneath it. Thus it is brought about that whereas below [nearer the earth's sur- face] mostly continuous changes are taking place in the temperature and the moment of rotation of the strata, on the other hand above, the boundaries of the broadening zones of calms (that have the great mo- ment of rotation that pertains to the equatorial air and which at 10° latitude must appear as a strong west wind, and at 20° latitude as a westerly storm), occur in direct contact with the underlying stratum that has less velocity of rotation and lower temperature. Evidently the upper side of this latter [lower] stratum can scarcely be changed as to the quantity of its contained heat and of its moment of rotation, while after the loss of its lower layer it is being pushed sidewise and towards the equator. As I have already shown in my communication to this Academy, April 23, 1868, on "Discontinuous Fluid Motions,"* such discontinuous motions can continue for a while, but the equilibrium at their boundary * [See No. Ill of this collection of Translations.] PAPER BY PROF. HELMHOLTZ. 91 surfaces is unstable, and sooner or later they break up into whirls that lead to general mixture of the two strata. This statement is confirmed by the experiments with sensitive flames and by those in which by means of a cylindrical current of air blown from a tube we make a sec- tion in a flame and thus make visible the boundary of the moving and the quiet mass. If, as in our case, the lower stratum is the heavier it can be shown that the perturbations must at first be similar to the waves of water that are excited by the wind. The process is made evident by the striated cirrus clouds that are visible when fog is pre- cipitated at the boundary of the two strata. The great billows of water that are raised by the wind show the same process which is different in degree only, by reason of the greater difference of the specific gravi- ties. The severer storms even turn the aqueous billows to breakers, that is to say, they form caps of froth and throw drops of water from the upper crest high into the air. Up to a certain limit, this process can be mathematically deduced and analyzed, on which subject I pro- pose a later communication. For slighter differences of specific gravity the result of this process must be a mixture of the two strata with a formation of whirls and under some circumstances with heavy rainfall. An observation of one such process under very favorable circumstances I once made accidentally upon the Eigi and have described.* The mixed strata acquire a temperature and moment of inertia whose values lie between those of the component parts of the mixture, and its position of equilibrium will therefore be found nearer the equator than the position previously occupied by the colder stratum that enters into it. The mixed stratum will descend toward the equator and push back the strata lying on the polar side. Into the empty space thus created above, the strata from which this descending portion has been drawn stretch upwards, and thus their cross section must be dimin- ished. Wherever the lower layers are pushed apart by descending masses of air, as is well known, there arise anti-cyclones ; wherever cavities or gaps arise by reason of ascending masses of air, there arise cyclones. Anti-cyclones and the corresponding barometric maxima are shown, with very great regularity, by the meteorological charts t aloug the very irregularly varying limits of the northeast trade in the Atlantic Ocean— in the wiuter, under latitude 30°; in summer, under 40° latitude. On account of the inclined position of the strata, the rain that frequently forms by reason of the mixture of air (Dove's Sub- tropical Rain) falls somewhat farther northward because the water must iall down almost vertically. J * See Proceedings of the Physical Society in Berlin, October 22, 1886. t Daily Synoptic Weather Charts. Published by the Danish Meteorological Insti- tute aud the German Seewarte, Copenhagen and Hamburg. i [The results stated iu the above paragraph were subsequently greatly modified by Helmholtz. See Section v of his second memoir, or page 98 of these Transla- tions.— C. A.] 92 THE MECHANICS OF THE EARTH'S ATMOSPHERE, Therefore the zone of cyclones begins there, but these become more frequent farther northward. We can certainly assume that the process of mixture is not perfected immediately at the exact border of the trade-wind zone, but that a part of the rapidly-rotating warm upper stratum remains unchanged or half mixed, which will presently bring about new mixtures farther on toward the pole. In general, in this zone of mixture, even below at the earth's surface, the west wind must retain the upper hand because the increase of the total moment of rotation which the mass of air, through friction, experi- ences in the east wind of the trade zone must finally rise to such a pitch that somewhere the west wind again touches the earth and experiences sufficient friction to entirely give back the increase that it had. The masses of air resting in the equilibrium of stratification can certainly have no long-continued motion of rotation that differs essentially from that of the earth beneath tbem. When therefore they are mixed with the stronger west wind of the air from above, they receive a movement toward the east. Moreover the falling rain that in great part comes from the upper west winds, must transmit its motion to the lower strata through which the rain falls. Eventually all zones that are pressed polewards by intermixed masses moving equatorially and descending from them will become west winds. Another permanent source of winds is the cooling of the earth at the poles. The cold layers endeavor to flow outwards from each other at the earth's surface and form east wind (or anti-cyclones). Above these the warmer upper strata must fill the vacancy and continue as west winds (or cyclones). Thus an equilibrium would come about, as is shown in Sect. II, if it were not that the lower cold stratum acquires, through friction, a more rapid movement of rotation, and is therefore competent for further advance. In doing this, according to the above given views this lower stratum must remain on the earth's surface. That in fact it does so is shown by frequent experiences during our northeast winter winds whose low temperatures frequently enough do not extend up to even the summit of the North German Mountains. Moreover on the front border of these east winds advancing into the warmer zone, the same circumstances are effective in order to bring about a discontinuity between the movement of the upper and lower currents, as in the advancing trade winds, and there is therefore here a new cause for the formation of vortex motions. The advance of the polar east wind, although recognizable in its principal features, proceeds relatively very irregularly since the cold pole does not agree with the pole of rotation of the earth, and also because low mountain ranges have a large influence. In addition to this comes the consideration that in the cold zone fog causes only a mod- erate cooling of the thicker stratum of air, but clear air brings about a very intense cooling of the lower layer. By such irregularities, it is brought about that the anti-cyclonic movement of the lower stratum PAPER BY PKOF. HELMHOLTZ. 93 and the great and gradually increasing cyclone of the upper stratum (that should otherwise be expected at the pole) break up into a large number of irregular, wandering cyclones and anti-cyclones, with a preponderance of the former. From these considerations, I draw the conclusion that the principal obstacle to the circulation of our atmosphere, which prevents the development of far more violent winds than are actually experienced, is to be found not so much in the friction on the earth's surface as in the mixing of differently moving strata of air by means of whirls that origiuate in the unrolling of surfaces of discontinuity. In the interior of such whirls the strata of air originally separate are wound in contin- ually more numerous, and therefore also thinner layers spirally about each other, and therefore by means of the enormously extended surfaces of contact there thus becomes possible a more rapid interchange of temperature and equalization of their movemeut by friction. The present memoir is intended only to show how by means of con- tinually effective forces, there arises in the atmosphere ihe formation of surfaces of discontinuity. I propose, at a future time, to present fur- ther analytical investigations as to the phenomena of such disturbances of continuity. VI. ON ATMOSPHERIC MOVEMENTS.* (SECOND PAPER.) By Prof. H. von ITelmholtz. ON THE TIIEORY OF WINDS AND WAVES. In my previous communication made to the Academy on the 31st of May, 1888, 1 endeavored to prove that conditions must regularly recur in the atmosphere where strata of different density lie contiguous one above another. The reason for the greater density of the lower stratum is conditioned by the fact that the latter has either a smaller amount of heat or a smaller velocity of rotation, if in fact both conditions do not work together. As soon as a lighter fluid lies above a denser one with well-defined boundary, then evidently the conditions exist at this boundary for the origin and regular propagation of waves, such as we are familiar with on the surface of water. This case of waves as ordinarily observed on the boundary surfaces between water and air is only to be distinguished from the system of waves that may exist between different strata of air, in that in the former the difference of density of the two fluids is much greater than in the latter case. It appeared to me of interest to investigate what other differences result from this in the phenomena of air waves and water waves. It appears to me not doubtful that such systems of waves occur with remarkable frequency at the bounding surfaces of strata of air of different densities, even although in most cases they remain invisible to us. Evidently we see them only when the lower stratum is so nearly saturated with aqueous vapor that the summit of the wave, within which the pressure is less, begins to form a haze. Then there appear streaky, parallel trains of clouds of very different breadths, occasionally stretching over the broad surface of the sky in regular patterns. More- over it seems to me probable that this which we thus observe under special conditions that have rather the character of exceptional cases, is present in innumerable other cases when we do not see it. * From the Sitzungs-iericlde of the Royal Prussian Academy of Sciences at Berlin, July 25, 1889, pp. 761-780. 94 PAPER BY PROF. HELMHOLTZ. 95 The calculations performed by me show further that for the observed velocities of the wind there may be formed in the atmosphere not only small waves, but also those whose wave-lengths are many kilometres which, when they approach the earth's surface to within an altitude of one or several kilometres, set the lower strata of air into violent motion and must bring about the so-called gusty weather. The peculiarity of of such weather (as I look at it) consists in this, that gusts of wind often accompanied by rain are repeated at the same place, many times a day, at nearly equal intervals and nearly uniform order of succession.* I think it may be assumed that this formation of waves in the at- mosphere most frequently gives occasion to the mixture of atmospheric strata and, under favorable circumstances, when the ascending masses form mist, give opportunity for disturbances of an equilibrium that had already become nearly unstable. Under conditions, such as those where we see water waves breaking and forming white caps, thorough mixtures must form between the strata of air. In the beginning of my previous paper I have explained how insuffi- cient are the known intensities of the internal friction and the thermal conductivity of gases in order to explain the equilibration of motions and temperatures in the atmosphere. Since now the mechanical the- ory of heat has taught us to consider friction in gases as the mixture of strata having different movements, but the conduction of heat as the mixture of strata having different temperatures, it is therefore in- telligible that a more thorough mixture of strata in the atmosphere must bring about, to a still higher degree, the effects of friction and conduction, t but certainly not in a quiet, steady progress, but pro- ceeding irregularly as is indeed the special character of meteorological processes. Therefore I have considered it important to develop the theory of waves at the common boundary surface of two fluids. Hitherto in studies on waves of water, so far as known to me, the influence of the air and its motion with the water has always been neglected, but this may not be done in the present work. The problem becomes thereby much more complicated aud difficult; and since even the simpler problem that takes no account of the influence of the wind has at the hands of many excellent mathematicians received only incomplete and approximate solutions, under assumptions chosen to simplify the problem, therefore I pray to be excused in that I also have at first treated the simplest case of the problem, namely, the movement oi rectilinear waves which propagate themselves with unchanged forms *This assumption of the formation of billows in the atmosphere that I recently briefly expressed in my first contribution has since then also been propounded by Jean Luvini (La Lumiere tUctrique. T. xxx, pp. 368, 617, 620). t Perhaps this would correspond to the assumptions that form the basis of the theory submitted to this (Berlin) Academy by Oberbeck, March 15, 1888. [See Nos. XII aud XIII of this collection. — C. A.] 96 THE MECHANICS OF THE EARTH'S ATMOSPHERE. and with uniform velocity on the plane boundary surface between in- definitely extended layers of two fluids of different densities and having different progressive movements. I shall call this kind of billows stationary billows, since they represent a stationary motion of two fluids when they are referred to a system of coordinates which itself advances with the waves. Since in the relative motion of the different parts of a closed material system nothing is changed when the whole receives a uniform rectilinear velocity toward any direction, therefore this rearrangement of our problem is allowable. Moreover I propose to-day to give only the results of my mathemati- cal investigations. The complete presentation of these I reserve for publication in another manner. Before I advance to the theory of atmospheric billows, I will however introduce a supplement to the considerations given in my communica- tion of May, 1888, by which the region in which we have to look for the conditions that give rise to atmospheric billows is better defined. V. THE ASCENT OF MIXED STRATA. In Section in of my previous communication I have shown what would be the law of equilibrium, in case such a condition should be attained, between atmospheric rings of different temperatures and dif- ferent speeds of rotation, which however are all assumed as being com- posed of mixtures that are similar to each other. I now return to equa- tion (ia, page 85). Let the location of a point in the atmosphere be given by the quantities p, the distance from the earth's axis. r, the distance from the center of the earth, Let oo0 be the angular velocity of the solid earth ; and /2i and il2 be the constant moments of rotation of the unit of mass of one or the other layer of air : Let #j and 82 be the quantities that I have called the contained ca- loric of the unit of mass of air, and that certainly may be better desig- nated by the term potential temperatures, so well chosen by Bezold, namely, those temperatures which the respective masses of air would assume when brought adiabatioally to the normal pressure. Let G =s constant of gravitation. In accordance with equation (4a) we now have at the boundary surfaces the relation ^ravL l-i ~^P4J (i) The ratio ^ indicates also the ratio of the sines of the two angles which the tangent to the curve in the meridional plai e makes on the one hand with the earth's axis, and on the other hand with the horizon. When, as is ordinarily the case, the warmer layer has also the greater PAPER RY PROF. HELMHOLTZ. !)7 moment of rotation, the ratio £y is then negative, and the tangent to the boundary surface cuts the celestial vault below the pole. The colder, more slowly rotatiug mass, which we will designate by the sub- script (2), lies in the acute angle betweeu the boundary surface and that part of the terrestrial surface which is ou the polar side of the given point. When now at the boundary surface of the two strata, a mixture takes place of the component masses »*, and m2, then will the moment of rotation (J2) of the mixed masses be given by the equation (mi-f //<,)/2=WilQi+m2l02, since the sum of the moments of rotation does not vary when no exterior rotatory forces are at work. Equally will the potential temperature 6 of the mixture be given by If now in equation (1), we at first substitute the mixture in place of the cooler mass (2), in order to find the direction of the boundary line between the mass (1) and the mixture, and indicate by dpi and drx the corresponding values of dp and dr, then our equation (1), alter an easy transformation, gives ,Grdn dr~\_ n>1d1 (nx-n2f pr2L^A~^pJ~Wi+W2 "0,-0, • • • • [M) Since in stable equilibrium 02<#i, therefore this equation shows that drx dr dpi dp dp] dp or drx dr that is to say, that the boundary surfaces betweeu mass (1) and the mixture must ascend more steeply with reference to the horizon than the boundary surface between (1) and (2). Similarlv it follows that the ratio -,-2 between the cooler mass (2) dp% and the mixture will be given by the equation — 3G dr2 dr dpi dp_ m2H2 (/2i — / k ) mi+m2 6x—B2 Therefore dfi >rJr; that is to say, the boundary surface between the dpi dp cooler mass (2) and the mixture must make a more acute angle with the horizon towards the pole than does the boundary surface between the mixture and the warmer mass (I). It is to be noted that the ratios ■£■ are positive when the tangent to the boundary line is more inclined than the line to the pole— in the other cases they are negative— and furthermore that the increase of a negative quantity means the diminution of its absolute value. un 4 7 98 THE MECHANICS OF THE EARTH'S ATMOSPHERE. But the required directions for the two boundary lines of the mixture can only exist when this mixture passes upwards between the two masses (1) and (2). Only thus can there be a condition of equilibrium. Hence results the important consequence that all newly formed mix- tures of strata that were in equilibrium with each other must rise upwards between the two layers originally present, a process that of course goes on more energetically when precipitations are formed in the ascending masses. While the mixed strata are ascending, those parts of the strata on the north and south that have hitherto rested quietly approach each other until they even come in contact, by which motion the difference of their velocities must necessarily increase since the strata lying on the equatorial side acquire greater moment of rotation with smaller radius, while those on the polar side acquire feebler rotation with a larger radius. If this occurs uniformly along an entire parallel of lati- tude we should again obtain a new surface of separation for strata of different rates of rotation whose equatorial side would show stronger west winds than the polar side, which latter might occasionally show east winds. On account of the numerous local disturbances of the great atmospheric currents there will, as a rule, be formed no contin- uous line of separation, but this will be broken into separate pieces which must appear as cyclones. But as soon as the total mixed masses have found their equilibrium the surfaces of separation will again begin to form below, and new wave formations will initiate a repetition of the same processes.* From these considerations it follows that the locality for the forma- tion of billows between the strata of air is to be sought especially in the lower parts of the atmosphere, while in the upper parts an almost continuous variation through the different values of rotation and tem- perature is to be expected. The boundary surfaces of different strata of air, along which the waves travel, have one edge at the earth's sur- face and there the strata becomes superficial. Experience also teaches, as does the theory, that water-waves that run against a shallow shore break upon it, and even waves which originally run parallel to the shore propagate themselves more slowly in shallow water. Therefore waves that are originally rectilinear and run parallel to the banks will * In the last section of my previous paper [see ante p. 91] I located the origin of the discontinuity principally in the upper strata of the atmosphere. But in that paper the point of departure was different from the present. In that the question considered was: If at any time the atmosphere has attained an initial stage of contin- uous steady motion without surfaces of separation, where will such a surface first form ? To this the answer is : At the upper houndary of the tropical belt of calms. At present the question is, Where in consequence of processes of mixture will the surfaces of separation necessarily he renewed ? But I must take back the proposi- tion on page 91 that treats of the descent of mixed strata, now that I have found the law expressed in this paragraph. PAPER BY PROF. HELMHOLTZ. !)«.) iu consequence of the delay become curved, whereby the convexity ot their arcs is turned toward the shore; in consequence of this they run upon the shore and break to pieces there. In the next paragraphs I will show iu what respects the movements and forms of water-waves must be changed in order to be applicable to the air. These relations are indeed not to be rigidly transferred from water-waves that break upon the shore to the air, and even the simpler theory hitherto developed, which neglects the influence of the air, gives no complete explanation on this point. But the conditions are not very different from those cases in which we can make a strict application, and I therefore believe there is no reason to doubt that waves of air which in the ideal atmospheric circu- lation symmetrical to the axis couhl only progress iu a west east direc- tion, must, when once they are initiated in the real atmosphere, turn down toward the earth's surface and break up by running along thisiu a northwesterly direction (iu the northern hemisphere). Another process that can cause the foaming of the waves at their summits is the general increase in velocity of the wind. My»analysis also demonstrates this : it shows that waves of given wave length can only co-exist with winds of definite strength. An increase in the differ- ential velocities within the atmosphere indeed ofteu happens, but one can not yet give the conditions generally effective for such a process. I will here also mention another point that may give rise to consid- erations against my explanation. Water-waves forced up to a great height always have narrow, strongly curved ridges and broad, fiat, curved troughs. Analysis shows that this feature is independent of the nature of the medium. Atmospheric waves have, on the other hand, rounded heads when they become visible to us as bands of cirri. But we must remember that according to the proposition first formu- lated by Beye, air that has formed cloud or mist is lighter than it was before. Therefore what we see as mist rises up and increases the size of the summit of the wave more than would be the case in transparent air. VI. CONSEQUENCES DEDUCED FROM THE PRINCIPLE OF MECHANICAL SIMILARITY. If we confine ourselves to the search for such rectilinear waves as advance with uuiform velocity without change of form, we may, as be- fore remarked, represent such a movement as a stationary one, by attributing to both the media a uniform rectilinear velocity equal aud opposite to that of the wave. It is well known no change is thereby introduced into the relative motions of the different parts of the masses. In this way the bounding surface of the two media appears as a sur- face fixed in space; above it the upper medium Hows iu one direction; below it the other medium in the opposite direction. At a great dis- tance from the bounding surface both movements become rectilinear 100 THE MECHANICS OF THE EARTH'S ATMOSPHERE. currents of uniform velocity, but in the neighborhood of the wavy boundary surface the motion must follow its direction. Designate by u and v the components of the velocities of the fluid particles at the point corresponding to the rectangular coordinates x and y; these velocities are by assumption, independent of the time, and (for an incompressible fluid whose current is free from vortices) can be presented in the form 11 — — / dy where ip is such a function of the coordinate as satisfies the differential equation ^+^"-0 (2) The equations ij-= const are in this case, as is well known, the streamlines of the fluid. The boundary line of both fluids must be such a stream-line, and we will give it for both sides the value V'i = =0 and c''2=<>- The above overscored letters will, in what follows, always indicate values on the boundary surface. The first boundary condition that we have to satisfy is therefore that, when we express fa and 4>% as functions of x and y. then the two equa- tions ^=0=^2 (2a) shall admit of an accordant solution. The second boundary coudition is that the pressure at the bounding surface shall be the same on both sides, or Pi=P-2 (2b) Now, under the adopted assumptions and when s is the density of the fluid and C is a constant, we have Therefore the equation (26) can be written : Const, = (.Sl-S2)^^+^^0'_^^y .... (3) PAPER BY PROF HELMHOLTZ. 1Q1 The equations (2) and (2a) remain true when we increase either the* values of the two coordinates x and y or those of i\,x or rp2 in any given ratio. Since the densities sx and s2 do not occur in these two equations, therefore also these can change to any amount. But equation (3) re- quires that the quantities jV&Y 1 and -*-(.* f ! shall remain unchanged. When therefore s, and s2 vary and we put their ratio »2 and when further the coordinates increase by the factor w, but tpx, by the factor a, and y>2 by the factor «2, then the quantities 1 — a n' l — ff n3 must both remain unchanged. Or when we, in the expressions for these quantities, put bl=ai and b2=?2 n n as the ratios by which the velocities are altered, then the above propo- sition becomes equivalent to saying that the geometrically similar wave- forms can occur when and 1—ff n 1—6 n remaiu unchanged. (1) If the ratios of the densities are not changed then in geometrically sim- ilar waves, the linear dimensions increase as the squares of the velocities of the two media ; the velocities therefore will increase in equal ratios. Therefore for a doubled velocity of the wind we shall have waves of four times the linear dimensions. This proposition is not limited to stationary movements, but is quite general.* The following propositions however will hold good only for stationary waves. (2) When the ratio of the density a is varied, the quantities A2_«i &i2 >b22~s2 b2 Gin— — rpconst. *See my paper "On a Theorem relative to geometrically similar movements of Fluid Bodies," in the Monats b. tier Akad. Berlin, 1873, pages 501 to 514 ; [or see No. IV of this collection of Translations.] 102 THE MECHANICS OF THE EARTH'S ATMOSPHERE. must remain constant; that is to say, the ratio of the living forces of the corresponding units of volume must remain unchanged. As correspond- ing- units of volume, those must be used that hold good in the region of rectilinear flow far from that of the wave surface ; but also for such units of volume as have centers that are corresponding images for each other the same proposition holds good. (3) If for a varied density the geometrically similar waves are to have the same wave-length, namely, n=l, then b, must increase as „ / — 1= /-- — - 62 must increase as /l — a= ——. For air and water at a temperature of 0° 0. we have the ratio 1 G- 1T6A For two strata of air whose temperatures are 0° aud 10° the ratio be- comes 273 °"=283 If both boundary surfaces are to show congruent waves and therefore also equal wave-lengths, and it I designate by /i( and /i2 the values of the quantities bi and b2 in this last case, then we have &! =145.21/?! &2=5.316/j2 therefore both the velocities, especially that of the wind relative to the waves of water, must be considerably diminished for the case of atrial billows. The value of the quantity . S2 ' °2 1 ~~«i : 6i* which is invariable for any change in the material for a given form of wave whose store of energy is equal to that of the rectilinear flow along a plane boundary surface is given at least approximately according to my computations, as p=0.43103. If by a wind-force w we understand the difference of the movement of the two media w=&i+&2 PAPER BY PROF. HELMHOLTZ. 103 then will for air and water -=0.069469 w , ... ..,. metres and it w =10 =- second A =0.208965 metre on the other hand for the two strata of air P* — n fi /*i+A 0.67135 audforzr=10 metres seeond A=549m.65 Hence it results that when we would obtain for this form of atmospheric wave the same wind velocity as for geometrically similar water-waves we must increase the wave-length of the air wave in the ratio of 1 to 2630.3. This ratio becomes somewhat smaller when we execute the computa- tion for the lowest waves for which This gives lor air and water p= 0.15692 &2=0.090776 w and for a wind velocity of 10 metres per second, A=0.n'S3222 The necessary magnification of the wave-length for equal strength of wind would be 1:2039.6 which gives a wave-length of more than 900 metres for a wind of 10 metres per second. Since the moderate winds that occur on the surface of the earth, often cause water-waves of a metre in length, therefore the same winds acting upon strata of air of 10° difference in temperature, maintain waves of from 2 to 5 kilometres in length. Larger ocean-waves from 5 to 10m long would correspond to atmospheric- waves of from 15 to 30 kilometres, such as would cover the whole sky of the observer and would have the ground at a depth below them less than that of one wave-length, therefore comparable with the waves in shallow water, such as set the water in motion to its very bottom. The principle of mechanical similarity, on which the propositions of this paragraph are founded, holds good for all waves that progress with an unchanged form and constant velocity of progress. Therefore these propositions can be applied to waves in shallow water, of uniform 104 THE MECHANICS OF THE EARTH'S ATMOSPHERE. depth, provided that the depth of the lower stratum in the image varies in the same ratio as the remaining linear dimensions of the waves. The velocity of propagation of such waves in shallow water depends on the depth of the water. For water waves of slight height and with- out wind it can be computed by well-known formula1. When we indi- cate the depth of the water by // and put «=""-, then is A which for h=& becomes f2_g e^—e"" ~n ' en7l+e "" 2 _ ff _ UX b* = •/ = H l7t and for small values of /* becomes b2=c/h When however the depth of the water is not small relatively to the wave length, then the retardation is unimportant, thus for = the speed of propagation diminishes as 1:0.95768 A 2 1 =4 =- the speed of propagation diminishes as 1:0.80978 = the speed of propagation diminishes as 1:0.39427 When it is calm at the earth's surface the wind beneath the trough of the aerial billow is opposed to the directiou of propagation, but un- der the summit of the billow it has the same direction as that. Since the amplitudes at the earth's surface are diminished in the proportion e-nh: 1 with respect to the amplitudes at the upper surface, therefore these latter variations can only make themselves felt below when the depth is notably smaller than the wave-length. Variations of baro- metric pressure are only to be expected when decided changes in the wind are noticed during the transit of the wave. VII. FUNDAMENTAL FORMULAE FOR THE COMPUTATION. 1 will here give the theory of the calculation only so far as is neces- sary, so that any investigator familiar with analytical methods can verify my results. I introduce two new variables, ?/ and 0, which are so connected with rectangular coordinates x and y that e»(»+*')=a[co8 (0+»t»)-cos e] (1) wherein n, a, and e are constants. The boundary line between the two fluids corresponds to a constant positive value of rf, namely : PAPER BY PROP. HELMHOLTZ. 105 Hence for this boundary line result the equations (■"■■ cos (, /(/! = « (cos ik cos 6— cos e) ) (■■■■ sin (iiy)=— ^sin (ih) sin 6 \ ' ' -' ' (la) By the elimination of 6 this gives an equation between x and y as the equation of the boundary line. Beside the constant a which deter- mines the initial point of thea? coordinate and the n which determines the wave-length this equation contains two arbitary parameters h and £ that determine the form of the curve. We take x vertical, increasing upwards, and then for the space oc- cupied by the upper fluid, for which we use the subscript j put <,"i + 9Pii=6i(//— h— id) by which ?/'+> i becomes simultaneously a function of (x+yi). When h = i/, then ^i=0, so the boundary line on the lower side coincides with the stream line. When 7/=+ then n(x+yi)=v—i0=T [fi+ a=1 la'e ' cos^M) J' Hence for >/=h there results 1 //•,= -nx+ log( " ) + //-2> a'^"' C0S (al) C0S a^]' When we determine the value of x from the equation (1) it is seen that for rf=h there results */-2=0, therefore it is seen that the boundary line for the second medium is also a stream-line. According to equation (1) for x=— co we have COS tf.COS 7/i=COS£ sin 6 . sin 7/i=0 106 THE MECHANICS OF THE EARTH'S ATMOSPHERE. The values corresponding to these are sin //i=0 COS #=COS£ In consequence of this the equation above given becomes 1 f=-nx+\og( % )+h-2 \ He-«\cos^(ai) ** l W /j , [a. cos (aW)_ (#= — co) The first term of the right-hand member is infinite, but all the others finite when h is a positive quantity. Therefore at great depths the value of ?/'2 reduces to i/-2=-nb2x that is to say that even there also the motion is a rectilinear liow with the velocity — nb2. The second boundary condition which has respect to the equality of pressure on both sides of the boundary surface can, however, by reason of the assumptions already made, be satisfied only approximately fin- waves of small altitude. The convergence of the series under consid- eration in this case depends upon the factor e~ah. When the quantity h is positive and not too small the series converges relatively rapidly and we obtain for this case sufficient approximation to the true value, in that in the value of the pressure as deduced from equation (3) we equate to zero the terms multiplied by the first to the third power of e-h, or of 7 ■• The terms that do not contain these factors serve only to determine the value of the constant of integration which forms the left-hand side of the equation. These terms just mentioned are linear functions of cos 8, cos 20, cos 30, and by equating to zero the co- efficients of these three quantities we satisfy equation (3) to terms that contain the fourth or higher x>ower of ,.- But this assumption cor- responds only to a single possible form of wave, not to the most general form. It has been chosen as an example on account of the simplicity of computation. The three equations that we obtain in this manner are those given below. For brevity we have put n = - «i-fri'-* + g.X.{s2 — «i) _ s2 b22 . n 9 • A(s2 — «i) COS 11% z = C cos e PAPER BY PROF. HELMHOLTZ. 107 The quantity z determines the altitude of the wave, which according to equation (la) is — * = 2£.-lognat.(j±£). The three equations referred to may now be written : I. z {£ [2 - 2z2 + i;sj + sp [2 + fC2] - (1 - C2)} = 0. II. D [2.c2 — l'2] - sp . c2 _ ^ + i^ = o. III. 0 ; D [2*2 - f ;«] + $ . I - ie2 + iC2 } = 0. Of the four quantities that occur herein any three may therefore in general be determined by the fourth. Only one system of values, namely, z = 0 and Q + $ = £, leaves £ undetermined. This solution holds good for the entire lower wave, for which z is to be neglected as compared with Q. Since in general one of the four quantities in the equations I to III remains undetermined, therefore for given properties of the medium and for a given strength of the wind, there remains always one varia- ble parameter of the stationary wave; and in fact the further investi- gation shows that this variable is connected with the quantity of energy that is accumulated in the wave. The simplest method of computation is to express the remaining quantities as functions of the cos £. ~ _ 7 COS2 £-— fV 36* cos2 £-§ m „ -v 1 (COS2 £ — £).(COS2 £ — #) sp = -icos2f + ££ = -^- -^e-f C2[£(cos2£-!)-n< + £] = Q + *-£ Since D and ^ must necessarily be positive, it follows from the first of these equations that or, cos2£> § = 0.666667; cos2 £<-& = 0.642857. The equation for ^ would also allow cos2 £>§, but 0.5 < cos2 £< 0.642857. Finally the equation for C2 can be written (0.68615-cos2 e) (cos2 £+2.18615) l _u.4 x (cos, £_0>G6537) (cos2 £+1.46537)' L08 THE MECHANICS OF THE EARTH'S ATMOSPHERE. Since C2 must be positive it follows that 0.66537 0.67264= -h + v7^- For this value the altitude of the wave will still be finite, namely : H= £- x 2.5112= A 0.39967. But the fact that the value of the coordinates can no longer be de- veloped in converging series, according to the powers of cos (ad) and sine (ad), shows that a discontinuity or an ambiguity of the coordinates must have come into existence. In fact the equations (la) also show that for small values of h h sin 6 tang (ny) = 2 te v a' cos 0— cos e e2nx_ a2(CQS #_C0S f)2. From the first of these it follows that wherever tan (n y) has a finite value then cos 0 must be nearly equal to cos £, and only at the points where tan (n y) is very small and passes through zero can 6 increase and rapidly pass through the interval to the next point, where cos 6 approaches again the value, cos €. Now for such values of h the diminution of the terms in the series expressing the value of the pressure will not be rapid enough, in order to express the value of the function sufficiently well by using only the first three terms of the series, and the true form of the wave curve for such values of h can only be obtained by further approximations. However, these relations show that waves which rise too high lose the continuity of their surfaces. But sharp ridges can not occur on the surfaces of the waves except when they are at rest relatively to the medium into which they protrude. For when the medium flows around the edge there would occur infinite velocity and infinite pressure at the place in question, which must violently draw up the other liquid, as in fact is occasionally observed in high and foaming waves. Iu the case of waves that advance with the same velocity as the wind the summits can in fact have a ridge of 120° before they break into foam. PAPER BY PROF. HELMHOLTZ. 109 The above given formulae show that when eos £ diminishes from its upper to its lower value, then both D and $ and C2 must continually increase. For waves whose lengths remain constant the increase of $ and D means an increase of the two velocities 6, and b2 as well as their sum, I e., the wind velocity w=bx+b2. If the latter remains constant, then the wave length must necessarily diminish with increasing cos e. It follows from this, that within certain limits the same wind can excite this form of waves of greater and smaller wave lengths. The longer waves will at the same time have a relatively greater altitude. This relation depends upon the store of energy that is accumulated in the wave. VIII. THE ENERGY OF THE WAVES. When we investigate the energy of the waves of water raised by the influence of the wind, and compare it with that which would be ap- propriate to the two fluids uniformly flowing with the same velocity when the boundary surface is a plane, we find that a large number of the possible forms of stationary wave motion demand a smaller storage of energy thau the corresponding current with a plane boundary. Hence the current with a plane boundary surface plays the part of a condition of unstable equilibrium to the above-described wave motion. Besides these, there are other forms of stationary wave motion where the store of energy for both the masses that are in undulating motion is the same, as in the case of currents of equal strength with plane bounding surfaces; and finally, there are those in which the energy of the wave is the greater. The reason for this is to be found in the following circumstances: In the undulating masses of water two forms of energy occur, namely: First, potential energy, represented by the water raised from the wave valley to the wave summit. This quantity of work increases with the increasing height of the wave, and must always be positive; it is only absent for perfectly smooth surfaces. Second, living force is common to the two forms of motion under com- parison, and according to the original assumption there is an equal quantity of it in the portions of the fluid masses distant from the boundary surface. The difference of the two modes of motion is not affected by the participation of the more distant strata of fluid, the difference between the two motions depends only on the strata that lie near the boundary surface. The wave surface which we again imagiue to ourselves fixed in space affords to the two fluids streaming along it an alternately broad and narrow channel ; where the bed is broader the fluid moves more slowly, the upper fluid above the wave valley, the lower fluid under the wave summit. Thereby the living force of the portion flowing through a broadening of the channel will be alternately smaller, while that flowing through a narrowing of the channel will be greater thau the living force in the corresponding part of the uniform 110 THE MECHANICS OF THE EARTH'S ATMOSPHERE. stream with the plane bounding surface. But the volumetric extension of the part with diminished liviug force, that fills the broader channel, is greater than the volume of increased velocity in the narrow channel. Therefore in the sum total the living force of the diminished portion prevails. Nevertheless only the terms of the fourth degree in Z which first oc- curred in the computation by considering the terms with r:' in the val- ues of x and y, give a basis for the computation of the difference of energy. This difference, as computed for one, wave-length according to my calculation in the class of waves discussed in Section vn, is as follows : E-^A: ^=&\ [5£2-2*2]+^ [^-15^C2-|C4] 1 =a z- 2 7T 0.675148 is smaller than that of the recti- linear How of both media with the same velocities. The difference of energy is zero for very low waves ; it is negative when we pass to rel- atively high waves; it reaches a maximum, then diminishes, and is again zero for the given boundary value. PAPER BY PROF. HELMHOLTZ. HI It is sufficient to have proven that for one form of wave billows due to wind are possible, which billows have a less store of energy than the same wind would have over a plane boundary surface. Hence it follows that the couditiou of rectilinear flow with plane boundarv sur- face appears at first as a condition of indifferent or neutral equilibrium, when we consider only the lower powers of small quantities. But if we consider the terms of higher degree, then this condition is one of unstable equilibrium, in view of certain disturbances that correspond to stationary waves between definite limits as to wave-length; but on the other hand is a condition of stable equilibrium when we consider shorter waves. This result is evidently of great importance for the origin of waves. It follows from this, as we everywhere see confirmed in nature, that even the most uniform wind can not blow over a plane surface of water without on the slightest disturbance causing waves of a certain length, which for a given height acquire regular form and speed of propaga- tion. If the wind increases then the heights of all these waves in- crease, the shorter ones among them break foaming, so that new longer ones of less height can be formed. The greater energy that is necessary in this case in order to push the shorter waves up higher becomes possible in that the previous feebler wind had already given a part of its energy to the mass of water, and the new stronger wind finds this part already present there. Breaking foaming atmospheric billows cause mixture of strata in the mass of air. Since the elevations of the air- waves in the atmos- phere can amount to many hundred metres, therefore precipitation can often occur in them which then itself causes more rapid and higher ascent. Waves of smaller and smallest wave-length are theoretically possible. But it is to be considered that perfectly sharp limits between atmospheric strata having different motions certainly seldom occur, and therefore in by far the greater number of cases only those waves will develop whose wave-length is very long compaied with the thick- ness of the layer of transition. The circumstance that the same wind can excite waves of different lengths and velocities, will cause interferences to occur between the waves, and also higher and lower wave summits to follow each other interchangeably. This is a process observed often enough on the shore of the ocean. But where two wave summits of different groups of waves reenforce each other a height will easily be attained at which they break into foam, and thereby, as in the analogous case of the production of sonorous combination tones, longer waves can be formed which, when they are favored by the strength of the wind, can also grow larger. This is one of the processes by which waves of great length can arise. VII. THE ENERGY OF THE BILLOWS AND THE WIND. By Prof. H. Vox Helmholtz. Iii my communication to the Academy on July 25, 1889, I called atten- tion to the fact that a plane surface of water above which a steady wind is blowing is in a state of unstable equilibrium an 1 that the origin of large waves or billows of water is essentially due to this circumstance. I have there also shown that the same process must be repeated at the boundary of two strata of air of different densities gliding over each other, but that in this case it can assume much larger dimensions and without doubt has an important meaning as a cause of nonperiodic meteorological phenomena. The importance of these processes has induced me to investigate still more thoroughly the relations of the energy and its distribution between the air and the water; at first, however, as before, with the limitation to stationary waves in which the motions of the particles of water only take place parallel to a vertical plane in which the coordinates are re- spectively (x) vertical and (y) horizontal. Since however we can only solve even this special problem by the development into a converging series whose higher terms rapidly diminish in magnitude but offer com- paratively complex forms therefore the conclusions that we may have drawn from a knowledge of the first largest term of such a serk s are necessarily always limited to waves of slight altitude and cause the correctness of many more important generalizations to appear doubtful. Many of these difficulties have been surmounted in that I have been able to reduce the law of stationary rectilinear waves to a problem of minima, in which the variable quantities are the potential and actual energies of the moving fluids. From this problem in variations many general conclusions can be deduced as to the decrease and increase of the energy, and the difference between stable and unstable equilibrium of the surface of water. Theoretically considered, there arises here a rather new problem in so far as we have to do, not with the difference between stable and un- * From the Sitzungsberichte of the Royal Prussian Academy of Sciences at Berlin. 1890, vol. vn, pp. 853-872. Wiedemann, Ammlev, 1890. xli, pp. 641-G62. 112 PAPER BY PROF. HELMHOLTZ. 113 stable equilibrium of masses at rest, but with moving- masses that are in steady motion. Some examples of such differences have indeed been already treated, as in tlie rotation of a solid body about the axis of its greatest or least moment of inertia, and in the rotation of a fluid ellipsoid subject to gravity. But a general principle such as is given for bodies at rest, in the proposition that stable equilibrium requires a minimum of poten- tial energy, has never yet been established for a moving system of bodies. The following investigations lead to such propositions, which more- over can also be considered as generalizations of the propositions that I have deduced from the general equations of motion given by Lagrange in their application to the motion of " poly cyclic" systems.* I. THE THEOREM OF MINIMUM ENERGY APPLIED TO STATIONARY WAVES HAVING A CONSTANT QUANTITY OF FLOW. As in my paper of last year,t 1 indicate by u and v the component velocities of the particles of water during any motion that is free from vortices by the equations: dy *' I (1) fc dy ' 3 I again assume, whenever the opposite is not expresshy stated, that the coordinate system for x i/ is at rest with reference to the wave, x being vertical, positive upward, y horizontal. Therefore the wave sur- face is at rest with reference to these coordinates while the two fluids How steadily along it. The wave curve will be considered as periodical with the wave length A. On the other hand, the flowing fluid will be considered as bounded by two horizontal planes whose equations are x=Hj and x=— H2 (1«) Corresponding to this, I indicate the remaining quantities that refer to the fluid which is on the positive side of x by the subscript 1 ; those that are on the negative side of x by the subscript 2. The wave lines and these two horizontal boundary lines must be stream lines— that is to say, >/• must have a constant value throughout their whole length. Since each of the functions ip can contain an arbi- trary additive constant, therefore we can assume arbitrarily both of the values of ip for one of the stream lines. I assume that for the wave line for which x = x we have the value */-=0 • (lb) * Kronecker und Weyeratruss, JovrnTf^Mathemai., 1W4, vof. xcvn, p. 118. t[See the previous paper, No. VI, in this collection of Translations.] 80A 8 114 THE MECHANICS OF THE EARTH'S ATMOSPHERE. On the other hand, for the boundary lins, for which x = Ex ) we have Vi — fa ) and for the other boundary line, whose equation is x = —H,) I (!<*) ■we have = fa ) The quantities fa and fa, as is well known, give respectively the vol- umes of the fluid that flow in the unit of time through every section between the wave surfaces for which fa = fa = 0, and through the upper or lower boundary surface. These are the quantities which I have above designated as quantities of flow. In taking the variations of these quantities, I shall, in this paragraph, consider fa and fa as invariable. That altitude will be adopted as the initial point for .r, at which the boundary surface of the two quantities of fluid under consideration would be at rest, which is expressed by the equation (2/o+A x dy = 0 (le) y° that is to say, x = 0 is a plane such that as much water is raised above it as sinks below it. Finally the space within which lie the quantities that are subject to variation is also bounded by two vertical planes that are separated from each other by one wave length. Since the movements are to be periodical and consistent with the wave length A, the velocities at the right vertical surface and at the left vertical surface must be equal or jx tu? therefore for the same values of x 'l'r=H'L (1/) and dy" dy W According to Eq. (1) this last equation can also be written dX }X or cpr— <^=constant (ity ^ow it is known that equations (1) are resolvable when {$+ply as are possible under the given circumstances, when at the same time the values \\ and p2 are considered as invariable. On the other hand the form of the wave line is not yet determined by the conditions hitherto given, except in so far that it must be period- ical with the period A. We can however determine the form of this boundary line corresponding to the physical condition that the pressure shall be the same on either side of it, in that we require that the varia- tion of the difference between the potential energy

previously existing here]. This prolongation of the function ip into the strip just described is, as well known, only possible in one manner without forming discontinuities. Only when a cusp of 116 THE MECHANICS OF THE EARTH'S ATMOSPHERE. the appropriate function ^exists in the original boundary, therefore, es- pecially when the boundary -line forms a sharp corner, is a continuous prolongation of the function excluded. The special physical signifi- cance of such a case we shall have to consider later on. By this first step in the variation of L we obtain But now the values of ip\ and y2 are no longer zero at the new boundary, but we have there, approximately *- $ SN and in order again to make these equal to zero we must execute a second step in the variation, such that the function ?/' shall so vary that these now again become zero at the new boundaries. Since according to the general laws of potential functions we have 8"L=-81 f^diPidssz f^S'hds therefore when we (as is necessary in our case) put sn=- * is we obtain the final value: **=**+*■*=-* jl*($y-*($y]**s. . <2e) Since finally the volume of each of the two liquids must remain unchanged during the variation, therefore it is necessary that f*N*=0 (2/) Hence results the variation, = -/ ds SN\p2 - 2h-} . (2g) Here p2 and ^ designate the fluid pressure on the upper and lower sides, respectively, of the bouudary surface as they result from Euler's hydrostatic equations. Since p2 and p^ contain arbitrary additive con- stants c can be omitted. PAPER BY PROP. HELMHOLTZ. 117 When, therefore, the equation (26) is to be satisfied, that is to say, when we must have 6 | 0 - L | = 0 then must jh=Pi throughout the boundary surface, which is the con- dition of a stationary surface. The stability of the steady motion.— For any form of surface that nearly corresponds to a stationary form, and which therefore still shows dif- ferences of pressure, it iollows from the preceding that such a surface when it changes with the differences of the pressures experiences there- fore a positive displacement 6N where p2 > plt therefore the quantity (0—L) diminishes and consequently approximates to a neighboring minimum of (0— L), and must therefore depart from the neighboring maximum of the same quantity. The hydro-dynamic equations show in fact that the equality of pres- sure in such cases can only be brought about by accelerations which act in the direction from the stronger to the feebler pressure and must disturb the steady motion. Therefore the stable equilibrium of a stationary wave-form must (among all possible variations of such a form) correspond to a minimum of the quantity (0—L), just as in the polycyclic systems for a constant velocity of their cyclic motions. When on the other hand this same quantity (0 — L) attains a maximum value or a cusp value for some other form of curve, then the condition of equality of pressure on both sides of the boundary surface is at least temporarily fulfilled; but individual or the very smallest disturbances of the form of equilibrium must continue to increase: the equilibrium will thus become unstable as is actually recognized in natural water-waves by the foaming and breaking of the crests of the waves. On the other hand it is to be remarked that these propositions hold good only when the functions X, and L2 are determined as minima in accordance with the boundary conditions of the spaces within which they hold good, and for every variation in the form of the boundary line the functions experience a change in accordance with this condi- tion that they shall be minima. Under the assumptions already made, the function 0 is certainly positive and finite, since only a finite quantity of liquid is present which can be raised up only through the finite altitude //,. L is also necessarily positive but can become +o>, since the summit of the wave can approximate to the upper but the trough of the wave to the lower boundary surface and the total constant quantity of moving fluid must then be pressed with infinite velocity through infinitely narrow crevices. The quantity (0—L) must therefore have a positive value for plane boundary surfaces where $=0, and it can become — a> for increasing wave altitudes. W hether a minimum occurs between these limits, and for what value of p this could occur, can only be decided by investigation 118 THE MECHANICS OF THE EAKTIl's ATMOSPHERE. of the individual forms <>l' the waves. At'leasl one cusp value occurs for a plane surface. Only this much can be at once seen, that when an absolute minimum exists there must be a transition leading from this to the infinite nega- tive value of ($— L), which transition at first begins with an ascending value and then again diminishes. There must then be a lowest value on the transition curve between the ascending and the descending values that corresponds to a maxinio-ininimum (absolute minimum) of the quan- tity (/7as also -^j- must necessarily be finite. PAPER BY PROF. HELMHOLTZ. 119 We now indicate by a a proper fraction and imagine that we have ex- ecuted a variation of Ly to the amount expressed by rv, such as would correspond to the variation a. 60. On the other haud we perform the variation 6L2, to the amount (1— a). Then the total variation for 0 is d&=[a+(l-a)] 60, 6L = a. 6LX -f (1— a). 6L2. If now 6L\ > 6L2 we obtain the maximum variation of 6L when we make a = 1; but for the opposite case we should have to make a = 0. Tims 6L attains the greatest value that it can have for the given value of 60 and the adopted form of wave. When the greatest positive value of 6L is smaller than 60 then a value for pf can be found that in any case will make \\26L>60 and therefore, for at least one method of change of form, which need not necessarily be a minimal form, will make the variation 6 (

f2 . : . . (3a) * My attention has been called to the fact that Sir William Thomson has already given this equation as the first approximation, taking into consideration the strength of the wind. Philos. May., 1871 (4), vol. XL, p. 362, where, moreover, ths influence of capillarity is also considered. PAPER BY PROF. HELMHOLTZ. 121 The quantities p and f are dependent on each other as soon as the form of the space is given for whose boundary they hold good; so that we can put p=f. 9t where )K indicates a value that depends only on the size and form of this space. Hence there results 2* 2 ' 2 W ' ' When therefore >H experiences a change S 8ft then if f remains un- changed we have 8L=* f2. -6 L = 0 dpl=dfo=0. which is the form of variation for the stationary condition where the variation of 3 L is deduced from the variation of the form of the region, also write o- €> +6 L = 0 6 f, =6 f2 = 0. The quantities f according to their definition have the value: pX, y + A V= / (u. dx+v. dy) x,y- the integral being taken for any value that leads from the point (x, y) to the point (x, y + A). When we choose the stream-line ?/'= constant for this path between these points then the integral also iudicates a path along which a series of material liquid particles would flow. The value of the integral f1? as computed for such a series of material flow- ing particles as is well known remains unchanged, whatever motions may otherwise be going on in the liquid, provided there are no differ- ences iu the sum total of the pressures and potentials of the exterior forces between the beginning and the end of the series, and provided there is no friction. This is the same sum that also remains unchanged in the vortex motion in every closed ring of material particles. We can therefore in fluid motions consider s, u and s2 f2 as the moments of 122 THE MECHANICS OF THE EARTH'S ATMOSPHERE. motion, which remains invariable except for the influence of direct ac- celerating forces, while the quantities of flow p, and ^thereby receive the significance of velocities. Thus the two problems in variations, here solved, are completely analogous to the propositions developed by me in the theory of polycyclic systems, that S{/•.,=(), and also #=0. we should also have hx and 7*2 both equal to zero, and the living force in this case becomes i21=-'M-p2. f2=ly a22. H2X When on the other hand billows have arisen, Lx is smaller for a con- stant value of «] and therefore also of h, since, as we have seeu then a negative value of 6L\ results from an increase in the altitude of the wave. We can therefore under these circumstances put a Lx=^ar {Bx—rx).X wherein rx has a positive value that depends on the form and height of the wave, but not on Hx. If we imagine Hx increased by the quan- tity D Hx and the quantity Lx correspondingly increased by D Lx then iu the strip thus added to the field the velocity is uniformly equal to ax and therefore DLx=%a\.DHx Lx+DLx="l-«\ [(Hx + l)Hx)-rx~^.X. Therefore the same value of rx also holds good for the greater alti- tude independent of the value of D Hx. The formula (4) gives directly t>i=-fi {Hi-r{) (4«) Compared with galvanic conditions, px measures the total flow or the intensity of the current; f, is the difference of potential between the boundary surfaces. Hence (Hx-rx) is the conductivity which is pro- 124 THE MECHANICS OP THE EARTH'S ATMOSPHERE. portional to the sectional area. Therefore rt corresponds to that con- stant diminution of the sectional area which causes the current to diminish just as the irregular obstruction by the waves does. For a constant value of ax and a2j respectively, since A, Hu and H, remain unchanged, the condition that a minimum of (& + L) should exist gives d{$+L)=d&-S^a\6rl—~a22dr2=0 (46) The other minimum condition in which the a are to be replaced by P a= u H—r is > *. *i , # '*i *2 ., $n which agrees perfectly with that first found. The quantities rx and r2 depend only on the form of the wave, and are generally found by simple computations as soon as we have found the form of the functions i/-x and i/-2. Horizontal transportation of the superficial layer. — The quantity of flow \\ and p2 of the two fluids is no longer the same as it would be over plane surfaces of water for equal values of the velocities ax and a2, but it is smaller than before in the upper medium by the quantity *"!«! and in the lower medium by the quantity r% a®. Imagine now the velocity (— a2) added to both sides so that the lower medium comes to rest, but the waves progress with the velocity {—(h). Theu beneath plane boundary surfaces all motion disappears, but be- neath billowy surfaces a general current is set up of the magnitude —a2 r2, and thus the wind in the upper region travels not with a uni- form velocity (a,i+a2), but just above the billowy surface there occurs a diminution of the flow of air to the amount of ax rx. These two currents cause the mass of air and water taken together to have a different moment of motion in a horizontal direction than if they flowed with the same velocities ax and a2 over plane boundary surfaces, and this difference of moment of motion, reckoned as positive in the direction of the wind, is M = s2a2r2 — s1air] (5). This can only be equal to zero when Si a2 r2=si oxti (5a), or, if we introduce w, the velocity of the wind, w = at + a2 . , . „ . . . . (56), PAPER BY PROF. HELMHOLTZ, 125 then equation (oa) becomes s2 r2 ic Si rx + s2 r2 sx r, to = «! = '_' Since now rx and r2 have values that differ but little for the ordinary waves (as the subsequent computations will show), and since for air and water s2 ~~~ 773.4' therefore this condition gives the rate of propagation of the wave against the water as approximately w <*2 = ™ 774.4 For waves of low altitude equation I, Section vn of my paper of the previous year,* neglecting the small quantities z and p, becomes «, a,' + W- **<;-*> lit If we put w=10 metres which corresponds to a rather strong wind, then for low waves of a constant moment of motion, we have ax = 9m.98709 a2 = 0m.(N291 A = 0m.082782 These waves of only 8 centimeters in length evidently can corre- spond only to the first crumpling of the surface, such as a strong wind striking upon it immediately excites. Only when the same wind blows for a long time over these initial waves, and gives them a part of the moment of motion of a long stretch of air, can waves be thereby pro- duced with greater velocities of propagation. Hence in accordance with experience it follows that wind of a uni- form strength striking a quiet surface of water can only produce more rapidly running waves, namely, those that are longer and higher, when it has acted for a long time on the waves that first arose, and has accompanied these for a long distance over the surface of the water. At the same time it also becomes clear that for a uuiform wind the waves can onlv increase in size when the wind advances faster in the same direction than the waves themselves. Energy of progressive waves on quiet water. — As in the case of the moment of motion, so also with the storage of energy in the wave. Our previous comparisons of the energy of different waves among themselves has reference to the energy of relative motion of the fluid with reference to the stationary wave. * [See page 107 of this collection of Translations.] 126 THE MECHANICS OF THE EARTH'S ATMOSPHERE. The well known proposition that the living force of any complex me- chanical system is equal to the living force of the motions relative to its center of gravity plus the living force of the motion of the center of gravity at which we imagine the ichole mass of the system to be concentrated, can, with only a small change in the method of expression, be applied to our case. For since the total mass of the system multiplied by the velocity b of the center of gravity, gives the amount of the total mo- mentum of the system in the direction of this velocity, therefore we can also put the living force 8 of the center of gravity !>> = £ M b = i m b3 (0), where M is the momentum of the whole system in the direction of b and SJt is the mass of the system. If we now compare with each other two different conditions of motion and configuration of the system in which Lx and L2 are the living forces of the motions relative to the center of gravity, #i and &2 are the potential energies, bi and b2are the parallel velocities of the center of gravity, then the difference in the total energy of the system in the two conditions is Ex - E2 = $! - $2 + Lx — L2 + I m. bi2- -i m. b22. If now, without changing the relative motions, I in both cases add the quantity c to the velocity of the center of gravity, then the above difference of energies changes into Ex'-E2'=Ei-L}2+c {Mx-M2). If {Mi — ilf2)=0, then the value of the difference in euergy is not changed by the addition of the velocity c. This must be true even when Hi and H2, and therefore the masses of the moving fluids, increase to infinity, since for our undulating fluids the differences (El—E2) and (Mi—M2) are finite for each wave length. Therefore the difference of the energy for stationary waves and for stationary deep water will be equally great only for waves that satisfy the condition (5a). According to the propositions above deduced, sta- tionary waves of this kind must have less euergy than smooth water, which is therefore also true in this case for this kind of waves above quiet water. For waves that have larger values of a2, the addition of a common velocity (— a2), which brings the deep water into rest, changes the dif- ference of energy between the two states, that of a smooth surface and that of a wave formation, by the quantity Ei1— E2'=Ei— E2-\-a2 {s2a2r2 — siaxri\ The index 1 refers to the billowy surface, the index 2 to the plane surface, the accented E' refers to quiet deep water, the non-accented E refers to stationary waves. PAPER BY PROF. HELMHOLTZ. 127 Hence it results that when waves of considerable progressive veloc- ity trench upon quiet deep water the generally very small differences (Ex— Hz) lose their negative and assume a positive value. Here also the energy that is given to the previously quiet water in the form of an elevation of its surface and the living force of its mo- tion must be abstracted from the atmosphere. In order to obtain a sufficient amount for the formation of large waves, it will on this ac- count be necessary that long layers of air shall blow over and shall give up a part of their living force. In the first moment when a new gust strikes the surface of the water stationary waves only can be formed for which 31=0 and JEi—E2=(i and a2 has the value given in equation (5a). The last condition shows that these waves will be near the point of spirting, as we in fact often see in the case of small ripples suddenly excited on the surface of the water. Moreover in these small ripples, as Sir William Thomson has shown, the capillary tension of the liquid comes iuto consideration, which somewhat increases the store of energy of the billowy surface. In general therefore, stationary waves are not formed immediately at the beginning, since the waves of constant momentum would leave behind an excess of energy. But when from the very beginning waves that have partly a positive and partly a negative difference of momen- tum and of energy are successively produced on the quiet water, then the sum of these differences can become zero. These systems of waves, having different wave-lengths and progressive velocities, cause mani- fold interferences as they progress, and, according to the principle given by me for combination-tones (which in its application to the tidal wave has already received a very beautiful confirmation by Sir William Thomson's analysis of the tidal observations collected by the British Association), waves of greater wave-length can gradually be formed. So long as the wind outruns the waves it steadily increases the store of energy and the momentum of the waves, and furthermore, so long- as the energies computed for stationary waves diminish and can form a still lower minimum, the inclination to attain the form of least en- ergy under the cooperation of all the small perturbations which the other concurrent waves bring about, in the case of nature, will develop still further. This will finally lead to the value corresponding to the formation of a cusp and to the foaming of the upper ridge in case this can be produced by the given wind velocity. In April of this year [1390] I endeavored by observations that I in- stituted at the Cape of Autibes [near Marseilles] to arrive at some con- clusion as to these consequences drawn from theory. With a small portable anemometer I measured the strength of the wind directly at the edge of the steep cliff of the narrow tongue of land which projects rather far into the sea. However, the observations showed that many times a stronger wind must have prevailed out on the sea than I had ; been able to observe on shore. I also counted the number of approach- ing billows. 128 THE MECHANICS OF THE EAKTll's ATMOSPHERE. With water-waves the same as with souud-waves it is to be assumed that, through all deviations, delays, aud diminutions that they experi- ence, the time of vibration remains unchanged. This time may there- fore be determined near the shore even though the progressive velocity in shallow water is changed and the form and the length of the waves change. The number, J\T, of the waves in a minute is expressed by N= 60.O, A When a.2 increases to na2 then A increases to n2\, as shown in my paper of a year ago, and therefore n A velocity (h=^ metres would give 0.4 waves per minute ; on the other hand a velocity a2=5 would give 18.8. The counting of the waves without registering instruments is now not to be executed with great accuracy, since on the sea, so far as I have seen it, there are always numerous adjacent waves of rather different periodic times which interfere and give phenomena corresponding to the acoustic beats. During the minimum of motion one can easily make errors in the counting; by repeated countings at the same place we obtain therefore variations of about one-tenth or even more of the desired number. The strength of the wind that I observed on the shore did not exceed 6.1 metres per second. This was on the evening of my arrival in Antibes, April 1, 1800; the wind was from east southeast ; I counted between 8.5 and 10 waves per minute. On the next morning, April 2, there were still 10 to 10.5 waves per minute, although the wind had almost entirely gone down. This number of waves would be expli- cable only when a wind about 10 metres per second had blown steadily over the open sea. On the 2d of April the wind rose in the course of the day to a velocity of only 4 metres per second. Yet on the 3d of April also the number of waves was still 0.5 with a very feeble wind ; on the 4th of April for the first time an increase was perceptible up to 12.3 waves per minute. During a series of quiet days the number of steadily diminishing waves gradually increased to 17 or 18. Finally on the 7th of April the wind began again to increase. In the morning I found a velocity of 3.3 metres per second, which in the course of the day increased to 5.5 and brought the number of waves down to 1L 5. This time, however, the location of the increased wind was demonstrable. In Marseilles during the previous night a severe whirlwind had prevailed and the larger waves excited by it stretched as a sharply defined dark-gray band from the sea horizon hitherward and reached Cape Antibes about midday, long before the stronger wind that had giveirrise to them and which had morever at the latter place by no means the same force as in Mar- seilles. PAPER BY PROF. HELMHOLTZ. 129 These few observations therefore show a connection between the number of waves per minute and the strength of the wind and even an agreement, at leastin the order of magnitude. But the numbers of waves are all somewhat smaller than they should be as computed from the strength of the wind on shore and leave us to conclude that a stronger wind must have prevailed in the open sea. They show however also that the reaction of a strong wind may last many days. For a progressive velocity of 10 metres the waves would in one day travel 7f degrees of longitude. Therefore, had the Mediterranean even to the Gulf of Sidra been on the 1st of April covered with waves ex- cited by a strong breeze of 10 metres velocity, these would need two and a half days before the last ones would reach the coast of southern Frauce. It will of course be possible to solve the problem more thoroughly only when we have at hand continuous registers of the billows and ex- tended observations of the velocity of the wind. These latter are un- fortunately not yet collected for the month of April of this year, or at least not yet published, and could therefore not be used by me. 80 A 9 VIII. THE THEORY OF FREE LIQUID JETS.' I'.v Prof. G. KlRCHHOFF. Helmholtz in his communication on discontinuous motions in liquids, Berlin, Monats-berichte, April, 1868,t has for the first time determined the form of a free jet of liquid in a special case. The method used by him in this determination can, as will here be shown, be so generalized that it leads to the solution of the same problem for a large number of cases. It is assumed that the fluid is incompressible, that no exterior forces act upon it, that its particles do not rotate, that the currents are steady, and finally, that the movement is everywhere parallel to a fixed plane. Let x and y be the rectaugular coordinates of any point of the space occupied by the flowing liquid reckoned parallel to the fixed plane and let cp be the velocity potential at this point, then cp is a function of x and y such that it satisfies the equation In this equation C^L anti iCf) are the velocities parallel to the axes of jx jy x and y and if p is the pressure and /? is the density, then we have further >=°-my can be any function of z. Therefore the equation of any curve of flow or stream line is f =constant, and we have dx ?

\JcpJ +VcW (d

. f£-VWl«Wt"> £X£)-' that is to say, the stream line corresponding to the value of tp can form a free boundary to the moving liquid in that portiou which corresponds to the range of cp. If there are many values of >p for which /(a?) has the described property then all the stream lines that correspond to these values can be free boundaries. In general co is defined by the equation above given for dz doo as a many- valued function of z for any definite assumption as to /(<»). Let the region of z, that is to say the space filled with the moving liquid, 132 THE MECHANICS OF THE EARTH'S ATMOSPHERE. be so bounded that, within it, no branch of go merges into another ; such a branch, therefore, represents a possible mode of fluid motion. The desired object will be attained when the region of oo is appropri- ately bounded. In reference to the boundary of the region of oo it is recognized, first, that it is a line that returns into itself and without cutting itself and that consists of parts for which >/• has a constant value and of parts for which

has a constant value. Furthermore yf f(oo)f(oo)—l should also be made a single- valued function of oo. in that through those points for which /(&?) = ± 1, the sec- tions pass for which >/• lias a constant value. For any point of the region of oo the sign of the radical quantity is still at our disposal. It' points occur for which /(ca) is infinite or infinitely great,* then for one of these points we may make Vf(G0)f(G0)-l=+f(G0) and assume that tbis equation holds good for them all. It is further assumed that the function /(co) is only infinite at its cusp points if it is so anywhere, and even here it is infinite only in such a way that if /(coo) is infinite then (go— oo0)f(oo) approximates to zero when oo has a value approximating that of oo0. Within the designated region of oo therefore z is a single-valued function of this variable and such that it is never infinite. Now consider oo as a function of z. The region of z that corresponds to the adopted region of oo does not extend through infinity, and is bounded by a line that returns into itself aud which is made up of the lines whose equations are cp=— ao and r^= + co and of stream lines; a certain portion of the latter can be considered as a free boundary of the moving fluid, the other part cau be considered as a fixed wall. Within this region of z, go has no cusp point, since at no point of it dz does -y- become zero. Therefore under the condition that the boundary of the region of z shall not intersect itself, go becomes within that region a single valued function of z. This function of z is completely determined as soon as one has found a single value of z corresponding to a given value of oo. (I.) An example that constitutes a generalization of the case treated of by Helmholtz is obtained if we put _ /(Gj)=7v + e-(» * By infinite, I designate the reciprocal of zero, but by infinitely great, the recipro- cal of an infinitely small quantity. PAPER BY PROF. KIRCHHOFF. 133 where, as also iu the following examples, Tc indicates a positive real fraction, and where the region of oo is bounded by the lines //<=0 q>=— oo f — Tt ^=+00 The expression adopted for /( oo) is single valued. The multiple points of Vf(oo)j\oo)—l that do not lie outside the region of go are the points p = n and

_ y/\k-e-+)%—l)& log [fA 9 where the root (as also hereafter every root of a positive quantity), is taken to be positive. By these equations the positive half of the axis of x is represented; this is to be taken as a fixed wall; at the initial point of coordinates it merges into the free boundary. For this free boundary, namely, for y=7t and cp > — log (1+&) we have -/ 4> (Jc—e-*)d cp l<>g(,-K) y=- / l—(lc—e-*)* dtp 1-K) Furthermore for tp=0 and cp <— log {1-lc) we have x= f(fc+e-*+ J \k+e-*)2-l)d — log (1— fe) x= I (fe+c-*) dcp+a J —log (I—*) y=- / Ji-{l'+e-*)2 dtp+b J -log(l-S) where a = A' loS iJEfc-2- n ^x~¥ b=—2nh The first part of the stream line ^=0 which is a straight line paral- lel to the axis of x and extending to the point x=a, y=b, is to be con- sidered as a fixed wall; the second part is to be considered as the free bouudary of the outflowing jet. 134 THE MECHANICS OF THE EARTIl's ATMOSPHERE. The approximate course of the Hues tp=n and //>=<> is shown iu Fig. 4. The completion of the boundary of the region of z is formed by the line (p=— go, namely, x=2 k (p—2e~'l'Gos f/'+«i y=2 k ip+2e~8imp-{-bi and by the line, 2 where ai, 6i, «2, 62 are constants whose values are easily obtainable and which are partly used in the computation of a and b. The first of these two lines can be defined as a half circle that is described with an infinitely large radius about the origin of coordinates; the second is astraight line that is perpendicular to the jet at an infinitely gre.it dis- tance from the origin; at this dis- tance the jet forms an angle with the positive axis of x whose cosine equals k. If we assume that k equals 1 then a becomes infinite and the point Fig 4 («, b) removes to infinity ; the region of oa cau in this case be bounded by the lines f=7t and f= — n instead of by the lines ^=7rand Fi-r. «. Fig. 5. If we make 7c equal zero then will b equal zero; iu this case the boundary of the moving fluid is represented by Fig. 6. PAPER BY PROF. KIRCHHOFP. 135 (II.) As a second example the case where will be treated and the region of go stretches indefinitely far in all direc- tions. In order to make/(&?) a single- valued function we draw a section from the point oo=0, for which section ?/=0 and

0 and assume that for y>= + 0 and '/=+0 the real part of V o is positive. The cusp points of the curve y/Tfoj) /'(<»)— I are the points for which go=0 , — _=1— Jc, 1 —. — =_(i + A:); therefore they all lie on the section already drawn VIP therefore do not require the making of anew section. As concerns the sign of Vf (&))/( go)— I it must be so determined according to the adopted rules that the real part of this radical quantity shall be positive for <£>= + (), and i/- = -\-0. Finally it is assumed that go and z disappear simultaneously. The line for which >/•=(), and

0, is the bouudary of the region of z. This line is composed of many parts which are to be distinguished from each other. For ?/>=+0, and 0</•=— 0, and Q n_fc\2 we have (to dqj ">+v?Z-^-(»hJ and for f/'= — 0 and cp> (1-,M2 we have dx dqj 1 dy /, /. 1 V The lines that are represented by tbe integrals of these equations, when we determine the constants of integration so that tbese lines start from the previously indi- cated termini of the fixed walls, are tbe free boundaries of the moving liquid. The other boundaries of the region of z lie at infinite distances, as is seen from the fact that when &?=oo we have dz JC doo = lc-iy/l-k2 Fig. 7. this equation shows at once that at an infinitely great dis- tance from the origin of coor- dinates the flow takes place with the velocity 1 in a direc- tion that forms an angle with the axis of x whose cosine is 7c. Figure 7 illustrates the boundary of the region of z ; besides this boundary the figure also gives the stream line for which >/'=0, and <^<0. (III.) Still one more example may be introduced. Let there be J v ' VI— er» and let ip vary between — n and +7r, but q> between — oo and +ao. From the point oo=Q draw a section for which ?/==0, and

0, and assume that for qj=-\-0, and j/<=+0, the real part of f(co) is positive. The points of bifurcation of \Z/(&? )/(<*?) — 1 are the two points oo=0, and G9=— log (1— W) both which are found upon the section that has been drawn. The sign of the radical quantity \/ f(oo)f( go) — I is determined by the rule that its real part shall be positive for ^= + 0, aud //<= + 0. PAPER BY PROF. KIRCHHOFF, 137 Finally we assume that go and z disappear simultaneously. At the boundary of the region of z we have, first the line for which ^'=0, and

0. This line is composed of the following portions : For ^=+0, and 0<(p<— log (1-F) we have, x -/0 (tt^^+Vi^-1 y* y=0 For if>=— 0 and 0<_log(l-fc2), therefore dx= h dV_ = _ f~T ~P~ d

_log(l-£2), whence — = _ ft ^ = _ /T~ ~^~ d/'<0, we have ■=- -= — A; — i Vl— ¥ and for /•>(), we have _=fc_« \Zi_F 138 THE MECHANICS OF THE EARTH'S ATMOSPHERE. For (p=—cc, we tberefore have y=+co, and the stream flows with a velocity of 1 in the direction of the negative axis of y; for <^=-|-gc, we have x= =p oo, and y=— oo, and the stream flows with a velocity of 1 in a direction that makes an angle whose cosine is ^pfc with the direction of the positive axis of x. In Fig. 8 the boundaries of the moving fluid are represented for this case. JO) FiK. 8. IX. DISCONTINUOUS MOTIONS IN LIQUIDS.* By Prof. A.. Oherbeck. I. It is customary to designate by the term discontinuous fluid mo>- tions, those phenomena of movement in which the velocity is not through- out the whole space tilled with the fluid a continuous fuuctiou of the location. Therefore iu such movements there occur surfaces within the fluid that separate from each other regions within which the veloci- ties differ from each other by finite quantities. The fundamental prin- ciples of the theory of these motions were first given by Eelmholtz.t If we assume that a velocity potential (tp) does exist for so-called steady fluid motions then the hydro-dynamic differential equations can be summarized in the one equation, Now Helmholtz has shown that the pressure^ and consequently the velocity can be discontinuous functions of the coordinates and that there are a great number of phenomena of motion for which the assump- tion of a discontinuous function is necessary. Especially has this theory been applied by Helmholtz and by Kirchhoff to fluid jets,! an(l tue boundaries of free jets can be given under the following assumptions: (a) That no accelerating force acts upon the fluid. (b) That the movement is steady. (c) That the movement depends only upon two variables, ./and y, and is therefore everywhere parallel to a fixed plane. If in other cases, for instauce for jets that are symmetrical about an axis or that are under the influence of the accelerating force of gravity, *Read at the session of the Physical Society iu Berlin, May 11, 1877. Translated from Wiedemann's Annalen der Physik unci Chemie, 1877, vol. n, p. 1-16. t See the Berlin Monalsberiehte, 1868, p. 215 [or No. II of this series of Translations.] t See Crelle's Journal vol. lxx, p. 289-299, [and Nos. Ill and VIII of this collection of Translations.] 139 140 THE MECHANICS OF THE EARTH'S ATMOSPHERE. it is not yet possible to determine the free boundaries by computation, then this is only because of the analytical difficulties. In general, bow- ever, one can judge of the nature of these boundaries from a considera- tion of the results already found. The mathematical investigations just referred to hold good equally well for liquid jets that are bounded by quiet air as for those that are bounded by similar quiet liquid. In the actual production of such liquid jets it of course makes a great difference whether we allow water to flow into the air or water to flow into water. In both cases disturbing circumstances occur of which the mathematical theory takes no consid- eration. The jets of water projected freely into the air have been most thoroughly investigated.* In these experiments the formation of jets occurs just as would be expected according to theory. On the other hand, however, it is known that water jets are influenced to an important extent by the capillary tension of the free surface, and that in consequence of this at certain distances from the orifice they break up into drops. If we allow a liquid to flow into a similar quiet liquid then these capillary effects do not occur; but in place of this another disturbing cause, the viscosity, influences the phenomena. Tue viscosity has hith- erto not been taken into consideration in the theory of the discontinu- ous movements of fluids. If we attempt to consider it we stumble upon a peculiar difficulty that has led the present author to experimentally investigate this class of fluid motions. ii. It is well known the theory of viscosity of fluids can be developed from the assumption first framed by Newton, t namely, that the retard- ing or accelerating influence of two portions of fluid flowing past each other with different velocities is proportional to their relative velocity. Especially has O. E. Meyer from this hypothesis developed the general differential equations for the motion of fluids.J If we assume that all parts of the moving fluid describe parallel paths, say in the direction of the axis of y, and that the velocities v are only functions of x and that finally /< is the coefficient of viscosity, then will the influence of two neighboring parts upon each other be represented by the expression dv dx' * Besides the older experiments of Bidone and Savart see especially Magnus, Pog- gendorff Annalen, vols, xcv and cvi. t Mathematical Principles of Natural Philosophy : German translation by Wolfers, Berlin, 1872, p. 368. t See Crelle, Journal, vol. lix, pp. 229-303, and Poggendorff Annalen, vol. cxm, pp. 68, 69. PAPER BY PROF. OBERBECK. 141 If v is a discontinuous function of x, then at such a locality the dif- ferential quotient will be indefinitely large. Therefore two neighbor- ing portions would exert an indefinitely great influence upon each other. If therefore one of the fluid portions is at rest while a neighboring por- tion that belongs to the jet flows by the first with a constant velocity communicated to it by some exterior influence, then the first or quiet particle must immediately begin to take part in the movement of the second, but the second on the other hand must begin to lose a definite fractional part of its velocity. The jet must therefore rapidly set the surrounding quiet fluid in motion with it. It would according to this appear to be doubtful whether sharply defined jets such as are de- manded by the above-mentioned theory of Helmholtz could be formed in a fluid subject to viscosity. The few experiments made hitherto upon this question appear to con- firm this suspicion. Especially notable is an investigation by Magnus (Poggeudovf£ Annalen, lxxx, pp. 1-40), who allowed pure water to flow from a cylindrical opening into a weak solution of salt and, by means of a glass tube drawn out into a fine point, led away a small quantity of the inflowing water in the neighborhood of the opening. The liquid thus caught was examined as to its salinity. From the latter one could calculate to what extent the inflowing liquid had besome mixed with that which was previously in the vessel. It resulted that pure water could not be caught at any point of the inflowing liquid; that therefore everywhere the original quiet liquid was carried along with the mov- ing liquid. The analogous case of jets of air and of smoke, as also that of the free jets of water in the air, demonstrates that in all these, we have to do with phenonena of very slight stability. It is well known how sensi- tive such jets frequently are with respect to the feeble periodic disturb- ances produced by waves of sound.* It seemed to me therefore of interest to investigate more accurately the formation of water jets in water and therein to utilize a method that allows of following the course of the phenomena of motion better than was possible in the experiments of Magnus. This object is most simply attained in that we allow feebly tinted water to flow into colorless water. Fuchsiu is used as coloring material. It is well known that with a very small quantity of this material an intense red color is produced with no fear lest hereby the specific gravity of the water be essentially changed. In the first experiments performed with this it resulted that the jet of colored liquid broke up at a very slight distance from the orifice into reddish clouds and drops that mixed with the quiet liquid and carried it along with them. By further investigation however, it became pos- sible to determine conditions under which real jets of considerable length and sharp boundaries were formed. These were of great sta- * See John Tyndall on Son^dTpp. 289-292 of the German edition edited by Helm- holtz and Wiedemann, Brunswick, 18(59. 142 THE MECHANICS OF THE EARTH'S ATMOSPHERE. bility, so that small disturbances had only a rapidly diminishing in- fluence upon their course. At the forward end of these jets there formed peculiar surfaces of flow that plainly allowed the influence of viscosity to be seen. These phenomena of motion are of remarkable beauty and delicacy, of which any one may convince himself who per- forms the easily repeated experiments. Since the theoretical investigations mentioned in the introduction treat of the modifications of jets by solid bodies, and Kirchhoff especially gives a series of interesting examples bearing upon this, therefore this question has also been taken into consideration in my ex- periments. Very stable forms of jets are also thus formed that have more similarity with those deduced by theory than one could have ex- pected. in. The experiments were made with the following simple apparatus: A cylindrical glass vessel (Fig. 9), of about 60 centimetres height and 12 centimetres diameter, was tilled with water. Into this there passed a flow of water from a filter through au India-rubber tube, a glass stop-cock, and a glass tube. The filter, as also the entire tubular system, was filled with the colored liquid. After filling with water the glass cylinder (in whose place one may also use any large glass vessel), one must wait a long time until the motion of the water has been destroyed by viscosity. The experiment succeeds best when the water has stood for many hours in tbe cylinder, since then cur- rents resulting from differences of temperature are no longer present. By a quick opening of the glass stop cock one can allow a definite quan- tity of colored liquid to enter into the quiet liquid, or by a longer opening one can attain a steady stationary current. By elevating or depressing the filter one can easily regulate the height of the upper fluid level. The use of a small difference of pressure was found to be the principal condition for the maintenance of regular current forma tions. The majority of the experiments if no other problem was on hand were executed with an excess of pressure of about 20 millimeters. By means of proper arrangements solid bodies could be opposed above the jet. For exact observation it is necessary to fasten a surface of white paper behind the glass cylinder. Fig. 9. IV. In order to understand the formation of jets it is advantageous first to learn the behavior of a definite quantity of liquid entering under PAPER BY PROF. OBERBECK. 143 a small excess of pressure iuto the quiescent liquid. I therefore begin with a description of the experiments relative to this. If we allow the stop-cock to be opened for only a short time, then even with the smallest differences of pressure of two or three milli- metres, a sharply defined mass of liquid penetrates into the quiescent liquid. The origiual form of this mass is soon modified by viscosity and by the participation of the hitherto quiet liquid iu its motion, in a peculiar manner, and finally it rolls itself into a ring. The colored mass of liquid goes through the series of forms presented iu Figs. 10, 11, 12, and 13. Of these drawings, as of most of the following ones, it is to be noted that they represeut a section of the mass of liquid by a plane that passes through the axis of symmetry of the formation. In order to find the true form therefore, one must imagine the figure revolved about this axis. O Fig. 13. Fig. 12. Fis. 10. Fig. 11. With the form of Fig. 13, the ring formation is completed. More- over iu general even for differences of pressure of 10 to 20 millimetres, the living force of the liquid was consumed so that this figure long- floated motionless in the colorless liquid. If we use somewhat larger differences of pressure we observe that the liquid within the ring continues rotating for a longer time. The original progressive movement has therefore been transformed into a vortex movement. The vortex movements have been theoretically treated by Helmholtz* and he has in the introduction to his memoir referred to the necessity of the transformation of any current or move- ment that has a velocity potential into a vortex movement in conse- quence of viscosity. Many other consequences drawn by Helmholtz in his memoir just referred to can be easily observed by the help of the apparatus used by the present writer. *Crelle's Journal lv. pp. 25-56, [and No. II of this collection of Translations. ] 144 THE MECHANICS OF THE EARTH'S ATMOSPHERE. If by alternately opening and closing the stopcock we allow two drops to enter into the colorless liquid in rapid succession, then there arises a ring formation for each drop aud the following one always catches up with the preceding one. Different cases are then possible, accordiug to the differences of pressure that are used ; if these are slight then the second ring is not able to penetrate the first one and a formation, as shown in Fig. 14, remains for a long time visible in the fluid. With greater differences of pressure, on the other hand, ring No. 2 passes through ring No. 1, since the former contracts while the latter expands. One can then observe that afterwards ring No. 1 en- deavors on its part to pass through ring No. 2. But generally the living force is by this time consumed, so that ordinarily the two rings settle into the formation shown in Fig. 15. This interchanging passage 2T Fig. 14. Fig. 15. of the vortex rings through each other was predicted by Helmholtz from theory in the memoir above referred to. Keusch has occupied himself experimentally with the formation of vortex rings.* After having described in detail the formation of smoke rings in the air, he passes to the formation of rings by "the sudden en- trance of a small quantity of colored liquid into colorless liquid. Al- though in his arrangement of the experiments the transition of the progressive iuto the vortex motion is very rapidly completed, still he also has frequently observed the intermediate stages shown in Figs. 11 and 12 and described them very appropriately as mushroom formations. The manner of this transition is seen directly from the examination of Figs. 10 to 13. Evidently there arise two currents in the quiescent liquid. The one current, indicated by the arrows A and B, is produced by the progressive movement of the drop, which moves forward in the liquid almost as a solid body. The other current, in the direction of the arrows C and 7), is principally produced by viscosity. The forma- tion of the spiral surface of rotation is finally the necessary consequence of these two opposite currents. Poggeudorff's Annalen, vol. ex, pp. 309-316. PAPER BY PROF. OBERBECK. 145 V. We can now pass on to the formation of jets proper by steady currents. If we allow the stop-cock to be open for a long time there arises (at first rapidly, afterward slowly) a jet whose upper portion has great similarity with the forms hitherto described. The jet soon attaius a certain altitude that depends upon the difference of pressure and above which it does not ordinarily go, or at least ouly with extreme slowness. Thus for a difference of pressure of 5 millimetres the alti- tude of tbe jet is about 20 millimetres ; for 10 millimetres pressure the altitude is about 80 millimetres; for 20 millimetres pressure the altitude is 200 millimetres; and for 30 millimetres difference of pressure the jet attains the upper limit of the water at an altitude of about 400 millimetres in about 80 seconds. The colored liquid spreads out over the surface of the colorless water and thence diffuses very slowly downward. The above given numbers do not present any general law, but only give approximately the connec- tion between the altitude of the jet and the difference of pressure. The former also depends some- what on the specific gravity of the inflowing liquid, which varies a little with thequantity of added coloring material, It depends also on the size of the discharg- ing aperture. Moreover the form of the front part of the jet is not always exactly the same; in the figs. 10 and 17 are given two of the ordinary forms of jet. In both these forms the jets proper are the same; the bell-shaped expansion, however, is formed in a somewhat different manner, perhaps conditioned by small variations of tempera- ture in the colorless liquid. By the avoidance of all disturbances the jets here described remain many minutes entirely unchanged, Only the bell-shaped portion con- tinues to extend slowly somewhat further downwards. Moreover, with respect to small disturbances the jets showed themselves by no means very sensitive. If by a gentle pressure on the India-rubber tube the velocity of the discharging liquid is diminished for an instant, then water presses from all sides into the jet; after the cessation of this pressure the original form of the jet is immediately resumed. Even 80 A — -1Q Fig. 16. rig. 17, 14(1 THE MECHANICS OP THE EARTH'S ATMOSPHERE. when the pressure on the rubber tube is periodically increased and diminished for a long time the continuity of the jet is not completely broken. Such a jet presents a very remarkable appearance, which is reproduced as well as possible in Fig. 18. The phenomena hitherto described occur with /"" ^\ differences of pressure of 60 millimetres at the /^\ /""">. maximum, but very different results are obtained I C\ t, ( /^)if larger differences of pressure are used. With 80 \ly \ I vJ-/or 90 millimetres we obtain jets of tbe greatest | | sensitiveness. By every small disturbance the con- | | tinuity of the jet is broken, and it must then form \ \ for itself a new path every time. Above 100 milli- metres difference of pressure there are formed only very short jets in the immediate neighborhood of the opening. These at a slight altitude break up into a cloud of individual small drops that under the rapid motion immediately mix with the color- less liquid. When (as an experiment) colored liquids were used whose specific gravity differed considerably from that of the colorless water, no regular dis- continuous currents could be obtained; thus in one experiment a solution of salt was added to the colored water, in another experiment some alcohol was added. The salt solution immediately after its discharge fell in thick, irregular, heavy drops back upon the discharge pipe, while the alco- hol moved in very thin threads, frequently broken up, toward the upper free surface of the water. From the experiments hitherto described it follows that in fact steady jets form with small differences of pressure. The viscosity therefore does not prevent discontinuous currents. Viscosity appears in general to exert so unimportant an influence upon the cylindrical portion of the jet formation that we are tempted to assume the real possibility of the sliding of moving particles of water past those at rest, as the simpler theory assumes to be the fact, without any consideration of viscosity. If, however, the transition from the finite velocity of the jet to the quiet fluid does not take place within the thickuess of a mathematical cylin- drical surface but gradually within a layer of a definite thickuess then this thickness can only be extraordinarily small and appears not to change with the time. That on the other hand the viscosity plays an important role in the origin of the jets is already mentioned above. The principal proof of this consists in the invariable formation of spiral sur- faces of rotation into which the jet is transformed. The origin of these assumes that the colorless liquid iu the neighborhood of the jet receives a certain velocity in the direction of the jet, PAPER BY PROF. OBERBECK. 147 The greater sensitiveness of the jets for large velocities of current, as also the impossibility of forming alcohol jets in water, is a direct consequence from the theory of discontinuous fluid motions. Since the difference of pressure in the moving and the quiet fluid is proportional to the square of the velocity, therefore for greater velocities the quiet liquid presses directly into the jet as soon as a slight disturbance occurs in its uniform course. When Anally, with more rapid outflow, such disturbances occur continually, then in general a jet can not form. VI. As already remarked above it is of interest to know the path that a jet will describe when it meets a solid body iu its path. The bodies used for this purpose by me were of different kinds, ami by a simple arrangement were brought in the neighborhood of the discharging aperture before the jet was produced by opening the stopcock. It is of course understood that at the beginning of the experiment one waited a long time until the movements of the liquid caused by these operations had subsided. Equally also was the solid body first freed from the air bubbles that adhered to it. The processes that occurred are most easily seen by considering the following experiment. If the jet strikes upon the sharp edge of a thin sheet of iron that passes parallel to the directiou of the jet and through its axis, then it is cut into two por tions which are deviated from the vertical directiou of the current. The angle between these side cur- rents and the original direction of (o\Tq) the jet becomes smaller little by little. The cause of this phenome- non consists in the fact that not only the solid body but also the fluid attached thereto force the moving fluid into a departure to one side. With currents of longer duration on the other hand a part of the quiet liquid is carried along so that the two upper branches of the jet slowly change their direc- tion of motion and more and more nearly approach rig. 20. the plane of the sheet-iron. Still one can always observe quiet colorless liquid between the moving colored liquid and the sheet-iron. The progress of this phenomenon depends upon the original difference of pressure or correspondingly upon the velocity of the flowing liquid. For a small velocity the current flows as shown in Fig. 19; for greater velocities, on the other hand, the two portious of the jet after a time take the position shown in Fig. 20, where the dotted part of the figure is intended to show the initial direction of the current. 148 THE MECHANICS OF THE EARTH'S ATMOSPHERE. The peculiar behavior of the originally adherent quiescent liquid, which is afterward carried along, explains the slow changes in the path of the current that is also observed with other solid bodies of different shapes. If a jet strikes upon a small brass sphere then with steady flow the stream path gradually takes the form shown in Figs. 21, 22, 23, and 24. W Fig. 21. Fig. 22. Fig. 23. Fig. 24. We see how at first the sphere and the adherent liquid force the moving- liquid to a deviation almost at right angles from its original course. Then gradually the quiescent liquid is carried along, the stream sur- face follows the surface of the sphere continually more and more closely. A consideration of the thin stream surface that finally en- closes the greater part of the sphere tempts one to assume that the moving fluid glides along the surface of the sphere. At least, by means of small solid bodies occasionally occurring in the liquid, one recognizes that in the immediate neighborhood of the fixed obstacle the liquid moves with finite velocity. The phenomena just described do not appear to depend especially on the substance of the solid body, assuming of course that it is provided with a smooth surface. Instead of the brass sphere an ivory sphere may be used. This is in the same way gradually covered over with a close-fitting 'stream surface. Similar to this was the process when the jet struck against the lower end of a test tube. With a steady current the lower part of the tube is slowly covered over with a thin stream surface, which at a distance of about 4 centimetres from the lower end of the glass surface bent away and ran into the spirals that here also perpetually recur. Of further special interest is the case where the jet meets a definite thin partition perpendicular to its own direction, since this current has been theoretically treated by Kirchhoff (Crelle's Journal, vol. lxx, p. 298), but under the rather different conditions already mentioned. Therefore, a small circular plate was placed perpendicular to the jet. The stream lines in this case depend essentially on the ratio of the radii of the plate and the jet. If the radius of the circular plate is PAPER BY PROF. OBERBECK. 149 Fig. 25. materially larger than that of the jet, then the latter will be deviated at the plate through a right angle and flows in a thiu layer radially along the plate, which it leaves in a horizontal direction, as in Fig. 25. If, on the other hand, the (cP ~^ / ==^ radius of the paitition is only a little larger than the radius of the jet, then will the stream lines be de- viated by a smaller an- gle from their original direction. This process is shown in Fig. 26, which has a great similarity to the drawing given by Kirchhort at the place just refered to.* A thin sharp-edged partition that extends to about the center of the jet exerts a very similar Fig.36. influence to the thin circular plate. In this case, while one part of the jet spreads in a thin layer along the plate the other part is deviated through an acute angle. In this experiment also the material of the plate appears to exert no sensible influence on the course of the stream. Disks of thin glass and of glazed drawing paper were used, while the above-mentioned thiu partition was replaced by a sheet of tinfoil, which was stretched over a glass frame, and one- half of which had been removed along a straight line. The stream phenomena remained exactly the same. The angle by which the J2t in this last case was deviated from its initial direction depended princi- pally upon the depth to which the thin partition penetrated into the jet. The phenomena here described of flow against solid bodies succeeded only for small velocities of the jet such as corresponded to differences of pressure of 20 or 30 millimetres. VII. Since it was the main object of the author to investigate discon- tinuous liquid motions in their simplest form, therefore he has for the present confined hims,elf at first to the above-described experiments. Still these shall be extended as soon as possible in different directions. As the next points for study the following especially commend them- selves: (a) The flow of a colored liquid into a colorless one through an open- ing in a thin partition. Some preliminary experiments with an imper- fect apparatus showed that the jets thus formed are similar to the above described under otherwise similar circumstances. (b) The discharge of a liquid into another liquid of equal specific gravity that is not iniscible with the first liquid. In this experiment * [See No. VIII of this collection of Translations.] 150 THE MECHANICS OF THE EARTH'S ATMOSPHERE. one could make use of the liquids employed by Plateau, namely, oil and alcohol of equal specific gravity. The question will here arise, in what manner the formation of a jet is modified by capillary action. (c) A stream of air in moving air, the latter being made visible by smoke. viii. The results of the present investigation can be summarized in the following theorems : (a) The viscosity of fluids does not prevent the formation of steady discontinuous fluid motions. In consequence of friction these motions in the beginning suffer important modifications by reason of simultane- ous spiral movements; but with long continued flow they form sharply defined fluid jets. (b) The jets thus formed are very stable for small velocities, and even after small perturbations again immediately assume their origiual form. For greater velocities, on the other hand, they become very sensitive. If the velocity exceeds a certain limiting value, then only very short jets form in the immediate neighborhood of the opening. (c) The jets are not only modified in their movement by solid bodies but also by the liquid adhering to these. The latter adherent liquid is slowly pushed aside by the jet. If then the body is bounded by a con- tinuously curved surface the flowing liquid surrounds it in a thin layer. If, on the other hand, the solid body is bounded by a surface that at certain points has an indefinitely large curvature, such as a sharp edge, then the stream lines follow it only up to this edge and from that point on leave the solid body. (d) The theory of discontinuous fluid motions, as Helmholtz and Kirch- hoff have thus far developed it [for perfect fluids], also gives in general the phenomena observed in a fluid subject to friction. The only differ- ence is the formation of vortex motions simultaneous with origination of the jets. In conclusion we may call attention to the fact that in nature we find a whole series of processes that have a common origin with those just described. These are to be observed in the currents in rivers and canals, especially at places where the banks have sharp corners or where solid bodies, like the piers of a bridge, retard the uniform movement. The eddying motions there occurring clearly show where the quiet and the moving liquids border on each other. Since as a specially noteworthy result of the investigation here communicated has been to show that discontinuous motions arise even for very small differences of pressure, therefore it is easy to see that they must occur often enough in the last mentioned streams. X. THE MOVEMENTS OF THE ATMOSPHERE ON THE EARTH'S SURFACE.* By A. Oberbeck I. INTRODUCTION. The investigations of Guldberg and Molint on tbe motions of the atmosphere certainly occupy a prominent place in the development of theoretical meteorology. If not the first they are at least the most ex- tensive and successful attempt to explain the most important phenomena of the motion of the air by the principles and fundamental equations of hydrodynamics. I would especially indicate as the special service of the authors that they have brought the problem of the motions of tbe air iuto a form amenable to mathematical treatment by simple but as I believe thoroughly appropriate assumptions. They themselves have already computed a series of interesting atmospheric movements that frequently occur in nature, especially the cases where the isobaric systems consist of parallel straight lines or concentric circles. I have attempted in the present work to go further on in the path laid out by Guldberg and Mohn, especially in that I have endeavored to apply to the atmosphere the methods developed in hydrodynamics for other problems. In the present memoir the steady movements of the atmosphere, or, as Guldberg and Mohn call them, " invariable systems of winds," are principally treated. It is natural to refer the movements of the atmos- phere back to the general modes of motion of fluids, that is to say, to motions that are characterized by a velocity potential and to vortex motions. In this way it is possible to attain solutions of great gener- ality that can be applied to any system of isobars whatever. By this method of treatment it is further possible to overcome a difficulty that occurs in the theory of cyclones of Gutdberg and Mohn, without as it would appear haviug been hitherto observed. These investigators dis- tinguish correctly an inner and an outer region for each cyclone, in "Translated from Wiedemann's Annalen der Physik und Chemie, 1882, vol. XVII., pp. 128-148. t " Studies on the Motions of the Atmosphere." Christiauia, Part I., 1876, Part II, 1880. 151 152 THE MECHANICS OF THE EARTH'S ATMOSPHERE. which the expressions for the velocity of the air at the earth's surface and for the pressure follow different laws. But they have not at- tempted so to deduce the expressions for the velocity and for the pres- sure in these two regions from one common principle, that these veloci- ties and pressures merge into each other continuously at the boundary. In the computation of numerical examples tbey have sought to help over this difficulty by not applying their formula to the region in the neighborhood of the boundary, but have here by interpolation intro- duced numerical values passably good, but therefore certainly rather arbitrary. Above all however it is a serious matter that according to their theory the direction of the wind at the boundary suddenly varies through a definite angle. The want of continuity here spoken of can originate either in the assumptions adopted as a basis or in the execution of the computation. I have arrived at the conviction that the latter is the case. I have therefore started from the same assumptions as Guhlberg and Mohn ; these are given in the following Section (n) and I add only thereto the following principle, about which there can be no doubt: " The pressure of the air, as also the velocity of the air and its direction, ought to experience only continuous variations throughout the whole region under consideration.''' By the application of this fundamental principle the theory of cyclones, even in the case of circular isobars, deviates not a little from the theory established by Guldberg and Mohn. II. ASSUMPTIONS THAT ARE THE BASIS OF THE PRESENT TREATMENT. The following assumptions form the foundation of my mathematical development : (a) The portion of the earth's surface coming into consideration is assumed to be a plane. A constant average value will be assumed for the geographical latitude of this region. (6) The air will be treated as an incompressible fluid. (c) The investigation here carried out refers only to a stratum of air of moderate height above the earth's surface. The latter surface ex- erts a retarding influence on the movements of the air that can be con- sidered as a force opposed to the movement and proportional to the velocity. ('/) The currents of air at the earth's surface are ordinarily directed toward a center or flow away from the neighborhood of such a center. Such currents can not be imagined without the existence of a vertical current in their neighborhood. If, therefore, we in general confine our- selves to the consideration of horizontal currents, still the consideration of vertical motion is not to be avoided for the neighborhood of such a center. We have, therefore, to distinguish between regions of pure horizontal motion and regions with vertical motion. As to the latter, PAPER BY PROF. OBERBECK. 153 the following simple assumption is made :* If we adopt a system of rec- tangular coordinates such that the plane of x y is the horizontal plane while the axis of z is directed vertically upward, then for the vertical component of an ascending current of air we have the expression w=c.z. If the boundary of the region above which this current ascends is known, while outside this boundary the movement is exclusively hori- zontal, then the whole system of winds (the cyclone) is thereby com- pletely determined. The quantity c can be designated as the constant of the ascending air. For regions with descending air currents the negative sign must be given to the constant. The region for which 1C = C.S will, for brevity, be designated as the inner region of the cyclone; that for which m?=0 will be designated as the outer region. The vertical component is to be considered only in connection with the equation of continuity. Therefore for the outer region this equa- tion becomes ^+^=0 (1 i)GO ,")&? y COS £ = fa»+ £_+„* +t£_ for which by introducing the notation we can write vcosf = fc(»+~. (8) dt v From these equations many consequences can be drawn that lead to specially simple theorems when the velocities of the wind are so small that the term u- — \-v - - dm dy can be neglected. But the following theorems will also be approxi- mately true even if the velocities are larger. (a) If we compare an invariable system of wind and one that is va- riable as to its intensity, and of which we will assume that at any given instant there prevails throughout it everywhere uniform velocities and 156 THE MECHANICS OF THE EARTH'S ATMOSPHERE. uniform gradients, then the angle between the direction of the wind and the gradient is smaller in the variable system than in the invariable when the intensities increase, and, inversely, larger when the intensities are diminishing. (6) If in one and the same system of winds having a progressive movement we compare two points that have equal velocities and equal gradients then the deviation of the wind direction from the gradient is smaller at the point where the wind velocity is increased than where it is diminished. Therefore in general the departures from the gradient will be smaller throughout the advancing half or front of a moving cyclone than within the rear half. (c) For steady motions of moderate intensity, where therefore dt the velocity is proportional to the projection of the gradient on the direction of movement. Furthermore for equal gradients and equal velocity the deviation of the direction of the wind is greater in propor- tion as the friction is less. Some of these theorems have been proven already for special cases by Guldberg and Mohn. The theorem expressed in paragraph (c) has also been attained in an entirely different way by A. Sprung.* I pass now to the investigation of the invariable systems of wind, and therefore assume that If further we put P=?+l(u>+v>) (9) then the equations (5) give jy+to=+(M-C)« \ If in these we introduce for u and v expressions of the form ordina- rily used in hydro-dynamics, namely : 3ee Wiedemann, Beiblatter, 1881, vol. v, page 24o7an7sprnng MeTeorologie, £ burg, 1835 am- PAPER BY PROF. OBERBECK. 157 and furthermore put f^P+kcp-XW; f2=lcW+\

/< satisfy the above given differential equations. Moreover, both functions stand in the following relations to each other. dq>_dtp. d/• into the right-hand side of this equation we obtain as the equation for the path described by the wind $—j- (p— constant (21) The path of the wind intersects the system of lines defined by the condition q>= constant at an angle that is everywhere the same. If we designate by s the angle that the direction of the wind makes with the normal to the curves cp= constant then we have tan € =-=- . K For currents of air of moderate velocity the term in equation (20) can be neglected in comparison with cp. In this case the isobars, for which p equals a constant, are identical with the curves f= constant and we obtain the following general theorem ; PAPER BY PROF. OBERBECK. 150 In regions of pure horizontal motion, and for moderate wind velocity, the angle between the direction of the wind and the gradient is constant and depends only on the constant of rotation and the constant of friction and is independent of the direction of the isobars. The above given relation had been found by Guldberg and Mohu * for the special cases of rectilinear and circular isobars. The general solutions contained in equations (18, 19, and 20) can now be so applied that we may adapt the function q> to any other given system of isobars. When this is achieved, then the motions of the air are determined by the first two of these equations. If, for instance, we have to do with a region that is under the in- fluence of numerous but distant maxima and minima of pressure, then we can approximately put /-=2xy leads to a special example already treated of by Guldberg and Mohn.t The potential curves and the stream lines x* — i/-=coustant 2xy — t {xl —y1)— con s tan t are systems of equilateral hyperbolas. See their Etudes, etc., Part I, pp. 23-26. \ Etudes, Part II, pp. 51, 52. 160 THE MECHANICS OF THE EARTH'S ATMOSPHERE. If we assume F{x+iy)=\og (x+iy) = (p+i>p and if we substitute x=rcos 6-, y=r sin 6 then follows cpz=:logr, ip=6. In this case the isobars consist of concentric circles. The paths of the wind are logarithmic spirals having the equation H— log >=constaut. V. STEADY SYSTEMS OF WINDS. It is certainly at present generally assumed in meteorology that the winds at the earth's surface owe their origin and maintenance to ver- tical currents of air that are limited to definite regions. Let us assume that there is given such a region having auy arbitrary boundary above which a current of air ascends whose velocity in the neighborhood of the earth's surface is determined by the constant (c). By this assumption the whole system of winds dependent thereon, as well as the distribu- tion of pressure, is determined for the whole region. It is therefore the province of mathematics to determiue all the quantities coming into consideration both for the inner and also for outer region. To this end the functions cp and to are to be propely determined. The first of these is found without further difficulty from well-known theorems in the theory of the poteutial. Since these functions must in the outer region satisfy the partial differential equation z/rp=0, and in the inner region must satisfy the equation Js(>+c«'>0 Kf'-C 17K0+C * )=° These equations are satisfied if we put /i— C Inconstant; /2+C <^=constant. By considering equation (12) there follows from the last equation especially Jc W+{X+Z) dr ' dr '•''"+(* + O^=0 (25) r dr \ dr J aud d x dW r' dr~ r dr therefore at the boundary we must have & z or fc > c, since otherwise for r = o f(r) would become infinitely great, and in the inner region a deviation of the wind from the gradient toward the left would occur instead of to- ward the right-hand side. The deviation of the wind direction from the gradient is constant in the outer region, but in the inner region it increases continuously and for r = o it attains the limiting value — A tan € = Tc-c' I pass now on to the computation of the pressure. According to equation (17) we have for the outer region — / A2\ Pa = constant — kcpa y 1 + , ., J 166 THE MECHANICS OF THE EARTH'S ATMOSPHERE. Consequently Pa= constant +^1+^2P log r. For the inner region the equation (24) is to be used. According- to it we have — ^ = _^+(A+C)^ dr dr v dr But according to equation (25) we have — dW A + C = — » d<£> Therefore, P,= const— Tap— 7c. The arbitrary constant cau be considered as determined in that the value of P is supposed to be given for r=0. (For the center of the depression we have r=0 and P=2.) Let P„ be this value. Then we 9 have— P^P^+Pir), Where *»-*■ \ ^^^KiT+w^rii)n \ ■ <31> Since Pa and Pt must at the boundary merge continuously into each other, therefore the constant in the expression for Pa is to be deter- mined in accordance with this condition, and we have — Po=P0+P(P) + |(l+^EMog| . . . (32) From equation 9 we obtain the expression for the pressure — P If we designate by p0, the pressure at the center of the depression, where ca=0, then in the inner region we have— l^l=F(r)-$Go2 (33) but in the outer region — *^-*W + *f(l+£)*.log£-i* (34) PAPER BY PROF. OBERBECK. 167 VII. NUMERICAL EXAMPLE FOR A CYCLONE: NOTE ON ANTICYCLONES. In order to show the applicability of the formulae -obtained in the last section to cyclones as they actually occur in nature, I have executed the following' computation of a numerical example: In this computation I have assumed X = 0.00012 This value corresponds to an average latitude of 55.5°. For Jc I have assumed the same value, whereby the value obtained for the influence of friction is rather large. For the complete determination of the system of winds the constant c of the ascending current of air and the dimensions of the inner region must also be known. We can obtain this in various ways. We can assume as given, a definite difference in pressure between the center and a circle of known radius ; or on the other hand, we can assume that the velocity of the wind is known at a certain distance from the center. I have chosen the last assumption. The wind system may therefore be characterized by the assumption that at a distance of 1000 kilometres from the center the wind velocity shall be 10 metres per second. According to equation (29) when we put A=A- we have c A'- 1 go = v2 If in this we put go = 10 metres and r = 1000000 metres we then have c E2 = 10000000 -v/> • Since furthermore c< A, therefore the same equation shows that we must have B > 343.3 kilometres. In the selection of appropriate values of c and B, another circum- stance is to be considered. The discussion of the formula? (30) for the velocity go shows that under the assumption here made of A = A-, the maximum velocity of the wind occurs at the boundary of the two regions. The smaller the inner region is chosen, by so much larger results the maximum velocity ooK. In the following table some coor- dinate values c, //, B, and gor are given. Table I. c M n IUK Kilometres. Metres per sec. i>- 5 383.8 26.06 3 3 420.4 23.78 lfc 2 4 485.5 20.60 3 G 594. G IG. 82 168 THE MECHANICS OF THE EARTH'S ATMOSPHERE. I have also executed the further complete computation for the first case where c=h ; the results of this work are given in Table 2. In this D computation the equations (29) and (30) were used for the determina- tion of the velocities gd and the deviations s of the directiou of the wiud from the radial gradient. Furthermore, the differences of pressure (?— Po) witu resPect to tuat at tne ceuter> iu tDe circles of radius r, were computed according to equations (31), (32), (33) and (34). These latter are however, converted from tbe units ordinarily used in hydro- dynamics into differences of barometric pressure (b— b0). This latter is easilv done if we recall that for fc=760 millimetres the ratio -is equal to the square of the Newtonian velocity of sound ; therefore we have the proportion [b-b0): 160=1 (p-p0); (279.9)2 The gradients y are in our present case tbe differences of barometric pressure for a horizontal distance of 100 kilcmetres. Table II. r 01 e (&-&„) Y Kilometres. Ah- ties per sec. O I Millimetres. Millimetres. 0 100 200 300 0 14.99 22,44 25. 53 78 41 71 19 64 40 55 39 0 2.37 7.01 12.04 2.37 4.64 5.03 383.8 26.06 45 00 15.88 U.78 400 25.00 45 00 16.82 ) 4.76 3.60 2.66 1.95 500 20.00 45 00 21.58 600 16.67 45 00 25. 18 800 12.50 45 00 30.50 1,000 10.00 45 00 34 45 From this table we see that the cyclone includes a broad storm region from r=200 to r=500 kilometres, of which » portion is in the inner region and another portion in the outer region. Of course the gradients are greatest in the inner region ; therefore there the isobars are most crowded together. From those values of the constant c that' are any way possible, it fol- lows that the velocity of the ascending current of air is extraordinarily small ; for the present example c equals 0.000096. If we assume that the formula w=cz holds good to an altitude of 1,000 metres, then the vertical velocity would at that height first attain the value of about 0.1 metre per second. Hitherto the discussion has exclusively dealt with regions of ascend- PAPER BY PROF. OBERBECK. 169 ing currents of air and the cyclones arising therefrom. It would be easy in an entirely similar way to develop the theory of descending currents of air and the anti-cyclones resulting therefrom, and here also, as an example, to assume an inner region bounded circularly. Before the actual execution of the exact computation I had believed that this was simply a case of the change of the sign of the constant c. But in this operation we stumble upon a peculiar difficulty. The function f (r) = l— -( -j-,) ( wherein /<=— ) which enters into the expression for the component velocities in the inner region becomes in- finitely great for negative values of c and ju and for r = o. The same is true of the function F (r) entering into the expression for the pressure. Hence it follows that the formula just given can not be applied to anti- cyclones with a reversed sign of c. Therefore minima and maxima of pressure show a characteristic dif- ference in their theoretical treatment. But this, as I believe, corre- sponds also to the real conditions of the true phenomena. Depressions are ordinarily confined to limited areas, but are of considerable inten- sity, while on the other hand the maxima of pressure extend with slight intensity over broad areas. Fig 27. Moreover, both phenomena stand in close connection, such that one can consider the ascending currents of air as the cause of the descend- ing currents. Hence to a complete cyclone there belong an inner region with ascending air current, a zone surrounding it of purely 170 THE MECHANICS OF THE EARTH'S ATMOSPHERE. horizontal movement, and at a greater distance from the center a ring- shaped region of descending currents. If we assume that the boundaries of the three regions consist of con- centric circles, it would not be difficult to compute the wind system for the whole region by the help of the potential theory as above employed. In this case, where we have to do with an annular region with a descending current of air, the use of the function / (r), even with a negative sign before the //, is allowable, and can be adopted in order to produce the necessary continuity of motion at the boundary of the two annular regions. If there are several regions of depression with ascending currents of air, as at A, B, C, fig. 27, then each of them is immediately surrounded by a zone of purely horizontal movement, which is bordered by an outside annular zone of descending movement. I have in the figure (27) distinguished the region of ascending and de- scending current by double and single shading. In the region where the different ring systems of asceudiug air currents merge iuto each other there will lie a region of highest pressure with anticycloual movement of the air somewhat as within the isobar M, A/, P. How- ever, the characteristic difference between ascending and descending currents of air always consists in this, that the former consist of defi- nite, simply connected areas ; the latter, on the other hand, of a net- work of several complexly couuected regions. Halle a. S., June, 1882. P. S.— After sending the above treatise to the editor of the Annalen, I found in tbe May number of tbe Zeitschrift of tbe Austrian Association for Meteorology (vol. xvn, pp. lb"l-17,r>) a review by Dr. A. Sprung of tbe second part of tbe collected memoirs by W. Ferrel, under the title of "Meteorological Researches." From this I perceive that the views expressed by me as to regions with high pressure had been already expressed by Ferrel. Therefore, although my point of view is no longer new, still I rejoice to see that it is shared by a prominent meteor- ologist. XL ON THE GULDBERG-MOHN THEORY OF HORIZONTAL ATMOSPHERIC CURRENTS.* By Prof. Dr. A. Obkrbkck, of the University of Halle. Starting from the generally known results of recent meteorological observations in so far as these relate to the distribution of pressure and the direction and force of the wiud, the author states that one of the most important problems of the mathematical theory of the motion of fluids is to explain quantitatively the connection of the above-named phenomena. The recently published investigations of Guldberg aud Mohn (Etudes sur les mouvements de V atmosphere. Christiauia, 1876 aud 1880) are to be considered as a specially successful attempt in this direction. It must be of interest also for the larger number of geog- raphers to know the most important results to which the Norwegian scientists have attained. In order to understand the horizontal movements of the atmosphere it is important for a moment to consider their causes. As such we con-, sider the differences of pressure at the surface of the earth as observed with the barometer. But whence do these arise f This question has been answered along time since. It is heat which is to be considered as the prime cause of the disturbance of equilibrium in the atmosphere. Because of the slight conductivity of the air the process of warming can progress only slowly from below upwards, so that as is well known the temperature of the air steadily diminishes as we ascend. The heated air expands. The pressure becomes less. If the heating takes place uuiformly over a large area there will be at first no reason for horizontal currents. But vertical currents can certainly be brought about by this means. If we imagine a circumscribed mass of air trans- ported into a higher region without any increase or diminution of its heat its temperature will sink because it has expanded itself propor- tionately to the diminished pressure. If its temperature is then equal to that prevailing in the upper stratum it will remain in equilibrium at this altitude as well as below. The atmosphere iu this case exists in a state of indifferent equilibrium. If its temperature is lower the * Translated from the Verhandlungen des Ziveiten Deutschen Geographentages. Halle, April, 1882. 171 172 THE MECHANICS OF THE EARTH'S ATMOSPHERE. mass of air will again sink down ; in the reverse case it will rise higher. The air in these cases is then in stable or unstable equilibrium respec- tively. In the latter case any vertical movement initiated by some acci- dental disturbance will not again disappear, but rapidly assume in- creasing dimensions. The current will also continue uniform for a long time. This is the explanation first given by the mathematician Reye,* of Strasburg, of the ascending air currents in the whirlwinds of the tropics. The winds of our (temperate) zone also presuppose such ascending currents whose origin must have been quite similar. The ascending current is in general restricted to a definite region that we can desig- nate as the base. Since the ascending current consists of warmer air, therefore above its base the pressure sinks. ' A barometric depression is inaugurated there. The pressure increases from this region outward in all directions. The isobars therefore surround the region of ascend- ing atmospheric currents in closed curves. At greater heights the up- per cooled air flows away to one side and in other regious gives occasion to descending currents of air. At the earth's surface itself, the air flows towards the depression ; its influence thus extends over an area much greater than that of the base. If we neglect the curvature of the earth's surface we find over this larger area only simple horizontal movements. Mathematical computations should now reveal to us the nature of such horizontal movements. To this end all the causes of motion, or the forces that come into consideration, are first to be col- lected. The differences of pressure have already been several times spoken of. We take as the measure of these differences, thegradient which gives for any point the direction and amount of the greatest change in pres- sure. In horizontal movements the effect of gravity can be omitted. On the other hand attention must be given to the rotation of the earth on its axis, since we are only interested in the paths of the winds on the rotating earth. This influence can be taken aceount of if we imagine at every point of the mass of air a force applied which is per- pendicular to the momentary direction of motion and is equal to the product of the double angular velocity of the earth by the sine of the latitude and by the velocity of the point. In the Northern Hemisphere this influence causes a continuous departure of the path towards the right hand side. Since the movement takes place directly on the earth's sin lace the direct influence of that surface, namely the friction, remains to be considered. Its influence diminishes with the distance from the earth's surface. Furthermore it depends on the nature of the earth's surface, whether sea or land, plains or wooded mountains. For this computation Guldberg and Mohn have made a convenient assumption in that they introduce the friction as a force which opposes the inove- I'l'liis explanation is of cmr.se much older tbau Reye (1864). who was preceded by Espy and Henry in the United States and by Wui. Thomson in Great Britain. C. J.] PAPER BY PROF. OBERBECK. 173 ment and is equal to the product of a given factor and the velocity. This factor can have different values according to the nature of the earth's surface [and will be called the friction constant]. All these forces are to be introduced into the general equations of motion of the air. If however one desires solutions of these general equations for special cases there is still needed a series of assumptions. Let there be only one single vertical current of air present. The to- tality of all the atmospheric movements depending upon this one verti- cal current is called a wind-system. If the strength of the ascending current is variable or if the base itself changes its place, then the wind- system is variable. In the first case the system stands still, in the second case it is movable. If on the other hand the ascending current of air retains its strength and location without change, or, which is the same, if the isobars for a long time retain their position, then the wind system is invariable. It is evident that the last case is by far the most simple. We will therefore begin with its consideration. In order to execute the calculation the location of the isobars must be known. Even in this respect also in a preliminary way, one must limit himself at first by simple assumptions. Let the isobars be either par- allel straight lines or concentric circles. In the first case the computation leads to the following simple results : (1) The parallel isobars are equally distant from each other. The gradient is therefore everywhere of equal magnitude. (2) The paths of the winds consist of parallel straight lines. The strength of the wind has everywhere the same value. (3) The direction of the wind forms an angle with the gradient whose tangent is equal to the quotient of the factor arising from the velocity of the earth's rotation divided by the friction constant. The deviation of the wind from the gradient is therefore greater in proportion as friction is smaller. If the earth's surface were perfectly smooth the wind would blow in the direction of the isobars. This result, following directly from the computation and at first sur- prising, finds its confirmation in a variety of observations. For exam- ple, in England we observe a deviation of 61° for land winds, but of 77° for sea breezes. From this it follows that the friction on the land is more than twice as great as on the sea. Conditions of pressure like those here considered frequently occur. In the regions of the trade winds and monsoons they ordinarily prevail either during the whole or about the halt of the year. The circular isobars to the consideration of which we now pass pro- duce systems of wind that can be considered as the simplest types of cyclones and anticyclones according as the pressure in the interior is a minimum or maximum. We confine ourselves here to the considera- tion of cyclones. As alreadv remarked cyclones are not conceivable without an ascend- 174 THE MECHANICS OF THE EARTH'S ATMOSPHERE. ing current of air, whose area in our case is defined by a circle. Out- side of this circle horizontal movements prevail exclusively ; inside of it there is also the vertical movement to be considered. Therefore the computations for the outer and inner regions are different. In this way we obtain the following results : (1) The pressure increases from all sides outward from the center ; the gradient increases also from the center out to the limit of the inner region : thence outward it diminishes and at a great distance becomes inappreciable. (2) The wind-paths in both regions are curved lines, logarithmic spirals, which cut the isobars everywhere at the same angle or make everywhere the same angle with the radial gradient. Therefore the movement of the air can be considered as consisting of a current toward the center and a rotation around the center, the latter in direction op- posite to the hands of a watch. This departure from the gradient is of different magnitudes in the outer and inner regions. For the former the departure has the same value as for straight-line isobars, that is to say, it depends alone upon the rotation of the earth and the friction. For the inner region the departure is greater, and depends besides upon the intensity of the ascending current of air. If both regions were sepa- rated from each other by a geometrical cylindrical surface then the wind- paths in these would not continuously merge into each other, but would form an angle with each other. This of course can never occur in nature. We must therefore assume a transition region in which the wind is con- tinuously diverted from one into the other direction. At any rate ac- curate and comparative observation of the wind direction in the inner and outer region of a cyclone would be of great interest. From these one could draw a conclusion as to the limitation of the ascending cur- rent of air. This limit is moreover also notable because at it the winds reach their greatest force. There are no other arrangements that have been discussed theoreti- cally as yet except the straight line and the circular and nearly circular forms of the isobars. We have as yet only spoken of the invariable systems of wind. In fact however their duration is relatively short, No sooner is a de- pression formed than it fills up. Furthermore the central region of depression generally does not remain long in the same place but wanders often with great velocity, drawing the whole system of winds with it. We must look to the density of the horizontal current flowing in to- wards the ascending current of air as the cause of these changes. The system of winds remains unchanged only when, as has hitherto been silently assumed, the temperature and density of the horizontal and vertical currents are alike. If the inflowing air is warmer the depres- sion increases in depth; in the opposite case it becomes shallower. Finally, if the inflowing air is not of the same temperature on all sides, but has on the one side higher and on the other side lower PAPER BY PROF. OBERBECK. 175 temperature than the ascending air, then it will on the one side be strengthened and its area increased, on the other side enfeebled and its area diminished. The consequence of this is that the current of air or the region of depression moves along; the cyclone progresses. Since in the cyclones of our north temperate zone the air entering on the east side comes from more southern — therefore in general — warmer regions, while the air entering on the west side comes from the north and is generally colder, therefore the cyclone progresses from west to east or from southwest to northeast. This is in fact the path of most cyclones in northern Europe. For a moving cyclone the isobaric curves must have a different shape than for one that is stationary; therefore one can inversely from the shape of the isobars infer the direction of motion. If the region of ascending air has a circular form the compu- tation can be rigorously executed. Without going into the details of this interesting problem in this place I will only remark that the isobars consist in closed curves similar to an ellipse. There is one direction from the center outward in which the isobars are most crowded together, while in the opposite direction they are furthest apart. The movement of the cyclone is in a direction at right angles to this line. With the solution of this problem we now stand about at the limits of wbat analysis has thus far accomplished. Still there is hope that it will make further progress so far as concerns the relation between the pressure and the motion of the air at the earth's surface. XII. ON THE PHENOMENA OF MOTION IN THE ATMOSPHERE.* (FIRST COMMUNICATION.) By Prof. A. Oberbeck, of the University of Greifawald, Germany. The meteorological observations of the last ten years have given a series of notable laws that principally relate to the connection between the currents of air and the pressure of the air in the neighborhood of the earth's surface. Of course one can only hope to obtain a complete insight into the complicated mechanism of the motion of the air when one understands more accurately the condition of the atmosphere in its higher strata. But difficulties that are perhaps never to be overcome oppose the observation of these strata. On the other hand, the completion of this and many other gaps in the theory of the motion of the air is certainly to be expected from a comprehensive mechanics of the atmosphere. The Treatise on Meteorology, by A. Sprung, Hamburg, 1885, gives a summary of what has hitherto been accomplished in this field, from which summary it is seeu that only special individual problems have found a satisfactory solution. The principal features of a rational mechanics of the atmosphere are given in the memoir by W. Siemens, "The conservation of energy in the earth's atmosphere." t It appears to me worth while to follow out mathematically the questions there treated of and to develop a the- ory of the motions of the air as general as possible. The results thus far attained by me, are collected in this present memoir. On account of the magnitude and difficulty of the problem to be solved, I have at first confined myself to the determination of the cur- rents of the air. A corresponding investigation of the distribution of pressure will follow hereafter. Moreover the phenomena of motion * Read before the Royal Prussian Academy of Sciences, at Berlin. March 15, 1888. Translated from the Sitzungsberichte Konigl. Preus. Akad. der Wissenschaften. 1888, pp. 383-395. t See Berlin Sitzungsoerichte, 1886, pp. 261-275. 176 PAPER BY PROF. OBERBECK. 177 will here be considered as "steady motion." On the other hand I have labored so to arrange the calculation that it can be applied to any condition of the atmosphere and to the general currents between the poles and the equator, or the atmospheric circulation, as well as also to individual cyclones or anticyclones. In order to test the applicability of the formula thus obtained, the first of the problems just mentioned is completely solved. I begin with an enumeration of the factors upon which the movement of the atmosphere depends, and with a description of the manner in which 1 have introduced these into the calculation. II (1) Since the ultimate cause of the motion of the air is to be sought in the effect of gravity and in the differences of temperature in the atmosphere, therefore the attraction of the earth must enter into the equations of motion as the moving force. But it is entirely sufficient here to consider the earth as a homogeneous sphere. (2) The temperature of the atmosphere is to be considered as a function of the locality, but entirely independent of the time. The last condition is necessary if one confines himself to steady motions. For the temperature T, the analytical condition A T— — -I- — -+- — — 0 must be satisfied. This equation, as is well known, follows from the assumption that the heat is distributed through the medium iu question according to the laws of the conduction of heat. Although I am by no means of the opinion that the conduction of heat principally determines the flow of heat from the earth's surface through the atmosphere into the planetary space, still it is very probable that the totality of all the phenomena here coming iuto consideration (conduction, radiation from the earth's surface with partial absorption in the atmosphere, vertical convection currents, etc.) will bring about a distribution of tempera- ture analogous to that due to the conduction of heat. (3) According to the rules of mechanics, the influence of the rota- tion of the earth can be expressed by a deflecting force, so that after its introduction the earth cau be considered as at rest. (4) Friction is furthermore to be considered, since without it the atmospheric currents under the continuous influence of accelerating forces would attain to indefinitely great velocities. In my opinion, the attempts made hitherto to give a correct theory of the motions of the air, especially one that can be developed analytically, have failed because of the insufficient or incorrect introduction of friction. I have adhered to the simplest assumption, namely, that the same law of friction holds good for atmospheric currents that has also been shown SO A 12 178 THE MECHANICS OF THE EARTH'S ATMOSPHERE. to be correct in the motion of liquids.* But I would not hereby assert that the same numerical coefficient is to be used as is given by the laboratory experiments on the internal friction of the air made under the exclusion of all attendant disturbing circumstances. More likely is it that along with the greater horizontal currents there will arise small vertical currents of a local nature which will iucrease the friction. The air can either be held fast at the earth's surface or glide with more or less resistance. This fact, as is well known, is expressed in the boundary equations of condition by a number, the coefficient of slip, whose value may lie between zero and infinity. (5.) The density of tbe air must be considered as dependent upon the temperature, since tbe effective cause of the currents results from this. But I have not objected to use, as the equation of continuity, that simpler expression that obtains for incompressible liquids. The error introduced hereby can be eliminated if, at places where the density is less than the average, one increases to a corresponding extent the velocity found for that locality, but considers the velocity as dimin- ished at locations where the density exceeds the average. (6) A hydro-dynamic problem is only perfectly definite when the fluid occupies a definite space, and its behavior is known for all limiting boundary surfaces. I have therefore assumed that the atmosphere is bounded both by the earth's surface and by a second spherical surface concentric therewith. The distance of the two spherical surfaces, which I will briefly designate as the height of the atmosphere, can remain un- determined. But this is quite small in comparison with the earth's radius. The above assumption just made however, only expresses the idea that for a given altitude above the earth's surface the radial or vertical currents are very sm,all, or rather that when they are present they exert an inappreciably small influence on the remaining motions. This is certainly the case, since at very large altitudes the density is very small. Since moreover it is assumed that the air can glide without resistance on the upper spherical surface, therefore in my opinion no limitation of the motions of the atmosphere, contradictory to the real phenomena, results from the introduction of such an upper boundary surface. in. The following notation will be used for the principal equations of the problem. The position of a point in the atmosphere is determined by the rectangular coordinates x, y, z. The center of the earth is the origin of coordinates and the earth's axis in the direction of the North Pole is the positive axis of z. The positive directions of the two other axes are to be so chosen that the axis of y as seen from the North Pole must be turned through an angle of 90° in the direction of the motion of the hands of a watch in order to be made to coincide with the axis of x. . | The term friction as here used therefore includes viscosity aud slip, hut excludes the resistance due to wave motion and to vortex motion and all the resistances implied iu turbulent flow of fluids— C. A.] PAPER BY PROF, OBERBECK. 179 Let tbere be furthermore — u, v, iv, the components of velocity ; p, the pressure ; //, the density; A-, the coefficient of friction ; 6r, the acceleration of gravity ; E, the radius of the earth ; r, the distance of any point from the center of the earth ; e, the angular velocity of the earth. Then we have — 1 dt t\r 1 1 Jx fx 1 dt dy >lM /' \ (i) dt & Pdz M dx jiy %& Since according to the law of Mariotte and Gay-Lussac we may put P=P° (1+aT) The zero point of temperature is arbitrary. It is most appropriate to assume for it the average temperature of the atmosphere. If c is the Newtonian value of the velocity of sound, then we have Mo After the introduction of these expressions into the above principal equations, imagine the latter divided throughout by 1 -f a T. Except- ing in that member in which the gravity occurs, one can omit from consideration the influence of the factor 1 , y- In the term just men- tioned one can, as a first approximation, put (1 — aT) for the value of this factor. Furthermore let k — — K 180 THE MECHANICS OF THE EARTH'S ATMOSPHERE. The first of the equations of motion now becomes du =ll-aT) GR> \- - c2 ^0fp+ k4u+26V. at Jx o® If the temperature of the atmosphere depended only on the altitude above the earth's surface and were therefore ouly a function of r, then would these equations be fulfilled by putting w, v, w respectively = 0 ; the atmosphere would then be in equilibrium. Therefore put T = To + T, wherein T0 is a function of r ouly, but T, is also a function of the longi- tude and latitude ; therefore ?l 0 dx~~ W r' dr Finally one may put 1 Jx ' d$ r 3x p =_p, . (1 + v). The quantity v in this latter equation expresses those changes of press- ure that are caused by the phenomena of motion. Since v is small in comparison with unity, therefore instead of log (1 + v) the quantity v itself cau be substituted. By this means the first principal equation becomes = GR> d- { 1— °^1 + a fT: dr \ - c*^-^ -c^ + uAu + 2 dx\ r J r2 S d% dx After transforming the two other principal equations in the same man- ner we can put cHogpt = constant + GR2 \ 1 °^ + a f~dr J (2) This equation gives the diminution of pressure at larger altitudes above the earth's surface, and can for smaller differences of altitude easily be transformed into the ordinary equation of barometric hyp- sometry. PAPER BY PROF. OBERBECK. 181 The following- system of equations relating- to the phenomena of mo- tion proper now remains : du aGR2 dT, 2>, . , 0 -51=- — . — — c2^ + «J»+2ev, dt r Jx dx dv aGR2 dTx 9dv , 0 . dt r tty dy (3) dw aGR2 tTlTi _ dv A dt=' r ■ -?z-c2K+"A"'' dx 2y dz One can now compute first those components of the current that de- pend only on temperature differences; after that those that are brought about by the rotation of the earth. If we put u=u1+u2; ti=vi + v2; w =Wi + w2; r=vl-{-v2-irv^ then will the following two systems of equa- tions be those that are first to be discussed: e2^1- __aGR2 dT, dx r ,\i- c2 dVx- dy aGR* .IT, , . dz = aGR> m r dz dx =2evi-\-x4u2; and c2 1^= — 2 f «., -f nAv% ; dy c2 dv:i=H/iw2. dz Thus there still remain the following equations which are no longer linear and which will serve principally in the computation of the varia- tions in pressure produced by the motion : c2— +m - — h v ~ -fw?^-=2 ev2', Jx^ dx^ dy dz ' o dV-i d® d® , dV o jy ^ aar dy dz d*A3 die , duo , dw _ /* — +u - — \- v ~ — \-ic --= 0. 6 dz^ dx^ dy dz The first two systems of equations are linear. When therefore Tt consists of a sum of terms we shall obtain corresponding sums for the 182 THE MECHANICS OF THE EARTH'S ATMOSPHERE. component velocities. The solution will be quite simple when T2 is developed into a series of spherical harmonics. If we put and for brevity /3=aGRz, and indicate by Q any term of the series with its corresponding con- stant then the solutions of the first two systems of equations are as follows : x ' dy dy S \ ^ h\ dz dz * In this E and F are functions of r only, and must satisfy the differ- ential equations f&F,2_dF\ dQ,2cW lQ=M(_l,a\ \dr2 r dr J dr dr t")r2 Jr\ r fd?F 2 dF\ Q iQ( dF.dE<_ft \dr*+r ~drj ^^JrK^dr J (5) The constant a must be added in order to obtain the number of con- stants needed in the consideration of the boundary conditions. The terms depending upon the earth's rotation are x2 » V tit 3y J dx s h* \ M- dx dV i Hl dZ c2v2=^JE u (6) Here also J and H are functions of r only, and must satisfy the differ- ential equations dr^r drjjr+^ Tr"^?^ f dlff, 2 dH\ „ ndff M) f ' ' ' (7) PAPER BY PROF. OBERBECK. 183 The constant b must also here be added for the same reason as above given. The function K is to be computed from the equation AK+(Y(M.*-W.y:\=o (8) dr V Jy r Jx r J v ' From this last equation it follows that the introduction of the func- tion K can be omitted when the temperature of the atmosphere is as- sumed symmetrical with reference to the earth's axis. In this case w2=0 and the [atmospheric] movement resulting from the rotation of the earth consists exclusively in a movement of rotation depending on the geographical latitude and the altitude above the earth's surface. In order to present iu the ordinary manner the currents of air for a given point in the atmosphere, the following components are to be in- troduced instead of u, r, w: V, the vertical component computed positively upwards; N and 0, the two horizontal components, of which the first indicates movement toward the north, the latter, movement toward the east; 6, the complement of the geographical latitude of a given place; //', the longitude counted from an arbitrary meridian; then we have \' — -\-(u cos ip-\-v sin //•) sin ft-\-w cos 6 ) \' = — (u cos ip+v sin //•) cos 0+w sin ft > . . . . (9) 0= — u sin if?+v cos ip. ) The formulae (I, G. and 9) contain the general solution of the problem so far as this is at present intended to be given, assuming the distribu- tion of temperature to be given and that the functions E, F, J, H, K are determined in accordance with the boundary conditions. IV. When one attempts to represent the distribution of temperature on the earth's surface by a series of harmonic functions then the most im- portant term is a harmonic function of the second order. Therefore as a first approximation we put T1= fAr2 + -J )(l-3 cos2 ft). This function, with a proper determination of the constants, ex- presses the great contrast in temperature between the equator and the pole. If now one would take into account the variation with the sea- sons one must next introduce harmonic functions of the first order. The consideration of the various peculiarities of the earth's surface will of course demand further terms that depend on the geographical longi- tude also. 184 THE MECHANICS OF THE EARTH'S ATMOSPHERE. I have at first limited myself to the computation for the above given distribution of temperature, and put Q=Ar2 (1-3 cos2 d) Q'=A' (1-3 cos2 6). JV = 0_ aGR,22e sip a ^l^ q nnot m f A„f :dS H2 The functions E,F,H,J are now to be computed with the help of this Q, and the corresponding E', F', H', and J' with the help of this Q'. We first obtain the general expressions : V=aGR\l— 3 cos2 6) I A | r2^+ 2r (*»+.#) J ^^2 G cos 0. sin 0 j Ar( F+E) + A' (F> + E<) \ in ^ T(i_3 cos2 8) J Ar ('/^ + 2 (JT+ J) ) + £(r^~3 (jH'+J,) ) I +G °°8' * I Ar (jff+ J) + £ (fl/J/) I ] The actual computation, haviug due reference to the boundary con- ditions, of the functions here introduced, gives results that are difficult to be discussed. But this is simplified when we make use of the cir- cumstance that the atmosphere fills a very thin shell in comparison with the terrestrial sphere, wherefore the distances from the earth's surface are all small in comparison with the earth's radius. If we put r=R(l+ff) then is a small with respect to unity. If we introduce these quantities in the above given equations and put r = Rry (a); then by restricting ourselves to the terms of the lowest order, we can obtain simple expressions for these functions. Primarily we iind that the functions /and/',

\) consist of one lower current toward the equator and of one upper current toward the poles. The distribution of pres- sure v0+vi existing in connection with this furnishes (by means of the equation (4), page 182 of the previous memoir) the anticipated result that on the surface of the earth the pressure increases from the equator to- ward the pole, while at a medium altitude the differences of pressure disappear, but that finally, at greater altitudes, the pressure is greatest at the equator and least at the poles. Since as above remarked, the actual distribution of pressure in no- wise agrees with the above, it must be concluded that the influence of the term v0-\-n on the pressure can only be slight. From the previous developments it results that the term r2 disappears under the assumption of a uniform distribution of temperature symmet- rical with the earth's axis, so that as was already indicated in the first memoir, vz will be the most important term. in. In the computation of this quantity va the system of equations pre- viously given is to be used, namely : <&* +%» +v^ +«&=_&* ay & dy d% Since according to the accordant opinion of meteorologists, as also according to my previous deductions, it is very probable that the inten- sity of the rotatory currents of the atmosphere materially exceeds that of the meridional currents, therefore I have only introduced into the further computation the rotation currents, whose components are des- ignated by u2 and v2. Since we have to do with a movement of rotation about the axis of z therefore we can put «*=— *y> «a=+^i w2=0, and these values can also be used for u, v, and w, in the above-given system of equations. The relative angular velocity x is to be deduced from the expression for the easterly component O (see equation (9), page 183). This is a func- PAPER BY PROP. OBERBECK. 191 tion of 6 aud of r or also of a the altitude above the earth's surface. The first system of equations is therefore transformed into the following: <^~ = &+X) x*, dz Since ^ is a function of r and 0, or of p aud z if we put z = r cos 6 p = r sin d; therefore, we can not find one function v3 that shall satisfy the three equations. If x were independent of z we should find c2v3 = constant + 1 (2e+x) XP &P- Since however this is not the case we must therefore conclude that J the above-given system of equations still needs a supplement; that therefore a movement of rotation of a fluid to the exclusion of all other movements can only exist when the angular velocity in the direc- tion of the axis of rotation is everywhere the same. If this is not the case then further currents occur perpendicular to the rotary motion. In our case these latter would consist of vertical and meridional move- ments. Their components may be designated by u3 v3 w3. These are to be introduced into the above system of equations as was done in the corresponding fundamental equations (3) of the first memoir which now become (?;)£=(2e+x)xy + »^ Oy c2— 3 = kAw3 dz dx + dy dz ' (2) J If the component motions indicated by the subscript 3 that directly depend on the movements subscript 1 are materially less in intensity than the movements of rotation, then in any computation of the pressure their introduction ought not to be omitted. The former memoir gave 192 THE MECHANICS OF THE EARTH'S ATMOSPHERE. a rather complicated value for the angular velocity X. I have intro- duced a simplified expression for this in that, while retaining the dependence upon the polar distance 0, as there given, I have tempo- rarily introduced a constant average value instead of the dependence upon the distance above the surface of the earth. According to this, one can put x=xi cos2 0-x-z (3) or with a slight difference X =& | Xiz2-X>r2 J (4) In these equations Xi and j2 a-re considered as constants. Therefore, as before found, the movement of rotation of the air in higher latitudes is positive, that is to say, has the same sign as the axial rotation of the earth. For a specific latitude the average value is 0, and at the equa- tor the movement has the opposite sign. Further computation shows that the relative angular velocity j is small in comparison with that of the earth £, so that the simpler equa- tions to be solved are as follows : c2j~ = 2e xx + h Au3 JV3 c\ — dy = ^xy + hAv3 hAw3 to ^dy __o (5) In solving these we first determine a function g that is of such form as to satisfy the conditions ?a n ^A n „ ~ = 2exx,^eXy. These conditions give ^ S v„2 Kir* \ *=%\ Xx* - % rZ j (6) Furthermore we put *=%>—$>--£ + * !7) where L and M are two new functions of x, y, and z, we can then write the system of equations as follows : C2p = S + K d_{AL) d% dx d% JJ/3 _ d% „ I C2^- + x L—{AL) dy dy dy PAPER BY PROF. OBERBECK. 193 The equation of continuity now becomes 3» (8) The three first equations lead to the two following c2 v-i = Constant + % — u — (9) AM = -.d3 (10) If the functions L and M are so determined that they satisfy the boundary conditions then the problem is to be considered as solved and equation (9) gives the desired distribution of pressure. As boundary conditions I have retained those previously laid down, viz, adhesion to the earth's surface, slipping on an upper boundary surface at au altitude R. h above che earth whereby h is to be considered as a small number in comparison with unity. For further calculation it is expedient to introduce the vertical and meridional components of the current or Fand N. These are con- nected with L and ill by the equations V=*^+ Mcoad ] > (11) N= -£^ + i¥ sin 0 I The equation of continuity now becomes £+?7-i{ «*«.*+£} (12) Jr r r i dV J The elimination of L gives the further equation ^ '-+ — n =r — sin 6+ -—y cos 6 (Id) The calculation gives the following values : V=2^R^Xi+^X2-Q(iXi+X2)GOS2d+S5XlGOSi6\.f(ff) . (14) F=2-R3 sin 6 cos &-Xi—2xt+1Xi • cos2 d7.cp(a) . . . (15) H ( ' In these /((r) aud q> ( V3=~J5Z~{ 31-295- 61.094 cos2 6 J =0.0413 cos2 #-0.0806 cos4 9 . . . . But the computation of v3 had already given v,=^cotf 0 \ 3-^+X2-Xi cos2 0 } wherein the appended constant can be omitted. Hence, the two expressions for r3 can be put equal to each other, and for the computation of the motion of rotation we obtain the two equations — r-*! =0.0806 If in these we put then we shall obtain E-#(f+*)=0.0413 i2=6379600m; c=280m; 6=0.00007292 Xi =0.0292 e ^=0.0836 Xi. Hence, the relative angular velocity of the rotary motion of the air is j=0.0292 s { cos2 0-O.OS36 \ . (20) This is small in comparison with e, the angular velocity of the earth, therefore it nowhere leads to improbably large movements of the at- mosphere. If we form the product xi Q we obtain for it the value 13.58 metres per second. But the true linear velocity corresponding to the rotatory motion is O = x> -R* sin 6. The maximum value of this occurs at S. latitude 56° 27' and amounts to 4.59 metres per second . From the S. pole to 1 6° 49' S. latitude the average 196 THE MECHANICS OF THE EARTH'S ATMOSPHERE. value of the rotatory motion is positive, that is to say, directed toward the east ; thence to the equator the value is negative, therefore directed toward the west. These results can easily be combined with the conclusions of my pre- vious memoir, according to which the motion of rotation can be consid- ered as the sum of two terms that are of entirely different natures. Of the second term it was remarked especially that the current correspond- ing to it first attains sensible values at great altitudes. This therefore becomes at that altitude materially larger than the above deduced av- erage value. The first term gave a movement entirely confined to the lower strata of the atmosphere: it is directed toward the east from the pole down to 35° latitude, but directed toward the west exclusively ill the equatorial zone and less in velocity than the first component move- ment. The numerical computation leads to the same conclusion, since X2 is small In comparison with Xi Since from 35° of latitude down to the neighborhood of the equator there are two currents of opposite signs flowing over each other, therefore the place where the average movement of rotation is 0° will lie nearer to the equator than to 35°. Therefore the conclusion of W. Siemens, which gave the first stimu- lus to the present investigation, has to be subjected to a modification only in so far as we must consider that the westward movement of the upper regions and higher latitudes has a predominance over the easterly movement of the lower regions and lower latitudes, because the former loses a much smaller fraction tha.n the latter of its living force in con- sequence of friction. The vertical and meridional components Fand iV'are to be added to the corresponding components that were computed in my first memoir. The vertical component is positive at the equator and at the pole, it therefore gives an ascending current at both places, whereas V is neg- ative throughout a broad central zone. Therefore at the equator the ascending current is strengthened, at the pole the descending current is enfeebled. The meridional component N is zero at the surface of the earth at the equator ; it is negative, i. - Here also the quantity on the left hand that is equated to zero represents the difference of the entropies between the final and the initial conditions of a kilogram of the mixture. The equation thus obtained can be used until the temperature attains the freezing point, then we arrive at the third stage. Third stage. — In this case, in addition to the vapor and the liquid water, the air contains also ice. By further expansion of the air, the temper- ature will now not sink immediately further, for the latent heat of the freezing water will, even without a lowering of temperature, furnish the force necessary for overcoming external pressure. But the heat of liquefaction must not be applied to this purpose only, but also to the evaporation into vapor of a part of the already condensed water. For since the volume increases during the expansion without allowing the temperature to sink, therefore at the end of the process again, more water is become vapor than before, therefore the weight of the ice that is formed will be less than that of the fluid that was present. Let now, again, v be that portion of yu that is in the form of aqueous vapor, 6 the part that exists as ice, and q the latent heat of liquefaction of a kilogram of ice. T, e, r are constants. Since therefore dT=0, we fill have now only to communicate to the air the quantity of heat A ART— and to the water that we evaporate the quantity of heat rdr, and to the water that we allow to freeze the quantity —qda. Therefore the quan- tity of heat given to the whole mixture is dQ=\ARTd-+rdv-qda. If we put dQ=0, divide by Tand integrate, there follows v \ARlog^-+^(r— n)-^{0— • zr—2 ' T~h yjr ( '' This equation connects the pressures p0 and jp„ at which respectively the third stage is attained and relinquished. It was not necessary to append an index figure to the quantities e and T since they are alike for the initial and final conditions. Fourth stage.— If now the temperature sinks lower, we have then only vapor and ice. The relations that we have to consider are the same as in the second stage, and the final formula is also the same. Only here the specific heat of evaporation has another value from that there given. Here, namely, it is equal to r+q since the heat that is necessary to immediately change ice into vapor must exactly equal the heat that is needed to first melt the ice and then chauge the water into vapor. If we would be perfectly rigorous we ought not to assume q as constant, but must consider it as slightly variable with the temperature, but the differences are so small that here they may remain out of consideration. In this fourth stage we may attain to those low temperatures at which the air itself can no longer be considered as a permaueut gas. The four stages that we have here distinguished, one can very prop- erly designate as the dry, the rain, the hail, and the snow stage. If one is now in a position such that he is obliged to exactly follow the changes that a mixture containing a considerable percentage of water must undergo, then nothing further remains than to abide by these more complicated formulae. In that case one proceeds in the following manner : First we substitute the values of A and fx in all the equations. Then we substitute the quantities p0 and T0 for the given initial condi- tion in equation (L). We then consider the resulting equation and the equation (16) as two simultaneous equations with the two unknown quantities p and T. Solving those equations with reference to these quantities, we obtain that condition through which we must go in pass- Atmospheric jPressi< AZZiluct ? (MiZ2i metres) sSxxzZe. PAPER BY DR. HERTZ. 203 ing from the first to the second stage. The values thus obtained are then to be substituted as p0 and T0 in equation (2). By substituting T=273° in the equation thus obtained, we obtain that p0 which occurs in the equations of the third stage. If now we further determine from the equation (3) the final pressure px of the third stage then this pressure and the temperature 273° form the p0 and T0 of the equations of the fourth stage. It will frequently happen that the temperature down to which the first stage holds good will lie below the freezing point ; in that case oue passes directly over to the fourth stage, omitting the second and third. After we have thus determined for all the equations the coefficients and the limits for which each equation holds good we can use them in order to determine the T belonging to any given p or inversely. All these computations can however only be executed by successive approximations, and one would do well to take the necessary approximate values from the accompanying diagram. If we have determined p and Tfor any special condition then the remaining characteristics are easily found. The density of the mixture follows from the corresponding equation of elasticity. The equation (la) gives the quantity of water still present in the form of vapor, and therefore also the quantity of water already liquefied. Frequently oue desires to know the difference in altitude h that corresponds to the different con- ditions ^o and p\ under the assumption that the whole atmosphere is found in the so-called condition of adiabatic equilibrium. If one de- sires the exact solution of this problem, it must be attained by the laborious mechanical evaluation of the integral h = / vdp ; J Pi but since it is precisely with regard to this point that an exact deter- mination never has a special value, therefore here one may always abide by the accompanying convenient diagram. IT. If we had to deal only with one mixture whose composition is exactly known for which we therefore can have only one value of the ratio // : A, then we could exactly re-produce the formula1 above developed by a graphic table that would enable us to directly perceive the adiabatic changes of the mixture for any condition. We should represent pressure and temperature by coordinates in one plane and cover this plane with a system of curves that should con- nect all those conditions together that can adiabatically pass from one to the other. It would then only be necessary to glide from a given initial condition along the curve going through the corresponding point in order to perceive the behavior of the mixture as it passes through all these stages. Since however the meteorologist must necessarily deal with mixtures 204 THE MECHANICS OF THE EARTH'S ATMOSPHERE. of very various proportions, therefore by this method a great num- ber of tables would be required. But it can now be shown that one can also manage with only one graphic table, if first we confine our- selves to those cases in which the weight and pressure of the aqueous vapor is small in comparison with the weight and pressure of the air, and if secondly we do not require of the results any greater accuracy than corresponds to the neglect of those quantities in comparison with these. If we neglect \x and eas compared respectively with A andj), then the form of the curves to be drawn is the same for all the absolute values of /a, therefore the same curve can be used for all the different mixtures. But the points at which the different stages pass into each other will be located very differently for different mixtures, and special devices will therefore be needed by meaus of which this point may be determined. The graphic table, Fig. 28, is therefore constructed in accordance with the following principles. The pressures are laid off as abscissas on the adopted scale for the interval between 300 millimetres to 800 millimeters of the barometer ; the temperatures are laid off as ordinates for the interval between —20° Cels. and +30° Cels. But as we see by the diagram, a uniform increase in the length of either of these coordinates does not indicate an equal in- crease of pressure or of temperature; on the contrary the diagram is so constructed that an equal increase of distance corresponds to an equal increase in the logarithm of the pressure and in the logarithm of the ab- solute temperature. The advantage of this arrangement consists in the fact that thus the curves with which we have to do become, some of them exact, and some of them approximate straight lines, which brings an important advantage in the accurate construction and use of the table. Now the adiabatics of the first stage (if we neglect /* with respect to A) are given by the equation cp log T—AR log p — constant (a) In this diagram the logarithms are always those of the natural sys- tem. With Clausius we put Calorie cp =0.2375 Cels# degree x kilogr. 1 Calorie #=29.27 423.55 Kilogram metre Kilogrammetre Cels. degree x kilogr. These adiabatics appear in our diagram as straight lines. One of them is distinguished by the letter alpha (a) and the whole of this system may be called by this letter. The individual lines are so drawn that PAPER BY DR. HERTZ. 205 from one to the next the value of the constant (which is the entropy) increases by the quantity 0.0025 Cal0rie Cels. degree x kilogram' These lines therefore appear at equal distances apart from each other. One of them is drawn to the point 0° Cels., and the pressure 760 milli- metres. The curves of the adiabatics in the second stage must satisfy the equation* — cp\og T-AR log p + j|.r.i= constant .... (/?) R In this equation -n- is the density of aqueous vapor in reference to the air, and therefore is equal to 0.6219. According to Clausius, r = 607 - 0.708 ( T - 273) . C.al°rie . v ' kilogram I have taken the value of e for the different temperatures from the table computed by Broch (Travaux. du Bur. Intemat. des Poids et Mes- ures, tome i). The curves run along with feeble curvature from the right hand above to the left hand below. One of these is distinguished by the letter beta (/?). They also are so drawn that the entropy in- creases from one to the next by a constant value of— 0.0025, Cal0ri6 Cels. degree x kilogram' or the same as before for the alpha system, and so that one of them passes through the point 0° C, 760 millimetres. The curves that correspond to the third stage coincide with the iso- therm of 0° C. Finally the curves of the fourth stage are entirely similar to those of the second stage, but are not exactly the same, for their formula is derived from that belonging to the second system by substituting r + q for r where q is equal to 80 calories per kilogram. They are distin- guished by the letter gamma (y), and are drawn according to the same rules as alpha (a) and beta (/3) curves. In general the gamma curves are not precise prolongations of the /? system. We have now to find some means by which the points of transition * Although jx is neglected in comparison with X, still it is questionable whether cp is negligible in comparison with cpX, since c is four times larger than cp. Even al- though within the limits of the diagram ju. does not exceed -fo A, yet the cp, is -fa cv*~ But in meteorologic applications we recall that in these extreme cases the liquid water is not generally wholly carried up with the air. Frequently so large a fraction of it falls from this air as rain that we keep nearer the truth when we entirely neglect the specific heat of the liquid water, rather than to introduce it with full value into the computation. 206 THE MECHANICS OF THE EARTH'S ATMOSPHERE. can be found for the different stages. The dotted lines serve to show the end of the first stage. These lines give the greatest quantity of water, expressed in grams and computed according to the formula 7? p v==~' J, that a kilogram of the mixture in the different conditions can contain as vapor. Thus, for instance, the curve designated by 25 con- nects all those conditions in which one kilogram of the mixture when saturated contains 25 grams of vapor. These curves are drawn from gram to gram. If a mixture contains n grams of vapor in every kilo- gram of mixture, then evidently we have to follow the curve of the first stage up to the dotted line n, but then we must pass either to the second or fourth stage. The limit of the second stage, with respect to the third, is given by the intersection of the corresponding adiabatic beta with the isotherm of 0° 0. By the pressure^ that corresponds to this intersection, and by the quantity jx of water, is determined the pressure ph at which the transition takes place from the third to the fourth stage. The small auxiliary diagram that is given beneath the main table of Fig. 28 serves for the graphic determination of px. This auxiliary diagram con- tains as abscissa the pressure arranged as in the larger diagram, and as ordinate the total quantity j.i of the water in all conditions ex- pressed in grams per kilogram of the mixture. The oblique lines of this small table are the curves that correspond to the equation (3) of the third stage, when in this equation we consider p0 as constant, but Pi and }.i as the variable coordinates. These lines are not perfectly straight, but are not to be distinguished from such in a diagram ou this scale. The highest point of each of these lines corresponds to the case px = jOo- The corresponding jj. is not zero, but is equal to the least value, v, that ).i must have in order that the mixture may be saturated at 0°C, and the auxiliary table come iuto use. If one wishes to find the px belonging to a definite value of p0 and /*, then we seek that oblique line whose highest point lies on the abscissa jp0, and then we pass along this line downwards to the ordinate //. The pressure at which we attain this ordinate is the desired pressure^. In this pres- sure we have the point of transition from the third to the fourth stage. Having in this way determined the totality of the stages through which the mixture runs, we find the remaining desired quantities for each stage in the following manner: (1.) The dotted line which one selects, (corresponding to the condi- tion given,) indicates directly the number of grams of water still remain- ing in the form of vapor. If we subtract this quantity from the original total quantity /<, we obtain the quantity of water that has already been condensed. (2.) The deusity d of the mixture can under the adopted approxima- tions be computed for all conditions by the formula gyp-, or log 6 = log p — log T— log E. PAPER BY DR. HERTZ. 207 These can also be read off graphically if the diagram is covered with another systim of lines of equal density. We see that these Hues will constitute a system of parallel degrees of density. Only one of these lines is in reality drawn on the accompanying diagram, namely, the line marked S (delta), in order not to confuse the diagram. But with the assistance of this one we can also compare the densities in any two conditions d and 02, according to the following rule: From the points 1 and 2, representing these conditions on the diagram, draw two straight lines, respectively, parallel to 3, until they intersect the isotherm 0° C, and read off the pressures pi and ^2for these points of intersection. The densities for the conditions Ci and G2 are in the ratio of the pressures px : p2 ; as is seen from the considera- tions that the densities for the condition (pu 0°), and. for (p2, 0°) are ac- cording to Mariotte's law in the ratio of px to p2, and are equal to the densities for the conditions C, and C2 since they lie on the same line of equal density with these. (3.) The difference of altitude h that corresponds under the assump- tion of adiabatic equilibrium to the passage from the condition p0 to the condition p is given by the equation dp. Up up 1 In using this equation we take T as a function of p from the diagram and then perform the integration mechanically. Actually however the assumption of adiabatic equilibrum is always so imperfectly ful- filled that it is not worth while to trouble about an exact development of its consequences. On the other hand, for moderate altitudes, we commit a relatively very unimportant error when we give T an average value, and consequently consider it as constant. Within the limits of the diagram T ranges only between the values 253 and 303; if there- fore we give it the constant value T0 = 273, then the error in h will scarcely exceed one-ninth of the whole value. If we are satisfied with this error, then we have h = constant— B T0 log p, and we now can, along with the pressure, directly introduce the altitude as abscissa. Consequently an equal increase in the length of the abscissa will everywhere correspond to an equal increase in altitude. The scale of altitudes is introduced at the base of the diagram. Its zoro point is put at the pressure 760, because this is usually taken as the normal pressure at sea-level. in. In order to illustrate the use of the table by an example, we propose to ourselves the following concrete problem : Given a mass of air at sea- level under the pressure of 750 milimetres, the temperature 27 degrees 208 THE MECHANICS OF THE EARTH'S ATMOSPHERE. centimetre, and relative humidity 50 per cent,, it is desired to find what conditions this mass of air will pass through wheu it is carried without change of heat into the higher strata of the atmosphere, and therefore into a lower pressure, and at what approximate altitudes above the sea- level the different conditions will be attained. We first seek from the diagram the point that corresponds to the ini- tial stage. We find it as the intersecting point of the horizontal iso- therm 27 and the vertical isobar 750. We remark that it lies almost ex- actly on the dotted line 22. This indicates that our mass of air must contain 22.0 grams of aqueous vapor in each kilogram of its owu weight in order to be saturated. Since however it has only a relative humid- ity of 50 per cent., therefore it contains 11.0 grams of water per kilo- gram. We note this for future use. Furthermore, we go along down the isobar 750 to the scale of altitude that is found at the lowest edge of the diagram, and here we read off 100 metres. The 0 point of the scale of altitude therefore lies about 100 metres below the sea-level adopted by us as a base, and therefore we have to subtract 100 metres always from all the direct readings on the altitude scale, in order to obtain the altitude above sea-level. If now we raise our atmospheric mass upward, then the series of conditions which it runs through will be directly given by that line of the Alpha system that passes through the initial condition.* An engraved line not being given for this case we therefore interpolate such an one (i. e., the — . . — . . line of the diagram). If the number of intersecting lines appears to be bewil ler- ing, then we take a strip of paper and lay it parallel to the system under consideration, when all confusion disappears. In order now to recognize the condition in the neighborhood of the altitude 700 metres we seek for the point 700 -f 100 = 800 on the scale of altitudes, and go perpendicularly up until we intersect our Alpha line. The intersection gives this point at pressure 687 milimetres, and temperature 19.3° C. But we ought to use the Alpha line only to that point in which it itself intersects the dotted line 11 (or the line of absolute weight of con- tained water). The attainment of this line indicates that we have ar- rived at a condition in which the air is only just able to contain 11 grams of water per kilogram in the form of aqueous vapor. Since now we have 11 grams per kilogram, therefore with any further cooling con- densation begins. The pressure for the point at which precipitation commences is 640 milimetres ; the temperature is 13.3° C. This is not the temperature of the original dew-point, but it is lower. The dotted line, eleven, intersects the isobar 750 at 15.8° C, and this is the initial dew-point. But since besides cooling our air has also experienced an increase in its volume, therefore the vapor has remained volatile to a *The letters a, /J, y, that designate the systems are to be found in the small circles at the edge of the diagram. For each of these there corresponds one line of the sys- tem that it designates. A line of special dots and dashes in the diagram indicates the change of condition of the air in our illustrative example. PAPER BY DR. HERTZ. 209 temperature 13.3. The altitude at which we now find ourselves corre- sponds to the lower limit of the formation of clouds, and is about 1,270 metres. Iu order to follow the conditions farther we draw a curve of the Beta (/i; system through the point of intersection. This curve is inclined much more slowly toward the axis of abscissas than the Alpha line hitherto used, therefore the temperature now changes with the altitude much more slowly than before, which is due to the evolution of the latent heat of the aqueous vapor. We have now risen 1,000 metres since the commencement of condensation, but the tem- perature has sunk only to 8.2°, or only 0.51° to each 100 meters. We now find ourselves on the dotted line 8.9, and perceive that 8.9 grams of water are still in the state of vapor ; that therefore in this first 1,000 metres of the cloud layer 2.1 grams of water have been condensed per kilogram of air. We attain the temperature zero degrees C. at the pressure 472 millimetres, and at the altitude 3,750 meters, whereas if the air has been dry, and we had not been obliged to leave the Alpha line, this temperature would have been attained at an altitude of 2,600 metres. It now appears that by this time 4.9 grams of water, or 0.45 per cent, of the total contents, have been condensed, and during further ex- pansion lhis portion begins to freeze and form hail [the reader will re- call that although 45 per cent, has been condensed into visible cloud, yet it has not separated from its original air and been precipitated as rain, but is still rising with the air and of course cooling with itj. But the temperature can not sink further until the last particle of water is frozen, and we therefore must retain the temperature 0° uniformly dur- ing a certain distance of further ascent. In order to ascertain this distance we make use of the auxiliary diagram between the scale of altitude and the larger diagram, we pass down the isobar 472 millimetres to the dotted line of this diagram; we draw through this intersection a line parallel to the inclined line of the auxiliary table, and go along this line until we reach that horizontal line that is characterized by the number 11, or the total weight of the contained water, and which we easily interpolate between the engraved lines 10 and 15. As soon as we have attained this line we read off the pressure p = 463 millimetres, and turn back to the larger diagram. At the pressure thus found the process of freezing is fin- ished, and the layer within which it all takes place has a thickness of about 150 metres. It must surprise one that, according to the dotted line, the quantitv of water in the form of aqueous vapor has again in- creased a little during the process of freezing. But this is quite cor- rect; in fact, the volume has increased without lowering the temper- ature. We leave the temperature 0° C. at the pressure 463 millimetres. The water which hereafter is precipitated passes directly over into the solid condition. Since there is now but little water as aqueous vapor, therefore the temperature again begins to sink more rapidly with the altitude. We ascertain the different conditions in that we make use of SO A 14 210 THE MECHANICS OF THE EAltTH's ATMOSPHERE. that special Gamma line tbat can be drawn through the point 4G3 millimetres on the isotherm 0° C. The temperature— 20 down to which our table can be used is attained at the altitude 7,200 metres, and at the pressure 305 millimetres, at which only two grams of water per kilo- gram remain as vapor, the other nine having been condensed. If it interests us to know how the density in this condition is related to the density in the initial condition, we draw through the corresponding points two lines parallel to the Delta line. These intersect the isotherm of 0° C. at the pressures 330 and G80 millimetres. The densities are to each other as these pressures, namely, as 33 to 68; and as 33 and US are to 76, so they are related to the density of the air in its normal condition of 0° C. temperature and 760 millimetre pressure. All these items are directly read off from the diagram. Errors that could be injurious certainly occur only in the altitudes. These latter refer strictly speaking to ascent in an atmosphere of a uniform temper- ature of 0° C. But it would have been generally better to have as- sumed that the temperature of the atmosphere is everywhere the same as that of the ascending mass of air. The resulting error can be ma- terially reduced by a very little computation. Thus we found that the point where condensation began, is at the pressure 640 millimetres. To this corresponds an altitude of 1,270 millimetres, provided that the temperature is 0°, but in our case this is between 27° and 13°, there- fore on the average about 20°. For this temperature the altitude must be about -2aT% or -^ greater, since the density of the air is by this same fraction smaller than for 0°. Therefore the altitude really lies between 1,350 and 1,400 millimetres. We must still supplement the above example by the mention of special cases : (1) We assumed in the above that during the hail-stadium the total quantity of water originally present in the air, namely, 11 grams, was still contained therein. This will certainly only be an appropriate as- sumption in the case of very rapid ascents. In other cases perhaps the greater part of the condensed water falls as rain, and therefore only a fraction of it remains to be frozen. If one has any estimate as to how great this fractional part is, then the diagram will always allow us to ascertain the correct conditions. Thus if in our example one had reason to assume that half of the water condensed at 0° were removed, then on attaining the isotherm of 0° only 8.5 grams of water per kilogram of air would be present. We should then in using the auxiliary table not descend to the horizontal 11, but only to the horizontal 8.5, and should have started from the temperature line of 0° at the point corre- sponding to the pressure 466 millimetres (instead of 4G3 millimetres) ; this would have been the only difference. (2) If we had assumed not 50 per cent, but 10 per cent, relative hu- midity in our example we should then have been able to use the Al- pha line only to the dotted line 2.2. This point of intersection occurs PAPER BY DR. HERTZ. 211 at pressure 455 millimetres, and at temperature — 13.6° 0., therefore considerably below 0. Therefore there would have been no formation of liquid water and therefore no stadium for the formation of hail but only sublimation of water from the vaporous into the solid condition. We should then from the intersection of the Alpha line with the dotted line 2.2 have followed directly the line of the Gamma system that might have passed through this intersecting point. The question is not uninteresting— what dew point is the highest that our mixture could have possessed in its initial condition as to pressure and temperature, in order that the condensation of liquid water, that is to say, the condensation at temperature above 0° C. should be just avoided ? In order to answer this we follow tbe Alpha line to the isotherm 0° and here find the dotted line 5.25. We therefore at the highest could have had 5.25 grams of water per kilogram of air. In order now to ascertain at what temperature the air would then have been saturated under a pressure 750 millimetres, we slide along the line 5.25 up to the isobar 750 and intersect it at the temperature 4.8° 0., and this is the desired maximum value of the dew point. Kiel, October, 1884. XV. ON THE THERMODYNAMICS OF THE ATMOSPHERE.* (FIRST COMMUNICATION.) Bv Prof. WlMIELM VON Bezold. In the application of the mechanical theory of heat to the processes going on in the atmosphere we have hitherto almost exclusively con- fined ourselves to those cases in which one can disregard the increase or loss of heat during the expausion or compression. The so-called convective equilibrium of the atmosphere, the unstable equilibrium in cyclones, the phenomena of the foehn winds have all hitherto been treated of under the assumption that we have to do with adiabatic changes of condition. In fact, especially in the last-mentioned phenomena, the quantity of heat used or produced by expansion and compression as also by the chauges in the physical condition of the water, are so prominent in com- parison with those that, in these rapidly executed processes, are intro- duced or taken away by other sources that the above-mentioned as- sumption may be said to be thoroughly allowable. Iu the investigation of the convective equilibrium we obtain, under this assumption, at least a glimpse of the special case that lies as a limiting case between the two greater groups that correspond to the loss or increase of heat. Not- withstanding these extremely restrictive assumptions, still through the above-mentioned investigations, the comprehension of meteorological processes has been furthered to such an extent that we must consider their introduction as one of the characteristic features of modern me- teorology. But the more valuable are the results that are already at- tained iu this manner, so much the stronger must be the desire to free ourselves from the above-given limitations, and to extend the applica- tion of the mechanical theory of heat to those atmospheric processes in which the increase and diminution of heat from without can be no longer neglected. That this generalization had not already been long before taken is certainly because the formulae are extremely complicated, so *Trauslated from the Sitzungsberichte der Konig. Preuss. Akademie der Wissenscha/len zu Btrlin: Berlin, April 26, 1388, pp. 485-522. 212 PAPER BY PROF. BEZOLD. 213 that one always runs in danger of losing the leading thought in the midst of the notation and signs. But in consideration of the fundamental importance that the applica- tion of the mechanical theory of heat in the most comprehensive man- ner possesses for the development of meteorology, one evidently ought not to be frightened by these extreme difficulties. This has induced me to make the attempt to introduce a method into meteorology that has proved so remarkably fruitful in the application of the mechanical theory of the heat to the theory of machines: I mean the graphic method that Clapeyrou* has invented in order to make the ideas first ex- pressed by Sadi Carnott visible and comprehensible. Already, some years ago, a step in a similar direction was taken by HertzJ in a highly meritorious work on a graphic method for the determination of the adiabatic changes in moist air; but the problem that Hertz had before him, as also the method which he adopted, were materially different from those that I have now in mind. On the one hand, Hertz confined himself, as his title states, exclusively to the consideration of the adia- batic changes, and on the other hand, his object was only by means of a simple graphic process to avoid the complicated computations that one has to execute in following these changes. My object, on the other hand, has been to give a method of presentation that can serveas a guid- ing thread in the still more complicated formulas with which one has to compute as soon as we disregard the restrictive assumption of adiabatic change, and that also allows one to draw certain important conclusions even from the form of the geometrical figures. To attain these objects however, scarcely any mental presentation is so appropriate as that in- troduced into science by Clapeyrou, of course with such extensions as are required by the condition that in meteorological problems we have not as there to consider only two independent variables, but three, or in special cases, even still more. But before I enter upon the subject itself I must touch upon another point on which notwithstanding its fundamental importance, remarkable to say, still perfectly clear views do not prevail. This has respect to the true reasou of the cooling that occurs in the ascent of air to higher regions as well as the corresponding warming for descending air. While Sir William Thomson, § Keye,|l Hann,fl Peslin,** and with these investigators probably also the greater part of all physicists and meteor- ologists, correctly consider the cooling of ascending air as a consequence of the expansion occurring therein, on the other hand, Guldberg and * Poggendorffs Annalen, vol. 59, pp. 446-566. \Reflexions sur la puissance motrico dufeu. Paris, 1824. XMeleorologisclie, Zeit., 1834, i, pp. 4-21-431. [See No. xiv of this collection.] § Proc. of Manchester Soc, 1862, n, 170-176. || Die Wirbelstiirme, Hannover, 1872. H Zeitschrift d. Oesterr. Ges. f. Met., 1874, Bd. ix, pp. 321, 337. Smithson. Rep. 1877, p. 397. * * Bull. held, de V Assoc, scientif. de France, 1868, Tome in. p. 299. 214 THE MECHANICS OF THE EARTH'S ATMOSPHERE. Molin* find the reason therefor in the work that is done in raising the air, and that is balanced by an equivalent quantity of heat taken from the air. Since by both methods of consideration the same value is fonud for the diminution of temperature with the height, therefore in the well-known excellent treatise of Sprung t both methods of consid- eration are presented beside each other as equally proper. But in fact only the first of these two is allowable, while that of Guldberg and Mohn contains in itself an error as to which one can only wonder that it could have escaped two such thoughtful investigators, and evidently also has hitherto not been remarked by others. In order to obtain perfect clearness on this point oue must first recall how it is that the ascending and descending currents in the atmosphere come to exist at all. This is, however, always brought about by differ- ences in specific gravity that cause an ascent at certain places, while a corresponding mass descends at other places. The work that is re- quired to raise the air, at the one place is therefore always obtained by the falling of an equally great mass at another place. If no friction occurs the corresponding rising and falling movements once started would continue without any further addition of energy to infinity, and such an external addition of energy is only needed in order to overcome these frictions. These latter, however, are left out of consideration in all the discussions that are here considered, and this will also be done in the present memoir. We can consequently then compare the process with which we have to do, with movements in closed systems of tubes, such as a closed series of hot water pipes, or the movements of a con- tinuous chain that hangs freely upon a roller. But it would never occur to any one to consider that the ascending water in the warmer half of a conduit, or the ascending portion of an endless chain must cool because of the work done in raising it. Similarly in the case of the ascending or descending currents in lakes or in the ocean, we must ex- pect cooling or warming in consequence of these motions, if the ascent is accomplished at the expense of the heat latent in the fluid. The tem- perature changes occurring in the vertical motious of the air are there- fore exclusively to be attributed to the work of expansion and com- pression, which is to be done or acquired respectively, and they would occur to precisely the same extent if the corresponding changes in pres- sure and volume occurred within a horizontal cylinder where rising and sinking was entirely out of the question. On the other hand if we have air compressed within a vertical cylin- der whose base is fixed, but which is closed above by a movable piston, and if we should now by a proper change in the load cause an expan* sion of the air then, besides the work of expansion, it would be neces- sary also to consider the work necessary in order to raise the center of gravity of the inclosed mass of air, and thus the cooling would be more * Zeit. Oesterr. Ges. Met., 1878, xin, p. 113. t Lehrbuch d. Meteorologie, Hamburg, 1865, p. 162. PAPER BY PROF. BEZOLD. 215 considerable than when the whole change of condition took place with a horizontal position of the cylinder. If the piston were without weight and without any loading, and if it were only at the beginning held fast but then suddenly loosed, and first held fast again at some other position at a greater distance from the base, then indeed the cooling would be attributable alone to the work which was necessary to be done in order to raise the center of gravity of the mass of air, since in this case no work of expansion is accom- plished. By the explanations that I have made in such detail, in con- sideration of the fundamental importance of the question, it certainly ought to be perfectly clear that the cooling and warming in ascending and descending currents of air in the atmosphere are to be considered only as consequences of the work of expansion and compression ; not of the work that is consumed in raising the air or that is gained by its descent, unless the ascending and descending masses belong perma- nently to one system. Since however the work of expansion and com- pression ought never to be left unconsidered, therefore in Guldberg and Mohn's method of consideration these, under all conditions, should have been further taken into consideration, and there would then have resulted for the rate of change of temperature with altitude a value exactly double that given by them. This being premised I will now pass to the problems mentioned in the opening paragraphs. For our purpose it is first necessary to establish the fundamental quantities that come into consideration in investigations into the change of condition of a mixture of air and water or aqueous vapor. If in this I do not accord wholly with the steps that Hertz has chosen, this is be- cause he has made various simplifying assumptions that are appropri- ate to the attainment of the end that he had in view, but that are not allowable in the general theoretical investigation that I contemplate. For the same reason I must again review the equations for the various conditions through which the mixture of air and water can pass, and which Hertz has developed in such a perspicuous manner, since not only by reason of the somewhat different notation, but also by the consider- ation of certain points intentionally neglected by Hertz, some material differences result. Hertz and others in their investigations have made the assumption ordinarily used in the mechanical theory of heat that the unit of mass of the substance under consideration is given, and that it in succession passes through the different conditions. This assumption can not be rigorously adhered to in the case of atmospheric processes. A kilo- gram of moist air retains unchanged its mass only so long as during the expansion no condensation of aqueous vapor occurs, but suffers a diminution as soon as the formation of precipitation begins and rain, snow, or hail falls from it. When therefore a mass of moist air that is rising within a depression, or on the windward side of a mountain during a foehn on the lee side, is followed on its way through the atmos- 216 THE MECHANICS OF THE EARTH -S ATMOSPHERE. phere until it finally, eitber within an anti-cyclone or under the well known conditions of the foebn wind on the lee side of a mountain, comes again to its initial level, it is not the whole mass that we again find there present, but only a portion, although it may be a very considera- ble fraction, since a part of the water has been lost. One can therefore begin the computation with the unit of mass of the mixture, but must consider the loss in mass that may occur in the course of the processes (a gain only occurs when the air passes over moist sur- faces). But in this we have to combat the difficulty that, according to the point of departure that we choose, or according to the prevailing absolute humidity of the air at the point of departure, we have present, not only different quantities of vapor, but also different quantities of dry air, since the sum of the two must be equal to unity. It is there- fore more appropriate to consider the unit of mass of dry air as given, and the water as an additional variable mixture. This being premised, we will now indicate by Ma, Mb, Mc, Md the masses of the mixture in the four stadia so well distinguished by Hertz, namely, the dry, the rain, the hail, and the snow stage, and will also attach to the other quantities similar subscript letters as indices, in so far as a distinction of the respective stages may be necessary. But in compu- tations that relate throughout to only one stadia these indices may be dropped, in order not to overburden the formula? too much. This being premised, we next find for the four stages the accompanying equations that may be temporarily designated as the equations of mixture. {A). — The dry stage : Ma = 1 + xa or abbreviated M = l+x where x or xa designates the mass of aqueous vapor that is mixed with the unit mass (one kilogram) of dry air. In this it is assumed that the air is not saturated with aqueous vapor, and therefore xa indicates always the mass of unsaturated (overheated) vapor that is contained in the mixture. This mixture remains, in general, constant in the free atmosphere, since in this stage precipitation is excluded and an appre- ciable introduction of aqueous vapor is only possible at the surface of the earth, and again since the quantity of aqueous vapor that is ex- changed in the atmosphere between masses of air of different absolute humidities can certainly at first be wholly neglected. (B). — The rain stage. Mb =1 + xh + x,,' or, when confined to one stage as before, M = 1 + x + x'. In this xh indicates the mass of saturated aqueous vapor that is con- tained in the air, x'h is the additional mass of water liquid that is present. PAPEK BY PROF. BEZOLD. 217 If we assume that by cooling, as for example through adiabatic ex- pansion, the air has passed from the dry stage to the rainy stage, then will M> < M . wherein the equality sign is the limiting case but iu general the in- equality is to be considered as the characteristic sign. The quantity x\ is always very small and can only assume a somewhat greater value in exceptional cases, as for instance in the case of a remarkably strong ascending current of air that hinders the fall of the rain or rather thai carries the drops upward with itself. How large this value may become we have as yet no indications whatever. (C). — The hail stage: for this case Mc = 1 +'xc+ x'c+ x"c wherein xc is the quantity of saturated vapor; x'c the quantity of water present in the fluid condition; x"c the quantity of ice that is present. Here as above, under the corresponding assumptions, we have Mc). — The snow stage; for this case where the notation is easily understood by what precedes and where again so far as the mixture can be considered as coming from the pre- vious stage, we must have Md < If.. In the most common case, where an ascending mass of air p by cool- ing gradually goes through all the different conditions, x' and x" are generally exceedi ugly small, so that the hail stage is entirely passed over, and iu all formulae only oue independent variable x appears. Iu this case If steadily diminishes. Hertz in his investigation has not considered the change of M, but has considered this quantity as coustant. This was allowable in view of his object, but here as already stated in the beginning, this limita- tion must be avoided. The present more general consideration leads first of all to the recognition of the fact that here we have to do with a class of processes which so far as I know have not yet been considered iu the mechanical theory of heat; such namely, as are reversible in their smallest parts but are not reversible as a whole. So long as the quantities x' and x" are not equal to zero but possess a finite value even though exceediugly small, then can the quantity of 218 THE MECHANICS OF THE EARTH'S ATMOSPHERE. vapor that is condensed by cooling, as in expansion, be again evapo- rated by warming or compression. But as soon as the small quantity of water is evaporated, then by a further warming, the air enters again into the dry stadium but with a different quantity of vapor than it originally had, so that now it will pass through other conditions than at first, when the air expanded under continued loss of water. In order now to be able to determine perfectly the condition of the mass of air, we need beside the variables that occur in the equations of mixture to know also the volume v that the mass .If occupies and the pressure p. The latter we measure by the pressure in kilograms per square metre, wherein we now have to understand by kilogram the weight that a kilogram of mass has at 45° Lat. The simple relation p = 13.6/i exists between the pressure p thus measured or the so called specific pressure and the barometric pressure (i expressed in millimetres of mercury; whereas expressed in atmospheres it has the value 10333 so that one can without difficulty pass from one mode of measurement to the other. This much being prefaced, we can now establish the equations for the gaseous condition [equations of elasticity] for the different stages. Their general form is f(v, p, t,x)=0 therefore they contain one variable more than we generally find in the equations of elasticity. The quantities x' and x" do not appear in these since in general they are so small that they exert no influence on p and v. If now we would geometrically picture a condition of mixture we must (besides p and v which will be represented in the ordinary method by ordinates and abscissas in a rectangular system of coordinates with the axes OP and 0 V) make use further of a third coordinate ; as such we advantageously choose the value of x, and lay this off parallel to the axis OX in a direction perpendicular to the plane PV. In this method of presentation, all conditions that correspond to any value of x find their representation in one and the same plane, which only slightly differs from the P V plane if we adopt the atmospheric pressure as the unit of pressure, and adopt lines of equal length in the direction of the axes of Y and X as expressing the units of volume (one cubic metre) and of mass (one kilogram). If now we imagine successive planes lying above each other, on which conditions are represented that differ progressively from gram to gram (that is, by a thousandth of the adopted unit), then these will lie PAPER BY PROF. BEZOLD. 219 like sheets above each other, and in the study of the changes in condi- tion one can simply adhere to the consideration of the curves described by the projection of the represented points upon the PFplane. There- fore, this plane will frequently hereafter be briefly designated as the coordinate plane. We can therefore execute the mental presentation of these processes in this plane, if only certain artifices are used, of which mention will be made hereafter, and when we consider the result- ing curves after a manner similar, as it were, to the lines on a Riemann surface. The most important result is, that thereby the external work consumed or expended finds its mental representation precisely as in the simple method of Clapeyron. The formula dQ = AdU+A pdv expresses the quantity of heat to be added for an infinitely small change of condition, under the notation* here adopted, and the special assump- tions here considered ; or if we pass from an initial condition over to the final condition Q = A [U2—U{] + A f pdv. In this equation the quantities x, x', x" are contained in the values for the energy, and indeed play a very important part therein ; moreover, pdv will be vi represented by the area included between the curved portion (more accurately, the projection on the PV plane of the curve) representing the change of condition, the initial and the final ordinate and the por- tion of the axis of abscissas lying between these ordinates. In the following sections the equations of condition for the individual stadia will now be considered, from them those of the characteristic curves (isotherms, adiabatics, and curves of constant quantities of sat- uration) will be deduced, and finally the course of these in the geo- metrical form of presentation will be investigated. A. THE DRY STAGE. If we indicate by^A the partial pressure exerted by the dry air, by ps the pressure resulting from the vapor and in general distinguish all quantities relating to the air and vapor, in an analogous manner by the same indices then we obtain directly RxT i RST p=Px+ps = ~ir+%-1f- or, Pv=(Rk + xRs)T (1) * I adopt Zeuner's method of writing as more familiar to me : that is, I assume that the energy is expressed in units of work. 220 THE MECHANICS OF THE EARTH'S ATMOSPHERE. where RK and Rs are the constants of the equations of elasticity* for air and vapor, namely, R*= 29.272 and Rs — 47.061. If now x remains con- stant, then for a constant T this equation becomes that for an equila- teral hyperbola. The isotherm for moist but not saturated air is there, fore, as for dry air, an equilateral hyperbola or a portion of one; but for certain values of v this equation loses its meaning. The fundamental condition of the dry stage consists simply in this: that the pressure p& shall be smaller than the pressure e corresponding to that of saturated vapor at the same temperature, or expressed alge- braically v where e is a quantity depending upon Tand rapidly increasing with T. The equation (1) holds good, however, only when V>XJ**1 (2) e and, therefore, the effective portion of the hyperbola begins with the point whose abscissa is XR& T /ox ^=-T"- (3) Since e increases more rapidly than T, therefore this initial abscissa diminishes with increasing temperature. The initial points of all isotherms belonging to one and the same quantity of vapor x lie there- fore on a curve whose course is to be seen approximately on the figures to be given hereafter, in which such curves are designated by 8 S with corresponding indices. Hence the area within which are represented the conditions of the dry stadium for constant quantities of vapor x is bounded by this curve on the side toward the co-ordinate axes. If this curve is so in- tersected by another curve, representing any change of condition that one passes from the side that is concave away from the axis over to its convex side then one leaves the dry stage and arrives in that of condensation, that is to say, in the rain or snow stage. I will therefore designate this curve as the curve of saturation or of dew-point. Points on this saturation curve are, in accord with the considerations just de- veloped, determined by the hyperbolic isotherm and the initial abscissa. We can, however, equally well utilize also the corresponding ordiuates and the initial abscissa. In the dry stage the following equation holds good for the ordinate : P=P\ +ps<%>\ + e consequently for the initial ordinate of the isotherm T, which will be * Throughout this work, the ' ' equation of elasticity " is used as a translation of the German Zustandsgleichung, as being preferable and more general than the ordinary expression '• Equation for a gas" or " equation of condition." — C. J. PAPER BY PROF. BEZOLD. 221 designated hyp, as being- located on the curve of saturation, the equa- firm is tion is Ih—Lh+c or RK T P*= v, or finally after substituting the value of v, R\+x Rx It is therefore easy to determine the correlated values of vs and ps for any constant quantity of moisture x and for any given temperature. On the other hand, only with the greatest difficulty and even then only by the use of empirical formulae is it possible to bring the curve of saturation into the ordinary form :* We also will therefore entirely relinquish all attempts in this direc- tion. By so much the more important is it therefore to show that from the curve of saturation for a given value of x one can with ease deduce such curve for any other quantity of moisture. If T and hence also e is constant, then it directly follows from the equation R&T i\=x e that the initial abscissas of isotherms corresponding to equal tempera- tures but different quantities of moisture are proportional to these quantities of moisture themselves, or if we indicate by vy and v2 the initial abscissas belonging to the quantities of moisture xx and j^, we have Vi : v2=Xi : x-2. If therefore we have any point such as Ni of the dew-point curve St corresponding to a given temperature Tthis will be the initial point of the isotherm (T, x}) if as in the above given manner we indicate the point corresponding to the temperature T and the quantity of vapor a?ij now draw the isotherm (T, x2) for the same temperature T but for another quantity of vapor x2, then we have only to increase or diminish the abscissa of Ni in the ratio x2: #1 in order to obtain the x2 of the * We see this from the following consideration: Since according to equation (4) e=

s=(fiA+£jRs). rp(p,) an equation which contains only vs andp,, hut not explicitly, as variables. 222 THE MECHANICS OF THE EARTH'S ATMOSPHERE. initial point N2 of the isotherm (T#g) originally considered as beiug unlimited; that is to say, in order to obtain a point in the dew-point curve S2 correspond- ing to the quantity of moist- ure x2. The dew-point curves #2, S3, of figure 29 therefore corre- spond respectively to quanti ties of vapor #2 = 2#i; xz = '5xx when Si corresponds to the quantity of vapor xx. The isotherms (T, x{) and (T,x2) run so near each other that they can only appear sep- arated in a figure drawn to a very large scale,* since be- tween the ordinates_pi and p2 of the two isotherms belonging to a given r, the following relations exist : Fig. 29. 2>i -i>2 = 0»t — X*) R&T v or also Pi __ Exjj-jCj_R^_ p2~ Rk + x2 Rs But this quotient is always very near unity, since all the values of a? that here come into consideration lie between zero and 0.03. In the majority of cases one can consider all the isotherms (T,x) corresponding to a given value T as coinciding with each other and have then only to re- member that according to the value of x they have their initial points at different places on the same hyperbola. Therefore from any one dew- point curve ^we obtain another one S2 in that as already done in figure 29 we simply go with a constant ratio of expansion or compres- sion further along an equilateral hyperbola drawn through Si. If we confine our consideration still to that portion of the plane of a constant quantity of vapor x that lies to the right (that is to say, on that side of the dew-point curve that is distant from the coordinate axes) that is to say to the dry stage, then in this region the same theorems will hold good for the characteristic curves as for the so-called perfect gas, and particularly as for air, Avith such very small changes iu the constants as depend on the mixing ratio [or the quantity x]. * It must here be expressly remarked that all the diagrams occuring in this memoir have a purely illustrative character. If we should introduce the separate quantities as they result from the computation the diagrams would lose perspicuity. The method here given therefore will need special modifications (as is hereafter to he shown) before it can be applied to graphical computations. PAPER BY PROF. BEZOLD. 223 In this stage the isodynaniic lines are also equilateral hyperbolas, and moreover the equation holds good also for the adiabatic liues, when pl and vx relate to a definite initial condition, but p and v to an arbitrary final condition. The constant n can be adopted without notable error the same as for dry air, namely, u = 1.41. The quantity of vapor therefore disappears entirely from the formula and the adiabatics have the same course in all the planes corresponding to the different values of x. If now the adiabatic curves are considered as lines of constant entropy and we therefore take the equation S-Si = 0 as the fundamental condition where - 8 is the entropy, then the equation of the adiabatic lines re- ceives the following form (cp + xc*) log % _ A (Rk + XBS ) log? = 0 U px where the capacity for heat of superheated aqueous vapor under con- stant pressure is indicated by c*. If one knows the path of any one adiabatic in the dry stage, then it is easy to construct any given number of others by means of it. To this end we consider that for any further progress along one and the same isotherm, according to well-known propositions, the following for- mula holds good for the quantity of heat needed in the expansion from Vi to v2 : Qiy2 = A>R* T\og^ where, for the sake of simplicity, we put R\ + xRs = R* Therefore we have % = 4iJ.log5 (5) But the quotient —m^ is nothing else than the diminution of the en- tropy in the isothermal expansion from the volume ^ to v2. If, there- fore, we start from a line of constant entropy (an adiabatic), and pro- ceed along various isotherms that cut this curve, so that the ratio of expansion remains constant, then we attain to points on a second adi- abatic. If now we put i\=v and v2=v+Av, and then make Av=vv, where v is a constant (an appropriate proper fraction), and if in a correspond- ing manner we put AQ for Q and AtS for the difference of the entropy, we find AS = ^=AR* log (1+r) Therefore as soon as the course of one adiabatic line is known (just " For the problems here presented, as is clone by Zeuner in the application of the mechanical theory of heat to machines, it is recommended to give the positive sign to 224 THE MECHANICS OF THE EARTH'S ATMOSPHERE. as in the case of the dew-point curve) one can by a simple method of construction cover the plane of coordinates with a series of such adi- abatics, each of which, with reference to its neighbor, shows a constant difference in the entropy by the amount AS. B. THE RAIN STAGE. For the rain stage, as already stated, there obtains the equation of mixture M=l + x + x', where x' is in general very small, but x, except in exceptional cases, can only diminish. The equatiou of elasticity, on the other hand, is P = — + e (0) where e is the vapor pressure, which in this stage, that is to say in the condition of saturation, depends simply and alone on the temperature T. Moreover, there obtains also the equation developed as a limiting condition in Art. 3 above, viz : e= (7) This last formula shows at once the above suggested fact, that here we have in general to do with changes that are reversible to only a very limited extent. If, for instance, T is put constant while v increases, then the equation can only be fulfilled when # increases. This same holds good (because c increases rapidly with increasing T) when v is kept constant and T increases, or, as expressed still more generally, it holds good for all changes in condition that are represented in the diagram by a movement toward the concave side of the dew-point curve. But an increase of a? is only to a very limited extent possible in gen- eral in the free atmosphere, namely, only when liquid water, in addi- tion to the vapor, is suspended in the air, and only so long as this store of liquid holds out. The latter in most cases is soon exhausted, since it is precisely the liquid drops of water that fail as rain as soon as their mass becomes considerable. Therefore in the rain-stage, changes of condition toward the concave side of the dew-point curve are possible only to a very limited extent and only until the condition of supersaturation comes to its end and the quantity of heat communicated to the air. Therefore au increase of the quotient Q -j corresponds to a diminution of the entropy according to the definition of entropy as given hy Clausius (see Clausius's Collected Memoirs, Brunswick, 1884) Memoir IV> PaS° 140, and Memoir vi, page 276. PAPER BY PROF. BEZOLD. 225 becomes that of simple saturation.* This occurs as soon as the curve of change of condition attains the dew-point curve x + x'. Having in mind the geometrical presentation one can express this proposition as follows : In the rain or suow stage, changes of condition are only reversible when and so long as they find their representation above the dew-point surface. If they fiud this in the dew-point surface itself, then only those changes are possible by which the representative point approaches the quasi horizontal coordinate plane, that is to say slides down toward the surface or in the limiting case becomes the dew-point curve itself. An ascent to the dew-point surface is in the free atmosphere only im- aginable in exceptional cases (as for instance in case of the falling of rain through other layers or the mixiug of other layers with moist air); a further progress toward the concave side of the dew-point curve or toward the lower side of the dew-point surface indicates a transition over into the dry stage. Therefore in making use of the graphic presentation one must always keep in mind that in the rain and snow stages the curves in general can only be travelled over in one direction best represented by arrows and that a backward movement on the same curve is an impossibility. Nevertheless for the forward progress in the one possible direction exactly the same formula) are applicable as for the reversible changes of condition. Therefore the case here occurring may with propriety be designated as " limited reversible." We now turn to the consideration of the isotherm and the adiabatic for the rain stage. The equation of the isotherm we obtain at once as soon as we consider the temperature T as constant in the equation of elasticity r V Since in this case e is also constant, therefore this curve as in the dry stage is an equilateral hyperbola, one of whose asymptotes, as in the dry stadium, coincides with the axis ofp, but the other is by the small quantity e shoved from the axis of v toward the side of positive p. At the same time however, in so far as we exclude super-saturation and starting from a given initial condition, this equation holds good only for diminishing values of v. Moreover a glance at the equations of the isotherms in the dry and the rain stages suffices to show us that the two curves for any given temperature differ from each other only very little and that in the transition from the dry to the rain stage only a very small indentation * In a certain sense the case where liquid water or ice is mixed with the air should certainly also be called that of super saturation, but of course with the reservation that any confusion with the condition of super-saturation properly so called, in which the excess above the quantity needed for saturation is present in gaseous form, shall be excluded. 80 a 15 226 THE MECHANICS OE THE EARTH'S ATMOSPHERE. caii be seen with the vextex toward the right and above. This results from the circumstance that the isotherm for the rain stage contains the initial points, of all isotherms for the dry stage, which points corre- spond to values of *0tliat are smaller than the value of xb from which one starts out. In order to obtain the equation of the adiabatic we must know the quantity of heat, dQ, that is to be communicated for a very small change in the condition. This dQ is composed of the quantity of heat dQK that is given to the dry air and of the quantity dQ& that is communi- cated to the intermingled water or aqueous vapor. The following equations hold good for these quantities:* dQ,= CvdT+ AR, T~ and dQs = Td Q£\ + (x + x') dT [Where r is the quantity of heat required to vaporize a uuit mass of water at the temperature T and the pressure p. J In these x' has values that lie between 0 and xa-x where xa indicates the quantity of vapor that was given to the original kilogram in its passage from the dry stage to the rain stage, x' is equal to 0 when all the condensed water immediately falls down and is thus separated from the mass; it is equal to xa—x when all such water is carried along with the mass. The two limiting cases will occur relatively quite seldom in nature, but since at present we have no basis for determining to what extent liquid water is suspended in the air or can be carried along with it, therefore one must in the theoretical investigation confine himself to these limiting cases. Expressed in the language of the graphic presentation one must content himself with investigating those cases in which the indicating point either remains in the same plane as in the dry stage or on the other hand goes further on over to the dew point surface itself. Hitherto the first case only has been taken into consid- eration, although in general the second better agrees with the conditions occurring in nature. Therefore the above given equation for dQs assumes different forms, according as we consider the one or the other limiting case and we have, either for the case where xa is constant when all the water formed by conden- sation remains suspended, or dQ6=TdOv^+xdT where „^ for the case when all this water immediately separates from the mass. * See Claueiua Collected Memoirs, Bfmtfwick, 1884, Memoir V, page 174, or BifMi'* translation of Claudius, pages l.v.i and ;jr>3. PAPER BY PROF. BEZOLD. 227 The first case corresponds to a super-saturation limited only by the original amount of water, or, as I will briefly call it, the " maximum super-saturation f the second case corresponds to the " normal satura- tion," rejecting any supersaturation. For the quantity of heat dQ=dQ\+dQi communicated to the mixture we obtain therefore two equations, namely: (1) For u maximum super-saturation:" dQ=(cv+xa)dT+Td(^\+ARKT(l* (8) (2) For the "normal saturation:" dQ=c4T+xdT+Tdf™\ + AtiKT(1*. ... (9) If we put dQ=0 then we obtain the differential equations of the adiabatics for the two limiting cases. But in doing this we ought not to overlook the fact that strictly speaking in satisfying the condition dQ=0 we have to do with an adiabatic in the ordinary sense of the word only in one of these limiting cases, namely, that of maximal supersaturation. For if we establish for the adiabatic the single con- dition that for the given change of condition heat shall be neither gained nor lost, then we have in both cases true adiabatics to deal with. If however we define the adiabatic change of condition as one in which not only all exterior work shall be done at the cost of the energy, but also where the whole loss of energy shall be consumed in exterior work then will the definition for the second limiting case and also for all intermediate cases corresponding to values of #'>0 and x'■' <.<„, that is to say, those changes for which the water wholly or partly separates the "pseudo-adiabatic," and especially that curve which obtains for the complete discharge of the water of condensation, the " pseudo-adiabat." Corresponding to this method of distinction the equation (cv+xjdT+Td*f+AR,Td*=i) (0) obtains for the adiabat and the equation {c.+x)dT+Td(^)+ARk3%=Q • • ■ ■ (10) obtains fur the pseudo-adiabat* 228 THE MECHANICS OF THE EaRTH'S ATMOSPHERE. From these two equations we see, first of all, that the pseudo-adia- bat descends more rapidly than the adiabat. Since for o*v>0 we always have dTKO and since moreover x < xai therefore the absolute value of dT in the case of pseudo-adiabatic expansion must be larger than for adiabatic; that is to say, the temperature must sink more rapidly when all the condensed water is immediately discharged than when it re- mains still suspended. Furthermore, both curves must sink more rapidly than the dew-point curve, or, in other words, for dv>0 we must always have dx2 — C2 J-l -Li -Ll or finally by consideration of equation (7) and by the substitution of the corresponding values of X\ and x2 ; A^log^+K+^log^+|^-|^)2=0 . . (12) or ^■og^+(o,+.^,o^;;+|[^^r2Y^)].o . (13) If we consider the final condition as variable and corresponding to this drop the subscript index 2 then the equations become the following : 00 f ARKlogv^(cv+xa)\ogT+^=C - (10a) ICY — AR,K\og (])-e) + (cp+xa)\og T+-T = G .... (11a) ABAlog«+(c1(+*a)logT+-g^2=C (12a) -ARJog (p-e)+(cp+xa) log T+H ■ T**-e) = G ' ' (13rt) PAPER BY PROF, BEZOLD. 229 Simple as are these collected equations in certain respects, still none of them allow us to express the relation between v and T or j) and T or even between p and v explicitly, and in using them we are obliged to proceed by trials. On the other hand one can, in comparatively simple manner, con- struct the curves in question when we remember that the left-hand side of equations (10) to (13), in all cases, even when they are not equal to 0, must still always give the value of 2) dQ when we take this integral from the initial condition Vip! to the final condition v2p2, and thereby apply the notation of the limits as here given, and as is easily comprehended. But this value is nothing else than the diminution of the entropy during the passage from the initial to the final condition. If therefore we compute this quantity for various properly chosen pairs of v2 and p2 we thus obtain the value of the entropy for the cor- responding points, excepting only a constant that holds good for the whole system. Thus we shall be enabled to interpolate the corre- sponding values for intermediate points and thus to draw lines of equal entropy, namely, adiabatics. It is especially desirable to so choose these points that they come to lie in regular succession on the isotherms. Then we have for the difference of the entropy due to the passage from a point 1 to a point 2 of the same isotherm, that is to say, for T, = T2=T i i2)dQJU=ARJVj+ ^ T u (1) J -L Vi where r = jj—ft,2 *nat *s to sa^? a quantity that remains constant for the same isotherm. This equation also teaches that the isentropic curves in the rain stage cut the isotherms at more acute angles than in the dry stage, for which latter the equation (5) holds good, namely, Q^2=AR*]ogV2 T in From the comparison of both equations, (5) and (14), it follows that a given change of the entropy in the dry stage corresponds to a greater change of v than in the rain stage. Since now the isotherms in both stages can be considered as having very nearly the same course and, when we consider a very small part of the coordinate plane, can be con- sidered as parallel straight lines, therefore for the given change of entropy in the dry stage one has to go a greater distance along the isotherm than in the rain stage. 230 THE MECHANICS OF THE EARTH'S ATMOSPHERE. Since, however, on the other hand, the dew-point curves descend more rapidly than the isotherms toward the positive side of the axis of abscissas, therefore the adiabatics must experience a bend at the dew-point curve in the manner shown in the figure 30. In this 8 # presents a part of a dew-point curve ; A A, A' A', etc., adiabatics ; T T, T T', etc., isotherms. The differential equation of the pseudo- adiabatic can be treated in a similar man- ner to that of the adiabatic, but whereas in the adiabatic the integration was pos- sible even when the connection of the independent variables was not explicitly given, on the other hand this is not the case for the pseudo- adiabatic. That is to say, instead of equation (10) we have for the pseudo-adiabatic the following: A£Alog + cv log ,., + / T + T - -T =0, Fig. 30. or, preferably, v T A Rx log -2+ {cv+xa) log ; I W(xa—x)AT x2r2 x-jYi =0 (15) If therefore the point (1) is at once located in the dew-point curve then will xx = #a; and if then we consider the point (2) alone as vari- able, that is to say, omit the subscript index 2 entirely, we obtain r-'i A 22A log- + (o, + xa) log T- J (i) or after further modifications (Xa — X) AT XT XaY\ T ^T Ti ~ (16) ARK\ogv + (cv + xa)\ogT+ xr T (2) (1) x) AT = C . (17) We omit the development of formula} entirely analogous to equations (11) etc., and it suffices to say that iu them all the integral occurs as a correcting term. Happily its value remains always within very moderate limits, so that iu the computation one can be satisfied with more or less perfect approximations. One can therefore omit the further considera- tion of the pseudo-adiabatic process and only call attention to the fact that it follows from equation (16) that the pseudo-adiabatic curve de- scends more rapidly than the adiabatic as was already pointed out above. For since when v2 >.«?i we always have AT < 0 therefore the definite integral that still occurs iu the equation has always a negative PAPER BY PROF. BEZOLD. 231 value and because of the minus sign before the integral it therefore always exerts its influence in the same direction as the term ARK log Vz. Therefore for the same starting point and for equal values of T2, we must have v2 in the case of the pseudo-adiabatic smaller than if we had gone along on the adiabatic. C. THE HAIL STAGE. The above given equations hold good for the value T> 273° ; as soon as the temperature 0° C. or the absolute temperature T= 273 has been attained, then very different equations replace these but only when liquid water is present. In this last case the following equation of mix- ture holds good, namely : M=l-\-x+x/-{-x", an equation that can only be true for the temperature 0° C. since only at this temperature can water and ice occur together. The equation of elasticity therefore then acquires the simple form aRK , while the equation a?=— _ becomes x=—-iJ . . . (18) Jxs J- alls wherein a=273, e0=62.56. But the one possible change of condition in i Ins stage consists in an isothermic expansion. For this case there- lore, the (IT also falls out of the equation for the transfer of heat and this takes the form, dQ=r0dx-ldx"+AEKa- (19) fr0=latent heat of evaporation at 0° O. ; /^latent heat of liquefaction of ice.] In this equation the first term on the right-hand side must be pos- itive, the second must have a negative sigu when dx and dx" are con- sidered as positive, since an increase in the quantity of vapor x makes an addition of heat necessary, but an increase in the formation of ice demands a withdrawal of heat. If we put dQ=0 then we have the differential equation of the adia- batic which in this case coincides with the isotherm and is moreover always a pseudo adiabat, since the ice that is formed falls away under all circumstances. If we consider that aR& then the differential equation of the adiabat takes the form ARKJv+r^dv-ldx"=0 (20) v uRs 232 THE MECHANICS OF THE EARTH'S ATMOSPHERE, hence we obtain by integration ABxa]oSVv2+^(T2-v1)-lx2"=0 (21) where we assume the integral to be taken throughout the whole stage from the initial value vx that corresponds the entrance into this stage to the final value v2 that refers to the exit therefrom, and remember that the initial value of x" namely, x\" is equal to 0 under these condi- tions. If however the integral extends only up to a value of v lying between these two limits and which v can then be considered as vari- able, then the equation can be again brought into a form analogous to that above given and we obtain ARia\ogv+r,,('"r-lx" = C (22) This equation allows us to see directly that for increasing values of v that is to say for continued progressive expansion the quantity of hail also steadily increases whereas on the other hand from [equation (18) or] the expression dx= %dv alts it follows that an evaporation goes hand in hand with the freezing of the water, so that at the end of the hail stage the quantity of vapor present is greater than it was at the entrance upon this stage. With the help of the above described geometrical presentation we represent these results in the following manner. The condition that must exist at the entrance upon the hail stage finds its representation at the termination N' of a straight Hue N0N' per- pendicular to the chief plane of coordinates and which rises up above the dew point surface. The length of this straight line is x+x'. It cuts the dew point surface at a point N that is distant from the plane of PV by the quantity x. If now the mixture expands along the isotherm then N rises along the dew-point surface slowly upwards, while the foot N0 of the straight line advances along an equilateral hyperbola. But at the same time, the total quantity x+x' diminishes in consequence of the discharge of the ice and N' sinks correspond- ingly down until N and N' coincide in a single point N2 and with this the had stage has reached its end. It is now of especial importance to learn how much water is thrown down in the form of hail ; this question is answered by the following con- sideration. At the beginning of this stage we have only water and vapor, at the end only ice and vapor while the sum of these in the first and in the second case remain the same, if we take the precipitated ice also into the computation. Let x{ be the quantity of liquid water present PAPER BY PROF. BEZOLD. 233 at the entrance into the hail stage, theu according to what has just been said, x'i-\-xi=xn2+x2 or X"2=X/l — {X2 — Xl) or finally, making use of the equation (18), x"2=x'l-~J^(v2-v1) (23) If we substitute this value in equation (21) then after an easy trans- formation we find ARxa\og ^^r°'\>l)eo(v2-v1)=lxf1 .... (24) From this we can now first find v2 by trial ; the value thus found can be substituted in equation (23), whence in this manner x"2 is found. If we are justified in the assumption that all the vapor of water originally present is also after the condensation carried along until the entrance upon the hail stage, as appears to be the case in heavy hail- storms, then we have x'i=xa, and this is certainly large with respect to x} and #2, and therefore so far as concerns the absolute value of x"2 we may briefly put x'i=x"2,s'uiue the difference x2—xx no longer comes into consideration. In cases in which this difference is appreciable, as for instance in the determination of v2, one can of course not make use of the above approximation. The equation (23) also shows in a very clear manner that in general the hail stage can only occur when liquid water is suspended in the air, that is to say, when x'^0 and that it acquires a greater extent the greater this value of x'h that is to say, the greater the quantity of sus- pended water that is present. Already, many years ago, Keye showed that on days of thunder storms the conditions are present in a con- spicuous degree for the suspension and carrying up of water. D. THE SNOW STAGE. If the air, saturated with aqueous vapor, be cooled below 0° C, theu a part of this vapor must be precipitated as snow. The same formula can be applied to this process as that which we have used in the rain stage if only in place of the heat of evaporation r there be inserted the sum r+l where I as above indicates the heat of liquefaction of ice. Therefore we can after small modifications apply to this stage all the equations developed in Section b. I confine myself to the re-writing in this modified form the two equations (10a) and (17); they thus become for the adiabatic ARK\ogv + (cv+cxc)\ogT+ir^V = C . . • (25) 234 THE MECHANICS OP THE EARTHS ATMOSPHERE, aud for the pseudo-adiabatic ABK\ogv+(cv+exc)logT+ KJ -Ja — t~ =C ' ' {2) where xe is the quantity of vapor at the beginning of the snow stage and the limits a and T are introduced into the integral, because in the hail stage, as in the beginning of the suow stage, T=a=213;c is the specific heat of ice. Since x is always smaller with diminishing T and finally approximates to 0, therefore in the snow stage the deeper the temperature falls the more does the adiabatic approximate to that of the dry stage. In the investigation just finished, attention has been especially di- rected to the course of the adiabatics, as had also been done in the above- mentioned older investigations. But in truth the adiabatic expansion and compression constitutes only a rare, exceptional case, as is already shown by the fact that the vertical temperature diminution computed under this assumption (according to the so-called convective equilib- rium) results considerably larger than is given on the average by ob- servations. It is therefore important to deduce the quantity of heat absorbed or emitted for given changes of condition, as determined by the values simultaneously observed of pressure, temperature, and mois- ture. In this process the method of geometrical presentation here de- veloped is applied with great advantage. First, a glance at the man- ner in which the curve representing any given change of condition cuts the adiabatic suffices to give a decision as to whether in this change one has to do with a gain or loss of heat. Moreover the curve puts one in a position to deduce the quantity of heat exchanged by graphic planimetric methods or by a combination of computation with plani- metric measures. According to what was said in the beginning the equation Q=A[U2-Ur}+A fmpdv holds good also for the processes here considered with three independ- ent variables, aud therefore also for a closed cyclic process Q = AF, where F is the surface inclosed by the projection of the points that are imagined to be upon the P "P" plane. Assuming that at . ai,y change of condition is given by its projection on this plane and is represented by the line between the points a and 6 in Fig. 31, then we obtain the ^ quantity of heat QaJ, involved in this change easily Fig.m. i" the following manner: One draws through a (Fig. 31) any curve of change of condition for which it may be easy to compute the increase or diminution of heat; also draw PAPER BY PROF. BEZOLD. 235 through b au adiabatic and prolong both curves until they cut each other in a point, c; then is Qbr = 0, and the quantity of heat is given by- or, Qa.>~ Ta holds good. Therefore after the passage over the mountain one has warm dry air, whereas at first it was cool and damp. At the same time we see directly from the diagram that the charac- teristic peculiarities of the foehn must stand out so much the plainer in proportion as the point a is nearer to the curve of saturation, that is to say, the warmer and moister the air is before its ascent and again, the longer the portion b c is, that is to say, the more extensive is the expansion in the rain stage, or in other words, the higher the summit is that has to be surmounted.- Therefore we understand also at once why it is that in the Alps, in- dependent of the prevailing conditions of atmospheric pressure, north- erly foehus are so much rarer than the southerly foehns, as also why descending winds that have surmounted no summit, but have only passed along over a plateau, as for example the bora, have not the char- acteristic warmth of the foehn. (2) The interchange of air between cyclone and anti-cyclone in summer. Between an anti-cyclone and the cyclones that feed it, similar rela- tions exist as between the masses of air on the two sides of a mountain range to be surmounted by them. In cyclones one has to do with an ascending current of air that afterwards descends in the anti-cyclone. Hence arises the precipitation in the region of the cyclone, the dryness and the clear sky in the region of anti-cyclone. But, whereas in the foehn the ascent and descent occur at points in the neighborhood of each other, so that in the short path there scarcely remains time for gain or loss of heat, but the whole process may in fact be considered as adiabatic ; on the other hand very different relations obtain for the ascent and de- scent in cyclone and anti-cyclone. These two opposite processes in gen- eral occur at places so distant from each other that in the transit from one to the other extended opportunity is offered to take up or give out heat. Iu this process during the summer season the increase of heat prevails, but during the winter time the loss of heat j the day-time also PAPEE BY PROF. BEZOLD. 239 in its relations follows more or less closely the summer, while the night- time is like the winter. Under the assumption of a prevailing- increase of heat the process pre- sents itself somewhat as shown in the diagram (Fig-. 34) ; starting with the condition a (in a cyclonic area) the expansion with a diminution of g , jr temperature proceeds according to the curve a />, which descends rather less steeply than does the the adia- batic curve. Corresponding to this, and also without reference to the initial quantity of moisture, the dew- point curve is first attained later, that is to say, at a greater altitude above the earth's surface than it would be in adiabatic expansion. In the rain stage, therefore, the IG" ' curve of change of condition experiences a deflection toward the upper side of the adiabatic, and therefore remains nearer the curve of satura- tion. If now there occurs a still further greater addition of heat, as must be the case during the period of insolation and at great altitudes, where the condensation is less and the density of the clouds is correspond- ingly diminished, then the air can again pass over into the dry stage as is indicated in the portion c d of the curve. Thus the upper limit of the first layer of clouds theu would be at c. At this limit, daring the summer days, more intense warming is in fact to be expected, which through a further expansion, that is to say at a greater altitude, on account of the diminished absorptive power of the atmosphere, again passes over into the approximate adiabatic c d, by which process, however, the dry stage is finally left and the snow stage d e is entered. To this greater increase of heat at the upper limit of the clouds the fact is certainly to be ascribed that the cirrus (or snow) clouds are not directly continuous with the (lower or) water clouds, but generally separated from them through a wide space such as corresponds to the expansion from c to d. During the descent in the anti-cyclone or by reason of the compres- sion the process must take place according to the curve e /, which in general nearly agrees with the adiabatic of the dry stage. As we approach the earth's surface however, on account of the strong absorp- tion of heat occurring there, then and for that reason this curve can depart to the right upwards from the adiabatic. This latter can how- ever only occur temporarily, since in such a case we should have to do with a condition, of unstable equilibrium. 240 THE MECHANICS OF THE EARTH'S ATMOSPHERE. The final pressure^, with which the sinking air reaches the ground in the an ti-cy clone, is greater than the initial pressure pa that prevails at the ground within the cyclone, and correspondingly /is higher above the axis of abscissas than a. In this case it may occur that the point/ comes to lie not only (as is self evident) above, but also to the right of «, so that vfy va or in other words that the air at the base of the anti- cyclone, notwithstanding the higher pressure, is specifically lighter than in the cyclone, because the temperature more than compensates for the influence of the pressure. This shows in a very clear manner that in the exchange of air be- tween cyclone and anticyclone we have to do not only with the specific weight of the mass of air, but that here dynamic relations are of first importance, a poiut to which Haun has called attention lately in the discussion of the observations taken on the Sounblick.* It will be well in the more accurate investigation of this question to give increased attention to the processes above the aqueous clouds especially at their upper boundary surfaces. As to the relations of the humidity to the processes just considered these are nearly the same as those in the case of the foehn. Here also, that is to say in the anticyclone, the air arrives in the neighborhood of the ground warm and dry, but in the immediate neighborhood of the ground the evaporation stimulated by unrestrained insolation will rapidly add moisture to the air, so that the indicator, which moving from b nearly to e has steadily approached the PV plane and from c on the, way towards /has remained a long time at the level of e, must now be imagined as rising immediately before reaching /. If now the air that has descended in an anticyclone again flows toward a new depression then will it (under the assumption of the same conditions in this as in the first cyclone), by reason of a continuous acquisition of aqueous va- por, pass through conditions that are represented in the diagram (Fig. 34) by the line fa. This line we have to imagine as slowly rising, so that the diagram here drawn presents in fact the projection of a closed line. (3) The interchange of air between cyclone and anticyclone in winter. In winter the diagram for this process of interchange has a figure essentially different from that in Summer. First, all changes in con- dition, at least insofar as concerns the initial and final conditions (see Fig. 35), take place nearer to the coordinate axes since the tempera- tures that come into consideration do not rise so high as in summer, and since, corresponding to this, the isotherms that lie far from the axis are not attained. Again, we have here lower pressure and higher tem- perature at the starting point a, but at the end d higher pressure and lower temperature, so that d is to be sought to the left and above a. * MeteoroloyischeJeitschrift, 1888, vol. v, page 15. L PAPER BY PROF. I3EZOLD. 241 Furthermore, the lines whose projections are here considered are not so far from the coordinate plane as in summer, because the absolute capacity for moisture remains always slight. If now we follow more accurately the change of condition during ascent in the cyclone, we may at first assume that the process up to the attainment of the upper limit of the cloud stratum very nearly agrees with the adiabatic expansion, since below this limit radiation, either to or from, can only play an unimport- ant part. If however a departure from the adiabatic process does oc- cur then it can be only in the oppo- site direction to that which occurs in summer, that is to say, the lines will descend more decidedly than in summer. In Fig. 35 this latter case is as- snmcd, as also that the passage out of the dry stage into the snow stage takes place immediately. From this point onwards the curve of condition again sinks more gradually, but with steadily increasing gradient in consequence of the overpowering cooling that certainly occurs at higher altitudes, until finally the turning point is attained and compression takes the place of expansion. The entire course of the change of condition to this point is presented by the curve abc. From this point onwards in consequence of the compres- sion, the curve of condition must gradually advance to the point d. So far as our knowledge of the actual conditions of the atmosphere has at- tained hitherto, this gradual return to the point d occurs in such a way that at greater altitudes the compression proceeds adiabatically accord- ing to the adiabatic of the dry stage, whereas on approaching the ground the cooling by radiation that prevails there causes a deviation of the curve of condition from the adiabatic toward the axis of ordinates, and corresponding thereto the curve shows a course like cd. This curve however is nothing but the graphic expression for the well-known in- version that occurs on clear winter days in the vertical distribution of temperature. By reason of this inversion the curve near d approaches the dew-point curve, aud can even pass it, so that condensation must occur and in the form of ground fog. But with the begiuning of the formation of fog the radiation increases materially and corresponding to it the temperature diminution becomes always more intense with the proximity to the earth of the descending current of air. Whether the passage from c to d be also possible by some other path by which from the very beginning of the compression the cooling and therewith the departure of the curve from the adiabatic makes itself felt, is a question that can be decided only after an accurate test com- 80 A 1C 242 THE MECHANICS OF THE EARTH'S ATMOSPHEKE. putation with the appropriate numerical data. At any rate such pos- sible process would assume that in the anticyclone, at a certain height above the ground, exactly the same pressure and the same temperature prevail as at less altitudes above the base of the cyclone, since the pro- jection of the curve of condition in this case must possess a double point. These few examples, given only in their outlines, will suffice to enable one to realize the varied and useful applications that the method of graphic presentation here developed is capable of. By a further com- pletion and development of the numerical side this method will give not only an excellent auxiliary means for the discussion and evaluation of existing data of observation, but above all will afford an indication as to the direction towards which materia) is to be collected in order to afford a deeper insight into the thermodynamics of the atmosphere. If anything should seem especially suited to enable us to recognize the importance of the method of consideration here developed, it is the abundance of questions that press upon us at the first step we take iu this way and that can at present be scarcely enumerated. 1 am think- ing now, not only of the further development of theoretical conse- quences, therefore especially of the meaning of the thermal changes that occur in the atmosphere (especially the application of the second theorem of the mechanical theory of heat to these processes which may be developed in subsequent communications), but also, above all, of the stimuli that are to be derived therefrom to the observations of mountain stations, and especially in balloon voyages. For the latter it is full of meaning, that in thermodynamic investigations the knowledge of the altitude above the sea can be entirely dispensed with and that it is en- tirely sufficient if we know the simultaneous values of the pressure, temperature, and moisture. XVI. ON THE THERMO-DYNAMICS OF THE ATMOSPHERE.* (SECOND COMMUNICATION.) Bj Prof. Wilhei.m vox Bezold. Ill a memoir published several months siuce,f I made an attempt to so extend the Olapeyron method of graphic presentation of thermo- dynamic processes as to allow of its application to atmospheric changes. At the same time I showed by some examples how with the assistance of this method of representation even complicated phenomena can be studied with comparative ease, and how by means of it we are put in the position of being able to draw most important conclusions almost like child's play. In the following, the same method will be applied to other questions not then or only lightly touched upon. First, I will treat of a conception that has lately been introduced into meteorology by von Helmholtz.f and which appears to me to possess great significance in this science. This is the idea of u warm egeh alt," or total amount of heat contained within a body. Helmholtz measures the heat contained in a mass of air by the abso- lute temperature that this same mass will assume when it is brought adiabatically to the normal pressure. The quantity that we here deal with is therefore not as one might easily have believed a quantity of heat but a temperature, and therefore it seemed to me, upon my first study of the memoir in question, desirable to replace the term " warme- gelialt" by another. In a conversation upon this matter von Helm- holtz recognized the objection expressed by me as proper, and proposed that the word " wiirmegehalt" should be replaced by the evidently much more proper expression "potential temperature." This latter expression will therefore be used exclusively in thefollowiug memoir, but at first this idea itself will be more accurately considered. Its presen- tation in a diagram will be attempted and a general theorem deduced from it. * Read before the Academy of Sciences, Berlin, November 15, 1888. (Translated from the Sitzungsberichte der Eonuj. Preuss. Akademiedtr Wissensch often zu Berlin, 1888, vol. xlvi, pp. 1189-1206.) t [See the preceding number of this collection of Translations. ] t "On Movements in the Atmosphere," Sitzb. Berlin A.kad., 1888, vol. xlvi, p. 647. [.See No. V of this collection of Translations. 1 •213 2 I I THE MECHANICS OF THE EARTH'S ATMOSPHERE. I. THE POTENTIAL TEMPERATURE. According to what has just been said the potential temperature is that absolute temperature that a body assumes wheu without gain or loss of heat it is adiabatically or pseudo adiabatically reduced to the normal pressure. I intentionally give this definition the form here chosen since we are here concerned with the application of the idea to meteorological processes, and since in our case the processes without increase or loss of heat do not need to be strictly adiabatic in the ordinary sense of the word. As I have shown in the pre- vious memoir we have only to do with adiabatic processes when the water formed by condensation does not fall to the earth but is carried along with the air, a condition that is only fulfilled in exceptional cases. As soon as water is lost, and this is generally the rule, even though no heat be gained or lost, we have to do with a process that is only pseudo adiabatic. When therefore in the following, mention is made of adiabatic changes, the pseudo-adiabatic will always be included therein in so far as this class is not excluded by the special term " strictly adiabatic." This much being premised we may now first investigate whether and how the potential temperature can be represented in a diagram. The answer to this question is extremely simple. From the equation of condition for the dry stage vp = R* T there results R* T V or if we substitute fovp the normal pressure j>0 R* T v = — . l. Therefore under constant pressure the absolute temperature is simply proportional to the volume, that is to say to the abscissa. But this absolute temperature under the pressure^ is the " potential tempera- ture" for all other conditions that find their representation on the adiabatic passing through the point whose coordinates arei>aud.p0. We t berefore obtain the following rule : If a condition is given that is represented in the diagram, Fig. 36, by the point a, then we find the corresponding potential temperature by draw- ing an adiabatic line through a and seeking its point of intersection N' with a straight line P0 N drawn parallel to the axis of abscissas and dis- tant therefrom by p0. The distance of this point of intersection N' from the axis of ordinate*, namely, the abscissa of N' (or N' P0) is now a meas- ure nj the potential temperature. We and the numerical values of v and T belonging to Po (and which . will now designate by v> and T> corresponding to the point #'. while I PAPER BY PROF. BEZOLD. 245 designate by va and Ta those corresponding to the initial condition a) by combining the equation of the adiabatic VaVKa=lKV'K, with the equation of elasticity PaVa T PoV' T = R* and we thus obtain K-l V = Po Pa T where x = 1.41.* But this simple method of consideration is only allowable so long as the changes of condition take place within the dry stage. If this stage is left then the potential temper- ature belonging to a definite iutial point has no longer a constant value, but it increases with the quantity of precipitation that is lost. A glance at the figure suf- fices to show this : Assuming that the adiabatic of the dry stadium drawn through a intersects the dew-point curve (which for simplicity is not shown in the figure) in b and that we now allow the air to still further ex- pand, then one has to pass from b down along the adiabatic ( or pseudo-adiabatic) of the rain or snow stage, that is to say along be. If now we seek the potential temperature for a point, c, of this line (in order to simplify the figure I have drawn the line be only just to this point), in that we bring it again adiabatically to the normal pres- sure, then one ought not to run back along the curve be, since on ac- count of the precipitated water the conditions represented by this por- tion of the line are not again attainable, but on the other hand one can only attain to the line of normal pressure by following the adiabatic cd corresponding to the dry stage, but a dry stage with less quantity of aqueous vapor than before. If we indicate by A" the point at which this occurs or at which the normal pressure is thus attained, then as the measure of the potential * In the previous memoir, in consequence of an oversight, k was used instead of k by von Bezold, but at his request this has been changed in the present translation. Fw. 3G. 24G THE MECHANICS OF THE EARTH'S ATMOSPHERE. temperature we have the length P0W>P0N' ; that is to say, the poten- tial temperature T", as attained by adiabatic change after passing into the condensation stage and after precipitation of some water, is higher than the potential temperature T of the initial condition and of all the conditions previously passed through in the dry stage 0. Analytically this may be proved in the following manner : For the transition from a to b the following equation obtains If this equation remains in force after crossiug over the curve of sat- uration, then we obtain for the pressure proper to the volume ve a value Pyv' and T">T' where v' and v" are the volumes corresponding to the normal pressure p0 on the adia- batics ab and cd ; hence, p0v'K = C and p0v"*=zC"\ beside which the following equation holds good: v':v"=T':T". Thus we attain to the theorem In adiabatic changes of condition in moist air the potential tempera- ture remains unchanged so long as the dry stage continues, but it rises with the occurrence of condensation and so much the more in proportion as more water is discharged. Since in the free atmosphere, in general, evaporation does not occur and since also the carrying aloug of all the water that is formed, at least in the case of heavy condensation, must be considered as an ex- ceptional case only, therefore, this theorem can also be brought into the following form : Adiabatic changes of condition in the free atmosphere, assuming that there is no evaporation, either leave the potential temperature unchanged or elevate it. PAPER BY PROF. BEZOLD. 247 From this theorem, which in its latter form retniuds one of the theorem of Clausius in respect to the entropy, "The entropy strives towards a maximum," though not identical with it, one can draw con- sequences of the greatest importance. The next two sections will be devoted to these. II. THE VERTICAL TEMPERATURE GRADIENT. All motions in the atmosphere can be considered as analyzed into vertical and horizontal components. The latter, in so far as they do not closely follow the irregularities of the earth's surface, are subject in only a slight degree to thermodynamic changes. On the other hand, in consequence of the expansion or compression in ascending and de- scending currents, the thermodynamic cooling or warming plays a very important role. The horizontal movements will therefore for the present be left entirely out of consideration, but the processes going on in the vertical currents will be thoroughly investigated. The changes of condition going on within ascending and descending currents must be considered in the free atmosphere as adiabatic so long as we con- sent ourselves with a first approximation, and that we must do at first, since in the free air there is only a small opportunity given for active radiation and absorption. On the other hand the increase and diminu- tion of heat will always make themselves felt decisively either where the absorbtivity and emissivity are remarkably increased or where the air comes into direct contact with bodies which themselves can strongly emit and absorb or otherwise take in or give out heat. This is the case: (a) In the neighborhood of the earth's surface, where besides the increase in absorbtivity and emissivity of the air due to cloud or fog, the warming and cooling of the ground by radiation, as well as the evaporation, the formation of dew or frost, the thawing and freezing, have a powerful influence. (b) In fog or cloud, which also possess a special power of absorbtion and emission, and where moreover evaporation can occur; and especially is it the upper limiting layer of clouds that one has to take into consideration. In so far therefore as one can leave out of consideration the special localities just indicated, as also the mixture with other masses of air, one can approximately consider the processes in ascending aud de- scending air currents as adiabatic. Even taking into consideration the special locations above mentioned, one can consider a scheme drawn up under the assumption of adiabatic change as to a certain extent an average or normal scheme, since such a scheme always occupies an intermediate position between those where the incoming radiation and those where the outgoing radiation prevails. How such a preva- lence of either radiation must show itself has already been indicated in the previous communication [p. 212J, where the interchange of air 248 THE MECHANICS OF THE EARTH'S ATMOSPHERE. between cyclone aud anticyclone in summer and winter was investi- gated, at least in its principal features. Bat in this study it is not necessary to limit oneself to the summer or the winter, but rather one can apply the scheme for the summer generally to all cases where the radiation is in excess, that is to say, not only to the summer time in general, but to the day-time and the hot zone; the scheme for the winter, on the other hand, is applicable not only to the winter season, but to the night-time and the cold zones of the earth. This normal scheme for the ascending and descending cur- rents will therefore appear as shown in Fig. 36. The portion a b has reference to the ascending current in the dry stage, b o is its continua- tion in the condensation stage, finally c d is the portion of the curve that corresponds to the descending current. This scheme differs only a little from that communicated in the first memoir. (For the case of the foehn, see page 240.) We can not expect it to be otherwise, since in the foehn one has also to do with an ascend- ing aud descending current of air in which the velocity with which the whole process goes on affords only a small opportunity for the gain and loss of heat. However, the diagram given in figure 36 as the " normal scheme" differs from that which obtains for the foehn in this respect, that the branch cd is longer. This is due to the fact that in the ordi- nary interchange between cyclone and anticyclone there always pre- vails a higher pressure at the base of the latter than at the base of the former; that is to say, the ending point d in the normal scheme must always lie higher than the starting point a, which is not the case in the foehn diagram. In general, one has to consider the process in the foehn as only a feature inserted into the normal interchange between anticyclone aud cyclone. In the foehn the passage over the mountain chain forces the air in its normal interchange to describe an antecedent ascent and a subsequent descent which is only then followed by the definitive ascent in the cyclone. This being premised, the processes in the interchange, according to the normal scheme, will now be more pre- cisely considered. If we introduce the conception of the potential temperature, we at- tain the following theorems without any difficulty : (a) In the ascending branch* the potential temperature increases steadily from the beginning of the condensation ; in the descending branch it remains constant at the maximum value attained in the whole process. This maximum value corresponds also to the highest point to which the air has risen in its path. (b) The potential temperature of the upper strata of the atmosphere is in general higher than that of the lower. The first of these^two theorems results directly from the diagram ; the second follows from the fact that in the lower -stratum the potential ¥ By the ascending branch is meant the portion ab which corresponds to the ascent in the atmosphere; the portion cd is considered as the descending branch. PAPER BY PROF. BEZOLD. 249 temperature must, in the continuous interchange between cyclone and anticyclone, retain an average value that lies between the maximum value T" and the smaller value T corresponding to the base of the cyclone; that is to say, to the point a on the diagram. This average value is, however, certainly smaller than the maximum value T" corre- sponding to the highest point of the path, and therefore to the condi- tion c, and thus the theorem (b) is proven. Hence it follows that in nature the diminution of temperature for a constant elevation, or we will rather say, for 100 metres, that is to say ; the so-called vertical temperature gradient, is, in general, smaller than results from the theory of the dry stage. As is well known, this gradient is 0.993 for the latter stage, that is to say, under the assumption of adiabatic change one would expect in the dry stage a diminution of 1° centigrade in temperature for an ascent of 100 metres. This value 0.993 I will call v. The above given theorems concerning the potential temperature show at once that under the assumption of adiabatic exchange the real value of the temperature gradient must be less than v. We reach this conclusion from the following considerations: Let ta and td be the temperatures at the bases of the cyclone and anticyclone respectively (that is to say, at the starting and resting poiut, of the ascending and descending currents) then, under the assumption of perfect adiabatic change, these will not greatly differ from the potential temperatures T and T", as these correspond to the ascending and descending branches in the dry stadium, that is to say, to the conditions represented by the curved portions ab and cd in figure 30. In this process the departures from these temperatures are always of such a nature that taT". For, since the pressure pa at the base of the cyclone is certainly smaller than the normal pressure, but the pressure pb at the base of the anticyclone greater than it (at least when a normal pressure is chosen appropriate to this case, and therefore lying between pa and pb), therefore the temperature ta is increased by referring it back to this pressure, while td by the cor- responding process is diminished. Since the statement is thus proven that taT", and since, moreover, T">T', therefore, by so much the more must td>ta. At the highest point of its path, such as corresponds to the point c of the diagram, the particle of air has a potential temperature T" that is to say, precisely the same as at the end. If now it be assumed that this point lies 100/t metres above the earth's surface, then there results as temperature gradient for the descending branch that is to say, as the increase of temperature for each 100 metres of descent, the well-known value h 250 THE MECHANICS OF THE EARTH'S ATMOSPHERE. On the other hand, for the ascending branch we obtain a value )/' — g c " h ' if for the sake of simplicity the difference of temperature prevailing above and below be equally distributed throughout the whole height. This simplification is, of course, not strictly correct since the ascend- ing branch of the two stages certainly includes in itself several stages, e. #., the dry stage, the rain or snow stage, and perhaps also the hail stage, or all together. Still the method of computation of the average gradient as given here in the formula is the only method that we can apply when we have only one upper and one lower station. The follow- ing considerations however remain applicable at least in a general way when we can apply more rigorous formula. Namely, for purely adiabatic change in any case we have taT; we shall always have »0 and in fact the differences Ta—Ti and r— n always increase simultaneously. A decided cooling in the lowest stratum alwajs causes a diminution of T; and with it also a diminution of n, whereby even a change in the sign of n may occur within moderate altitudes. In the latter case, the temperature below is lower than in somewhat higher layers, and in that case we have the so called inversiou of temperature. If the cool- ing is not sufficiently strong to bring about an actual inversion of the temperature, still it causes a diminution of the gradient. Such decided cooling always takes place in the lowest stratum at the time of increased radiation, therefore especially in the region of the anti-cyclone, i. e., uuder a clear sky, in winter, and in the night time. Therefore in the winter and in the night-time the vertical temperature gradient must be smaller than during the summer and day-time, even if inversion in the distribution of temperature does not occur. This result agrees perfectly with observations, as is especially proven by the many facts that Harm and others have collected from the Alpine regions. On the other hand the investigation here carried out teaches that the inversion of temperature and the diminution of vertical gradient con- nected therewith are to be treated not as phenomena peculiar only to the mountain regions, but that we are to expect them also above the plains, and even above the ocean, at least insofar as the more violent movements of the air do not interfere therewith. We are therefore obliged to agree with Woeikoff* when he from a few data draws the conclusion that this inversion is also to be expected in the region of the great winter anti-cyclone of eastern Siberia. On the other hand I can not agree with him when he deduces from this the consequence that Messrs. Wild and Hann should have consid- ered this circumstance in drawing their isotherms, and I consider the standpoint taken by them as perfectly justified. t * Woeikoff", Klimate der Erde, German edition, 1887, Bd. n, p. 322; Mcteorolo- gische Zeitschrift, 1884, Bd. I, p. 443. t Hanu, Atlas der Met., 1887, p. 5. Wild, Repert., 1888, Bd. xi, Nr. 14. 252 THE MECHANICS OF THE EARTH'S ATMOSPHERE. A direct proof of the inversion of temperature above the lowlands can only be expected from balloon observations. To what extent radiation causes the inversion or at least the dimi- nution of the gradient we shall learn from a work now soon to be pub- lished, that Siihring* has executed at my recommendation, audin which the vertical gradients of temperature between the Eichberg and the Schueekoppe, as well as between Neuenburg and Obaumont, are inves- tigated according to the separate percentages of cloudiness. It is not improbable that also above the ocean, and even at the time of the stronger insolation, a diminution of gradient, if not even an in- version of temperature, occurs, since over the sea the rapid evapora- tion in connection with the mobility of the water puts an impassable limit to the rise of temperature. The stability of the Atlantic anti- cyclone during the summer mouths may be based upon this circum- stance. The cases in which an increase of heat occurs at the earth's sur- face need no special consideration in the questions here considered. The gradient can only for a short time exceed the value r, as deter- mined for the expansion or compression of dry air. If this case occurs, then, according to the investigations of Keye and others, we have unstable equilibrium or a condition that can only exist temporarily, as in whirlwinds or thunderstorms. Therefore, even for the strongest insolation, the considerations above given continue to hold good. On the other hand the fact must excite great consideration that, not only on the average of all cases, but also when we investigate only the region of ascending currents (and of these only those that are below the limit of clouds, that is to say, for moderate elevation of the upper station) we find that the vertical gradient is always decidedly smaller than v. The reason of this is principally to be sought in the fact that the above views as pr< sented by me, as also by other investigators in this direction, all rest upon an implied assumption that is only allow- able to a very limited extent. They are based namely upon the assumption that the air ascending from the earth experiences no change in its constitution, except that due to the loss of water consequent on the adiabatic expansion, i.e., that it experiences no mixture with masses of air of other temperature or other degrees of moisture, as also that every particle of air considered in the interchange between cyclones and anti-cyclones describes the whole path from the earth's surface to the limit of the temperature and back again. But this is by no means the case. Only a small fraction of the air under consideration actually comes in contact with or even in close proximity to the earth's surface; and similarly with the ascent to the limit of the atmosphere or at least to the highest stratum that at any time takes part in the process under consideration. Moreover in the Siilning, Dievertilcah Temperaturabnahme. Inaugural Dissertation d. Universitiit, Berlin, 1890 PAPER BY PROF. BEZOLD. 253 ascending whirl, masses of air are always drawn in from one side that had not yet sunk to the earth's surface and had remained correspond- ingly unaffected by the radiation and absorption that have their seat in that stratum, and which also had had no opportunity to take up water from the earth's surface. Since these masses of air coming from the upper portions of the anticyclone have in general higher potential and therefore also higher absolute temperature than the portions of the cyclone lying at equal altitudes above sea level, therefore the mixture of these will diminish the cooling of the ascending air and both there- by as also by reason of the lesser quantity of water that they possess, will delay the occurrence of condensation. Therefore in the cyclone itself the vertical temperature gradient even beneath the clouds will not be so large as one would expect according to the law of the adiabatic changes for the dry stadium without mix- ture of foreign masses of air. Similar relations obtain, although not to an equally great extent, with regard to the descending current, which in its upper half is also fed by portions of the cyclone in which the con- densation has not yet gone so far and has not yet attained the high potential temperature of the highest stratum concerned in the whole process. Therefore in reality both the ascending and the descending branches of the curve deviate from the schema of Figure 36, and in both of them the vertical gradient will more or less approximate the average as we find it when we consider the ascent and descent as a connected whole. These considerations are entirely in accord with observed facts. Even when we deduce the vertical temperature gradient from observa- tions at stations of which the upper one is not so high that it is fre- quently within the clouds, we attain to temperature gradients that in general are far less than that computed for the dry stage ; this result is in great part only explicable as due to the above described mixture. The observations of the clouds also agree perfectly with what has been said, both with regard to the temperature conditions and the moisture. Only the central part of the cyclone is to any considerable extent fed by masses of air that have flowed along the surface of the earth itself, as one can easily convince himself by a simple diagram ;* whereas the periphery receives more and more air from the higher strata, whereby its lower boundary surface is raised but its power must be diminished. In fact also the clouds at the center of the cyclone hang down the lowest and are higher near the circumference, exactly as is demanded by the moisture conditions and the higher potential temperature of the intermixed masses of air. The friuge of clouds that we perceive beneath the layer of clouds that covers the sky (especially on wooded hills during the prevalence of a cyclone) and in which we can clearly follow the ascent of air in inclined paths, gives in connection with the See, for example, Mohn, Grundzuge, 3d edition, ISS3, p. 261. 254 THE MECHANICS OF THE EARTH'S ATMOSPHERE. ragged clouds surrounding the border of the continuous cloud cover, an excellent picture of the mixture just described. Of course it is understood that all these considerations relate only to the conditions that ordinarily occur in the interchange of air between cyclone and anti-cyclone. Processes in which we have to do with unstable equilibrium (such as occur, for instance, in the great thunderstorms in front of an advanc- ing current of air, where a whirl *with a long horizontal axis rolls rapidly forward and brings simultaneously on the side of the descend- ing current heavy rain-fall and great cooling with higher barometric pressure, while on the front or ascending side the cloudiness is just beginning) — such processes demand a very special investigation that may be postponed to some future occasion. At present only one more consequence will be drawn from the propositions relative to potential temperature which seems to me calculated to throw a new light on the interchange of heat in the atmosphere, and that especially demands consideration from a climatological point of view. III. ON COMPLEX CONVECTION. It has been shown above that in the adiabatic transfer of air out of the cyclone into the anti-cyclone, the potential temperature in the de- scending branch is higher than in the ascending. Hence it follows that in the descending branch a higher temperature prevails after attaining the initial pressure than prevails at the initial point, and a still higher temperature prevails at the end of the descending branch, that is to say on the ground in the anti-cyclone where, according to experience as well as for mechanical reasons, the pressure is always higher. Therefore in this transfer of air we are concerned not only with a simple transfer of the quantity of heat belonging to the air at the base of the cyclone, which we can here temporarily call the original quantity of contained heat, but this quantity of heat is increased by that heat of condensation which in the condensation stage did a part of the work of expansion and thereby diminished the cooling to a smaller quantity than it otherwise would be. Even when in consequence of the stronger abstraction of heat at the base of the anticyclone the air is fiually colder than it would have been in purely adiabatic interchange ; and even when temperature inversion has occurred, still the temperature at the end of the process is still always higher than if the transportation of the air had taken place at the level of the earth's surface and the cooling influences had remained the same. The heat of condensation or negative heat of evaporation, or as it teas formerly called the liberated latent heat, accrues to the advantage of that region in which the descending current has reached the carWs surface. We can therefore compare the whole process with that of a steam heater. PAPEE BY PROF. BEZOLD. 255 Moist air rises in the cyclone, attains the condensation stage and cools from that time on less rapidly since the heat of condensation does a part of the necessary work. The heat thus saved then enters into the descending current and finally is carried to the point at which the descending current reaches the earth's surface. I consider it proper to designate by a special word those transfers of heat in which, besides the transport of warm or cooled bodies, changes of the condition of aggregation also occur, and therefore propose the name "complex convection" or "complex transfer." iSuch complex convection is met with when vapor is formed at one place and precipi- tated at another, or when ice falls as snow or hail, or when it is trans- ported in the form of icebergs by ocean currents. If we apply this designation to the above-given considerations we obtain the lollowing proposition : uln consequence of complex convection the temperature in anticy 'clonal regions is always higher than would he the case in simple convection.v The application of this proposition to the warm zone is of very special interest (I designedly avoid saying Tropical Zone since I can not con- sider the warm zone as limited by the Tropics) that is to say to the calm zone and the rings of higher atmospheric pressure that border it on either side, of which rings however the northern one is frequently interrupted. The proposition just enunciated teaches that these two rings in con- sequence of complex convection are much warmer than would be the case if in the whole interchange one had only to do with dry air or with movements on one level. The warm zone is therefore hereby broadened and at the same time there is found within it a diminution of the tem- perature gradients. In the calm zone itself much heat is used in evaporation and hence, in connection with the diminution of insolation by the covering of clouds, as also by reason of the water precipitated from colder regions above, the rise of temperature above a given limit is prevented. The heat consumed by evaporation at the earth's surface or at the ocean's surface does its work at a greater altitude in the region of the clouds when liberated by the condensation, and thus diminishes the cooling of the ascending current only to again reappear below in both the belts of descending currents. A further development of the climatological consequences deducible from these considerations does not belong here. But this much we see at once, that the conclusions drawn from the mechanical theory of heat without any hypothesis whatever stand in direct contradiction to the older meteorological views. Formerly it was taught that the descend- ing trade wiud by cooling delivers to higher latitudes the water brought with it from the calm zone. Similarly it was taught that the heat lib- erated during the condensation raised the temperature, and that this 256 THE MECHANICS OF THE EARTH'S ATMOSPHERE. higher temperature inured to the places at or above which the con- densation occurred. The mechanical theory of heat shows that the current ascending in the calm zone must precipitate its water right there in the form of tropical showers, and that then it must descend as a drier and also as a warmer current (except in so far as it does not experience any mate- rial cooling, especially at the earth's surface). This theory further shows that the heat of condensation, in so far as super-saturation proper does not come into consideration, never shows itself as actually warm- ing but only as diminishing the cooling that accompanies the ascent of the air, so that the current arrives at the upper limit warmer than it would without the accompanying condensation, and that the heat thus economized benefits the point at which the descending current reaches the earth's surface. The considerations here developed can of course only be considered as approximate steps that still await additions and corrections. To my eye they play a role similar to that of the investigation of the so- called solar climate in climatology. Moreover some of these have no claim to complete novelty, but will be found here and there in connec- tion with other special investigations. On the other hand, they have never as yet been developed in such general — and never in such a simple — manner as is here done with the help of the idea of " potential temperature " and of the theorems that it was possible to deduce from this as to the potential temperature of the different layers of air. The consequences that can be deduced from this as to the static relations of the atmosphere, especially with refer- ence to the fundamentally different behavior of cyclones and anti- cyclones in winter and in summer, both in respect to their intensity and their duration, are delayed to a later communication. [An Appendix as published in the original memoir by von Bezold is omitted from this translation, as it has been at the author's request incorporated in its proper place in the latter portion of his first com- munication.] XVII. ON THE THERMO-DYNAMICS OF THE ATMOSPHERE. (THIRD COMMUNICATION. ) By Prof. Wilhelm vox Bezold. In the two papers previously published on the above subject the re- strictive assumption has been always made that the masses of air under consideration experience no mixture with similar masses having other temperatures and other degrees of moisture. At the same time how- ever it was shown that such mixtures must frequently occur in nature and that the investigations in question could possess only a restricted application so long as we neglect these processes. For this reason therefore it is now necessary to extend the previous investigations in this direction. But investigations on this subject have also a special interest because for a long time we formerly attributed too much importance to the mix- ing of masses of air of unequal temperature and near the point of sat- uration, whereas in more recent times we have gone to the opposite extreme and attributed to it scarcely any importance at all. Following the example of James Hutton,t the mixture of such masses of air was, until within a few decades of years, considered as the prin- cipal cause of atmospheric precipitation. Wettstein was (so far as I know) the first to antagonize this view| which however even to-day is still widely accepted. He however fell into the opposite error in that he contended tha% in general, precipitation never could occur by mixing. Here, as in so many other points of modern meteorology, Hann§ first made the matter clear in that he, in the year 1874, proved that by mix- ture condensation could be indeed produced, but that the former method of computing the quantity of precipitation was affected by an error in principle after correcting for which the values obtained are so small * Read before the Academy of Sciences at Berlin, October 17, 1889. [Translated front the SitsungsbericMe der Konig. Preus. J had. der Wissenschaften zu Berlin, 1890, pp. 355-390. J t Uoy. Soc. Edinh. Travs., 1788, Vol. i, pp. 41-86. t I'ierleljahrs.s. d. naturf. Gesell. Ziirich, 18<>9, XIV, pp. 60-103. \n Ztschft. Oeaterr. GeselL Met., 1874, Vol. IX, "pp. 292r29C. [Hep. Smithson., 1877, p. 385.1 257 80 A 17 258 THE MECHANICS OF THE EAKT H S ATMOSPHERE. that the production of a moderately heavy precipitation in this way is impossible. At the same time he showed that the adiabatic expansion in this re- spect played an entirely different and much more important role, aud that, in it we have to recognize the source of all considerable precipi- tations. In this paper, so far as it concerned mixture Haun confined himself to the computation of an example from which it appeared that even under very improbable assumptions there could in this way only be lealized very slight quantities of precipitatious. Pernter many years later* contributed to the solution of the prob- lem in that he brought it into an exact mathematical form and at the same time also computed small numerical tables in order to facilitate the comprehension of the quantities that euter into the question. But since the empiric formula for the tension of aqueous vapor en- ters into the expression given by Pernter, therefore the latter is rather complex and is not especially clear. It seems therefore to me not only desirable but really necessary to take up the question anew and if possible prosecute it to a definite conclusion. This is the object of the following lines. It will be shown how graphic methods give with extraordinary ease an insight into the whole theory of the mixture of air and how in such methods we possess at the same time the simplest means for the nu- merical evaluation of the quantities that enter into the question. Various tables — some of which may also be welcome for other investi- gations— will also facilitate a general survey as well as the exact com- putations. After these preparatory sections there will be considered the various causes of the formation of precipitation, namely, direct cooling, adiabatic expansion, and mixture, in their relative importance and it will be shown how that only by the consideration of all these causes is it possible to obtain a deeper insight into the methods of the formation of clouds. (a.) THE MIXTURE OF QUANTITIES OF AIR OF UNEQUAL TEMPERATURE AND MOISTURE. Before we proceed to the mathematical treatment of this problem we must first come to a clear understanding as to whether definite masses or definite volumes shall be made the basis of the computation. At the first view it would seem appropriate to adopt the volume, since we can from well known tables obtain directly the quantity of water which corresponds to the saturation of one unit of volume. This is doubtless the reason why in the older investigations of this subject based on Hutton's theory, one always started with the con- si deration^ of the unit of volume, and why Hanu— when he would * Zeitschft, Oesterr. Ge&elf, Met., 1882, VoTTxvn, pp. 421-4267 PAPER BY PROF. BEZOLD. 259* demonstrate the imperfections of this theory in his considerations on this subject, followed the earlier method of treatment, and adopted the volume as a basis. This is also quite justifiable so far as concerns the first estimates, aud I also recently have made the same application in a popular lecture. But when one wishes to obtain exact formuke this method brings him into difficulties. These arise from the fact that the capacity for he!it of a unit of volume, the so-called volume capacity, even without the consid- eration of the intermixed vapor of water, is to a high degree affected by pressure aud temperature, so that no forms of approximation are al- lowable. The capacity for heat of the unit of mass of moist air, there- fore its capacity for heat in the ordinary sense of the word, is entirely independent of the above mentioned quantities and is also so little in- fluenced by the contained water within the limits that occur m meteor- ology that, as will later be more accurately shown, we can in the pres- ent question simply consider it as constant. In order however not to lose the advantage that inures from the utili- zation of existing tables, I have computed for different pressures and successive degrees the quantity of aqueous vapor that is contained in a kilogram of saturated moist air for such pressures and temperatures as occur in the atmosphere and have communicated the table thus formed in an appendix to this paper (see page 287). This table not only facilitates very considerably the solution of the questions that refer to the mixture of moist air, but it can also be ap- plied with profit to many other investigations. After this preface the problem itself is to be considered more closely, and to this end an appro- priate notation is first to be introduced. Let there be nil and m2, the quantities expressed in kilograms, of air to be mixed together; t{ aud t2, their temperatures; i/i aud y2, the quantities expressed in grams, of vapor actually con- tained in a kilogram of moist air; y'i and y'2, the corresponding values of contained moisture in a kilo- gram of air at ti and t2 in the saturated coudition ; Bx and R2, the accompanying values in per cent, of the relative hu- midity. pi and p2, the same quantities expressed as fractions of unity, that is to say p.= Rl and p2=*^. . " 100 iuo t3, 2/3, y'3i E3, and p3, the various values of the same above-named quantities in the mixture, in so far as the limit of saturation has not been exceeded, or at least no water has been lost, that is to say, true sat- uration exists. 260 THE MECHANICS OF THE EARTH S ATMOSPHERE. t, y, y', R, and p, the corresponding values after mixture and after the loss of the quantity of water that exceeds the normal quantity for saturation, or also, in general, any given group of the same quanti- ties belonging together. The pressure expressed in millimetres of mercury will as before be expressed by /?; the maximum of the elastic force of the vapor will in a corresponding manner be expressed by €. The pressure /i can be considered ascoustaut during the process of mixing. This is allowable since, where mixture actually occurs, the two masses of air must nec- essarily exist under very nearly the same pressure and must also retain this [in the free air] even when on account of the mixing a change oc- curs in the total volume, which in general is very unimportant. The problem of mixture becomes extremely simple so long as no pre- cipitation of water occurs, that is to say so long as the quantities ob tained by the mixture are to be indicated as in the above notation by the subscript3. In this case y3{m1+m2)=y1mi+y2m2 or ml(y3—y1)=m2{y2^if3) (1) aud further cinh(ts— ti)=m2c2(t2— 12) where by ct and c2 we understand the thermal capacities of the quan. tities of air to be mixed,* or since these quantities are to beconsideied equal wi,(/3-*i)=»i2(*2-*3) (13) Jf we combine the equations (i) and (2) we obtain (the mixing ratio) y.-i— y\_h— t1=mz 2/2—2/s t2—h Ml which is the well known equation that holds good for the mixture of two r. quantities of the fluid in question, having two different temperatures. Since the graphic m ithod will be chosen in the further development, therefore first of all this simple for- mula must be translated into a geo. metrical form. To this end, in a rectangular sys- tem of coordinates, Fig. 37, the tern, peratures (t) are taken as abscissas, the quantities of moisture (y) as ordi- nates, and these are designated in the ordinary manner by OT^OT, .... "Strictly speaking we should use mean values computed by a special formula be- tween the above named Cl aud c2 and that of the mixture v,. Since, however, the values of c scarcely differ from each other for the different temperatures and pressures, we. can therefore omit this rehnement. PAPER BY PROF. PEZOLD. 261 T1F1, T2F2, etc.; in the figure the origin 0 is omitted. We see at once that F-i lies on the straight line drawn through FY and F-, and that TlT3_T3F3-TlF1 m, T2T3-T2F2-T3F3~m1 In order now to obtain a decision as to the degree of saturation, we must also introduce into the diagram, as ordiuates, along with the values of?/,, y,, and y3, also the values of?//, y2', and y/, corresi)onding to complete saturation. The ends of these ordinates, which are repre- sented by Fi', F2', and Fz' in the diagram, all lie upon a curve that with increasing t rises rapidly, and the equation* of which is 2/ = 623/i~-(U77, when for ft we insert the proper constant pressure. With the assistance of this equation, or with the approximate for- mula obtained by development, y= 023^ + 234.88 (J J the tables communicated in the appendix [page 287J have been com- puted, by the help of which the curves can be easily constructed di- rectly for the pressures therein considered, and which we can designate as curves of the quantity of vapor needed for saturation at the pres- sure ft [or for brevity, tlie saturation curve]. It will now suffice to cast a glance at the figure in order at once to obtain the following propositions: (1) So long as for given temperatures ty and t2, the values * ii = pl and J'2/ = p.,, remain within given limits, the straight lino Fx F2 passes entirely beneath the saturation curve, aud therefore there can be no mixing-ratio for which conden- sation can occur. (2) When px and p2 increase so much that the straight line Fi F2 touches or cuts the saturation curve, as in figure (38), then there occurs either one or many mixing-ratios that may bring about condensation. , r (3) When JR, = 7?,=100, ie.,when the J? J two quantities ol air to be mixed are J% saturated, then the straight line F{ F\ coincides with the curve Fx' F2, and then for every mixture theie occurs -^ / super-saturation or condensation. Fi.J. 38 Harm, Zeit. Oeaterr. Gesell. Met, 1874, vol. ix, p. 324. [Smithson Rep., 1«77, p. 399.] 262 THE MECHANICS OF THE EARTHS ATMOSPHERE. The investigation of the cases included in 2 can always be referred to case 3, since the points Fi* and F2*, in which the straight line Fx F2 cuts the curve, JY F2' play precisely the same role in the second case as Fx and F2 in the third case. If we consider more closely the propositions just enunciated, then we shall involuntarily be led to seek certain limiting values, the knowledge of which leads to the solution of the fundamental question whether, under given conditions, condensation will be possible or not. The questions that obtrude in this connection are as follows: (1) Whatlimit must the relative humidity exceed for a given tempera- ture of the components, or at least for one of them, in order that con- densation may be possible for a properly chosen mixing-ratio ? (2) What limiting value must the relative humidity of one component exceed when the value of the other is given, aud when also condensa- tion is to become possible for a properly chosen mixing-ratio? The first of these two questions can be expressed in the following form : When the limit of satura- tion is to be attained for an appropriate mixing ra- tio, aud the relative hu- midity of both components is to be the same, what is the minimum value of this relative humidity ? That the knowledge of this minimum value is also a solution of question 1, we see most easily when we more accurately examine the answer to the question as last formulated. We obtain this latter answer very easily through the following con sideration : If R{ is to equal E2, then the straight line Fx F2 must cut the axis of abscissae at the same point P Fig. 30, as does the prolonga- tion of the chord Fx' F2'. For if this condition is fulfilled then— Fig. 3D. but now and and consequently, also TXFX._T2F2 TXF,'- T2F2' rI\ Fx< ~ yi> Pl ~ 100 T*Ft y2 B, T2F2'~y2'~P2-W(] R\ = R2. PAPER BY PROF. BEZOLI). 263 If now for a given value of R1=R2, which may be called #0, the point of saturation is to be just attained by proper mixing, then the straight line P Fi F2 must just touch the saturation curve Fx' F2'. The point of tangency 8 gives therefore the temperature of the mix- ture for which saturation will be just attained, and hence also the mixing ratio. But the value K0, as the figure shows at the first glance, must be exceeded by at least one of the components when condensation is to become possible, and it therefore is precisely that limiting value that is desired in question No. (1). It is easily seen that the knowledge of these boundary values is of high importance, it is yrv therefore carefully considered in tables to be /f subsequently communicated. Equally simple J$ is the solution of the second question, which, however, will here be considered only under J^" the special assumptions that Ex or E2 is equal fig. 40. to 100. If #1 = 100, that is to say, if the cooler of the two components is in the state of complete saturation, then we obtain the minimum value of E2, when we, as in Fig. 40, draw at F}' a tangent to the saturation curve, and prolong this until it cuts the ordinate F2' T2 at the point F2. The desired value is #9=100 J8, * • As soon as i?> exceeds this limit con- F2'T2 densation occurs on mixing, provided that there is sufficient of the colder component, that is to say, provided — is large enough. If, however, we consider the other case as given and assume that #2=100, that is to say, that the warmer component is saturated, then we find Ei when at T2' we draw a tangent to the saturation curve and seek the intersection of it with the ordinate Fv' T\. Thus it becomes at once apparent to the eye that Ev is always smaller than E2, so that for sufficiently great distance between Tx and 1\ the quantity Ex can even attain a negative value, if such were imaginable. The physical interpretation of this is that when warm saturated air is mixed with colder the latter can have a high degree of dryness and still condensation may occur for a proper mixing ratio; in many cases even the cooler air may be absolutely dry; it might even have a nega- tive Rx corresponding to its containing a certain mass of hygroscopic substance, if only there is sufficient quantity of warmer air, that is to say, if only — 2 is large enough.. In such cases, therefore, in place of the minimum value #i there occurs a limiting value of // = -?' which must be exceeded if conden- sation is to occur. 264 THE MECHANIC'S OF THE EARTH'* ATMOSPHERE. These considerations show that mixtures of saturated warmer with unsaturated cooler air gives rise to condensations much more easily than do mixtures of saturated cooler with drier and warmer air. The flow of a jet of saturated warmer air into a cool space must therefore be accompanied by much more powerful condensation than is the inflow of saturated colder air into a space filled with unsaturated warmer air. The fact that clouds of vapor so easily arise over every open vessel tilled with warm water, while the formation of fog near very cold bodies in warmer regions is much more rarely to be observed, gives an assurance of the correctness of this principle. Whenever during moderately cool weather the door of a wash-house is opened great clouds of vapor pour out, but the opening of an ice cellar on a hot day has not a similar result. Now that the limits have been determined within which, in general, mixture can occur, it is proper to give the quantity that can be precip- itated by the condensation. Such precipitation occurs whenever the point Ft lies above the saturation curve. For then the limit of satura- tion is exceeded, and by a quantity that is represented by the length This quantity, which will be designated by a3, is that of which, before the writings of Wettsteiu and Hann, it was assumed that it was pre- cipitated as water as the result of the mixing. To what extent one was led into error by this assumption is most easily seen from the figure by the following considerations: Let it be assumed that at first actual saturation occurs in the mix- ture, and let the whole quantity #3 be actually present in the form of vapor or aqueous gas, then will the gradual precipitation of the vapor be accompanied by a simultaneous warming. The increase of temperature hereby brought about is found from the equation 1000 cdt = - rdy, where c is the capacity for heat of the moist air under constant pressure, and r is the latent heat of evaporation, and where c is to be multiplied by 1,000, since we have taken a kilogram of the mixture, whereas y is expressed in grams. Since now, as will subsequently become evident, the temperature t rises only a few degrees even for a very considerable supersaturation, therefore we can consider c as constant in each individual case, and T corresponding to this we obtain 103c yi-y = -T-(t-ta) (3) in which y and t represent those values that are obtained after the precipitation of the water that is in excess of the limit of saturation. PAPER BY PROF. BEZOLD. 265 Iii Fig. 41, therefore, we find this temperature t in a very simple man- ner in that we draw through F3 a straight line that makes with the axis of abscissas an angle 10V a= arc tang -. The point F, in which this straight line cuts the saturation curve, has the desired coordinates t and y, whereas the quantity of precipitated water a = y3 — y is a quantity that is represented in the figure by the short line F3i. According to the old theory t3 and f, as well as y3' and y', or, what is the same, y3' and y, were considered respectively as the same. But now we see, as Hann had already shown in a special example, that this is not the case, but that t>t3 and y 0°. If t <0°, then we have to add the quantity 80 [calories] U the value of r. If we consider this and then compute K for 0° and for — 30°, first for /3 = 760 milimetres, and next for fi = 400 milimetres we obtain as extreme values 2.87 and 2.98. so that here with even more right we can assume K to be constant and as we in fact will do equal to 2.9. According to this, without important error, we may consider the lines F3 F, in general, as parallel straight lines which experience only a slight bend at the point corresponding to 0°. In the actual application of the above-explained graphic method we do best to place upon the system of coordinates, on which we have en- tered the saturation curve, a group of straight lines representing the series Fs F, of which those on the left of the zero coordinate are inclined to the axis of abscissas so that tan «=on, but those to the right of the zero coordinate have tan a=?r^ 2.5. * Hann, Zeit. Oeat. Gesell. Met., 1874, vol. ix, p. 324. [Sinithson. Rep., 1877, p, 399.] t According to the investigations of Dieterici (Wiedemann's Annalen, 1889, xxxvn, pp. 494-508), as well as according to those of Ekholm (Bihaiig K. Svenska Vet. Akad. HavdL, 1889, xv, Part I, No. 6.) ; these numbers are indeed not quite free from criti- cism. Since however on the one hand, the correction of these numbers scarcely comes into consideration in the final result here desired, and since on the other hand the value of the capacity for heat of dry air here adopted is based on the caloric used by Regnault, it appeared to me proper, if not even necessary, also to make use of the older value for r. PAPER BY PROF. BEZOLD. " 267 Special interest attends the question : In what ratio two quantities of air of given temperature and humidity must be mixed in order to obtain the greatest possible precipitation \ The solution of this prob- lem is given by a glance at Fig. 41. Since the quantity of precipita- tion is a = F3F sin a, therefore a will be a maximum when F3 F has its greatest value. But this is evidently the case when the tangent at the poiut F on the curve is parallel to the straight line Fi F2, or Fx' F2. The point at which this tangent touches the curve cau be determined either by construction and trial or, in case we have at hand a table of quantities of saturation, such as that in the appendix, computed for the barometric pressure in question, we have then to seek from it a value of t such that dt t2—ti which is not difficult to do after constructing a corresponding supple- mentary table of differences for each tenth of a degree. Having found the point F we move further parallel to the previously mentioned group of straight lines until we strike the line Fi F2, and thus determine the point F3, which on its part gives the point T3, and thus the distances Tx T3 and T3 T2, whence results the mixing ratio that corresponds to the maximum precipitation. The precipitation itself we obtain from the above-given formula, a = Vs — y- But we can also adopt another and purely numerical method for obtaining these quantities. For it is not difficult to see that FL (Fig. 41) is also a maximum at the same time with F3 F, where we designate by L the point in which the prolongation of the ordinate FT intersects the straight line Fx F2. Moreover when we represent the line FL by Z, we have f='/i + (*-Mtau/i-i/ =zt tan ft—y+y\ —U tan ft, where ft represents the angle that the line F, F^ makes with the axis of abscissas, that is to say, tan ft = *=? t2—t\ Since the value of y is not difficult to compute, when not taken directly from the table, one is therefore in condition to form a small auxiliary table for the value of the quantity I for certain values of t, such as lie in the neighborhood of the one desired, and from it take out 26-S THE MECHANICS OF THE EARTH'S ATMOSPHERE. the maximum value of I or the value of t corresponding thereto. Then the value of a is given by the formula , tan a tan a + tan ft whose deduction may here be omitted. Thus both a numerical and a graphic method are at our disposal. If we follow the former, we can easily perceive that an extremely accurate knowledge of the quantity of vapor contained in a kilogram when in the condition of saturation is presupposed for an even nioder ately accurate determination of the value of a and t, as well as of the ratio -1. Because of the unreliability of the data at hand the values obtained by computation haveiu themselves a rather high degree of uncertainty, so that one can equally well make use of the far more convenient graphic method without thereby in fact losing anything in accuracy. Iu this latter way the following small tables have been computed, which give the limiting cases above treated as especially interesting for the pressures 700 and 400 mm. and for temperatures that proceed by steps of 10° and thereby makes possible a quick review of the various questions relative to mixtures of air. The first horizontal line of each of these twelve tables relates to the case where both component masses are completely saturated, and gives iu the columu a the greatest precipitation that can occur* under these AH circumstances and under the most favorable mixing ratio ' . mz Therefore the a on the first line of each table, gives the maximum possible precipitation that can be brought about by mixture at the given temperatures. The second line of each table gives the value of the relative humidity which must (at least for one of the components) be exceeded if precipi- tation by mixture is to be auy way possible. We also find on this line under the headings t and —L the mixing temperature and the mixing ratio for which the point of saturation will be just attained when in both components the relative humidity has the minimum values, given under R{ and Rz. The third line shows the value of R_> that must be exceeded by the relative humidity of the warmer component, if the cooler component is completely saturated and if precipitation is to become possible by mixture. The fourth line gives the mixing ratio which must be exceeded if precipitation is to become possible by means of any proper mixing [Expressed in grams of water per kilogram of moist air.] PAPER BY PROF. BEZOLD. 269 ratio when the cooler component is perfectly dry and the warmer com- ponent perfectly saturated. The fifth line shows, under a, the maximum precipitation that is con- ceivable under the last mentioned condition of the components as to humidity as well as the mixing ratio and mixing temperature at which this maximum precipitation is attainable. In many cases no precipitation is possible with perfect dryness of the cooler component. In such cases the fourth line is the analogue of the third since it gives the minimum value which the relative humidity of the colder component must exceed if in general precipitation is to be- come possible by mixture. Under these conditions in the nature of the case the fifth line becomes a blank. The tables as here given relate only to the two piessures 700 and 400 millimetres. Since however these include all altitudes between 080 and 5,150 metres, that is to say, those altitudes in which the forma- tion of cloud or at least precipitation proper principally occurs, and since the supplementing of these tables by means of the table given in the appendix is not difficult, I have thought that I might confine myself to these special cases. At any rate these will suffice in order to give a general orientation as to the quantities coming into consideration, and therefore the tables themselves are now given, and it need only be stated that the figures must be considered only as approximations, since iu general they are has- d upon the first differential quotients, but occasionally on the second differential quotients of the curve of vapor- pressure, so that very small changes in the experimental data or iu the method of interpolation must make themselves very sensible. «1 t2 n, R., a t m4 '■ m2 6=700°" 0 ; t2—t,= =20°. ( ■ 100 100 0.4 —9.0 102 : 98 I 70 76 l/oo —14.0 140 : 60 —20° o^l 1 100 • 52 J 00 ]/00 >0. 0 —20.0 >— 11.8 1 : 0 <118: 82 t 0 100 <0. 13 >-5.5 >60 : 140 r 10!) 100 0.55 1.0 106 : 94 81 81 1 OO —2.8 128 : 72 —10 +io| 100 0 61 100 l/oo >0.0 —10.0 >— 0. 1 1 :0 <1 : 1 0 100 <0.2 >0.5 >54 : 146 100 100 0.75 11.0 108 : 92 86 86 1/x 6.2 138 : 62 0 +20, 100 02 0 100 l/oo >0. 0 0. 0 >12. 2 1 : 0 <80 : 120 i ! 0 100 ! | <0.2 >16. 7 >37 : 16H 270 THE MECHANICS OF THE EARTH'S ATMOSPHERE. h <2 J\>\ li-i a f mj : OTj 6=700n"n; h— ti=10°. | 100 [00 0.04 i —15.5 57: 43 1 92 92 l/oo —16.0 60 : 40 -20 -10 < { r ICO 55 100 82 100 1 oo l/oo —20. 0 —10. 0 1 : 0 0 : 1 100 0.11 —4.0 43 : 57 04 91 l/oo -5.5 55 : 45 —10 o< 100 47 85 100 1/oC 1 CO —10.0 0.0 1 :0 0 : 1 f 100 100 0. 19 5.0 54 : 10 94 01 1/O0 4.5 55 : 45 O1 +10 <| 1 100 64 87 100 l/oo 1 00 0.0 10.0 1:0 0: 1 I r ioo 100 0.21 14. 5 55 : 45 | 94 94 1 X 14.0 60 40 + 10 + 20-^ 1 100 70 87 100 1 00 1 00 10.0 20.0 , 1 : 0 0: 1 i I b=400m"> ; tj—t,= :20°. 100 100 0.50 —9.5 108 : 92 : 76 76 l/oo —14.0 140 : 60 | -20° 0°-. 1 100 0 58 100 l/oo >1 -do —20. 0 >— 11. 8 1:0 < 1 18 : 82 1 0 100 <0.2 ^-5.4 < >54: 146 < 100 100 0.75 1.2 110: 90 80 80 1/oc —3.3 133 : 67 —10 +10 « 100 0 65 100 1/O0 >l/oo -10.0 >(). 3 1 : 0 <97 : 103 0 100 <0.2 >6. 0 ^45 : 155 &.=400"""; t2— «, =10°. 100 100 0.12 —15.5 58: 42 96 96 l/oo —16.0 60: 40 -20 -10 < 100 85 l/oo —20. 0 1:0 48 100 1,00 —10.0 0 : 1 r 10° 100 0.17 —4.5 50: 50 94 94 1/00 -5.5 55: 45 —10 o. ; ioo 88 l/oo —10. 0 1 : 0 52 100 l/oo —0.0 0: 1 f 100 100 0.20 CO 47: 53 93 93 l/oo 5.0 50 : 50 0 10 { 100 86 1/00 0.0 1 : 0 65 100 l/oo 10.0 0 ! 1 In agreement with the previous results by TTann and Pernter, these tables show how small is the precipitation attainable by mixture when we consider components whose differences of temperature are even greater than ever occurs in nature. PAPER BY PROF. BEZOLD. 271 Since on the other hand, according to the data recently collected by Hann,* quantities of water considerably greater than these can remain suspended in the air (as mist, fog, and cloud), therefore we see very plainly that, while the formation of cloud can be caused by mixture, yet the precipitation of rain or snow in any appreciable quantity can scarcely be brought about iu this way. At the same time the following diagram, uhich we here make use of for graphic computation, enables, iu the most simple manner, to com- pare the quantity of precipitation formed by mixture with that which is produced by direct cooling as well as that produced by adiabatic expansion. If we assume that by mixture under a favorable mixing ratio of sat- urated air at the temperature t2 with other saturated air at the temper- ature tu the quantity of water a is precipitated (see Fig. 42), then we obtain the same quantity of precipita- tion when we directly cool the com- ponent y2, from its temperature t2 to a new temperature td, for which we have. y'il = y'2 — a, but y'd is the ordinate whose foot is Td in Fig. 42. A glance at the general saturation curve suffices to show at once that the J-J^ difference t2 — td is very much smaller than the difference t2— t ; that is to say, that a very slight direct cooliug affords as much precipitation as a considerable cooling by mixture with colder air, even when the latter is completely saturated. The effect of adiabatic cooliug is seen wheu iu the diagram we draw the adiabatic curve as a function of the temperature and quautity of water contained in a kilogram of moist air. Such an adiabatic curve sinks, as we easily perceive, rather more slowly from the right toward the left than the saturation curve. For since in this case the diminution of temperature goes hand iu hand with the increase in volume, therefore, the quantity of moisture neces- sary for saturation will for falling temperatures be greater than it would be if the initial pressure were maintained ; that is to say, than it would be by progressing along the saturation curve. The adiabatic (which without any difficulty can be introduced into the diagram with sufficient accuracy with the aid of Hertz's Graphic Method*) will therefore have a path similar to that shown by the curve F2 A in Fig. 42. * Meleorologische Zeitschrift, 1889, vol. VI, pp. 303-306. * Meleorologische Zeitschrift, 1884, vol. I, pi. VII. [See No. XIV of this collection of Translations.] 272 THE MECHANICS OF THE EARTH'S ATMOSPHERE. But in this case the lowering of the temperature must be forced down to ta, if the quautity of precipitated water is to be equal to a, since then the equation holds good for y'a, which represents the ordinate erected at Ta. Here also the general course of the curve again shows that the fall of temperature necessary in order that a definite quantity of precipita- tion may be caused by adiabatic expansion is very much less than when the same quantity is to be produced by mixture. A numerical example will best illustrate this principle: From the above-given small tables we see that at 700 millimetres pressure satu- rated air at0° C. mixed with saturated air at 20° can precipitate at the most only 0.75 grams of water per kilogram of the mixture and that the final temperature of the mixture will be 11°.0; that is to say, for a cooling of the warmer component from 20° down to 11°. By direct cooling, on the other hand, the same quality of water would be precipitated from 1 kilogram of the warmer component when it is cooled from 20° down to 19°.2; whereas by adiabatic expansion a cooling of from 20° down to 18°.4 would be necessary; that is to say, a vertical ascent through a distance of about 31.0 metres. This example shows in a very striking manner how slight need be the direct cooling by contact with cold objects, or by radiation, or even by adiabatic expansion, in order to produce quantities of precipitation, such as would by mixture be only obtainable in the extremest, scarcely imaginable cases. "With this the consideration of the mixture of masses of moist air may be brought to a close and only the single remark be made that the difference t — ts is smaller as the quautity a of the precipitated liquid decreases. The amount of this difference will therefore only exceed the value of 1° or 2° in such extreme cases as are assumed in the pre- vious tables and generally will remain far within this limit. Therefore in the majority of cases the mixing temperature may, with- out important error, be put equal to that which we obtain by mixing equal masses of dry air, whereby many computations experience a great simplification. (6.) SUPER SATURATED AIR. In the foregoing solution of the problem of mixture it was assumed for the sake of simplicity, that in the cases where the formation of precipitation in this manner is really possible, super-saturation must first occur, and then precipitation follows. This assumption was implied by Hann in his above-mentioned memoir* at a time when we still knew nothing as to whether aqueous vapor could actually exist in a supersaturated condition. * Zeitschrift Oest. Gesell. Met., 1874, vol. ix. [Smithson. Rep., 1877, p. 397.] PAPER BY PROF. BEZOLD. 273 But since the possibility of this has been demonstrated by the inves- tigations of Aitken, Coulier, Mascart, Kiessling, and especially by Rob- ert von Helmholtz,* it has some interest for us to make the precipitation from supersaturated air the object of a special investigation. This precipitation, as is well known, occurs when super-saturated air (which can only exist when perfectly free from dust) is suddenly mixed with very fine particles of solid bodies, or possibly, also, when electric discharges take place through such supersaturated air.t We obtain directly from the above-given rules the amount of the precipitation, as also the rise in temperature. We have only to omit the parts designated by the indices 1 and 2 in Fig 41, and to consider the condition indicated by the subscript iudex 3 as the starting point, then the ordinate T3 F3 = y3 gives the quantity of water in the state of supersaturation, while y again indicates as before the tiual remaining moisture; y3 — y indicates the quantity of moisture precipitated and t — t3 the consequent rise of temperature. This is, therefore, a method of formation of precipitation, in which one can actually speak of a liberation of latent heat (the latent heat of evaporation), as was formerly done in explaining the formation of pre- cipitation in general. In a certain sense this usage is allowable, even in the formation of precipitation by mixture, in so far as the temperature of the mixture comes out higher when water is precipitated than when this, under otherwise similar conditions, is not the case because of the iusufficieut quantity of water. This rise of temperature is however always a very unimportant one in consideration of the small quantity that can be con- densed by mixture. It is otherwise when true super-saturation is present. lu such cases the rise of temperature can, according to the degree of super-saturation, be very considerable, as is easily seen from Fig. 41. Still more considerable must the precipitation be that is caused by the sudden cessation of the super-saturation, namely : So soon as a sud- den development of heat occurs at any one place in the atmosphere there follows a powerful ascent of the air which then, by adiabatic cool- ing, must always produce new formation of precipitation. Under those conditions, when the vertical distribution of temperature approximates even distantly to that of convective equilibrium, then the sudden cessation of the condition of super-saturation causes this equilibrium to become unstable, and thus this cessation then affords the key to the explanation of a series of phenomena. I consider it probable that it is in such processes, which indeed deserve a thorough investigation, that we have to seek the reason for the ''cloudbursts" properly so called. Of course, to establish this * Wiedemann's Anvalen, 1886, xxvii, p. .r>27. t R. von Helmholtz, Wiedemann's Annalen, 1887, xxxu, p. 4. Si) A 18 274 THE MECHANICS OF THE EARTH'S ATMOSPHERE. view tbe proof must first be given that the super-saturation, which we have hitherto only known in laboratory experiments, also occurs in the free atmosphere. The mixture of super saturated air with other quantities of air scarcely needs a special consideration, since we at once see the result of such mixture when we imagine, in Fig. 41, one of the points, Fi or F2, trans- posed to the upper side of the curved' F', and then execute the further constructions according to tbe rules previously given. (C.) MOIST AIR WITH INTERMIXED WATER OR ICE. Water occurs in the atmosphere not only as vapor, but also in the form of drops of rain, crystals of ice, and particles of fog. The psy- chrometer and hygrometer teach us that the air is not necessarily sat- urated with vapor when water is mixed with it in this manner. Unfor- tunately we possess only very imperfect data as to how great a quan- tity of water can in this way be mechanically mixed with the atmos- phere.* But there can be scarcely any doubt that the sum of the water mechanically mixed and that which is preseut in the form of vapor may together be smaller, or equal to, or even greater than the quan- tity corresponding to saturation for the given temperature. Corre- sponding to this statement, I will designate such mixtures as air which is " partly saturated mechanically," " wholly saturated mechanically," or " super-saturated mechanically." And now, first of all, we will inves- tigate how such masses of air behave when mixed with ordinary air more or less moist. By this investigation we shall come to learn the conditions under which the dissolution of fog or clouds or the evaporation of falling rain- drops may occur. Such dissolution is, as we at once see, to be attained by mixture ouly when the intermixed air, which at first may be as- sumed to be the warmer component, is relatively dry. Therefore, we will at first investigate the phenomena of mixture under the following conditions: Let R{ > 100 and composed of two parts, of which the one Ri is in the form of vapor and the other .K, is liquid, and moreover let Ri < 100 while Furthermore let R2 < 100 and t2 > tu This being assumed, the follow- ing formulae hold good, using a notation which by analogy is intelligi- ble of itself: y\>yi ii\2. In this latter case, which is represented in Fig. 44, all the water is not dissolved but only a portion {y'—y3) as is represented in Fig. 44 by the distance F3 P. The first of these two cases can be easily handled numerically, since under these conditions we have t=t3 — Ky3 = t3 — E^ nu mx -f m2 niitx + m2t2 mA 4- m2 -Kh Wo mi -r «*2 nijti 4- m2i2 — Kyxm2 w'i 4- m2 But the computation is as simple as this only when all the water is really evaporated; in tbe second case where mechanical supersaturation still continues it is better to apply the graphic method. An especial interest; pertains here again to the investigation of the limiting cases for which in general there can occur a complete dissolution of the water originally present as liquid in one of the components. Of such extreme eases there is an extraordinary variety according as we are at liberty to assume arbitrarily either the mixing-ratio or the humidity of one or the other of the components. At present we shall consider only the question, what is the initial limit of the mixing-ratio for given components in order that complete dissolution must always follow. This limit is evidently obtained when F' Fn lie at the same altitude above the axis of abscissas, that is to say, when ii = y'=yZ) or when F and F' coincide. In this case F is the PAPER BY PROF. ISEZOLD. 277 apex of a right-angled triangle whose vertical side is F3 F3 and whose hypothenuse is parallel to the guide-line. If now we imagine the point T:{ moving to and fro along the axis of abscissas, then the apex of the triangle erected in the given manner upon the vertical side F3 F3 will describe a straight line passing through the point F2, which line we easily find when we erect such a triangle on the portion cut off by the straight lines F} F2 and Fx F2 from any arbitrary ordinate and then join this apex with F2. We can, for instance, as in Fig. 45, choose for this pur- pose the ordinate erected at Tx. Then 2^-FoFi is the triangle described and F0 F2 is the straight line on which the de- sired point F must lie; but since it must also lie on the saturation curve, therefore it is at the intersection of F0F2 and the curve Fi'F2', and the desired limiting value of the mixing ratio is Fig. 45. 1\T3- 1Uo 1 When the mixing ratio attains this limit or exceeds it on the side toward m2, that is to say, as soon as — 2; = or >/i a complete dissolution of all the suspended water occurs. Mi In such mixtures it can happen that the line F F3 cuts the curve JV F2' on the left-hand side of 1\ Fx. In such cases the temperature resulting from the completion of the mixture is lower than that of either compouent. The mixing ratio for which this phenomenon begins to occnr is easily found by d raw- ing, as in Fig. 4G, through F (which is in this case identi- cal with Fx') a line parallel to FQ Fx (which is a guide line), and find its intersection with F{F2. The abscissa of this point is then the temperature t3, which is producrd by 278 THE MECHANICS OF THE EARTH'S ATMOSPHERE. mixture in this ratio before the subsequent dissolution. From this value of ta this mixing ratio itself can be determined. We find by a very simple consideration that for this special value of t3 the following equation liolds good : But since and therefore and consequently K F-SF3= F\t\=y,-yx h—h h—h yi—Thg-ts—ti^nh U—U U- h «»i Whenever /K/<8, that is to say, when the cooler " mechanically super- saturated air," or at least the saturated component enters into the mix- ture with greater weight, then will t. 100 2.0 2.0 100 2.0 2.0 100 2.8 2.8 100 3.5 3.4 98 4.5 4.0 93 0.7 0.2 93 Remarks. Oberstdorf.* Starting point, Riezlern ; fog. Upper itnit of fog. Faulhorn. Faulhorn; fog rises; upper boundary of fog attains and surpasses the sum- mit of the mountain. Faulhorn ; fog siuks. Fog sinks ; upper boundary descends to the summit of the mountain. Descending ; fog still continues. Fog. Fog. Above the lower limit of fog. Below the lower limit of fog. At Riezlern.t * According to Trautwein ("Southern Bavaria, etc.," seventh edition, Augsburg. 1884) the altitude above the sea of Oberstdorf, which is that here adopted, is 808 metres; that of the summit of the Faul- horn is 2,033 metres, so that the value 2,031 metres is an excellent testimony to the reliability of the data. tThe morning observation gave the altitude of Riezlern as 1,058 metres, whereas the observation at 5 hours 10 minutes p. m. gave 1,078 metres, making use of the barometric pressure observed in the morning in Oberstdorf. But if we assume, as is required by the observations at Augsburg and Munich, that this reading had, during the intervening time, diminished by 1 millimetre, and further- more adopt the very probable assumption that the aneroid could not perfectly follow the rapid changes of pressure during the descent, and theiefore read about 2 millimetres too low, we shall obtain for Riezlern the altitude 1,062 metres above the sea, or a figure that agrees almost perfectly with that de duced from the morning observation. The numbers above tabulated were given on the one hand with a well compared, quite reliable aneroid, and on ihe other so far as con- cerns the temperatures with an Assmann's aspiration psychrometer of the older construction. From the above table we see very clearly that the upper boundary of the stratum of fog always shows a lower temperature than the neigh boring strata above and below. But whether this is as above assumed essentially the cold due to evaporation can not properly be decided. The high relative humidity 282 THE MECHANICS OF THE EAETH'S ATMOSPHERE. which was fouud even in the highest layer of fog raises some doubt in this direction. Some observations made by First Lieutenant Moedebeck aud Lieutenant Gross on the occasion of a balloon voyage made on June 19, 1889, and which Lieutenant Gross has recently published * in a very interesting essay, apparently speak more clearly on this point, and certainly deserve a thorough scientific analysis, f Here also the passage through thick clouds showed that the temper- ature at the upper boundary of these fell very low but immediately above this it rose at once suddenly. The observations of humidity also agree better with the theoretical views developed above. On this point Lieutenant Gross says with reference to a diagram given by him which shows the changes of the dry and wet thermometers, " We see from the •comparison of the carves of the dry and wet thermometers that the moisture of the air rapidly increases with approach to the cloud, and that in the cloud itself where both curves coincide the air is completely saturated with aqueous vapor. But it is only in the lower part of the cloud that this is the case, and the moisture diminishes towards its upper part, an observation that I have already frequently made. This is certainly also explicable: In the upper part of the cloud the sun acts again as at first. Immediately above the cloud the wet thermometer makes a sudden rise. The air becomes suddenly very dry, as results without anything further, from the heat reflected back from the cloud." That the lowest temperature should be observed immediately under the upper boundary of the cloud in spite of the influence of the sun seems to me explicable only by means of the cold due to evaporation in accordance with the mauner above theoretically predicted. One ought to be able to observe with all sharpness on the Eiffel tower the questions relating to the behavior of the upper surface of fog since it must frequently happen there that the boundary floats but a short distance above the meteorological instruments. Perhaps also it will be possible there to establish at different heights self registering thermometers and psychrometers or hygrometers in order to obtain truly simultaneous observations immediately above aud below the upper boundary of the fog (or mist). ((h) THE FORMATION AND DISSOLUTION OF FOG AND CLOUD. The preceding investigations into the formation of precipitation by mixture of quantities of air of unequal warmth and moisture show that * Zeitschrift fur Luftschiff fahrt, 1889, vm, p. 249. t In referring to this essay I might also mention that Lieutenant Gross has also in the meantime confirmed the expectation expressed in my former communication [see pages 251-953] according to which the inversion of temperature in the region of the winter anti-cyclone is not a peculiarity of mouutainous regions. On the occasion of a balloon voyage undertaken on December 19, 18t8, from Berlin under the influence <>t such an anti-cyclone, the sling thermometer gave an increase of temperature of fi in 1,000 metres of ascent between Ip.ji, and 4 p. m. PAPER BY PROF. BEZOLD. 283 although such mixtures cau not produce heavy rain or snow yet they can be of great importance in the formation of fog and cloud. In accordance with this there are three processes that can, either by themselves alone or in conjunction, cause a condensation of the aqueous vapor in the atmosphere: (a) Direct cooling, whether by contact with cold bodies or by radia- tion. (/>) Adiabatic expansion, or at least expansion with insufficient addi- tion of heat. (c) Mixture of masses of air of different temperatures. In a corresponding manner the dissolution of fog and cloud already present may take place through the following processes: (a) Direct warming, either by radiation or by contact with warmer bodies. (b) Compression, whether adiabatic or at least with an insufficient abstraction of heat. (c) Mixture with other masses of air having sufficient temperature and moisture. Of these three different processes the one first mentioned is always the most effective. In order to condense or dissolve a given quantity of water there need be only a relatively slight direct cooling or warming. When the con- densation or dissolution of a certain quantity is to be accomplished by adiabatic expansion or compression the cooling or warming must be greater, that is to say, must cover a wider range of temperature than for direct cooling or warmiug. Still larger temperature differences must come into play when the same quantity is to be condensed or evaporated by the process of mix- ture, in so far as this is any way possible. The first pair of these processes, namely, the direct cooling or direct warmiug, comes especially into consideration in the formation of fog proper, which beginning at the earth's surface, extends upwards to greater or less altitudes. At times of excessive radiation the earth's surface first cools. When the cooling has reached the dew point there occurs condensation in the very lowest layer. Hereby the emissivity of this layer itself is increased. It then cools in its upper portion also by radiation, and thus the layer of fog grows upwards more aud more until subsequently, at the time of increased inflow of heat, it dissolves itself in a precisely inverse manner. No other considerable precipitation is formed by this method of con- densation except the so-called drizzle. The reason of this undoubtedly is that the growth of the layer of fog upwards removes the possibility of further more intense radiation by the lower stratum. In the higher strata of the atmosphere such condensation by direct radiation can certainly only occur when cloudiness has already been produced in some other way, whether by mixture or by expansion or possibly by smoke. 284 THE MECHANICS OF THE EARTH'S ATMOSPHERE. At the upper limit of the cloud, especially in stratus clouds, the processes of growth and dissolution of the cloud by direct loss or gain of heat by radiation are carried on like the formation and dissolution of fog in the lowest strata of air. The formation of clouds by adiabatic expansion as well as the disso- lution by compression occurs wherever we have to do with ascending or descending currents of air. This process has in recent times been so frequently treated that the subject may here be treated very briefly. The cumulus clouds of summer with horizontal bases, the thunder cloud and the rain cloud, properly so called, owe their origin to this process. To what extent "nocturnal radiation " influences the upper layers of such clouds can only be made clear by further investigation. Still more complicated than the two methods hitherto considered in the formation and dissolution of clouds and fog are the processes that accompany mixture. In both the above mentioned pairs of processes a steady increase of cooling or warming is accompanied by a steadily progressive condensation or dissolution. It is quite otherwise in mix- tures. A process of mixture can progress in the same direction and yet cause at first condensation and in its subsequent stages dissolution. The breath which we exhale into the cool air leaves the mouth satu- rated but not yet in the condition of fog ; only after the beginning of the mixing with the colder air does the formation of the cloud of vapor begin, which then through further mixture with colder, drier air, again dissolves. We see this process depicted in a strictly mathematical way in Fig. 38. Jf for instance we assume that a small quantity of air at the temperature t{ is mixed with a larger quantity at the higher temperature % AM t2, then all possible mixing-ratios will occur trom — =0 up to the final result, which we will assume to be greater than that which corresponds to the higher value y2*. In this case the quantity of contained water y passes through all values belonging to the ordinates of the line F{F2 until reaching the final value y>y2*. In this process condensation must occur as soon as the mixing-ratio exceeds the value which corresponds to the ordinate yx*; if it increases still further then beyond a definite point as it approaches towards the ordinate y2* dissolution again begins, which becomes complete for a mixing-ratio corresponding to the ordi- nate y2* and thus again results an unsaturated mixture. If a smaller quantity of nearly saturated warmer air mixes with a larger quantity of colder air then will the mixture pass through its conditions in an inverse order, and again the initial condensation and the subsequent dissolution will occur under the conditions assumed in Fig. 38. Although now in both cases condensation occurs first and then dis- solution, still there is an important difference between them. For if we imagine the mixing-ratio to undergo steady change between the points PAPER BY PROF. BEZOLD. 285 of condensation and of dissolution, that is to say, between the ordiuates 2/i* and y2*, then will the resulting mean temperature t=^1* be attained quicker when we go from yv* towards y2* than when we go from y2* towards #,*. For since t > t3 therefore for t = £ (£i*+<2*) the mixing-ratio 1 > 1, that is to say, the mixture shows the average m2 temperature, although so far as mass is concerned the colder component is in excess. According to this, if we mix saturated cooler air with steadily increasing quantities of saturated warmer air, then the warm ing of the mixture proceeds more rapidly at first than subsequently, whereas in the reverse process cooliug proceeds more slowly at first aud then steadily faster. The quantity condeused has also a similar relation ; it also attaius its maximum wheu there is an excess of the cooler component. " Therefore condensation begins sooner when a jet of cold moist air pen- etrates a large mass of warmer air than when a jet of warm moist air is blown into cooler air." Therefore by the outward appearances of clouds that are forming and dissolving in this manner, one perceives whether warmer or colder ail1 predominates. From all the preceding we conclude that the following forms of fog and clouds may be considered as originating by mixture : (1) The fog above warm moist surfaces, under the influence of colder air, therefore especially the fog over the sea iu the cold season of the year or during the occurrence of cold winds. (2) The "rank and file1' clouds occurring on the boundary between two different strata of air flowing rapidly above each other, which von Helmholtz* has first recognized as a consequence of wave motion and designated by the name, atmospheric billows, in which however adia- batic condensation also comes into consideration at places where the air is thrown upward after the manner of the formation of crests and foam on ocean waves. (3) The layers of stratus that also form at such separating surfaces and which frequently first appear as atmospheric billows aud subse- quently become denser. (4) Cloud streamers that form and again dissolve at the summits of mountains or in narrow mountain p sses'when the form of the moun- tain is such as to make it possible lor jets of warmer or colder masses of air to penetrate into similar masses of other temperatures.t (5) The ragged clouds, or the disconnected clouds, such as one fre- quently observes during rapid motions of the air, perpetually changing * Sitzungsberickte, Konig. Preus. Akad. Wisstnsch . sn Berlin : Berlin, 1888, p. GG1, ami 1889, p. 503. [8re also Nos. VI and VII of this collection.] t Von Bezohl, Himmel und Erde, 1889, vol. II, u. 7, 286 THE MECHANICS OF THE EARTH'S ATMOSPHERE. their form and appearing ami disappearing, and such as also occur with clouds formed by adiabatic expansion, especially during thunder storms. These different methods of cloud formation by direct cooling, by adia- batic expansion, and by mixture can of course also occur side by side in the most varied combinations, as is expressed in the extraordinary diversity of cloud forms. It seems to me very important in the study of these forms to keep these different processes in view, since only then can we hope finally to attain a thorough knowledge of these forms. Above all, as Hellmann has appropriately expressed it, it is necessary to lay the foundation for a "physiology of the clouds" before we can hope to attain to a truly satisfactory arrangement and nomenclature.* But further work will still be necessary before this problem is solved, since on the one hand the question becomes more complicated the nearer we approach to it, and siuce on the other hand it appears so extraordinarily difficult to realize experimentally even approximately those conditions under which the formation and dissolution of clouds take place in the atmosphere. Beautiful and praiseworthy as are the experiments that Vet tin has made with clouds of smoke, still we must be very careful about the con- clusions which we would draw from them as to the formation of the real clouds. All experiments with smoke, when looked at properly, give only pictures of the movements in dry air, since the condensation and evaporation as well as the processes of compression and expansion are excluded, and we therefore are working under conditions such that in the real atmosphere no formation of clouds would occur. But it is precisely because of these processes (condensation, evapora- tion, compression, and expansion) that we can not consider the motion of a cloud as a measure of the motion of the air, for not only do clouds hang apparently motionless on the mountains, whereas in fact strong winds are streaming through them (e. g. Fohn cloud-bank, the Table- cloth of the Table mountain, the Cloud-cap of the Helm-wind) but it even happens to aeronauts that they pass through clouds while moving in a horizontal direction. This latter is however only possible when the cloud has a velocity different from that of the air in which it floats, since the balloon itself has only the power of vertical motion. The cloud is in fact not a body that can be driven forward as such by the air unchanged, but is a form in a process of continuous forma- tion and disappearance, and can have as a whole motions entirely dif- ferent from those of the particles of which it consists. On account of the increased interest with which at the present time we are studying the forms and motions of the clouds, it seemed to me important to call attention to all these points sin«e we must have these in mind when we attempt from the external appearance of the clouds to draw any conclusion as to the processes which in individual cases de- termine their growth or dissolution and therefore also their form. Compare also 0. Volger in Gaea, 1890, vol. n, pp. 65-75. PAPER BY PROF. BEZOLD APPENDIX. 287 Table giving the quantity of water in grams that is contained as vapor in a kilogram of saturated air. t b=760"»" 6=700°"" 6=600""" 6=500""" 6=400mm 6=300""" &=200"m" -30 0 31 0.34 0.39 0.48 0.60 0.80 1.20 29 .34 .37 . V.l .52 .65 .87 1. 31 28 .38 .41 .48 57 .71 .95 1.43 -27 0.41 0.45 0.52 0.63 0.78 1.04 1.56 26 .45 .19 .57 .69 .86 1.14 1.71 25 .49 .54 .63 .75 .94 1.25 1.88 24 .54 .59 .C9 .82 1.03 1 37 2. 06 23 .59 .65 .75 .90 1.13 1 50 2.25 -22 0.65 0.71 0.82 0.99 1.23 1.63 2.46 1 21 .71 .77 .90 1.08 1.34 1.78 2.69 ! 20 .77 .81 .98 1.18 1.46 1.94 2.94 1 19 .84 .92 1.07 1.28 1.60 2.12 3.21 18 .92 1.00 1.16 1.39 1.74 2.32 ; 3.50 -17 1.00 1.00 1.26 1.52 1.90 2.53 3.81 16 1.09 1. 18 1.37 1.65 2.07 2.75 4.14 15 1.19 1.28 1.49 1.79 2.24 2.99 4.49 14 1.28 1.39 1.62 1.94 2.43 3.24 | 4.87 13 1.39 1.51 1.76 2.11 2.64 3.52 5.28 —12 1.50 1. G4 1.90 2.29 »2.86 3.82 5. 73 11 1.63 1.77 2. 06 2.48 3.10 4.13 6.20 10 1.76 1.91 2.2} 2. G8 3 35 4.47 6.72 i 9 1.91 2.07 J. 41 2.90 3.62 4.84 7.26 8 2.06 2.24 2.61 3.13 3.92 5.23 7.85 -7 2.23 2.42 2.82 3.38 4.24 5.65 8.49 6 2.40 2.61 3.04 3.65 4.58 0.10 9.16 5 2.59 2.81 3.28 3.94 4.94 G. 58 9.88 4 2.79 3.03 3.54 4.25 5.32 7.09 10.66 3 3.01 3.27 3.81 4.58 5.72 7.64 11.49 ._o 3.24 3.52 4.10 4.93 6.16 8.23 12. 37 -1 3.48 3.78 4.42 5.30 6.63 8.85 13.32 0 3.75 4.07 4.75 5.71 7.13 9.52 14.33 +1 4.03 4.37 5.10 6.13 7.67 10.24 2 +3 4.32 4.64 4.70 5.04 5.48 5.88 6. 58 8.24 11.00 7.07 8.85 11.81 4 4.98 5.41 6.31 7.58 9.49 12.68 5 5.34 5.80 6.77 8.13 10.18 13.60 6 5.71 6.22 7.26 8.72 10.91 _ t 6.13 6.66 7.77 9.34 11.69 +8 6.56 7.13 8.32 9.99 12. 52 9 7.02 7.63 8.91 10.70 13.40 10 7.51 8.16 9.53 11.44 14.33 11 8.03 8.72 10.18 12.24 15.32 12 8.58 9.32 10.88 13.08 16.38 +13 9.16 9.95 11.62 13.97 17.50 14 9.78 10.62 12.41 14.91 18.69 15 10. 43 11.34 13.24 15.91 19.94 I 16 11.13 12.09 14.12 16.97 17 11.86 12.89 15.05 18.10 288 THE MECHANICS OF THE EARTH S ATMOSPHERE. Table giving the quantity of water in grams that is contained as vapor in a kilogram of saturated air —Con tin tied . t + 18 19 20 21 22 +23 24 25 26 27 +28 29 30 6=760"""' 12.64 13. -40 14 33 15.25 16.22 17.24 18.32 19.47 20.68 21.95 23. 29 24.70 26.18 6 = 700""" 13.73 14. 62 15.57 16 57 17.63 18.75 19.93 21.17 22.48 23.80 25.31 26 84 28.47 6 = 600""" 16 01 i7. 09 18 20 19.37 20. 59 21.90 2::. 28 24.73 6=500""" 19.29 20.55 21.88 6=100""" 6 = 300""" 6=200""" In computing this table the vapor tensions of aqueous vapor have been adopted as given by Broch, Travaux et Memoires, Bur. Internat. des Poids et Mesures, 1881, tome I. XVIII. ON VIBRATIONS OF AN ATMOSPHERE.* By Lord Rayleigh. In order to introduce greater precision into our ideas respecting the behavior of the earth's atmosphere, it seems advisable to solve any problems that may present themselves, even though the search for sim- plicity may lead us to stray rather far from the actual question. It is proposed here to consider the case of an atmosphere composed of gas which obeys Boyle's law, viz, such that the pressure is always propor- tional to the density. And in the first instance we shall neglect the curvature and rotation of the earth, supposing that the strata of equal density are parallel planes perpendicular to the direction in which gravity acts. If p, a be the equilibrium pressure and density at the height z, then I--* "> and by Boyle's law, p=a2ff, (2) where a is the velocity of sound. Hence adz a and (3) G=a,e a\ (4) where a0 is the density at 2=0. According to this law, as is well known, there is no limit to the height of the atmosphere. Before proceeding further, let us pause for a moment to consider how the density at various heights would be affected by a small change of temperature, altering a for a', the whole quantity of air and there- *From the London, Edinburgh and Dublin Phil Mag., Feb., 1890, fifth series, vol. xxix, pp. 173-180. 80 A 19 %9 290 THE MECHANICS OF THE EARTH'S ATMOSPHERE. fore the pressure p0 at the surface remaining unchanged. If the dashes relate to the second state of things, we have — gz ~9± —gz —gz V =p0e a'\ p'—po <' °'\ while a2 Ga~an 6'Q. If an _ a2 _ ^ we may write approximately p'—p 8 a2 qz zJF p0 a1 a2 The alteration of pressure vanishes when s=0, and also when z=a2, p is negative when z — 0, and becomes + go when z = go. The transition p = 0occurs when «L = 1, that is, at the same place where a2 p' — p reaches a maximum. In considering the small vibrations, the component velocities at any point are denoted by u, v, w, the original density g becomes (ff -j- ffp), and the increment of pressure is dp. On neglecting the squares of small quantities the equation of continuity is dp , du , dv die da V or by (3) ffdt+Gd-*+%+ffdz + Wdz==(J dp du dv die 0w_n .p.. di+~dx+dlj+~dz ~ ¥~" (5) The dynamical equations are ddp du ddp dv ddp dw dx = ~ffv -d^=-ffrv -&=-'*?-*-&'> or by (3) since 6p = a?ffp, 2dp_ du Ap dv .dp dw ,„, *T*—W Ty=-W dz=~~dt ' ' ' ' () PAPER BY LORD RAYLEIGH. 291 We will consider first the case of one dimension, where u, v vanish, while p, w are functions of z aud t only. From (5) and (G), dp dw_gw_ dt+dz a2~^ (i) or by elimination of p, a2' dt2 dz2, a2 dz The right-hand member of (9) may be written dp_ dw. a dz~ St' (8) 1 d2w d2w q dw ,_, (9) 0 \dz~2azJ 4a4 w' and in this the latter term may be neglected when the variation of w with respect to z is not too slow. If X be of the nature of the wave- (llll 74) length, — is comparable with — ; and the simplification is justifiable when a2 is large in comparison with gX, that is when the velocity of sound is great in comparison with that of gravity- waves (as upon water) of wave length A. The equation then becomes d2w -a*( d - ^Yw S#-a\Jz 2tf)W' or, if w --= We* , (10) d2W_ 2d2W. (U) the ordinary equation of sound in a uniform medium. Waves of the kind contemplated are therefore propagated without change of type except for the effect of the exponential factor in (10), indicating the increase of motion as the waves pass upwards. This increase is necessary in order that the same amount of energy may be conveyed in spite of the growing attenuation of the medium. In fact w2 (13) then the solution of (12) is w = Aemiz + Be™** , (14) A and B denoting arbitrary constants in which the factor eint may be supposed to be included. The case already considered corresponds to the neglect of g2 in the radical of (13), so that _#±2 nai and W '"■- 2a> 111 in(t+^j, |L*l(|-i) (15) we a '• -Ae x a'+Be A wave propagated upwards is thus w=e coswf t ) . . . . (16) and there is nothing of the nature of reflection from the upper atmos- phere. A stationery wave would be of type w=e cos w£ sin — (17) a w being supposed to vanish with z. According to (17), the energy of vibration is the same in every wave length, uot diminishing with ele- vation. The viscosity of the rarefied air in the upper regions would suffice to put a stop to such a motion, which can not therefore be taken to represent anything that could actually happen. When 2 na<.g, the values of m from (13) are real, and are both posi- tive. We will suppose that m, is greater than m2. If w vanish with z, we luwe from (14) as the expression of the stationary vibration e —e Ji (18) which shows that w is of one sign throughout. Again by (8) a2p=n sin nt \ e_ 1_ > (10) ( »»i w?2 ) Hence J*, proportional to w, is of one sign throughout; p itself is negative for small values of 2, and positive for large values, vanishing once when e<-»-*'=«i (20) m2 PAPER BY LORD RAYLEIGH. 293 When n is small we have approximately 5 9 m2= - 0 so that p vanishes when gz or by (4) when n2d2 (23) Below the point determined by (23) the variation of density is of one sign and above it of the contrary sigu. The integrated variation of ,.CO density, represented by | a p dz, vanishes, as of course it should do. « J i> It may be of interest to give a numerical example of (23). Let us In suppose that the period is one hour, so that in c. G. s. measure n= -r-^-r-- 3600 We take «=33x 104, #=981. Then an equation which may be solved in the same form as (12). One obvious solution of (27) is of importance. If ^ =0,so thatw=0, the equations are satisfied by tf^tfd2 (28) 294 THE MECHANICS OF THE EARTH'S ATMOSPHERE. Every horizontal stratum moves alike, and tbe proportional variation of density (p) is the same at all levels. Tbe possibility of such a motion is evident beforehand, since on account of the assumption of Boyle's law the velocity of sound is the same throughout. In the application to meteorology, the shortness of the more import- ant periods of the vertical motion suggests that an "equilibrium theory" of this motion may be adequate. For vibrations like those of (28) there is no difficulty in taking account of the earth's curvature. For the motion is that of a simple spherical sheet of air, considered in my book upon the " Theory of Sound," § .'J33. If r be the radius of the earth, the equation determining the frequency of the vibration corre- sponding to the harmonic of order h is n2r2=h (h+l) a2 (29) n the actual frequency being -9— . If r be the period, we have _ 2?rr ~aVh {h+l) (30) For h=l, corresponding to a swaying of the atmosphere from one side of the earth to the opposite, 2nr r'=^' <31> and in like manner for h=z2. 2 nr ri T'= -T*~7l (32) To reduce these results to numbers we may take for the earth's quadrant _ 7rr=10e centimeters; and if we take for a the velocity of sound at 0° as ordinarily observed, or as calculated upon Laplace's theory, viz, 33 x 103 cent'meter.% we shall find 4xK>9 ri= 777TT q.» w nn seconds=23.8 hours on the same basis, t2=13.7 hours. It must however be remarked that the suitability of this value of a is very doubtful, and that the suppositions of the present paper are inconsistent with the use of Laplace's correction to Newton's theory of sound propagation. In a more elaborate treatment a difficult question would present itself as to whether the heat and cold developed during atmospheric vibrations could be supposed to remain undissipated. It PAPER BY LoRb rAyleigM 295 is evidently one thing to make this supposition for sonorous vibrations* and another for vibrations of about 24 hours period. If the dissipation were neither very rapid nor very slow in comparison with diurnal changes (and the latter alternative at least seems improbable), the vibra- tions would be subject to the damping action discussed by Stokes.* In any case the near approach of rx to 24 hours, and of r2 to 12 hours, may well be very important. Beforehand the diurnal variation of the barometer would have been expected to have been much more con- spicuous than the semidiurnal. The relative magnitude of the latter, as observed at most parts of the earth's surface, is still a mystery, all the attempted explanations being illusory. It is difficult to see how the operative forces can be mainly semidiurnal in character; and if the effect is so, the readiest explanation would be in a near coincidence between the natural period and 12 hours. According to this view the semidiurnal barometric movement should be the same at the sea level all round the earth, varying (at the equinoxes) merely as the square of the cosine of the latitude, except in consequence of local disturbances due to want of uniformity in the condition of the earth's surface. Terling Place, Witham, Bee, 3889. *Phil. Mag., 1851 (4), vol, I, p. 305. Also, Raylfeigh; "Theory of Sound/' vS 247. XIX. ON THE VIBRATIONS OF AN ATMOSPHERE PERIODICALLY HEATED.* By Max Margules. The computation of the variations of pressure iu the atmosphere arising from periodic changes in temperature has a certain interest in connection with a problem of meteorology that, like all dynamic prob- lems in this field, necessitates very extensive computations. The daily variation of the barometer, freed from all non-periodic influences, can be represented very satisfactorily by the super-position of two waves, one of which has a whole day as its period; the other has the half day. The diurnal wave is undoubtedly an effect of the varia- tion of temperature. It appears much stronger on clear days than on cloudy days; it is very slight at sea and shows on the land notable inequalities. The semi diurnal wave is on the o" her hand of a regular- ity that is uncommon in meteorological phenomena. At places of the same latitude it is of very nearly equal amplitude and of the same phase iu reference to the local time. If we consider this wave also as a consequence of the variations of temperature, then the connection seems to be obscure. The mean daily temperature represented for any place by a curve, can like every such curve, be analyzed into a series of waves of twenty- four, twelve, eight, and six hour periods. Does the twenty-four-hour wave of pressure originate from the corresponding wave of tempera- ture?- Does the twelve-hour variation of pressure depend on the twelve-hour temperature variation? Why is the amplitude of the twelve-hour pressure term so large in comparison with the twenty-four- hour term, whereas the reverse is true for the temperature? Whence come the regularity of the one and the local variations of the other? These questions have been asked repeatedly. In a memoir recently published,! Hann has given the most comprehensive and thorough de. scriptiou of the daily oscillation of the barometer, utilizing the rich * Translated from the Sitzungsberiohle dtr Koniglich Akademie der fVissenschaflen zu Wien (Math.), 1890, vol. xcix, pp. 204-227. See, also, Exner's Bepertorium der Phgsik. 1890, Band xxvi, pp. 613-633. + " Unters. u. d. tagliche Oscillation d. Barometers," Vienna Denk., vol. 55, 1889. 296 PAPER BY MAX MARGULES. 297 observational material from all lauds and oceans with the object of establishing a basis for a further matbematico-physical theory. In order to attain this, one must first treat the phenomenon under assumptions that simplify the labor. I believed that some computations as to the variations of pressure in air that is periodically heated would contribute to the better understanding of the diurnal variation of the barometer. In the course of the work, it appeared that the computa- tion must not be confiued to the si.nplest cases if one would make it useful to a certain degree. For this reason the investigation has grown to a larger size than was desired by me. Before giving the detailed computations let the review of certain results take precedence. Let T0 and p0 indicate the absolute tempera- ture and the pressure of the air when at rest; T0'(l + r) and p0 (l + £) indicate temperature and pressure of air in motion. When r is given as a periodic function of the time t and of the locality x then e will also appear as such a function. Let a wave of temperature with constant amplitude move in the direction— x in a plane layer of air upon which no other forces are acting. This will produce a wave of pressure L2 „•_ 9 /' t , x where c indicates the velocity of propagation of a free vibration when the process is strictly isothermal; in air at the temperature 273° we have c=280 metres per second. If we assume the length of the wave to equal the circumference of the equator then for a period whose duration is one day and for a pres- sure p0 expressed as 760 millimetres of the barometer a variation of temperature of one degree will produce a variation of pressure of 4.4 millimetres. Both temperature and pressure vibrations have the same phases when their velocity of propagation fv—^ J is greater than c, but op- posite phases when it is smaller than c. If V=c then will e be indefi- nitely large, as must occur in the case of a frictionless medium when the forced vibrations have the same period as the free. Again, let a wave of temperature similar to the preceding advance in a plane stratum of air, subject to the influence of constant gravity. The air now moves horizontally in the direction of the progress of the wave and also vertically. The pressure wave on the ground is given by an equation similar to the preceding only in the numerator c2 &2 is to be substituted for JA For the equator, the day and 760 millimetres, a 298 Tin: mechanics of the earth's atmosphere. temperature variation of oue degree gives a pressure variation of 1.3 millimetres. If however tbe amplitude of the temperature variation is not uniform throughout the whole height, but dimiuishes with the height so that it diminishes by one-half for each ascent of 1,000 metres, then a tem- perature variation often degrees at the earth's surface gives a varia- tion of pressure at the same level of only 2.4 millimetres. In respect to tbe whole-day wave for continental tropical regions one could be satisfied with this result. The agreement, however, is only accidental. Tbe twelve-hour wave of pressure at sea still remains entirely inexplicable. Even on the land one should expect that the amplitudes of the whole-day and half-day waves of pressure would have the same ratio as the corresponding temperature amplitudes, since s remains unchanged when we put £ L and £ 0 in place of L and ©. Tbe computation would hold good for a cylinder of great diameter equally as for a plane ; even under certain restrictions it would also hold good for a mass of air within a circular boundary. But it can only be applied to the atmosphere when the air is divided into a num- ber of zones by vertical walls parallel to the circles of latitude. The zones in the neighborhood of the latitude of 50° would have enormous variations of pressure, and there also two neighboring zones would have opposite phases; the amplitudes diminish thence toward both the pole and tbe equator. From the great differences in pressure that are thus obtained for dif- ferent zones, we see the necessity of reducing to calculation the condi- tion of the air over the whole sphere without any partition walls. I pass over tbe formulas for the sphere at rest in order to report upon that part of the computation that apparently offers useful results for the elucidation of the half-day wave of pressure. First, I will present some passages quoted already by Hann from a memoir of Sir William Thomson's. After speaking of tbe disproportion between the whole and half day variation of the temperature on one hand and the pressure on the other, Thomson says:* " We must consider the atmosphere as a whole and investigate its vibrations with the help of the formulas that Laplace has developed for the ocean in the Mecanique Celeste, and which, as he has shown, are also applicable to the atmosphere. When in the calculation of the tide-producing force, one introduces the influence of temperature instead of attraction, and develops the oscillations corresponding to the whole day and half day terms of the temperature curve, it will proba- bly be found that in the first case the period of the free oscillations de- parts more from twenty-four hours than in the second case from twelve hours, wherefore for a relatively small amount of tide-producing force, " "On the thermo-dynamic acceleration of the earth's rotation," Proc. E. S. Edin- burgh, 1882, vol. xi. Sir William Thomson. "Mathematical and Physical Papers," Loudon, 1890, vol, in, page 344. tAPER BY MAX MARGULES. •_".!! I the amplitude of the half day term will be greater than that of the whole-day term." This prediction is completely verified. When we execute the compu- tation for an atmosphere considered as a rotating spherical shell in which waves of temperature advance from meridian to meridian ac- cording to the equation t=C sin go sin (nt-\- A) (where go = Polar distance, A = geographical longitude, n = velocity of the rotation of the earth), then we find for T0=273° the wave of pressure €—C sin (nt+\) [1.146 sin a?— 0.423 sin3 go— 0.370 sin5 go— 0.106 sin7 go -0.018 sin9 &3-0.002 sin11 go- ... ] When however at every place the wave of temperature repeats itself twice daily and we assume t = G sin2 go sin (2nt+2\) then there results €= — C sin (2nt+2\) [37.99 sin4 05+23.06 sin6 go +5.75 sin8 cy+0.81 sin1" gj+0.07 sin12 <»+...] The law according to which the amplitude of the temperature wave diminishes from the equator toward the pole has been assumed differ- ent in the two cases only because of the easier computation; this how- ever is of slight influence in the general result which is, that for equal variations of temperature the resulting variations of pressure become much greater in the double daily wave than in the single wave. The coefficients of the first sine series vary only very slowly with T0 (or with n when we, as Thomson does, consider the period as the variable). It is otherwise in the half-day wave; here the factor of sin4 go in the neighborhood of T0=268°passes, from — oo over to + co precisely as in the plane wave before considered when the ve- locity of propagation of the forced vibration is made equal to that of the free vibration. Thus slight semi-diurnal waves of temperature of scarcely appreciable amplitude are sufficient to produce great waves of pressure in frictionless air if we assume the temperature of the spher- ical shell to be iu the neighborhood of 268°. Thus far the computation. Its application to the daily variation of the barometer is only clear as to one point. The semi-diurnal wave of pressure may be considered as a consequence of a semi-diurnal wave of temperature of small amplitude. Thus is explained the relative magni- tudes (of the diurnal and semi-diurnal temperature and pressure waves) but not the uniformity of the semi-diurnal variation of pressure over the land and the ocean. This uniformity has led Hann to seek the ori- gin of the phenomenon in the absorption of heat by the upper strata of air. But the lower strata have also a semi-diurnal temperature va- 300 THE MECHANICS OF THE EARTH'S ATMOSPHERE. riation and one that varies with locality and with the condition as to cloudiness. It is a question whether the variations of pressure thence resulting are so small iu comparison with the regular variations that they are not very noticeable iu the averages. The neglect of the friction and the vertical motion of the air in our last calculations, the assumption of a constant mean temperature for the whole mass of air, and the assumption that for equal latitudes we have equally large ranges of temperature and pressure, allow us to make only the most general application to the case of nature. A more perfect calculatiou, taking account of the actual distribution of laud and water, would be as difficult to execute as would be the computa- tion of the rise and fall of the tides for an ocean of irregulir shape, or even for one bouuded by meridians. I. MOVEMENT OF THE AIR IN VERTICAL PLANES. Notation, u = horizontal velocity along the axis of x ; w = vertical velocity positive upwards along the axis of z; ju= density; p = pressure T = absolute temperature ; t = time ; g = the acceleration of gravity B = a constant. We imagine the earth as an infinite plane above which, iu alJ east west vertical planes, the movement of the air occurs in a similar manner For slight velocities that allow us to neglect terms in the equations of motiou that are of the second degree in u and w, these equations, to gether with the equations of continuity and of elasticity are as follows : * du _ _ 1 dp dt fX clX dw _ 1 dp dt '' '' ix dz ) (1) dt + dx- + dz p = B pi T. If the atmosphere is at rest thenp, ju, T, have the value p0, /^0, T01 which are functions of the altitude only, 1 dp0 g p0 dz ' ' BT0 1 dfA0 _g_ l^aZT, > (2) Mo dz - BT0 ~ T0 dz p0 = B /u0 TQ [* The expression "Zustands-Gleichung der Oase," which is applied in Germany to the equation p v = RT has, I believe, no single equivalent in ordinary English sci- entific phraseology unless we adopt the very inelegant historical title Bojle-Mari- otte-Gaylussac- Charles-Law. It is the law connecting density, temperature, vol- ume, or pressure, and expresses the simple fact that the substance is truly gaseous. But the characteristic of a gas is its elasticity, and the equation gives the elastic pressure. — C. A.] PAPER BY MAX MARGULES. 301 The motion is caused by small variations of temperature. Such vari- ations will, as a rule, produce only slight variations of density and of pressure. If we put V = Po{\ + 0 M = /'«. (1 + ff) T = T<> (1 + t) then e, RT0jz RT0 dt% I (4) g_dt 9(9 .1 *To\r- *-ft| " .BT0 d* RT0\RTo T0 dG = G> To dr " RTo d€ where the change of kinetic energy is omitted, as being a quantity of 302 THE MECHANICS OF THE EARTH'S ATMOSPHERE. the second degree, in u ; dQ= the heat imparted to the unit mass of air during the time dt ; C, = specific heat of air under constant vol- ume; Cp = specific heat under constant pressure CP=CV+R Qv T0 Gv G„ T0 Gp By combining this last equation with (4a) we obtain *l-BT.°*.*=-±-™ («) Jt> Cv dx2 GVT0 dt2 which converts into the Laplacian equation when Q = 0. In this the temperature variations of the air for rapid acoustic vibrations produced by adiabatic compressions and expansions are considered, and the velocity of propagation is therefore '-JbT.% For our purpose it will be more convenient to consider the pressure variations as a consequence of the temperature variation snot as aeon- sequence of the variable flow of heat. We therefore return to equation (4a). III. WAVE OF TEMPERATURE. A progressive wave of temperature t — A sin (nt + nix) =A sin 2tt (*+?£\ .... (5) causes a wave of pressure s = B sin (nt -f mx) ~) p- if ± I- (6) advancing in the same direction. = V is the velocity of the progress of both of these waves. The vy phases of the waves are the same or opposite according as V is larger or smaller than c. But V=e leads to an infinitely large value of B, a result to which we must always come when in a frictionless medium the period of the forced vibrations agrees with those of the free. For the atmosphere we have l?__10333x9.80C 9Q7n jK-T73xh29r=287-°' PAPER BY MAX MARGULES. 303 Here, and in the following, we adopt as units the metre, the kilogram, the second of time, the degree of the Centigrade thermometer, and for pressures the barometric scale. For T0=273° we have c=279.9. The values 7/=4xl07 or the circumference of the equator, @= 24x60x60 or 1 day and T0=273° gives a wave of pressure whose max- imum coincides with the maximum of temperature, and also gives 5=1.576 x A. A temperature variation of 1° C. produces a pressure p x 1.576 variation ° " — or 4.4 millimetres of mercury when p0 is 760 on the barometer scale. When we desire to obtain pure horizontal vibrations in a layer of appreciable altitude without neglecting force of gravity we should have to introduce a function (A) of the altitude as we see from the equations (3), tliat shall satisfy the condition ldi g Lz - c2 02 A dz c2 I? For isothermal vibrations in a vertical column of air the conditions are r = 0 ¥=<> and equation (4) becomes f £ g d « l f * d z* ~ RT0 d z ~ BT0 d t2 = 0. This equation or the corresponding equation in w has recently been discussed at length by Lord Rayleigh (Phil. Mag., Feb., 1890).* IV. VIBRATIONS OF THE AIR WHEN A WAVE OF TEMPERATURE AD- VANCES HORIZONTALLY, TAKING INTO CONSIDERATION THE FORCE OF GRAVITY. With a constant value of T0 and putting r = A sin (mx + nt) the dif- ferential equation (4) becomes t¥+Tt>-arz-g-Ji>--gJt>-aT ' ' ' ' (4C) [" = wJ The wave of pressure will be of the form e = F (z) sin (mx + nt). * [See also Nor XVIII of this present collection of Translations.] 304 THE MECHANICS OF THE EARTH'S ATMOSPHERE. The notation and solution are as follows : d2F_(dF dz2 dz hF=( an2- a2 \ A h=(*n2 — ni2 L g J F (z) ^B+K^+K^* A/IX,^ 2 B=^ ( -n2 — a h\9 ^-~a— A*2 7, fr>— a4- la2 // In order to determine the constants of integration Ki and K2 whose factors in the expression for s represent free vibrations we note that ic=0 when z=0 and also when z has a very large value=Z which cor- responds to a fictitious upper plane bounding the atmosphere. From the second of equations (3) we obtain w = -g— (Kx~kxekiZ + K2lc2ek2Z — aA) cos (mx+nt) an The bound ar y conditions give Rik1-\-K2h=cxA KlkieklZ+K1k2ek*z=aA KJn = aA K2h=(vA ek"-z— I \-e^z If now, as in our example (where the wave length is the circumfer- ence of the earth and the period is one day), h is very small compared with a2, then is 1c very small, aud Tc2 nearly equal to a. Hence, K2 will be smaller in proportion as Z is larger. If we desire to apply the re- sulting formula only to altitudes that are slight in comparison with Z, then will K2ek*. With this limitation we put K2=0 and K1fc1=aA, and obtain w=Al(eklZ— 1) cos (mx+nt) PAPER BY MAX MARGULES. 305 Under the assumption that —= is a small quantity we have or h . h2 a a2 k~h 1' and when we retain only the first two terms of the exponential series we obtain e=AQK +az ) siU (>™+nt)=A (l^i+«) sin 27r (1+Xl)' For J>=4xl07, 0=24x00x60, we obtain e=A (0.576+0.000125*) sin (mx+nt). The relative variations of pressure near the earth's surface increase very slowly with the altitude. At the surface of the earth itself the variations of pressure are appreciably smaller in the ratio of i:-^? than in the example of the third section, where purely horizontal vibrations occurred. A daily variation of temperature of 1° C. would in the present case cause a pressure variation of 1.6mm. The phases of both vibrations occur simultaneously when L > c@. V. A SIMILAR COMPUTATION FOR THE CASE WHEN THE AMPLITUDE OF THE TEMPERATURE VIBRATION DIMINISHES WITH THE ALTITUDE. The differential equation (4) becomes *? + #* ajz gjt2-a?z gd# To the assumption r = Ae~sz sin (mx + nt) there corresponds € = (Be-S2 + Kekz) sin (mx+ nt) B(s2+as + h) = A (— - a2 - as ) fe=2~V4-ft h has the same meaning as before. K stands for ffj and K, disappears under the same limitations as before, (namely, that the result is to be applied only to altitudes that are slight in comparison to Z), . 80 A 20 306 THE MECHANICS OF THE EARTH'S ATMOSPHERE. From the condition w = 0 when z = ®, there follows KJc — Bs + Aa, hence A i fa „ „ \ „ . a /'n2 € = s2 + «s + - j (~n2 — a2— as) er"+ £ ( - s + h ) e** J sin (mx + nt) If — 2 is very small, and s of the same order of magnitude as a, or even much larger, then for values of z that are not too large, this last equa- tion becomes , f a m2 , \ . , 6 = A \7+~a lb + aZ) S1D (mX + ^ A (' a cl G2 , \ n / X . t = A \j+^ U-dX dt (8) These equations are applicable to the motion on a sphere at rest. In order to investigate the relative motion on the rotating terrestrial sphere, we modify equation (7) in that we put vt + X in place of X where v is the velocity of rotatiou of the earth. In place of -j-. in equation (7) there now occurs — iff~- !£ again, we put c in place of the new r sin gt-jt, if we retain the products ra, vb, re, and if on the other hand we omit the terms in v2, which indicate only a slight change in the force of gravity, then we obtain the equations for the motion of a fluid on a rotating sphere. On the right-hand sides of the equations (8) the terms — 2vc sin co, —2rc cos go and -\-2vasiu. Go+2vb cos go are to be added respectively. The equation of continuity has the same form for the sphere at rest as for the rotating sphere. jgj. ?(^r!8ft) ,i d (t* b siD ^ I d(/*c) __0 ,9) dt r2dr r sin go doo r sin go d X .... \ > Introducing the notation p=p0(l+s), T=T„(l + r) allied to that above used, we obtain the following differential equations 308 THE MECHANICS OF THE EARTH'S ATMOSPHERE. for tbe motion of the atmosphere on the rotating sphere that result from small variations of the temperature r (10) -RT0 ^ - = C-r-2vGG08 GO _7?y — rJl£ — -=^+2fasin oj-\-2v b gosgj de_dr.{2_ g x Ja 3 (ft sin a?) ?c =Q dt dt \r R%) Jr r sin oodoo rsmcod^ If j/=0, these give the corresponding equations for the sphere at rest. VII. THE ATMOSPHERE WITHIN A SPHERICAL SHELL AT REST. As in the first computation in the second section for the case of a plane we shall assume only horizontal motions. Moreover the radius of the sphere 8 will be assumed very large in proportion to the height of the stratum of air. If in equation (10) we substitute 8 instead of r, put a=0 and v=0 and eliminate ft and c from the last three equations, there results Single daily wave. The wave of temperature r = A sin oo sin {nt + A) causes a wave of pressure e = B sin &> sin (nt + A) where A and B have the relation \BT0 —^) ~AB T0 7T With T0 = 213°, n = 04 y 60 y 60 ^ = ra,hus °f tne earth, andjp0 = 760 mm., a variation of temperature of 1° on the equator will produce a variation of pressure at the equator of 10.4 mm. B will be equally large for the spherical shell as for a plane wave of the same periodic time, when we assume the wave length for the plane to be equal to the circumference of the circle of 45° latitude on the sphere. Double daily wave. For the temperature wave r = A sin 2co sin (2 nt + 2 A) we obtain the pressure wave e = B sin 2co sin (2 nt + 2 A) PAPER BY MAX MARGULES. -J09 with the following relation between A and B R T0 V RT0 With the same constants as before 1° variation of temperature on the equator gives 6.2 mm. variation of pressure. On the occasion of the computation for the rotating sphere we shall again have opportunity to explain that the particular integrals that we, in both cases, have given as the solution of the differential equation (11) contain the complete solution for the whole spherical shell. If we put ®i for the duration of the vibration for siugle waves for which B is infinitely large, and similarity 02 for the double wave, then we have 2 n 2 n 8 © = 2/t 2ttS 2ir2-: vfur These are the values of the periods of free vibrations of a spherical shell. Lord Kayleigh (L. E. I). Phil Mag. Feb. 1890) investigates only such and finds (by putting-^/ -RT0 — ^~ for the velocity of propagation instead of V R T0) for the atmosphere on the earth at rest Qy = 23.8 hours and Q2 = 13.7 hours ; therefore the first is much nearer to 24 than the second is to 12 hours. He remarks however that it is doubtful whether one ought to adopt the Laplacian velocity of propagation for vibration of such long duration. Therefore the relative magnitudes of the semi-diurnal variation of the barometer still remains a riddle. But this is so only so long as we confine the calculations to the sphere at rest. VIII. CALCULATION FOR A ROTATING SPHERE. Diurnal wave. — In this case also the calculation will be carried out only for air in a spherical shell whose thickness is small in comparison with the radius S of the sphere, and also under the further assumption that the movements are horizontal, and that therefore a— 0. [This lat- ter assumption and the omission of the first of equations (10) are cer- tainly uot unobjectionable; they are imitated from the analogous pro- c. sses in the theory of the tides.] The difference between the sidereal day and the solar day is not considered, and v=n _^ii=^_2wc cos co 8 doo dt _BT0 d£ 30 +2nb coa co \ • • • (10a) 8 sin cod A Jt \Jt d- E sin go nS 1—4 COS2 GO These latter values substituted in the preceding equation lead to a relation between E and A only, or between e and r. It will be con- venient for the further computation to introduce an auxiliary function, -(*+!>-*°-0 1 (s+3)a,- ( Jh-+ 4+2 )«,+ ifca1 = 0 V 7/ v 5 7 y 5 ,-2 + _f_ W - ( °.k + _5_ + i - 3 ) a, i H- 2 7 l i i + 2 ' 4 + -.+ A-«,- 4 = 0 (11a) i = 5, 7, 9, . . .3 Apparently ax remains undetermined ; for the computation of the others, following the lead of Laplace, we write a, 4 k (i + 2) a,i~i 3 k (i + 2) + (i -2 ) i (i + 2) - (%- 1) i (i + 1 ) 0i By the interchange of i with t + 2 a similar expression ir. formed for ^i— and then ^±2 and in a similar manner for the subsequent terms of a{_2 at the series, and by substituting these values in the above equation we obtain a continued rapidly converging fraction. N1=3 k. 7 + 3. 5. 6, «3 = «5 ~a3 4 fc 9 'N3-Z5 #5- ^7 #7- «5= rt7 ~a5~ 4 fell " #5- ^7 #7 -^9 JVT9- • • ■ #3=3 /»•. 9+5. 7. 8, Z3=4 jfc. 4. 5. 6. 9 #5=3A\11+7.9.10, Z5=4 k. 6. 7. 8. 11, a3 If in the second of equations (11a) we pat a5=q3 a3, then will - also be determined, and the quotient has the same value as if it were com- puted from the serial fraction a3 4fc. 7 N3-& o 0 O 812 THE MECHANICS OE THE EAETIi's' ATMOSPHERE. By the first of equations (11a) we obtain also the value of ax) conse quently that of (h=q\ fli «5= Jt = 2.7352 A 1.119 1.146 fa - 0.448 - 0.423 fa - 0.326 - 0.370 fa - 0.090 -0.106 fa - 0.013 -0.018 With the value of Jc = 2.7352 we obtain as the sum of the series of sines within the [ ] in the value of e : On the equator 0.23 At latitude 30° 0.50 At latitude 45° 0.58 At latitude 60o 0.51 Therefore the variation of pressure has a maximum in the neighbor- hood of 45° when we assume the variation of temperature to be pro- portional to the cosiue of the latitude. For 20 = 273, *• The ratio — is given from the first equation, but q3, q5, etc., retain the same values as before. The secoud equation determines the value of ax. As before we have — t = C sin3 go sin (nt -|- A) ) € = G sin (nt + A) [fa sin go + fa sin3 go + fa sin5 go + . . .J J (126) For Jc = 2.7352 we have— fa = 0.601 fa = - 0.172 fa = 0.316 fa=~ 0.030 fa = - 0.566 fax = ~ 0-003 314 THE MECHANICS OF THE EARTH'S ATMOSPHERE. The sum of the series of sines in the value of s is—1 For the equator * 0.15 For latitude 30° 0.38 For latitude 45° 0.42 For latitude 60° 0.32 Again we find a minimum at the equator ; the maximum of the press- ure amplitude lies between latitudes 30° and 45°; the diminution in the higher latitudes is greater than in the previous examples, but still slow in comparison with the diminution of the temperature amplitude. Ac- cording to equations (12) aud (126) the greatest pressure and highest tem- perature occur simultaneously. IX. ROTATING SPHERE : SEMI-DIURNAL WAVE. If in the differential equations (10a), for the horizontal motions on a rotating sphere, we put t—A (go) sin (2nt+2X) s=E(go) sin (2nt+2\) b=q> (co)cos(2n«+2A) c=ip (go) sin (2nt+ 2A) there results : dE _2 cos go _RT0 d,Go sin go ™~2nS sin2 go dE 2E lilodGO siu GO "~~2nS sin2 co n282 After the elimination of cp and ip, and when we again put Jc=„7fr there remains d?E . . dE j^ sin2 oo— j^ sin gocos go+ E(4ksinico+2 sin2 co-S)=ikA(Go)smi go . (13) If we assume that A(go) = G sin2 go, we have then to do with the same problem as in the computation of the .semidiurnal tide in an ocean of constant depth. Assuming E( go)= a0 -f a-z si n2 go -f a4 sin4 &?+a6 sin6 go + there results a0=0, «2=0, a4 apparently undetermined, (4x6-8)a6-(3x4-2)fl.4-4fcC=0 ) (i2+6i)ai+i— (i24-3i)Oi+2+4ftaj=0 > .... (13a) t==4, 6, 8 . . . ) _ai+2_ 4fe ai i(i+3)-i(i+G)v William Ferrel. In this paper (supplementary to that under the same heading in vol. ix, No. 6, of the Astronomical Journal), it is proposed to explain more fully a certain point in the latter (which did not appear clear to a correspondent some time since), by presenting the matter more in detail, and also to clear up some doubts held by some with regard to theconver- gency of the series in the tidal expression. In Darwin's Equation No. (34), t we have the following differential equation to be satisfied, which is equivalent to that of Laplace : vi (l_v2)_^J __ v ^—(8— 2k2— fiv*) u+ft Ev6=0 .... (1) dV dV [Darwin's Eq. (33).] in which u is the difference between the real amplitude of the tide and that given by the equilibrium theory, r==sin 3" is the sine of the geo- graphical polar distance 3, Evz is the amplitude of the equilibrium tide, and gl n2 1 in which '- = -—— and I is the depth of the ocean, supposed to be uni- g 289 form, in terms of the earth's radius. Putting u=K2v2 + K^ + K6 v6 Kuv» (3) in which n is any even number, corresponding with the exponent, and substituting this value of u and its derivatives in (1) above, we get, by equating the coefficients of like powers of v to 0, K2=0. 12iT4— 12^4=0. 16K6+fiE=0, etc., * From Gould's Astronomical Journal, 1890, vol. x, pp. 121-125. t Encyclopedia Britannica, 9th ed. art. " Tides," § 16, vol. xxm, p. 359. u=(K--E) v'2+K4 vt+Ke, ve+ . . K2i v* . . . (34.) 31 J 320 THE MECHANICS OF THE EARTH'S ATMOSPHERE. and generally after E6. {n («_2)— 8) Kn—[(n—2){n—3)— 2] En_,+pEn^=0. From these equations we get the following expressions of Kn : KA=K, I (4) and generally, after 7T,;, This general expression is equivalant to Laplace's and Darwin's law as given in my preceding paper, equation (2), but is more simple and convenient in deducing any coefficient En from the last two preceding. The one is reducible to the other by putting n=.2i-\-i. The general law of (5) does not hold until after E6, but E4 and K6 being obtained from the direct equation of the coefficients of y4 and rG, then by means of these, Es is obtained, either directly from the equatiou of the coefficients, or from the general expression of (5), and this law can be extended forward, but not backward. For instance, EG is not obtainable from Ki and E2. As is usual in such cases, the general law is not ob- tained until after several equations of the coefficients, and when the values of En are given directly in this way, and not by the general law, the former must be taken, and the general law, which is a relation found between the coefficients after E6 only, can not be extended back. Putting h for the amplitude of the real tide, we have, from what has been stated above, h=zEv'z+u=Ev2+Eivi+E6v6 - +Envn . . (6.) Laplace extended the relation above, found to exist between the co- efficients of v in (.3), and after E6 only, back so as to make it, by means of the continued fraction, determine the value of 2f4 and so the relation between Ev2 and u. This makes K4 a determinate quantity, whereas the equation of the coefficients of v* gives ^L4=K4, an indeterminate quantity. It is evident that any value of K4 satisfies the differential equatiou, and so, with the other coefficients depending upon it, is a so- lution of the tidal equation. The extension of the general relation of (5) back so as to make it de- termine K4, and the relation between Ev1 and u in (6), was regarded by the writer in his previous paper as an extension of the law back where PAPER BY WILLIAM FERREL. 321 it is not applicable, and this is what was not clearly understood by his correspondent. From (4) it is seen that the tidal expression consists of two parts, one of which depends upon JT4, aud is independent of the tidal forces contained in E, and the latter depends upon these forces. It is evident that the former can exist without the latter. Also that being inde- pendent of the forces, and dependent simply upon certain initial motions which the sea may be supposed to have independent of the forces, it must vanish when there is friction, and so K4 must be put equal to 0 in the real case of nature. We come now to the second part of what we have proposed to con- sider here, namely, the convergency of the series in the expression of u in (3). Inasmuch as the vanishing ratio between consecutive values of Kn is unity, as is readily seen from an inspection of (5), it has been said that the device of Laplace in the use of the continued fraction was necessary to make the expression of u convergent at the equator where v = 1, so as to give a finite value of u. It is true that the expression at first is more convergent with a large value of K±, such as is given by the continued fraction, but still the vanishing ratio in any case is unity. But it can be shown that the expression gives a finite value of u when we put 2f4 = 0. We get by development, (i_v*)i=i-Av2_Av4_iLv6 .... -AnV«=i+2?AnVn:.(7). in which the relation between each coefficient An and the preceding one> commencing with — -, is •j A»=~An_2 • • • (8). Hence we have, when > = 1 2»An=-l (9). 2:,+2A»=-(l + XAn) (10). in which n' is the exponent of any assumed term in the series. The expression of (5) above may be put into the form, ^=^^+^^-(^4^)^ ■ • (11)- From this, bv means of (8), we get for any coefficient for which the characteristic is w', Kn,=^An,Fn (12). -»n'-2 80 A 21 322 THE mkciianicm OF THE BASTE'! ATMOSPHERE. in wliirh, V -( 1 I 6 "v \Ss ' (131 " V ' (n' fli)( n'-3) (/t' + 2)(///— 1 )(■«'-: :i) //r„. , and putting n'+2 tor »' In (12) vreget A.. /I A' " I /'' A "+* /I /'' a », ( i • Phis becomes i».v substituting o>i ," fc? its value derived from the pre ceding expression becomes. A „ i • A /l " 1 /'' /' in ilk*-, manner we get generally A' " /I f /' /' (I!) ' n - in which the ralues <»r the factors /''., .,„ /''„,», F„ ,„ :"<- given bj (18) by adding 2, i, and J respectively to "' in that expression. Now, all these factors are finite, and beoce putting now /r„ for its equivalent, /»',. ,, :m Therefore, we have, put I Oik « loin' i /, • /■'.. -•■■:,- '] A„,orby (10),<-^ (I i :■:■; a„ ) (17) when /; o (18) lincc this Is what (lo) becomes when I Is infinitely great. This Is simply the limiting condition ■ " nil cases, mi> <>m ( \) ■ V /'., I (K (19) r^s'i.r. ~-'i "111 ■ .. : Zl :11 ." - ; - - - in which W::h the values of P* and i . pea ». and this in (S) the am plit ode of the tide. Laplace computed the values of 24, that is. the range of the the equator, at the tines of conjunction of the non and 60% several value* -qual 1 ' which, by - 1 1 rial values of J. the depth of the ocean, equal to — - of the earth's radios, or apwioaitateiy 1.4. 5-5, and U ■ i, at Inking as an example the ease in wi " by patting K = z. z -.-. Inst eofatmn of the following table, and : ing valnes of JLT in the second column. :=1 " - . ►f i,in - I ■ X. - ( umrnt - B9 1 H - ■ Hi .9z~ .- . . ^B ■ Cfl- ". .:: < . ■ . i. m .MSI a : - .' ■ 48 : A 1 «0 j. : "- „