~ i Leys u abe RD i } 0 ; me ae pS LY -_ he fo Paap HO 3 1924 003 912 874 Cornell Aniversity Library BNA RBOB occ cde cr UNITED STATES COAST SURVEY. CARLILE P. PATTERSON, SuPERINTENDENT. METHODS, DISCUSSIONS, AND RESULTS. . ~ a“ FOR THE USE OF THE COAST PILOT. . PART I. °° > ¢€ S ss | WASHINGTON: GOVERNMENT PRINTING OFFICE. 1877. METEOROLOGICAL RESEARCHES UNITED STATES COAST SURVEY. | CARLILE P. PATTERSON, SUPERINTENDENT. METHODS, DISCUSSIONS, AND RESULTS. METEOROLOGICAL RESEARCHES FOR THE USE OF THE COAST PILOT. PAH YT 1. WASHINGTON: GOVERNMENT PRINTING OFFICE 1877. mM tS OSE t4— Abe aos Vie UNITED STATES Coast SURVEY OFFICE, Washington, D. C., July 21, 1877. The great storms which yearly traverse with terrible energy some portion of the vast extent of sea-coast that bounds the United States on the Atlantic and Pacific Oceans and the Gulf of Mexico have caused in the aggregate the loss of many thousands of lives and the destruction of an immense amount of property. The most violent of these storms are the cyclones that originate commonly in the Atlantic Ocean near the equator. They pass over the West Indies, curve around by the Gulf of Mexico and Florida, and then sweep the entire coast of the Atlantic in a northeasterly direction. The frequency of recurrence, and the marked force exerted along the same general course, sug- gest that these cyclones result from the operations of laws that control the general motions of the atmosphere, and that some understanding in regard to the cause and time of their occurrence might be gained by research, extended so as to include an area commensurate with the developed phe- nomena; but, until now, it is believed that no attempt directed to that end has been made. Any knowledge in advance respecting the direction and rate of motion of these storms would be of incalculable benefit to commerce and navigation, as well as to the people living along the sea-coasts. Their-investigation has, therefore, been undertaken in the hope that the exact knowledge of the configuration of the coast and of its dangers, given to the mariner in the Coast Pilot, may in time be supplemented by information concerning the atmospheric disturbances to which the same region is subject. Mr. William Ferrel, to whom this discussion has been intrusted, and whose previous studies and special ability peculiarly qualify him for the proposed research, presents, in the follow- ing paper, the results of an investigation of the mechanics and the general motions of the atmos- phere. This preliminary inquiry will be followed by his investigation of the effects of various dis- turbances that® are local, as compared with those upon which general motions depend. Such disturbances give rise to local cyclones, due to unequal distributions of temperature in the northern and southern hemispheres. It will be. shown also that local disturbances of equilibrium are the occasion of progressive cyclones. The principles and results developed in the leading part of the discussion will, in a separate paper, be applied for perfecting the formule and methods to be used in the determination of barometric heights; and, as all the general principles applicable to atmospheric motions probably apply also to those of the ocean, the discussion will be supplemented by a chapter on-the subject of ocean currents. These researches will embody conclusious from the most important parts of a memoir pub- lished by Mr. Ferrel in 1859 and 1860 in the ‘‘ Mathematical Monthly.” The memoir appeared neces- sarily as small detached papers, separated by other matter, in two volumes of that journal, and hence the views of the author attracted little notice until extracts from his memoir were quoted in the publications of the United States Signal Service. The reader of the first paper by Mr. Ferrel upon this subject will notice a few slight changes; but it was to be expected that, in a matter so complex, additional years of study would offer both new and improved views. CARLILE P. PATTERSON, Superintendent. METEOROLOGICAL RESEARCHES. By WILLIAM FERREL. A.D 1. ON THE MECHANICS AND THE GENERAL MOTIONS OF THE ATMOSPHERE. CHAPTER I. GENERAL EQUATIONS OF THE MOTIONS AND THE PRESSURE OF THE ATMOSPHERE. 1. In making out equations of condition for the motions and pressure of the atmosphere or the sea, where the part under consideration comprises the whole or a considerable part of the earth’s surface, it is very important that the rotation of the earth on its axis should be taken into account. This can be most conveniently done by referring each particle at any time to three rectangular co-ordinates whose directions are fixed in space, and thus making out equations between the motions, forces, and pressures for each of these directions. These co-ordinates thus become a function of the earth’s rotation and the motion of the particle relatively to the earth’s surface; and by transforming the rectangular to polar co-ordinates, and expressing the equations in functions of the latter, we obtain equations containing very important terms, depending upon the earth’s rotation on its axis. 2. Let x, y, and z be three rectangular co-ordinates having their origin at the center of the earth, # correspondiug with the axis of rotation; also let— V = the potential of the attractive force of the earth; P = the pressure of the fluid; and k = its density: then DzV, D,V, and D,V are the accelerating forces arising from the earth’s attraction, and 7 DeP, £DyP, and 4 D-P those arising from the pressure of the fluid in the reverse directions respectively of x, y, and 2; and hence we have, fo: the equations of the absolute motions of the fluid, regarding the center of the earth at rest,— | Déw+ DV +7 DsP =0 ; 1 a (DY. ul es a RS ae ge EY .<{ Déy+D,V+7DyP =0 | pes4v.v4ip.p =o Putting P = 0, these become the equations of a projectile. ' 3. Let— y = the distance from the earth’s center; 6 = the polar distance; y = the longitude; a = the angular velocity of the earth’s rotation on its axis. We then have— y =r sin 0 cos (né+ 9) =F Sin 0-co8 w zg=rsin 0 sin (nt+ ¢) =r sin 6 sin w (2) fy Srsin by putting, for brevity, n¢+ ¢ =, and making the origin of ¢ such as to make sin (n¢-+ ¢) vanish in the plane of «, y. From these expressions of x, y, and 2 we get— Div= cos 0 Tyr — 7 sin 0 DyO Dy = sin 0 cos w Dur + 7 COs 0 cos w D,0 — 7 sin 0 sin o Dyw D,2 = sin 0 sin o Der + 7 cos 6 sin w Did + 7 sin 6 cos Dw Takiug the second derivatives, we get— ‘D?x = cos 6 D?r — 28in 0 D.r D,6 —r cos 0 (D,6)? — r sin 9D2Ze D?y = sin 0 cos w D2r + 2 cos 0 cos w D,r D,6 — 2 sin 6 sin w D,r Dw +r cos 9 cos w D0 (3) —rsin cos w(D,0)? — 2 r cosdsinw D,0 D,w — rsinésinw D2 gy — r sin @ cos w(D,w)? } D2 z=sin 0sinw D2 r + 2 cosé@ sin w D,r D,6 4+ 2 sin 6 cosw Dr D,w + 7 cos 6 sin wo DZO —rsin 0 sin w (D,6) + 27 cos 0cosw D,0D,# + rsin @coswD?79 —rsin@ sin w (D,«)* Since 2, y, and z are functions of 7, 6, and g, we have— (D.V=D.V. Der+DoV.Dz,0+D,V.De¢ (4) - - 2-4) SD,V=D,V.D,r+DeV.D,0+ Dy V.D,¢ eo ae We also have— r= erty + 2 tan@= vey “a tan w Il Wl From these, we get— D,7r =— = cos 0 818 D,r = = sin ¢cos w r D.r =< = sin Osinw Yr D eet = sin 6 x v2 Yr vy __ COS 6 COS w DOI= Rye te FT Lz __ cos @ sin w DiS ayece | Ff Dg =0 @ sin w De =— page rain d - D yY _ cosw -P=P42 Tsind By means of these equations, equations (4) give— ae cos 0D, V — — Ds V 5) D, V = sin 0 cos w D, V 4 £98 2 €08 _ Sine ( y : ce De V ry sind De V , D,V = sing suo D, V+ OS 2S2 OD vy cos wo oT o" + yang Do V 9 By putting P for V in (4), we obtain in like manner— ohuea D,P— mi DeP (6) D, P = sin 6 cos w D,P + 8 98 & p, p_ S44 pn p r rsiné D, P = sin ¢sin w D, P + 28 9 84 p, p 4 C08 w r r sin 6 By substituting the values of the first members of equations (3), (5), and (6) in equations (1), and then multiplying both members of the equations respectively by cos 0, sin 6 cos w, and sin 6 sin w, and adding, we obtain the first of the following equations. Again, multiplying them respectively by 7 sin 6, —r cos 6 cos w, and —r cos 6 sin w, and adding, we obtain the second of those equations. Finally, multiplying the last two respectively by rv sin 6 sin » and—r sin @ cos w, and adding, we get the last of the following equations : — , DP = —D2r+r(D,0" + r sin? a(n + D,«) D,¢g +7 v sin? 6 — D,V (7) ;DeP =— D20—2r D,r D,6 + 7’ sin 0cos6 (n + D,w) D,g + 7? n? sin 6 cos 0 — De V 1p, P = — r sin’ @ D2 9 — 2r sin? 9 D,r D,w— 27° sin 0 c0s0 D,6 D,w — Dg V 4. Let us now put— h = the height above the earth’s surface ; u = linear distance south ; » = linear distance east;. g = the accelerating force of gravity. By regarding the cosine of the angle between the directions of r and h, which is extremely small, equal to unity, aud neglecting the small terms depending upon the motions of the fluid, which are multiplied into the sine of the very small angle between the directions of r and h, and putting for D,w its equal, n + D,¢, we shall have— {DP =—D?Ph+ (Du) + sin 6(n+ D,¢)D,0 +77’ sin? 6 —D,V D, P = — D?u—2D,hD,u + coso (2n + D,g) D,v + rn? sin 6 cos @ — D,V D,P =— D?v — 2sin dé (n+ D,g) D,h — 2 cos 0(n + D, ¢) D,w— D,V ae Sle 5. In the case in which the fluid is at rest relatively to the earth’s surface, we have— r D,P =r resin? ~D,V=—g9 i Du P = rn?sin # cos 0— Dy V =0 ,D P=—D,V=0 Hence, in the case of the motions of the fluid, we have — ( D,P = — D2h+ (DM? + sino (n + D,g) D,v —g k r (8) 48 Jz D, P= — Diu — 2D,h Diu + cos 0 (20 + Dig) D, 0 (3p, p=— Dp 0—2.sin on + Dig) Dh Bem (m+ Dig) Di “2 10 6. In these equations, there are several terms which are so small that they may be neglected in all cases without sensible error. The value of the term D/?h is of the same order, in comparison with g, as the rate with which an ascending or descending particle of air is accelerated or retarded in comparison with the rate with which a falling body in vacuum is accelerated ; since, putting P = 0 in the first of the equations (8), we get g=—D/h. In all ordinary conditions of the air, the ver- tical velocities are so small, and these velocities are accelerated or retarded so gradually, that the value of D?h corresponding to these accelerations or retardations, in comparison with the rate of acceleration of a falling body, is so small that it may be neglected. In the case of a tornado, the value of this term might be sensible through some part of the height of the atmosphere; but, as the velocity of an ascending body cannot be accelerated in one p*rt of its ascent without being retarded in another part, the integration of this term through the whole height of the atmosphere would be naught, and hence it would not affect P at the surface of the earth. If any part of the atmosphere has a vertical acceleration or retardation of velocity equal tog in 100 seconds, or a little less than two minutes, then the pressure or value of P arising from this part of the atmosphere is affected the one-hundredth part; aud, in such a condition of the atmosphere, in determining heights from barometric pressure, the results would be seriously affected. But such conditions could only take place under some extraordinary disturbances of the atmosphere, when any such determinatoins would not be attempted. It is seen from the first of (8) that the pressure is diminished when the ascending velocity is accelerated or descending velocity retarded, and vice versa. : 7. We have, neglecting D, ¢ in comparison with n,— sin? 6 (n + D,g)D,v _ rsin’énD,g rn. Dig ain? d= an sin? @ De? g g g n 289 nN But D,¢, the angular eastward motion of the air relatively to the earth’s surface, is always very small in comparison with n, and hence, even at the earth’s equator, where sin? 0 = 1, the ratio between sin’ @ (n-+D,¢) D,v and gis extremely small, and the former may in all cases be neglected 2 in connection with the latter. The value of the term eda! is evidently very much smaller than sin’ 0 (n+ D,¢)D,v, and hence its effect is always entirely insensible. In the last two of equations (8), the terms containing D,h as a factor may both be neglected in comparison with the terms which follow them, since the vertical velocity D,h is always very much smaller than the horizontal velocities D,u or D,v, where these latter are such as to give a sensible value to the terms containing them as factors. 8. The density of dry air is as the pressure and inversely as the amount of absolute tem per- ature. But the atmosphere always contains a certain amouut of aqueous vapor which is lighter than air, and affects the density of the atmosphere in proportion to the amount of this vapor con- tained init. This effect is expressed by (1+,/(e)) in the denomiuator of the expression for the density, in which ¢ is the relative amount of aqueous vapor contained in the air. We can therefore put— (QD) ew ee A we we we Be we ee eee in which, putting a, for the value of « when t= 0 and e= 0,— 273° 1 ~ (QT +H (LFF) (+ 0.36630 (T+ Fe)” The average value of e varies with the temperature, but is very different at different times and in different localities for the same temperature. As an average value of f(e) for all localities and seasons, we can put— (40)... . a F (e) = 0.00154 4+ 0.000341 ¢* With this value of f (e), we get— be foe en ee (11) . . . . . . . . . . a= 1.00154 + 0.004¢ * See a paper by Dr. J. Hann, entitled “Zur barometrischen Héhenmessung,” in * Band LXXIV der Sitzb. der kaiser- lichen Akademie der Wissenschaften.” 11 This latter expression of « may be used in all cases in which the preceding equations are applied to the atmosphere generally, in which local and temporary deviations must be neglected; but, in local applications, the former must be used, the value of e being determined from observation for the particular time and locality. 9. So far we have neglected to consider the effect of friction, which is an important element entering into equations where the motions of fluids, either elastic or inelastic, are concerned, and oue which is most difficult to treat. In the preceding equations, therefore, we must have a term to represent the resistance which each particle suffers from friction, and this term cannot be expressed by any function of the velocity simply, as is sometimes supposed, but it depends rather upon the differences in the velocities of the different strata, and upon the differences of pressure. If a stratum lie between two other strata, all having the same velocity, it suffers no resistance from friction, however great the velocity may be; and the same is the case where the relative velocities of the strata are such that the action of the stratum above upon the intermediate stratum is exactly equal to the reaction of the lower one upon it in the contrary direction. This may require the rel- ative velocities of the different strata to be different on account of the differences in the amount of pressure, and it, no doubt, requires them to increase as the pressure diminishes, that is, with the height. The amount of resistance, therefore, which any particle suffers, requiring extraneous forces to overcome, is generally an unknown quantity, and all that we can do, therefore, is to intro- duce unknown functions into the equations representing the resistances from friction in the direc- tions of the co-ordinates, and leave these to be determined approximately, where it can be done, from a comparison of the final results deduced from the equations with observation. If we, there- fore, put F,, and F, for the forces acting in the directions respectively of w and v necessary to overcome the resistances of friction, and substitute for & its value in (9), we get from (8), neglecting the insensible terms pointed out in §§ 6 and 7,— (12). D, log P= —aD?2u+acos0(2n+D,¢) usaf, D, log P=—ga }D.log P= —aDivt cose nt Die) Derek, In the case of a homogeneous fluid, it is readily seen that we must put P for log P and & in- stead of a The value of g is nearly constant; but, when great accuracy is required, it must be regarded as a fanction of h and 6, and we may put— 2h GBe: e089, Se od g = 9 (1— | + 0.00284 cos 2 0) in which g’ is the value of g at the sea-level and on the parallel of 45°. 10. By regarding g as independent of h, equation (12) gives, by integration,— (14). . log P’ — log P = a (gn +F(h)) + aif (Deu — cos 0(2n+Di¢) Div + F,) +4 { (Div + 2008 0(n-+Dig) Din + F) ) in which P’ and a’ are the values of P and a respectively at the earth’s surface, and /f (hk) is a small function of h depending upon the decrease of temperature with the height, which may always be neglected, especially when h is small, except in accurate hypsometrical determinations. Where common tabular logarithms are used, the last number of this equation must be. multiplied into the modulus M = 0.4342945. The preceding expression gives the difference of the pressure between any two assumed points. If these two points are in the same vertical, the terms depending upon D,u and D,v vanish, and the expression is coufined to the first term; but, if these points are in verticals a considerable distance apart, the integration of the last two terms, depending mostly upon the earth’s rotation and the motions of the atmosphere relative to the earth’s surface, may give a considerable difference of pressure at the sea-level, or for any two assumed points at equal heights above it. 12 11. Where P and P’ are in the same vertical, we get from (14), by neReenne the small term f (h) in connection with g h, and regarding g constant,— D, log P’ — D, log P = gh D,, a’ D, log P’ — D, log P = gh D, a’ By means of these equations, we get from (12)— +, D, log P= —D?2u-+cos 6 (2+ D.g) D,v —F, + gh D, log a! Th. ick . o?) 1 p, log P’ = — D?v — 2cos 6 (n + D,y) Dw —F, + 9h D, log a! a At sea-level, we have h = 0, and consequently the last term of these equations vanishes, and they ‘then give the gradients of barometrical pressure in the directious of u and v, depending upon the motions of the atmosphere at the earth’s surface, friction, inertia, and the earth’s rotation on its axis. In this case, the small neglected function f(%) also vanishes. 12. If we put h/ for the value of h belonging to a stratum of the atmosphere of equal density, or pressure, we shall have, for this stratum, D, P = 0 and D, P = 0; and we get in this case from the first of (14), neglecting the small function f (4) in comparison with g h,— D, log P’=gh’ D,a’ + ga’ D, h’ D, log P’ =gh’ D, a’ + ga’ D, hl’ With these equations, we get from (15), by putting h = 0,— at (16) Vea cos 6 (2n+ Deg) D vo! — Fy! — gh’ Dy! log a! gD, =—D?v' —2c08 6( n+ Dig) Dt w —F,! —gi’ D,’ log a! in which w/ and v! are the values of u and v at the earth’s surface. These equations give the gradients of the strata of equal pressure in the directions of u and v; and, by integration, they give the differences of level of any two points in such a stratum. This, at the earth’s surface, when h’ = 0, depends simply upon the motions of the atmosphere at the earth’s, surface, friction, and Inertia; but the last terms of these equations show that these gradients are increased or dimin- ished, as the case may be, in proportion to the height. 13. From (12) we get, for the part depending upon the earth’s rotation, D,log P= 2uncosé6D,v D, log P= —2ancos60D,u If we now put s for the resultant of « and v, and q a perpendicular tos on the right-hand side of the direction of motion, we shall have— D, log P = D, log P. D, w+ D, log P. D,v =D,logP. cos — D, log P. sin 4 D,s =D,usin 2 + D,v cos 4 » From these and the two preceding equations, we get— q (17) . . D,log P=2ancosoD,s The direction of s is entirely arbitrary, and D,s represents velocity in that direction, and D, log P represents an ascending gradient in the direction of q to the right of the direction of s, and consequently the force depending upon the earth’s rotation which causes this gradient, expressed by the last member of (17), is a force acting in that direction, and is positive in the northern hemisphere, and the contrary in the southern, 13 according to the sign of cos 6. Hence, in whatever direction a body moves upon the surface of the earth, there is a force arising from the earth’s rotation which tends to deflect it to the right in the northern Lemiephiors, but to the left in the southern hemisphere. This important principle, useful in explaining so many of the relations between the motions and the barometric gradients of the atmosphere, was first published in my former paper on this subject in the “‘Mathematical Monthly” in the year 1860. It is a generalization of the principle upon which the theory of the trade winds has been based, according to which this deflecting force, arising from the earth’s rotation, takes place only in the case of motions north or south. But, by the true and more general principle above, it is seen that this deflecting force is exactly the same for motions in all other directions. Influenced by the usual theory upon this subject, observers have imagined that they have seen evidences of a tendency in rivers running north and south to wear away the banks, and also to deposit drift-wood on the right- rather than on the left-hand side, and likewise a tendency in the cars of railroads extending north and south to be thrown off the track on the right- rather than the left-hand side, while in the cases of rivers or of railroads extending in other directions no evidences of such effects could be seen. But we now know that if any sensible effects of this sort arise from this deflecting force in the case of rivers or of railroads running north or south, the very same effects must take place where they run in any other direction. The amount of this force as deduced above, from the true principles of mechanics, is exactly double of that which has been obtained from the erroneous principle adopted by Hadley, and brought down through text-books to the present time. This latter principle assumes that the moving body in approaching or receding trom the earth’s axis must retain the same linear motion east or west, whereas, by the principle of the preservation of areas, which must hold in this case, the linear motion east or west must be increased in the former case and diminished in the latter in such proportion as to make the deflecting force double of that given by the principle of equal linear east or west motions for all distances from the earth’s axis. This matter has been explained in detail in “‘ Nature,” vol. v, p. 384. 14, The accelerating force in the direction of g which is adequate to produce a gradient rep- resented by D, P is— 1 1 2 cos 6 2cos¢9 1 2cos@e D,s (18) 7 D,P =+D,log P =2.n 008 #D,¢ = = 8" rn? Dp = 2208", “5399 Da The coefficient of g in this expression shows the ratio between this deflecting force and the force of gravity. a Let us put at sea-level, on the parallel of 459,— rv = 6366252" logr = _ 6.8038838 22 = ___ "ss = 0,000072924 ] = — 5,86287 . "= (23 x 60 4 56) 60 eee ; g = 9".805307 : logg = 0.9914612 r 0 = 4647.25 logrn= 2.66675 With D,s = 13.889, which is equal to a velocity in the direction of s of 50‘ per hour, and with @ = 45°, corresponding to the parallel of 45°, we get gqs5 9 for the accelerating: force in the direc- tion of q arising from the earth’s rotation, aud hence the lateral pressure of a body moving with the assumed velocity in any direction on that parallel is equal to ;j55 of its weight. In the southern hemisphere, where cos @ is negative, of course the lateral pressure is in the contrary direction, that is, to the left of the direction of motion. Where a body is free, this deflecting force produces motion in a curved line. In the case of a fluid, as air or water, this force causes a disturbance of - the static level surface, and the amount of gradient resulting from it is expressed by the coefficient of g in (18). For instance, with the assumed velocity above and on the parallel of 45°, the gradient would be one meter in the distance of 6839 meters. If a river has a velocity of 5*" per hour, the ascending gradient to the right of direction i in the 14 northern hemisphere is one meter in 6839 meters; and hence, if the river is one kilometer in width, the water stands about 2, of a meter higher on the rignt than on the left bank. 15. In the case of the atmosphere, in which the gradient is usually measured by the differences of barometric pressure, we get from (9) and (18) for this gradient— (19)... «. . . . . . D, P= 0,00014595 2 P cos 6 D,s In order to have a numerical expression of the gradient in terms of the pressure and tempera- ture, and independent of the density, it is necessary to determine the value of a, in (11), from which we thus obtain the value of « corresponding to any given temperature ¢, and that of a! corresponding to the value of « at the earth’s surface, which occurs in the most of the preceding expressions, and of which, therefore, it is important to have a determination. The value of a, is the value of « in (9) belonging to dry air, with the temperature at zero (32° F.). If we put J equal to the height of a homogeneous atmosphere of temperature 0°, and pressure, measured by the height of the barometrical column, of 0°.76, we have for such an atmosphere— P=ghkl=gk' x 0°.76 in which k, is the density of dry air under the assumed pressure, and k’ the density of mercury. Putting k, = a P (9), we get, since the heights of the homogeneous atmosphere and mercury are inversely as their deusities— I! L=gal=ga 7 x 0".76 0 * With k’ =13.6001 and k, = aaa we finally get— lt 1 (20) CE SS = og ee With this value of a,, we get from (11) the approximate value of a for any given temperature, to be used in (19) for determining the barometric gradient belonging to the deflecting force arising from the velocity D,s. For the temperature of 0°, a in (19) becomes a,; and, with the preceding value of a, we get in this case— (21) 6. . eSoft D,P = 0.0000000014162 cos 6 D, s in which D, s must be expressed in meters per second. If we wish to express the gradients by the change in millimeters belonging to the distance of a mean degree of the meridian, equal 111111111 millimeters, we must multiply the second member above by 111111111, and then, representing the gradient by G, we get— (21). 2 6 1 ww ee ee G=0.15893 cos 6 D, 8 @ 16. In many cases, it is necessary to have the equations of the motions and pressures in terms of polar co-ordinates, in which the pole does not coincide with the pole of the earth’s axis, * Prof. F. A. P. Barnard, Metric System, p. 171, has obtained, from the average of the weights of a cubic inch of dry air given by several authorities, 0.0012228315 for the specific gravity of dry air at a temperature of 62° F. and a baro- metric pressure of 30 inches. This, reduced to the temperature of oe C. and barometric pressure of 0™.760, gives the value above. 15 Let— =the arc between the pole of the earth’s axis and the new pole of the co-ordinates ; ‘ p = the distance in arc from the new pole; # = the angle between p and the meridian ; u = linear distance in the direction of ; and v = linear distance in a direction perpendicular to p. In the case in which the earth has no rotation on its axis, the pole of the co-ordinates in the preceding equations can be taken at pleasure, and therefore, instead of N, the pole of the earth’s axis in the annexed figure, we can put it at P, and hence by putting p, », u, and v for 6, ¢g, u, and v respectively, we get from (15) in this case, by putting 1 = 0,— 1 ~7D, log P! = —D?7u+ cosgD,4D,v —F, + 9h D, log a 1 77D, log P’ = — D?.v — 2 cos ¢ D,u D,u—F, + 9h D, log a’ We must now add tothese equations the terms arising from the deflecting forces depending upon the earth’s rotation belonging to the velocities D,u and D,v, which are forces in directions respectively perpendicular to u and v. From (17) we get, for the parts of = D, log P’ and = D, log P’4,— 1 a D, log P’ = 2" cosé D,v : D, log P’ = 2 cos 6 D,u a Adding the right-hand members of these equations to those of the two preceding ones, we get for the equations in the case in which the earth rotates on its axis— { 1p, log P’ = — D2u + (200s 6 + cos p D,») D, v — Fa + gh Dy log a’ (22)... z 5, D, log P’ = — D?v — 2 (n cos 0 + cos p D,z) D,u — FY + gh D, log a We have, from the trigonometrical relations of a spherical triangle,— COS 0 = COS y COS p — Sin y SiN p COS p Where the range of p is so small that we can neglect the second term of this expression in com- parison with the first, and put cos p = 1 without material error, (22) becomes— 1p, log P’ = — Du + (2ncos » + Dix) Dov — Fu tg h Du log a! 93)..22% | ai 1 p, log P! = — D7 v — 2 (neos » + Dia) Du — Fy +g h Dy log @ a 17. Besides the preceding equations (15) and (23) showing the relations between the motions and pressure of the fluid, there is still another condition, which must always be satisfied in the case of motions of the atmosphere, which is, that the volume of air which occupies any given space must be the same, and the amount of air directly proportional to its density, and the motions of the atmosphere must always be such as to satisfy these conditions. The mathematical expression - of this condition in any case where such can be formed, is called the equation of continuity. 16 CHAPTER LI. THE TEMPERATURE AND PRESSURE OF THE ATMOSPHERE AT THE EARTH’S SURFACE OBTAINED FROM OBSERVATION. 18. In most of the equations‘of the preceding chapter, there are found the two functions a’ and P’, which it is necessary to determine for all parts of the earth’s surface from observation. It is seen from (11) that a is a function of t the temperature, a’ being the value of a corresponding to the temperature of the atmosphere at the earth’s surface. With the value of ¢, therefore, for all parts of the earth’s surface, we obtain from (11) the expression of a’ iu a function of @ and ¢, and consequently of wand v, from which we obtain the functions D, log @ and D, log a’. The general equations of motion and pressure of the atmosphere contained in the preceding chapter are of such a character, on account of their complexity, and the unknown friction terms entering into them, that they do not admit of a complete solution, and it is therefore important to obtain from observation as many as possible of the fanctions entering into those equations. If the general equations could be completely solved, we should need only the temperature from observa- tion, and the solution of the equations would then give the atmospheric pressures for all parts of the earth belonging to this temperature, and we should not need the pressures from observation, except for a verification of theory. But as it is, we also need the observed pressures in the different parts of the earth’s surfa-e, in order to enable us to obtain other unknown quantities depending upon these pressures, which cannot be obtained from the solution of the equations. . The following tables, I and II, of approximate temperatures for the two extremes, January and July, have been obtained by interpolation from Buchan’s Charts of Isothermal Lines, with some cor- rections first applied, to make them agree with recent observations. The authority for these cor- rections is derived mostly from the “‘ Contributions to our Knowledge of the Meteorology of Cape Horn and the West Coast of South America,” published by the authority of the Meteorological Com- mittee of London. These contributions give temperatures generally from four to six degrees higher for these parts than those indicated by Buchan’s isothermal lines for July, and three or four degrees higher than those indicated by his lines for January. As the temperature of the latitude of Cape Horn must be very nearly the same for all longitudes, the temperatures.cf the extreme southern latitudes in the following tables have been increased accordingly all around the globe. The last columns of these tables contain the means of the temperature for all the different longitudes. Although the numbers given for the different longitudes are only approximate, and may in some instances be considerably in error, yet the means of so many longitudes must give very nearly the averages of the different latitudes of the globe, and will be sufficiently accurate for our purpose. And the local variations of temperature independent of latitude will be only needed approxi- mately for explaining in a general way the phenomena depending upon them, and not for any accurate comparisons of theory with observation. 17 TABLE I.—Showing the approximate mean temverature in degrees of Fahrenheit for the different parts of the earth’s surface in JANUARY. Ss Longitudes west. = 3 180 | 170 | 160 | 150 | 140 | 130 | 120 | 110 | 100 | 90 80 70 60 50 40 30 20 10 0 80 | —33 | —32 | —32 | ~33 | —34 | —35 | —37 | —39 | —40 | —39 | —36 | —30 | —24 | —22 | —20 | —18} --15 | —12 | —10 70 | —26 | —24 | —22 | —23 | —24 | —26 | —28 | —30 | —32 | —31 | —30 | —23 | —15 | — 5 4 8 12 18 22 60 10 11 12 12 iL 5 | — 5 | —13 | —20 | —24 | —22 | —16 0 12 20 28 36 36 34 50 40 40 40 40 40 36 32 22 10 5}, 0 5 10 | . 26 38 42 44 42 38 40 50 49 50 51 50 50 46 42 36 32 |, 30 32 38 48 54 56 54 52 48 30 60} 59 58 57 58 5T 56 o4 54 54 54 58 €2 66 66 66 64 62 60 20 72 WU 70 69 68 66 66 66 66 70 rR 71 74 4 74 74 4 74 74 10 16 76 16 76 76 76 6 76 16 76 78 78 78 76 76 78 80 82 85 ‘0 80 80 80 80 80 80 80 80 22 82 82 82 82 81 80 80 80 30 80 —10 82 82 81 80 80 80 80 80 80 80 80 82 84 84 84 82 78 78 78 —20 7B 78 78 78 7% 77 76 76 16 6 78 80 82 82 80 78 76 15 74 —30 74 7 B 72 72 72 7 7 70 69 69 72] (74 74 4) 74 ve) TW 70 —40 64 64 64 64 64 62 62 62 62 62 62 64 66 67 68 48 66 62 62 —50 52 52 52 51 51 5! SL 51 50 50 55 54 55 55 55 55 55 53 52 —60 39 39 39 38 38 38 38 38 37 37 38 40 41 42 42 42 42 AL 39 s Longitudes east. = do 2 7 . oS 3 10 20 30 40 50 60 wil 80 90 100 | 110 | 120 130 140 150 160 170 5 so | —10 | —10 | —10 | —10 | —10 —10 | —12 | —15 | —20 | —24 | —28 | —32) —33| —34|) —34) —34) —33 | —25.0 70 25 18 5|—6| —10 | —10 | —12 | —15 | —1g | —22 | —27 | —32] —36) —38) —38 | —36 | —30 | —15.5 60 30 24 16 10 6 0|—4 |— 8| —12 | —16 | —24 | —30 | —30] —24| —16 0 12 1.7 50 34 28 24 18 12 6 4 0 0 0 0 0 4 10 18 24 34 21.3 40 46 44 42 40 35 30 30 30 30 30 26 20 22 30 36 40 44 40.0 30 58 56 55 54 53 52 51 50 47 44 42 40 44) 50 53 56 60 55. 2 20 wu 4 72 68 66 68 70 72 TR 72 72 10 70 72 72 72 72) 71.0 10 88 90. 86 80 80 80 82 82 80 78 76 76 16 76 716 16 16) 78.7 0 84 90 90 86 84 82 82 82 81 80 80 80 80 80 80 80 80 81.2 —10 86 90 90 86 84 82 82 82 82 82 82 82 82 82 82 82 82) 82.2 —20 82 86 |, 86) 84 83 82 82 82 82 82 82 82 82 82 82 80 80 80.0 —30 16 78} Wy 78 78 78 78 76 74 14 73 72 72 72 )° 12 7 74 734 —40 62 €4 64 64 64 64) 64 64 64 64 64 64 64 64 63 62 63 63.6 —50 $2'| 627° 51 51 51 51 51 SL 51 51 51 51 51 50 50 50 50} 52.0 —60 37 36 35 35 36 36 37 37 38 38 38 38 38 39 39 39 39 38.5 18 TABLE II.—Showing the approximate temperature in degrees of Fahrenheit for. the different parts of the earth’s surface in JULY. oS Longitudes west. 3 = 5 180 | 170 | 160 | 150 | 140 | 130 | 120 | 110 | 100 90 80 70 60 50 | 40 <0 20 10 0 80 34 33 32 32 32 32 32 32 32 33 34 34 34 32 30 30 30 31 32 70 40 40 40 46 50 50 46 44 42 40 33 40 42 42 38 38 38 40 43 60 48 50 52 54 58 62 62 60 56 52 48 45 46 | 47 48 50 54 55 66 50 60 58 56 56 60 62 68 70 70 68 60 60 60 60 | 60 62 64 64 64 40 66 66 64 62 60 62 68 74 84 84 76 rR 70 710 70 70 70 72 74 30 72 69 66 66 66 68 70 80 88 88 84 82 80 78 17 78 73 81 81 20 80 6 76 76 76 76 16 80 84 85 84 82 82 81 80 82 84 88 90 10 80 80 80 80 80 81 82 82 a2 83 84 84 84 82 81 81 82 84 88 ' 0 76 TW 78 78 78 78 78 79 80 81 82 82 84 82 78 78 78 78 8 —10 74 74 4 3 6 74 73 74 6 74 6 76 716 75 74 74 74 14 4 —20 66 68 68 68 6 }- 68 68 69 68 67 66 68 70 72 72 10 68 68 68 —30 60 60 | 60 60 60 60 60 60 00 59 58 58 60 64 64 64 64 66 | 62 —40 52 Sl] 52 53 54 54 54 53 52 50 50 48 48 49 50 52 54 55 56 —50 45 45 45 45 45 44 413 43 43 43 43 43 43 44 45 45 45 45 45 —60 | .34 34 33 33 33 32 32 32 3L 31 32 33 32 33 34 34 34 33 33 1 Longitudes east. 10 20 30 40 50 60 | 70 80 90 100 | 110 | 120 130 140 150 160 170 Latitude. Mean. 80 33 34 34 34 33 32 32 32 33 34 35 36 37 38 37 36 35] 34.1 70 46 50 48 44 40 38 40 42 46 50 50 52 52 52 51 50] 46) 443 60 60 62 63 64 64 64 64 65 65 65 65 65 64 60 58 54 52) 57.0 50 65 66 68 70 72 74 74 74 74 72 vee 10 67 6f 63 60 60 | 65.5 40 76 73 wie} 78 78 78 80 82 82 82 80 78 74 10 68 66 65} 73.0 30 82 83 86 88 90 90 90 89 82 87 85 82 80 ets] 16 74 71] 80.0 0) St 92 92 91 90 90 89 88 87 87 86 85 83 82 81 80 80} 84.2 10 90 90 90 88 86 84 82 382 82 83 83 83 82 82 82 82 81} 83.2 0 80 84 84 80 78 78 78 78 719 80 81 80 78 76 16 76 76 | 79.0 —10 78 80 80 78 WT 76 6 76 76 716 76 16 6B 4 14 74 13 | %.2 —20 70 14 14 14 74 14 72) 70 69 68 68 68 68 68 68 68 69} 69.5 —30 61 60 60 60 60 60 60 60 60 60 59 58 56 54 56 58 59 | 60.1 —40 55 54 54 54 53 52 52. 52 52 52 51 50 50 50 51 52 52] 52.0 —50 45 45 45 45 44 43 43 43 42 41 41 41 41 41 41 42). 43] 43.5 —60 32 31 31 30 30 30 31 31 32 32 32 32 32 32 32 31 32 |) 32.0 ~19 Tables III and IV give the approximate mean annual temperatures for each tenth degree of latitude and longitude and the approximate mean annual range of temperature, and likewise the means of all the longitudes. The former have been obtained by taking simply the mean of the extreme mean temperatures of January and July, and the latter by taking the differences of these extremes. Any one must be struck, from an inspection of Table IV, with the great differences between the mean annual ranges of temperature of the northern aud southern hemispheres, arising from the unequal distribution of land and water in the two hemispheres. TABLE III.—Showing the approximate mean annual temperature in degrees of Fahrenheit for the different parts of the earth’s surface. 4 (January + July). Longitudes west. 180 | 170 | 160 | 150 | 140 | 130 | 120 | 110 | 100 | 90 80 70 60 50 40 30 20 10 0 80 0 0 0 0;—-1)/—1}/-—-2}/-—3]/-—4]-3]/-1 2 5 5 5 6 8 10 il 70 7 8 9 il 13 12 9 7 5 4 4 8 13 19 21 23 25 29 32 60 29 30 | (32 33 35 34 29 24 18 14 13 14 23 29 34 39 45 45 45 50 50 49 48 48 50 49 47 46 40 36 30 32 35, 43 49 52 54 53 51 40 53 57 57 56 56 57 57 58 60 58 53 52 54 59 62 63 62 62 56 30 66 64 62 61 61 61 62 67 71 W 69 70 7 72 72 712 7 72 7 20 76 74 73 Wl wal 7 73 wis) 82 78 77 78 77 17 78 79 81 82 10 78 78 78 78 78 78 79 79 719 80 8L 81 81 80 80 80 81 83 86 0 78 18 79 79 79 79 79 80 81 81 82 |" 82 83 82 79 79 719 79 79 —10 B 78 78 77 77 q7 77 77 7 77 78 719 80 80 79 78 76 76 76 —20 3 3 B 73 73 3 72 72 72 72 72 74 76 77 76 74 72 vp) 7 —30 67 67 66 66 66 66 65 65 65 64 64 65 67 69 69 69 69 | 69 66 —40 58 58 538 58 59 58 58 58} 57 56 56 56 57 58 59 60 60 59 59 —50 48 48 48 48 48 48 47 47 47 47 49 49 49 49 50 50 50 50 48 —60 36 36 36 36 35 35 35 35 34 34 35 37 37 37 38 38 38 37 36 Latitude. Longitudes east. Mean. 10 20 30 40 50 60 70 20 90 100 | 110 | 120 130 140 150 160 170 ow a wo n wo wo _ 30 11 12 12 12 12 11 10 8 6 4.5 70 35 34 27 19 15 14 14 13 14 14 12 10 8 7 6 7 8] 14.4 60 45 43 40 37 35 32 30 28 26 25 21 17 17 18 Qi 27 32] 29.3 50 50 47 46 44 42 40 39 37 37 36 35 35 35 37 40 42 47 | 43.4 40 61 61 60 59 57 54 55 56 56 56 53 49 48 50 52 53 54 | 56.5 30 70 70 70 mW 71 7 71 70 68 66 64 61 62 64 65 65 65 | 67.6 20 82 83 82 80 78 79 79 80 80 80 79 78 76 7 7 76 76 | 77.6 10 89 90 88 84 83 82 82 82 81 80 80 80 719 79 79 719 78 | 81.0 0 82 87 87 83 8 80 80 80 80 80 80 80 719 78 78 78 78] 80.1 —10 82 85 85 82 80 79 79 719 719 79+] 79 719 79 78 78 78 77 | 78.7 —20 16 80 80 719 79 8 vt 76 wis) 5 wh) 75 5 wt) wi) vi) 14 | 74.7 —30 68 69 69 69 69 69 69 68 67 67 66 65 64 63 64 65 66 | 66.7 —40 58 59 59 59 59 58 58 58 58 58 58 57 57 57 57 57 57 | 57.9 —50 48 48 48 48 48 47 47 47 47 46 46 46 46 46 46 46 47 | 47.8 —60 35 33 33 33 33 34 33 33 33 34 35 35 35 34 34 34 35 | 35.3 20 TABLE IV.—Showing the approximate mean annual range of temperature in degrees of Fahrenheit for the different parts of the carth’s surface. Ww uly — January.) 3 Longitudes west. 2 3 180 | 170 | 160 | 150 | 140 | 130 | 120 | 110 | 100 90 80 10 60 50 40 30 20 10 0 80 67 65 64 65 66 67 69 a vp) R 70 64 58 54 50 48 45 43 42 70 66 64 62 69 14 76 14 74 4 71 68 63 57 47 34 30 26 22 21 60 38 40 40 42 47 57 67 TS 16 6 0 61 46 35 28 22 18 19 22 50 20 18 16 16 20 26 36 438 60 63 60 55 50 34 22 20 20 22 26 40 16 17 4 1l|; 8 10 22 32 48 52 46 40 32 22 16 14 16 20 26 30 12 10 8 9 10 13 16 26 34 34 30 24 18 12 il 12 14 19 21 20 8 3 6 7 8 9 10 15 18 15 12 11 8 7 6 8 10 14 16 10 4 4 4 4 4 5 6 6 6 7 6 6 6 6 5 3 2 2 3 O);-4/-3)/-—2);-—-2;-—~2)}—2}/—2;-—1}-—2]/-—1 0 0 2 1}/—2]}/—2]/—2}]—2]—-—2 -|—-10;/—8/)/—8)/-~7}/-71|—5]}/—6]}/—7)/-—6)—5]-—6|;—5]/—6]—8)—9]-10;/—8/-—4]-—4]-—4 —20 | —10 | —10 | —10 | —10; -10} ~ 9] —8}—7}— x8] —9] ~-12) —12} -12} -10!—8/-—8/-—8/—7]-—6 —3s0 | —14 | —14 |} —13 | —12 | —12 | ~—12] —11 |] —11 | —i0 | —10 | —1L | —14 | —14 | —10 | -—10] —10} —8|/-—5/]~—8 ~40 | —12 | —13 | —12 | —11} —10} — 8] — 8] — 9] —10} —12} —12 | —16 | —-18 | —18] —18] -—16] —12] —7] -—6 —50/-—7);-—6}—6|]-—6;-—6|]—7]—8;—9] —10] —11] —12 oH, —11 | —12; —11 | —10 | —10/ —8|]—7 —60;-—-5}/-5;/;—5|/—~5)/—5/—6|—6]/—6]/~6]—6;-—6]/—7}/—7]/—9}~8]/—8]—8/—8/]-—6 3s Longitudes east. 3 ‘ a s 10 | 20 | 30 | 40 | 50 | 60 | 7 | 80 | 90 | 100 | 110 | 120 | 130 | 140 | 150 | 160 | 170 é ° 80 43 44 44 44 43 42 44 AT 53 58 63 68 70 72 val 70 68} 59.1 70 al 32 43 50 50 48 52 57 64 72 V7 84 8b 90 89 86 76] 59.8 60 30 38 47 54 58 64 68 73 17 81 89 95 94° 84)" 74 54 40) 55.3 50 3L 38 44 52 60 68 10 4 74 72 71 70 63 54 45 36 26} 44,2 40 30 34 36 38 43 48 50 52 52 52 54 58 52 40 32 26 21] 33.0 30 24 27 31 34 37 38 39 39 41 43 43 42 36 28 23 I8 11} 24.8 20/ 17{ 18] 20] 93] 2] 22] a9] a6] 15] 15] 14] 15] 33} ww 9 8 8] 132 10 2 of 4 8 6 4 0 0 2 5 7 7 6 6 6 6 5] 45 o{/—-4}/-6/—6]-6}/-6|/—4/-~4]-4}/~2] of 1] of ~2} ~4] ~4] ~4] ~alee —10|—8|~10}-10]~8]—7]/-6|-—6/-—6|-—6|]--6)-6;-6| —7| ~s} ~e] —8} —~gl_70 —20 | —12 | ~12] -12] -10} — 9] — 8 | —10 | ~12] -13 | —14] -14] -14) -14] -14] ~14] -212] —11 |-10.5 —30 | ~15 | —18 | —18 | —18 | —18 | -18 | —18 | -16 | —14 |. —14) -14] -14! ~16] -18] -16] ~14] —15 |_13.3 ~—40 | — 7| —10 | —10 | —10 | —11 | ~12 | —19 | ~12 | -12 | -12 | -13 | -15 |) -14] +14] -12] ~10] —11 |-i1.8 —s0|-—7/—7]/-—6]/-6]/—7/-~8|/—e8}—8}/-—9]-10|-1]|-10| -10} ~9] ~9] —~s] ~7las —60}—5/—5/—5/—5|-—6/-—6/-—6/—6;—6/-—6|/-6]-~6]| —6; ~7| ~7] ~8| _vl_6s5 The numbers in Tables I and II being averages for the months of January and July, and not the extremes of the average mean daily temperature, the results of this table require a sma’ cor- rection, which is the difference between the averages of the month and the extreme of the average mean daily temperatures, in order to obtain the absolute mean range of temperature; but this cor rection is quite small. 21 19, By reducing the mean temperature in the last columns of Tables I, II, and III to centi- grade degrees, we get the second, third, and fourth columns in— TABLE V. Temperature. Mean. 8 . 3 & o 3 q A 3 3 g ® 5 ° 5 a ° G °o ° ° oO 0: reeweacee: | ceaasct lbargacsous AIO beceswnets io | —31.9 | 1.0 | —15.5 15.8 | +03 20 26.5 6.9 9.8 10.2: | +04 30 169 | 138 | —~16 | — 22 + 0.6 40 | — 6.0 Bn6 | + 63 | + 65 —0.2 50 | + 4.5 | 228 13.6 14.4 —0.8 60 12.9 26. 6 19. 8 20.4 = 06: 70 21.7 | 29.0 25.3 24.3 +10 80 25.9 | 28.4 27.2 26.4 +0.8 90 27.3 | 26.1 26.7 26.8 | -—0.1 100 27.9 | 24.0 25.9 28.0 —0.1 e 110 26.6 | 20.8 23.7 23.8 —0.1 120 23.0 | 15.6 19.3 20. 2 —09 130 17.6 | 14 14.4 14.9 —0.5 140 Ld 6.4 8.8 8.2 + 0.6 150 | + 3.6 00} +18) +09 +0.9 160> | reemeveeeslleatemes daemons S598. | ccuecees TTO® diesaecebesaclll pean, eee 10:6° Veen ven BO | darren corcer [penemters arate! ADA acces If we put for the mean annual temperature— (24) . . . . . t=h+acos4+ bcos 26+ ¢cos 30+ deos4 o we get from this equation, with the fifteen observed values of ¢, and the corresponding values of 6 in the first column, fifteen equations of condition for dbtarnitine: by the method of least Banas; the values of t, a, b, c, and d. With the values so determined, we get— (25). . t = 8°.50 — 1°.75 cos 0 — 209.95 cos 2 a ee ee From this expression of t we get the computed values of ¢ in the third column of the preceding table, which. satisfies the observations with the residuals contained in the last column. Although this expression may represent the observations best for the latitudes for which they have been made, yet it cannot be regarded as representing the temperatures very accurately at or near the poles, especially the south pole. = 20. In order to obtain the mean temperature of the earth’s surface, we must integrate the expression obtained from (24) by multiplying it into sin @ and integrating. with regard to 0, by which we get— fresine= fisin a(t +acos 0 + Dos 20 + 0.cos 30 + d.cos 40) 6 d =—(%-5>) cos 0— (ac) cos20— 7 (d— 2) c0s30— Locos 46 1 a CG io? cos5 6 + - 22 in which— . 1 1 1 1 1 1 1 3 J. C=(t— —b = _— (2-2) a —d= = 2 a SG a (% 5 y+i(a J+; b—ad Pe OF aye f+ 44 3? 5° ip” This integral gives for the northern hemisphere, using the values of t, a, b, c, and d@ in (25),— ‘ 1 1 J ts = a2 be ——d = 15°. fo sin 6 = ty) + at 5 Cc is? 5°.30 1 Zz and for the southern hemisphere,— peso 0 = ty — a2 b+ he — Fd = 160.05 The mean of these two results gives 159.67 for the mean temperature of the whole surface of the earth. From Dove’s Charts of Isothermal Lines, which do not extend beyond the middle latitudes in the southern hemisphere, it has been inferred that the southern hemisphere is colder than the northern, and this has been the accepted view ever since his charts were first published, in the year 1852; but, from the results obtained above, itis seen that the mean temperature of the southern hem- isphere is the greater of the two. If, however, we compare the values of ¢ in the preceding table for the two hemispheres between the parallels of 30° north and south, we find that the southern bem- isphere is the colder of the two between these parallels; but beyond the parallels of about 35° the temperaturgs of the southern hemisphere become greater than those of the corresponding latitudes of the northern hemisphere, so that the average temperature is also greater, as shown above. The cause of this is found in the unequal distribution of land and water in the two hemispheres; for we now know, both from theory and observation, that there is a constant, though very slow, inter- change of the water of the ocean between the equatorial and polar regions, which tends to diminish in some measure the difference of temperature between these regions, so that in the southern hem- isphere, where there is mostly water, the temperatures of the higher latitudes must be greater than those of the same latitudes in the northern hemisphere, and the reverse for the lower latitudes. The small differences above hetween the mean temperatures of the two hemispheres is perhaps only of the order of the possible errors of these results, so that we cannot infer that there is any real difference in the averages of the two hemispheres.* 21. We now come to the subject of atmospheric pressure on the different parts of the earth's surface, upon which the values of D,,P’ and D,P’ in the general equations of the preceding chapter depend. This pressure cannot be determined from theory, on account of the complexity of the equations and the uncertain element of friction entering into them, and we shall, therefore, en- deavor to determine it, so far as possible, from observation. The very valuable and exhaustive paper by Buchan on this subject, the ‘‘Mean Pressure of the Atmosphere and the Prevailing Winds over the Globe, for the Month and for the Year,” published in the year 1869, left nothing undone which could have been done at the time in the way of determining the atmospheric pressure from observation on the different parts of the globe; but since that time there has been so great an accumu- lation of barometrical observations from almost all parts of the world, that it does not seem proper, in our present researches on the subject, not to avail ourselves of these observations, at least in some measure, in determining this pressure still more accurately for all places from which additional and more recent observations have been obtained. It is also thought that the knowledge which we now have of the relations between the barometric pressures and the velocities and directions of the winds, obtained from theory and corroborated by observation, can be now used in laying down isobaric lines for those countries and the vast expanse of ocean from which we have none, or at least only a very few, observations, much more accurately than has been done heretofore, since much * See two papers on this subject by Dr. Hann iu the “ Zeitschritt der dsterreichischen Gesellschaft tiir Meteorologie,” Band vii, S. 261, and Band xii, S. 100, which were not seen until after the preceding results were obtained, and in which thése results are corroborated. If, however, the observations upon which the results obtained by Dr. Hann are based had been on hand at the time, the values of ¢ in Table V would most probably have been diminished a very little in the extreme southern latitudes, and then the results obtained for the average temperatures of the two hemispheres wight have been about equal. + Transactions of the Royal Society of Edinburgh, vol. xxv. 23 may be inferred with regard to the barometric pressures from the known forces and directions of the winds. We shall, therefore, undertake to make out new charts of isobaric lines, not for each month, but simply for the mean annual pressures of the globe, and for the two extremes of January and July in the northern hemisphere, using for this purpose all the observations on hand, except in some parts of Europe, where there is so great an accumulatiou of them that all were not con- sidered necessary for our purpose, since, in a general system of isobars for the whole globe, in which, from the scarcity of observations in many parts, there is much that is necessarily hypothetical and conjectural, it is not worth while to aim at extreme accuracy in certain comparatively very small parts of the surface of the globe. We shall also make out charts showing the annual variation of the atmospheric pressures over all parts of the globe, so far as this can be determined from obser- vation, so that, with the mean annual pressure and the annual variation, the mean monthly press- ures may be readily obtained, even with greater accuracy than they can be obtained from charts laid down from monthly averages, for it will be shown that all other neglected inequalities are gen- erally either insensible, or at least smaller than the probable errors of monthly averages. These charts will be given on polar projections of the two hemispheres, since each hemisphere contains a complete system of winds and barometric pressures which are similar, and because in such projec- tions there is less distortion of the parts in the higher latitudes, where, in the northern hemisphere, there are some features in the winds and barometric pressures, and their annual variations, which it is desirable to have more accurately represented than they can be in the usual projections. 22. For our purpose, we have put for each station for which we have observations of the pressure P— (26) . . . . P=B.+ B, cos(g—«:) + B, cos (2 g—e,) + &e. = B, + M,cos¢+ N, sin g + M,cos29 + Nz sin2 9 + &e. a in which— ; M, = B, cos ¢,; N, = B, sine, M, => B, cos &o5 N, = B, sin €9 N N. tan «; = —*; tan a= a. M, M N M. N. Bis = or 22s By Sor COS €] sin 4] COS €2 SiN é5 In the preceding expression of P — By = the mean annual barometric pressure ; B, = the coefficient of the annual inequality ; B, = that of the semi-annual inequality ; yg = an angle increasing in proportion to the time at a rate, on the average, of 30° per month ; e, = a constant which is the value of ¢ at the time of the maximum of the annual in- equality; and 2, =a constant which is the value of 2 ¢ at the time of the maximum of the semi- annual inequality. If we put S, for the monthly mean of the barometer for the mth month and the epoch of ¢ in the middle of December, we have, by regarding each month as the twelfth part of a year,— 2,8, 23 —(Si-S: 30° (; a) cos 60° (s -S: ) a) es 6N,= S:—S:) gin 309 +58) sin 60° 4+ (8-5, ) “2 ~\S;—8n S4—Bro S; —Sy Sg 81-8, Ss, —S, 60° a 2 GN, ={ %&—Ss sin 60° 6M=( sg, POO’ Ts, 8 )) | ss. Sn = S Ss—Si 24 With the values of M,, Mz, Ni, and N,, given by these formule, we get from the preceding expressions the values of B,, B,, «, and ep. The preceding expression of P is simply a transformation of what is usually called Bessel’s formula, in which sines are used, instead of the cosines in the form of expression here adopted. This foim is preferable to the other, since from the values of ¢, and e, the times of the maxima of the in- equalities are more readily obtained, and this is made still more convenient by having the constants e, and ¢, of the angles in the expression negative, as is usual in all tidal expressions of this sort, for in this case we have merely to add to the assumed epoch of the time-angles the time which is required for the angles to change by the quantity ¢, or e. in order to have the time of the maximum of the inequality. For the annual inequality, we divide ¢,, expressed in degrees, by 30, ang for the semi- annual inequality <2. by 60, in order to get the time in months by which the maximum follows the assumed epoch of the middle of December. The preceding formule for obtaining M,, M,, N,, and N, follow directly from the determination of these quantities by the method of least squares in the special case of twelve equal divisions of the period of the principal inequality. The values, therefore, of the constants B,, Ba, «1, and «, thus obtained are the most probable values of these constants. The form of (26) can likewise be applied to represent the temperature of the earth or atmos- phere at any place, in which case the principal inequality has an annual period. In the case of both atmospheric pressure and temperature, more than two inequalities need not be considered ; for if those of a lower order have sensible coefficients, they are of an order much smaller than the probable errors of the constants determined from observation, unless the period of observation embraces a very long series of years. 23. More convenient formule for the determination of the constants in (26) may be obtained upon the principle of averages, and the results obtained upon this principle, though differing a little from the most probable results given by the preceding formule, are sufficiently accurate in all meteorological researches, and the differences between the results of the two sets of formule will generally be found to be much less than the probable errors of the determinations. If we put— a = the average of all the observations of P within the limits of g = ¢, and gy = ¢,, g' = the mean value of ¢ between the limit g’ and ¢”, by considering only the first two inequalities in P (26), and supposing the observation ¢ equally distributed in time, we have— (27) 2 6. «+ @= By +k By cos (g/ — 1) + h By cos (2 g! — «) in which— ames 8 eo: ene sin 5 (¢ — ?,) asin, © (28) 2. 6 2 ee ee Pi — Fy ~2nr+e vy — Sie (Pn — 2) sine Pu — $1 Ann +e c in the last form of expression of k and k’ being the excess of ¢,,— ¢, over an equal number of periods 2 n x of the inequality.* Where ¢,, — ¢, = 2 =, that is, where any number of even multiples of the period of the angle ¢ is used in taking the average, a, the arc c, and consequently k and k’, vanish, and we havea = By. The same is sensibly true in the case of a long series of observations, since ¢, and consequently 2 sin 5 ¢ and sinc, then becomes very small in comparison with 2 = in the denominator, in which n denotes the number of periods of the inequality used. If we take the observations within the limits of the half-period ¢,, — ¢, =, since sin z = 0, we then have k’ = 0, and (27) becomes— a = By + kB, cos (¢/— «1) * The preceding results of this section are demonstrated in my Tidal Researches, p. 158, published as an appendix to the United States Coast Survey Report for 1874. 25 The same is of course true if we take any number of successive half-periods within the same limits of y. If we take the average for the other alternate half-periods, we get— a= By — k B, cos (¢’— «) If we therefore reverse the signs of the observations for the latter, we get for the average— (29). 6. 6 ee ee ee ww GH KB, COS (g/— 1) =k M, cos g/+ kN; sin ¢’ If we put 8, = the monthly average of the nth month, and assume the epoch of the augle ¢ at the beginning of January, and take the limits ¢, and ¢,,, so that g’ = 0, and put— A=8, +8 +8; B=S, +8s + 8 C =S, +8; + 8 D=Syp + 8n + Sp (30) . we get from (29), since in this case cos g/ = 1 and sin ¢g/ = 0,— 12a@=12kKM,;=Syp+ Sit S2+8:+8).+ 83 — S,- Ss; — Ss — S,— S;— 8S =(A+D) —(B+0C) If we now assume the limits so that ¢’ falls on the Ist of April, we have in (29) sin g’=1, and cos g’ = 0, and we get— 12a=12-¢N,=8,+8.+8;+5,+ 5; + 8 — 8; — 8S. — 85 — Sip — Si — Se = (A + B) —(C + D) 1 | bo With the value of k == from (28), since ¢,, — ¢, in this case 1s equal to 3 z, these give— Tv eee (64) N, =0.1309(A + B—O-—D) ; With the values of M, and N,, we obtain B, and e by the formule in § 22. If we now take the averages for intervals of the angle, or ¢,, — ¢, equal to 4z, or 90°, changing the signs of the observations for each second alternating interval, it is readily seen that the first inequality iv (27) is eliminated, since in this case (28) gives k = 0, and we get, instead of (29),— (32). 0. 6 6 eee ee GEM, Cos 2g’ + I’ Nz sin 2g’ In this case, the half-period of the angle embraces only three months; and, in order to obtain the value of M,, we must use intervals of two mouths only, so as to have g¢’ fall in the middle of the interval, and we thus get, since cos 2 g/ = 1 and sin 2 g/ = 0,— 8a=S8k' Mz =Spe+ 8; —S3— Si + Se + Sy — Sp — Sin In this case, we have the range between the limits g,, — ¢g, = 60°, and hence we get from (28),— if = 80 60° _ 0,827 If we now take the limits of the alternate intervals g, and ¢,,, so that ¢’ of the first falls on the middle of February, we shall have sin 2 ¢/ = 1 in (32) and cos 2 g' = 0, and consequently — 12a4=12h/ N, =S, + &. + 8; — 8, — 8; — So + Sz + 8p + Sy — Sip — Su — Sve Since we have in this case ¢,,— ¢, = 90°, (28) gives— ; wai? gm iQ 26 We therefore get from the preceding expressions of 8 k’ M, and 12 k’ N,, with the corresponding values of k’ for each,— 2 ee { M, = 0.1511 [(S; — 83) + (Se — Sx) + (S7 — So) + (Siz — Sto)] ” N, = 0.1309 [(A + C) —(B + D)] With the values of M, and N, given by (33), we get, by the formule in § 26, the values of B, and e. It is very convenient in practice to obtain by (81) and (33) thie values of M,, N,, and M,, N,, from observation, and the values so found are very accurate when the expression of P (26) is so conver gent as to make the inequalities after the first two very small or entirely insensible, as is the case in meteorology. In a comparison of the values of B, for thirty cases, obtained from series of mete- orological observations by both of the methods which have been given, the average of the differ- ences by the two methods, taken without regard to signs, was only 0.13", the, maximum being 0.39™™, and in the case of the values of B, the differences were of the same order. Even in cases in which it is desirable to use the former more accurate formule of the preceding section, these latter will be found a very convenient check within very narrow limits. Where the range of angle ¢,,—¢,, belonging to a group of observations of which the average a is taken, is small, the values of k and k’ (28) do not differ sensibly from unity, and, by comparing (26) with (27), it is seen that the average can be used as the value of the function belonging to the middle of the interval, in which the value of ¢ is g’. If, however, the range should be consider- able, this average must be corrected in order to obtain the value of the function for the middle of the group of observations. From (26) and (27) we get— (84) . . . P=a-+ (1—k) B, cos (g/— «) + (L —X’) By cos (2¢/ —e) : The last two terms, therefore, will be small corrections to the average a iu order to get the value of the function P for the middle of the group, for which g= ¢’. For instance, if we had monthly averages, we should have ¢,, — ¢, = 30°, and (28) would give 1—k = 1 — 0.9886 = 0.0114, and 1 — k/ = 1 — 0.9549 = 0.0451, and the correction is— P — a= 0.0114 B, cos (g—e;) + 0.0451 By cos (2 g! — e2) The last term, on account of the smallness of By, is generally very small in monthly averages. When the range ¢,,—¢, is larger, of course these corrections hecome of more importance. 24, The constants in the last columns of the following table have been deduced, by means of the preceding formule, from the monthly means reduced to sea-level and the gravity of the parallel of 45°, contained in Rikatcheff’s paper entitled ‘ La Distribution de la Pression Atmosphérique dans la Russie @’Europe,” published in the Repertorium fiir Meteorologie, Band iv, Heft 1, 8. 46, 47. ~ 27 TABLE VI. , Pi | : ace, | Lati = atitude. Longitude. Altitude S g i : eee es i \ ge 1 x} ROM teeveerneeene | : Mahar eta a na Pat i | | pArohangel wvecsssesssctntsnseneesenenteneenesenn | E oo MEE raren eros . le ees ae wa) "a) "af ; t. Petersburg Shane tesa oes aes At : , s| mis| or] 5) , .. Boyolovak. .. 60 10] ; 24 57 slaee| 0 : 7 = Sn os a , ae 15 | 759.61) 05] O5+ 86 |! ‘or sens ress eet a 59 45} 60 1/ 1 ml alsa heehee FE ee ae ao % 93:71 16 | 761.6] 36 F 0:5 fo | Bee : , | a a oe ox ae 17| 759.7] 03] 41 us : oe ba ~3| 143 | 24 areata leaker anc aid ct nde, teed haat aie is ses ‘ =| mea] o2/ 7 3 = So oe) . a a2 |. 760.3] 02 0.3 oe ne i. ve ner a 26 |, 762.3) 44 | 17 r ul} 180 aGiaen 56 58 24 6 cat feet pe tee : ) «it m. | mm. | mm. | mm. o ° Armagh, Ireland. .........-..----2----222-- 54 21]—6 49 64 1L 752. 7 1.4] 0.9 191 312 Belfast, Ireland .......----+----+--ee-eee eee 54 36] —5 56 oO} 11 759.0] 1.3] 10] 174} 315 Corky Welanid: « ceccidemces cee So elewe Se veig ese 51 53 )— 8 28 8} 11 759.5) 1.4] 1.0] 190] 347 Aberdeen, Scotland .. 57 9/—~2 7 34] 11 754.9] 1.8] 1.4] 170] 992 Glasgow, Scotland .............-206+ --ee2ee- 55 3/—4 18 55) 11 752.6] 1.5) 1.3] 179 | 296 Milne-Graden, Scotland . aeisbyatacoeetieewee 55 0} —2 12 31] il | 755.9) 16] 1.3] 173] 298 Liverpool, England .......-.....20-2-.-.-+-- 53 25 | —2 59 oy} i 759.2] 121 1.3] 178] 326 Norwich, England 2... .o.20-0-s006 ccnsoceoes 52 36 1 18 Oo; i 760.5] 0.6] 1.5] 198] 321 Helston, England ........ 50 7/—5 16] 32] 20 | 759.0] a6] o7!] 163] 49 Geneva, Switzerland 4h 12 6 9] 407] 25 726.4] 11] 11] 273 36 . Puvin Ttaly sesekesent Maeeisalapeewereiseta 45 4 7 41) 279) 74 | 7391] 1.7] 06] 214 56 : Rome, Italy...-....-..- dinidla sitelelemiommca ceo 41 54 12 28 Oo; 15 | 71.7} 0.2] 1.8] 301 58 ‘Malta, Tal ynisso: anu conn aerascaecs ielenade 35 54] 14 3h o| 6 | 7625] 03] 02] 276] 68 Bologna, Italy 4430] 11 211 4] 40 | 755.1] 0.6] 06] 350] 393 Krakau, Austria 50. 4 19 55] 216 19 742.5 1.3] 0.6] 290} 336 Kremsmiinster, Austria......---....--+-.--- 48 3 14 6} 283] 19 728.0) 1.1] 0.9] 285 20 Szegedin, Austria....-....... dies eomaccp iar 46 15 20 6 84] 12 754.0) 20] 0.2] 324] 312 Tesina, Austria .............22.-e0ceeeeee ees 43 11} 16 2]{ i9] 9 | 759.1] 0.5] 0.3! 300 0 Munich, Bavaria 48 9 11 34] S511] 10 | 716.0} 1.2] 1.1) 254 2 Kénigsberg, Prussia 54 43 20 29 22) 10 758.5} 0.1] 1.0] 225] 939 Datei, Pris seccsca. casas cecaseercenewns 54 21 18 41 9] 32 760.3] 0.4] 0.2 45] 148 Corfu, Greece ....-..-2--+e0- bs acuigduhdtcaidy 39 39] 19 si] of 6 | 761.8] 1.2] 0.4) gat] ary Barnaul, Russia 53 20 83 +57] 122] 19 749.3) 81) 1.5 15} 183 Jakutsk, Russia 62 2] 129 14 87] -13 |] 753.8) 7.2) 0.3 0; 120 Bogolovsk, Russia 59 45 60 2] 181] 26 741.4) 3.2] 1.0 1] 181 Ayansk, Russia...... 12.222 -eeee eee e ee ee eee 56 27] 138 26] (2) 2 | 163) 20] 20] 33] 168 ]........)...... Peterpaulshaven, Russia...........---.+---- 53 10] 158 32] (2) 1 | 753.8] 3.2] 23) 176] 56 ]........ 0.4 ‘Transactions of the Royal Society of Edinburgh, vol, xxv. 33 TABLE VII[I—Continued. Saint Helena...-------+-- aia a ivi nieaipemeneataisis 2 we é r ace 44 wef 6 a _Ie23 Place. egal 3 Bo. B. | B. | «. | |8 5 | 4B, ; 3s ee he g 3 eis : | . & % | #-l 3 # | 2/8 -&B A A | 4] ae : ; ; 0 f oo m. mm. mm. jmm. ae mm. | mm. Irkutsk, Russia .........2....2+. Beesewpere we s2 17] 192-11] 362] 15 | 7242] 77 |, 14] 14) 195] 760.1] 29 Udskoi, Russia : BS a, 54 30| 134 a8] (2) 1 | 7541) 59] 0.6] 27] 308 }.--.---)..0. Nerchinsk, Russia ‘5L 19] 119 36] 650}: 18 705.1 | 5.0 15, 3] 120 165.6] 5.4 Pekin, China:........ 22. ..000..20.000. 39 54) 116 26]. (2) | 14 | 759.2] 97] 1.0 Wf QO | ceccaeslacen ve Canton, China 93 12) 113 17] () | 10 | 159.3] 70] 11] 14] 330 Shanghai, China 30 4] 8 33] o| 2 | wi7] 72] 0.5] 6| 206 Tien-Tsin, China. 39 9] 117-16 9] 1 | Le] 76] 35] 45] 417 Hong-Kong, China 22 16] 114 10/ 11| 6 | 760.6] 61] 0.9) 15] 299 Macao, Pelew Islands 22 15] 113°36) () | 1 | 732) 531 03 8. 170 Hakodadi, Japan 41 48} 140. 47| 46] 4) 755.4] 23] 24] 13] 194], Mooltan, Hindostan.. ‘31 11] mm 33} 137| 6 | 745.7] &3] oO8 3] 213 Bombay, Hindostan : .18 54 72 «48 11} 14 WL1) 35] 04 5} 194 Madras, Hindostan.......-2-2.0c0002.-. ‘13 4] 80 19 8| ar | 784] 39; 04] 10] 69 Colombo, Hindostan...........2-...-.- bead 6 56] 79 50 o| 6 | 758.7] 09] 03] 352] 69 Trivandrun, Hindostan 8 31| 77 of 40] se] 41] 11] 04] 349] 62} Upernavik, Greenland.....:......... (72:48 |— 55 53; 5] 5 | 28] 15) 08] 160] 234 Jacobshaven, Greenland. . 69 12/—51 0 3 94 155. 6 22) 1.3) 159] 213 Godthaab, Greenland .. 64 to|- si 53] 5] 5 | 65] 25] 18] 174] 924 Baffin’s Bay (Arctic)...-...--.....---- 72 30 (Various) 0 1 155.6 | 3.7) 3:0] 146] 215 Van Rensselaer (Arctic)...........-.. rant 73 37/— 73 0 o} 2 | 7563] 19] 12] 90] 251 Port Foulke (Arcti¢).....: 7 18\—73 o|, 2) 1 | 25) 18] 35] 47] 267 Port Kennedy (Arotic) ........2..00..0.-06. 72 1\—94 Of O}.1 | 160.4] 39] 26] 72] 213 Boothia Felix (Arctic).....-.-- bac aedcaee 7 3/95 0 0} 2 | 70.5) 1.7) 17] 108] 257 Mellville Island (Arctic) ......-.-----.--++- .% 40/112 3] oO} 1 | 7584] 27] 18] 7] 281 Port Bowen ‘(Arctic)..---.....-.--------.---| 73 13 |— 88 54 o} 1 759.1] 28) 31 73) 235 Sitka, Alaska .... Brotis hte cine 56 50 [135 . 0 6| 17.) #47] 39] 0.6] 188] 36 Esquimaux Harbor.....---.--..-----+-----. 48 25 |—123 27 Oy 1 yf. 7633} 02) 08 “124 83 | Astoria, Oregon..----.--2---eeepeeeeeneeeeee ‘46 8|—193 48] (2) | 2} 7627] 09} 02] 147} 114 Saint John’s, Newfoundland <.-....-..-..--. 47 35 1— 52 43 0) 6 | 159.6] 25! 09]. 217] 301: Halifax, Nova Scotia....... Lives] 44 39/63 37] 0] 4 | 760.0] 22] 1.2] 214] 323° DOie-se2eccddvtos 44 39) 63 37) 0] 2 | 756.3) 1.6] 03] 190]. 239 Albion Mines, Nova Scotia............------ 45 34/— 62.42) 0} 10.) 7547] 0.9] 0.2.) 234] 148 } Quebec, Canada.........-- ated aes eee de 46 438 \— 71 12 o| 33.) 761.5] 1.0] 0.6] 354] 94.) 761.6]...... Kingston, Canada... 44°14 |— 76 31 0. 44 | 761.3) 0.7] 0.4] 353 105°] 761.2]... . Hamilton, Canada ' 43 15/— 79 57] 99] 11 | 753.5] 1.2] 0.3] 242 9| 7624! 0.4 Gardiner, Me -....2-.:2--0-e0eeeeeeeeeeeetee] 4411+ 69 46] 28] 5] 1570] 12] 0.5) 290] 350} 759.8) 0.1 Steuben, Me ..-.. 44 28 |— 67 50| 15] 6G | 759.6) 1.1] 0.7] 265) 19] 761.0]...... “Amherst, Mass _ @ ai 72 34) sl) 6 | 755.1] 16] 03] 355) 53) 7623] 03 New Bedford, Mass .--..--..-- oe vanity pinata! “41 39|/- 7 56) 28] 6 | 7595] 11].0.7] a77| 51] 76L9].-.... Nantucket, Mass......-.-.222--+-0006 Edegasen 4 t6|—70 6] 9] 6 | 76i.6] 1.0] 0&6] 289] 40) v021]...... Burlington, Vt 44:99 (— 73° 12] 106] 5 | LO] 11] 06 261} 20) LE! 06 Rochester, N.¥..-.-----0tee---00- sree cees 438 81 77 SL] 158) 4. | 746.7) 0.9} 06) 354) 5L) W6l1) Ld Harrisburg, Pa...-c2.--5-2 ' 40 16|- 76 50| 85] 6 | 755.5] 16 0.5| 314] 62) 762.9] 0.5 Washington, D.C ...-..---2002--20+- ' 38 36[-76 58] 22) 11 | 761.7) 1.5} 0.5 | 339} 51} %61.7)...... Savannah, Ga 2 wee cne eee cece eee terre een ee: ' 32 51-'Bl 7 13}. 6 763.3 | 1.4) 0.6) 345 88 | 762.5} ....- Jacksonville, Fla..------------2+0-00ee2 eee ' 30 .30|— 82 0 4] 6 | 7646) 16] 08] 359] 99) 7640) ..... + Columbus, Miss....--..------------- seme ees 33° 30 |—'88 29) 70 4-| 757.9) 20] 08) 347 30°] 763.5] 0.2 Glenwood, Tenn ..-.-2..-----eeee ee eee Leet] 36 98/87 13] 140] 6 | 751.0) 1.5] 0,8] 339] 55) 7641] 0.4 Cincinnati, Ohio ..-.-.- echacuarieaaik aumneih / 39 6 \— 84 28] 155}, 4 | 749.0] 1.8) 05] 345] 57] 7025) 0.8 New Harurony, Ind:.......----- a cccceealys gaa )—s7 50] 98] 6 | 7535] 16] 0.7] 346] 68) 7618] 05 Dubuque, Lowa .---.--- yaebe ea pais nedeaee P42 30-90 52] 207) 6 | 7447/13] 0.8] 325), 70] 7631) 1.1 Nassau, West Indies | 2 4/— 77 22 4| 6 | 7637) 09/ 1.3] 56} 20) 762.8) ..... Up Park Camp, Jamaica... eee f 18 OF W 56] 0] 6 | 761.7] 0.8] 0.8) 56) 50} 760.1}...... Georgetown, British Guiana .....-..-.. ---- 6 50/58 8 3| 11 | 760.4} 0.6] 0.6) 135). 7%) 7588 ]...... Cayenne, Freweh Guiana....--.------ veces] 4-56 |— 55 39 2] 6 | 760.0] 07) 04) 182] 90| 7582]...... Rio de Janeiro, Brazil tee |= 22 BT | 43 TY) «69, 6 | 757.9) 3.0) 0.5] 186] 42) 762.5 )...... Monte Video, Uruguay :....--:0.eceereeree- [= 94 54 | 58 383 8; 10 | 760.4} 20] O07] 196} 170) 760.4 ]...... —15 si 5 42 5 | 7648 | wewes| covers [eee well etecelesceneenes lee 5 34 27. The results of the following table have been deduced from the monthly means of barometric pressure given in the reports of the Chief Signal Officer of the United States for the years 1872-76 inclusive. The means as given in the reports are reduced to sea-level, but the results here given are also reduced to the gravity of the parallel of 45°. TABLE IX, Place. Latitude. | Longitude.| Altitude. Bo. By. | Bo | «&- eg. } p 4 o 4 Feet. mm.-|mm.| mm. | © ° Augusta, Ga.............- 33 28 | —8l 53 172 763.6 | 1.9] 0.8] 350 20 Baltimore, Md...-........ 39 18 76 36 45 1#63.2| 1.8] 0.2] 346] 33 . Boston, Mass 43 20 il 3 142 761.8 | 0.9] 0.4} 308 44 Breckenridge, Minn...... 46 16 96 38 966 762.1 | 39] 0.8 5 93 |, AUG, DN Ecce meen saw 42 53 78 55 666 W141) 1.2} 0.3] 351 | 126 Burlington, Vt......---..- 44. 29 73 «#11 241 761.5 | 1.6] 0.3] 329] -58), Cairo; Wl secs exceevercoses 370 89 0 367 763.3 | 26] 0.5| 346] 291 Cape May, N.J ........-.. 39 «(0 74 58 14 163.0 | 1.5] 0.4] 344 24 Charleston, 8.C...-..----- 32 45 19 57 61 763.5 | 1.6] 0.8] 353 20 Chicago, Ill.........-----. 41 52 87 35 663 761.9] 20] 0.5] 354 65 Cincinnati, Ohio ......-... 39 «6 84 30 ‘596 763.1) 2.4] 0.4] 346 38 | Cleveland, Ohio .......... 41 30; a1 36 688 wig | 1.5] 0.3] 358] 82] Davenport, Iowa........-. 41 30 | 90 36 603 702.4) 28] 0.6] 352 55 Detroit, Mich............. 42 18 f 83 0 644 15) 17] 0.5] 359 94 | Duluth, Miun............. ; 46 48 | 92 6 643 761.5] 23) 0.4 10 | 134 Escanaba, Mich........... 46 36; 8 u7 6 619, |. 7613] 1.4] 0.6 1| 104 | . Fort St. Michael’s, Alaska ! (2 years’ observations). . 63 28 lvl 45 0 759.8.) 21] 1.5 34.| 70 Galveston, Tex ...-.-..--. 29 19 94 46 39 762.3} 1.9] 0.8] 353 7 Grand Haven, Mich ...... 43°05 86 13 616 761.4] 16] 0.3] 350 90 Indianapolis, Ind ......... 39 42} 8686 G 17 762.2] 23] 0.3} 346| 57] Jacksonville, Fla .......-- 30 15; g 0 23 763.5] 16] 09) 2] nH ‘Keokuk, Iowa -- 40 18] 91 30 584 761.4] 28] 0.4] 353] 68 Key West, Fla............ 24 36] Bl 48 32 7623] 11] 1.0] 35] 95 ; Knoxville, Tenn ......-... 35 56 L 83 58 993 763.2 | 23] 0.6] 347 12 | Lake City, Fla, (3 years’ j observations) ....-...--- 30 «6 ] 82 42 |----....---- 762.9} 1.4] 0.9] 352 0 | Leavenworth, Kans....... 39 21) 94 44 813 761.5 | 3.6] 0.5] 350] 49 Louisville, Ky ...-- Rete aces 38 (0 85 25}; 496 762.5] 2.41 0.4] 348] 46 Lynchburg, Va.....-.---- 37 18 85 54 651 763.3] 21] 0.5] 344 12 Marquetta, Mich.......-.. 46 33 87 23 666 761.3] 1.5] 0.7 Q7 95 Memphis, Tenn .........-. 35 8 88 (0 299 763.5] 26] .0,5| 347 20 Milwaukeo, Wis .....--... 43 3 87 57 672 762.0! 1.6] 0.4] 346] 84 Mobile, Ala.....- Banectatery 30 42 87 59 39 763.6 | 1.9] 0.8 0 5 Nashville, Tenn .--.--.... 36 10 -86 49. 504 763.4] 2.6) 0,5} 346 24 New London, Conn ......- 41 22 2 9 38 762.5) 1.0] 0,3] 323 63 3 New Orleans, La.......--. 29 57 909 0 56 762.9] 1.8] 0,7] 358 5 New York, N.Y ..--....-. 40 42 4°61 166 762.7] 1.3} 0,4] 339 49 Norfolk, Va .....---.----+- 36 51 76 19 56 763.2) 1.4] 0.4] 330] 308 Omaka, Nebr... 41 26 96 0 1055 760.6] 3.6] 0.5] 356 62 Oswego, N.Y ...---....--- 43 28 76 35 299 761.7) 0.9} 0.5 | 343 94 Philadelphia, Pa ....... aa 39 57). 75 12 47 763.2{ 1.6] 0.3] 344] 35 Pittsburgh, Pa....-....-.. 40 32: 80 2 791 762.0] 2.2] 0,7] 357 63 | Portland, Me.......-...-. 43 40 ; 70 14 584 761.2} 0.4] 0,4] 997 56 . Portland, Oreg.....--.---- 45 30| 192 27 90 764.4] 0.4] 0,6] 45) 260 Punta Rassa, Fla....-...-. 27 «(0 82 18 17 763.2} 1.1] 0,8 24 24 Rochester, W. Y...----.---| 43 8 Tt 51 584 761.2) 1.2] 0.4] 340] 320 San Diego, Oal.........--- 32 44] 17 6 62 | 761.8! 1.8] 0.4] 37] 298 San Francisco, Cal........ 37 «48 122 26 60 762.5} 1.6] 0,2 32) 290 Savannah, Ga. - ae 32. =«5 81 8 7 763.6] 1.8] 0,8] 356 9 Shreveport, La.........--. 32 30. 93 45 229 762.5 | 22) 0.5] 359 1 Saint Louis, Mo .......-.. 33 37 90 16 557 762.5} 25] 0.5| 348 55 Saint Paul, Minn ......... 44 53 93 5 794 761.0 2.7 1,0 358 116 St. Paul’s Island, Alaska | ; | (4 years’ observations) .. 57 3] 170 0 755.9 | 3.0] 1.6] 159 73 Toledo, Ohio......-.------ 40 39 83 32 531 761.7! 1.9] 0.3] 346] 36 Vicksburg, Miss..-.....-. : (32 24 | 91 0 280 764.1) 24) 0.6] 350] 351 | ‘Washington, D.C......--- 38 53: 1 1 106 763.0} 2.0] 0.3! 355] 340 Wilmington, N.G......... | 34 11 —78 10 14 763.5 | 1.6] 0.7 | 353 0 30 28. With the values of By in the preceding tables, Charts | and II have been constructed, show- ing the mean barometric pressure, reduced to the gravity of the parallel of 45°, for the northern and southern hemispheres, by giving the positions of the isobars for each 2™™. These isobars represent the mean pressures very accurately in Barope aul the United States of America, the greater part of the North Atlautic, aud many other places where the observations suffice to lay them down accu- rately; but throughout all the interior of Asia, Africa, and South America, and the greater part of the great oceans, where there are but few observations, atid these not reliable in many cases on account of the uncertainty of altitude above sea-level, and the lack of comparisons of the barometers used with any standard of comparison, of course the true positious of these lines are uncertain, but nowhere entirely conjectural, since we can derive much aid from theory and analogy in laying down these lines for those parts of the earth’s surface for which we have few or no observa- tions. As reliable observations multiply, and are obtained for those parts of the earth for which we have yet no observations, of course the positions of the lines as laid down in tiese charts will be found to be somewhat in error for all places for which we have not yet sufficient observations to determine them; but it is thought that the errors in geueral will be found to be small. The arrows on these and the followiug charts denote the prevailing directions of the wind. These are given, not from observation, but from theoretical considerations of the relations between the gradients and the directions and velocities of the winds, to be explained in a subsequent part of this work. The winds, as represented on these charts, are the resultants of all the winds for the whole year, which can now be laid down more accurately from a knowledge of the isobaric lines than from observations, which in most parts of the earth consist merely in the observation of the: relative frequency of the winds from the different points of the compass. The prevailing direction of the wind, as obtained from such observations, may be very different from the resultant obtained by Lamnbert’s formula trom observations of the true velocitiesand directions of the wind through the year. 29. With the values of B, in the preceding tables, reduced to sea-level by means of the values of 4B, where the monthly means of the observations were not given for sea-level, Charts III and IV have been constructed, representing the coefficients of the annual inequality of barometric pressure over the whole globe. These coefficients are accurately represented by the charts for all portions of the earth where the observations were sufficient for their determination ; but, of course, there is the same uncertainty with regard to them where few or no observations have been made which there is with regard to the mean pressures. Where the signs of these coefficients as given on the charts are positive, the maximum of the barometric pressure occurs in the winter and the minimam in the summer. It is seen from the chart of the northern hemisphere that these signs are mostly positive on land and negative on the ocean, especially on the middle parallels of latitude. This arises from the higher tempera- ture of the air in summer and lower temperature in winter on land than on the ocean. The line of no annual inequality of barometric pressure passes over Norway and Sweden and a little east of London, touching upon France and Portugal, having its-most southern point in the middle of the Atlantic, a little south of the parallel of 20°, and then, curving northward, passes over the east- ern part of New England in America. On account of the great extent of continent and the great extremes of temperature in the interior of Asia, this coefficient of the annual inequality amounts to about 10™, or a range of 20™™ between winter and summer, while in America it amounts at the maximum to only about one- third us much. This difference between Asia and North America does not depend so much upon the difference in the extremes of temperature of the two countries, which is inconsiderable, as upon the difference in the extent of the two continents. The lines on Charts III and IV represent the gradients which in winter have to be added to the gradients given on Charts I and II to obtain the gradients for that season, and these in summer are completely reversed, aud hence the steeper these gradients the greater are the monsoon infla- ences in the different parts of the globe. These, it is scen, are very great in the southern part of Asia. In the southern hemisphere, on account of the small extent of land and the small range of temperature between winter and sammer, the coefficient of the annual inequality of barometric pressure is small and negative so far as we have observations to determine it, and nearly the same : 36 in all longitudes on the same parallel of Jatitade. Its being negative shows that the maximum occurs in winter, that is, during the suinmer of the northern hemisphere. About the parallel of 559, this coefficient seems to vanish, beyond which it probably becomes positive, making the maximum of pressure in the summer as in the northern parts of the Atlantic and Pacific Oceans of the northern hemisphere. 30. The values of « in the preceding tables are very various, depending upon the want-of sufficient observations in many cases to eliminate the abnormal inequalities and bring out the true value, especially in such cases as give a small value of the coefficient B,, which is often less than the possible error of observation, when the value of ¢, is of course indeterminate, ¢ and may have any value whatever. Taking the average of all the values of ¢, in Table VI belonging to coefficients greater than 3.0", we get «, = 9°, which makes the maximum of barometric pressure in Europe occur about the 9th of January, a little carlier than the minimum of temperature. This is perlaps the most probable value of ¢, for all stations in Europe, the variations from this value of in the different stations being merely possible errors, of observation, though they may possibly depend in some measure upon local causes. To this value, the values of ¢, for other places in Europe in the other tables seem also to point in cases in which the coefficient is sufficiently large for the values of <, to be determined approximately from observation. / Where stations have great altitude above sea-level, as in the case of Héhenpeisenburg in Table VII, the value of ¢, is such as to make the maximum of the barometric pressure occur in summer instead of winter, in which case we can change ¢, by 180°, and consider the value of B, as negative. Applying the correction then in the column headed by 4B, to reduce this coefficient to sea-level, it becomes positive, and makes the maximum of pressure fall in the wiuter. In the case of Hohen- peisenburg, we get — 1.9™™ + 3,32" = 1.4™™ for the coefficient of aunual inequality, with the value of ej = 2289 — 180° = 48°, making the maximum occur after the middle of February. In applying the reduction to sea-level, 4 B,, itis supposed for convenience that the maximum of pressure coincides with the minimum of temperature, which, we have seen, is not strictly correct, but the errors in these small reductions arising from this cause are wenerally very small. : In all cases in which the values of ¢, are such as to throw the maximum of barometric pressure into the time of summer of the northern hemisphere, as in the northern parts of the Atlantic and Pacific Oceans and in the southern hemisphere geuerally, the values of B, as entered:in the Charts | HI and IV are considered negative, and in all such cases the values of ¢, must be diminished by 180°, or the negative sign of the coefficient on the chart changed to the positive sign if used with the average value of «, given by the tables in such cases, which does not generally differ much from 200°, throwing the maximum of barometric pressure in July. If we take the average of all the values of ¢, in Table IX, deduced from the observations of the Signal Service of the United States, giving them weights in proportion to the magnitudes of Bi, and excluding the stations on the Pacific coast, we get ¢; = 353° for the stations north of the par- allel of 40° and e; = 3519.1 for the stations south of that parallel; and hence we may put for the United States, except the Pacific coast, e; = 352°. This makes the mabeint ts of barometric pressure occur about the 23d of December, and about sixteen days earlier than. in Europe, and in both places considerably earlier than the time of the minimum of temperature. This is most probably caused by the greater amount of aqueous vapor in the atmosphere in the spring than in the fall, which causes the maximum of barometric pressure to be earlier. ‘ 31. In addition to the annual inequality of barometric pressure, there is 3 also @ very small semi- annual inequality, as may be seen from av inspection of the columns in the preceding tables headed by B, aud ¢. Tae values of B, as given are mostly of the order of the probable, or, at least, pos- sible, errors of the results, as may be seen from the scattering. values of eg, and hence do not indi- cate real terms; but if we examine all the larger values of Bz, we find that the corresponding values of ¢, are such as to indicate real terms, and give a maximum of inequality in the middle of winter and summer, and a minimum in the spring and fall. The coefficient of this. inequality seems to be gener- ally Jess than 1.0™™, but in some places, especially toward the north pole, it appears to be much greater. The average. of these coefficients in Table 1X, from the observations of the Signal Service of the United States, excluding the stations on the Pacific coast, give for the stations south of” the parallel of 35°, B, = 1.02", and the corresponding average of the values of e, giving each weight in proportion to the magnitude of the coefficient, is 8°, indicating maxima about the 4th of January and July. Inthe same manner, we get for the stations between the parallels of 35° 37 and 40°, B, = 0.53", and «,=19°.6. For those stations north of the parallel of 40°, we get B, = 0.48™" and «, = 45°.4. The coefficient, therefore, seems to diminish in the United States with the increase of latitude, and the times of maxima to become lator, being, in the northern part of the United States, about the 23d of January and July. 82, If we take from Charts I and I[ the mean barometric pressures for each fifth parallel of latitude and each tenth degree of longitude, and take the averages with regard to the different longitudes, we get the results contained in the first column headed B, in Table X, which are the mean pressures for the corresponding latitudes in the first column. It is seen that there is a minimum pressure about 6° north of the equator, a maximum in the northern hemisphere near the parallel of 35°, and in the southern hemisphere near the parallel of 28°. There is also a minimum at the parallel of 65° in the northern hemisphere, arising from the two great depressions of barometric pressure in the northera parts of the Atlantic and Pacific Oceans. From the differences of Bo, with the irregularities a little smoothed off, the gradients in the column headed G are obtained, which express, in millimeters, the differences of barometric pressure, corresponding to a distance of one degree of a circle having the mean radius of the earth, or 111111 meters. By taking the averages of all the values of B,, taken from Charts III and IV, for each tenth degree of longitude, we get the results contained in the fourth column of Table X, from the differ- ences of which, smoothed off a little, we get the corresponding gradients in the column headed by g. These latter results, added to those of the mean pressures, give the pressures and gradients for January, and subtracted give those for Jaly. From what we have seen in § 30, the maxima and minima for the average of all longitudes occur about the first of these months, and before the maxima and minima of temperature. TABLE X. Annual mean. Annual inequality. January. July. Lat. : Bo G. B. g- Bo. Ga. Bo. G. 2 mm mm. mm. mm. mm mm. mm. mm, 80 760.5 | ......--- —0.06 }........- 760, 4 | ceezjeicses TOC | sweeasees 5 760. 0 —0. 19 +0. 19 +0. 04 760. 2 —0.15 758. 8 —0. 23 70 758. 6 —0. 14 0. 36 0. 05 759. 0 —0. 09 758. 2 0.19 65 758. 2 +0, 01 0. 63 0. 06 758. 8 +0. 07 757. 6 —0. 05 60 758. 7 0,15 0.97 0. 06 759. 7 0. 21 757.7 +0. 09 55. 159.7 0. 20 1, 26 0. 05 761.0 0. 25 758. 4 0.15 50 760. 7 0.18 1.41 0. 03 762.1 0. 21 759. 3 0.15 45 761.5, | 0,15 1.53 0. 02 763. 0 0.17 760.0 0. 13 40 762. 0 +0. 07 1.61 +0. 01 7163. 6 +0. 08 760. 4 +0, 06 35 162.4 —0. 03 1. 66 0. 00 764.1 | ~—0. 03 760.7 —0. 03 30 761.7 0. 18 1. 66 —0.01 763. 4 0.19 760. 0 9.17 25 760. 4 0. 25 1.61 0. 03 762. 0 0. 28 758. 8 0. 22 20 159. 2 0.21 1,41 0. 06 760. 6 0, 27 157.8 0.15 15 7158. 3 0.13 1. 05 0. 09 7159. 3 0. 22 157.3 —0. 04 ‘ 10 157.9 —0. 03 +0. 50 0.11 758. 4 0.14 757. 4 +0. 08 5 758. 0 +0.01 —0. 05 0. 12 758. 0 0.11 157.9 0.13 0 758. 0 0.04 0. 63 0.12 757.4 —0. 08 758. 6 0. 16 —5 758. 3 0,11 1.18 0.11 7.1 0. 00 759.5 0. 22 10 759.1 0. 20 1.70 0. 08 157.4 +0. 12 760. 8 0. 28 , 15 160. 2 0. 26 2. 00 0. 06 758. 2 0. 20 762. 2 0. 32 20 761.7 0. 29 2. 22 —0. 03 7159. 5 0. 26 763.9 0. 32 25 163. 2 +0, 18 2. 36 0. 00 760. 8 +0. 18 765. 6 +0. 18 30 763. 5 —0. 08 2. 22 +0. 03 761.3 —0. 05 765.7 —0. 11 . 35 7162. 4 0. 30 1. 85 0. 06 760. 6 0, 25 764, 2 0. 35 40 160.5 0. 51 1.41 0.07 759.1 0. 44 761.9 0. 58 45 157.3 0. 73 1.00 0. 09 756. 3 0. €4 158. 3 0. 82 50 153, 2 0. 91 —0. 50 0. 10 752. 7 0. 81 153.7 1.01 55. 7148.2 0. 97 0. 00 4+-0. 10 748, 2 —0. 87 748, 2 —1.07 60 743.4 65 739.7 —70 738. 0 2 38 With the values of G in this table, the values of D,P contained in equations (15), and in expressions deduced from them in the following chapter, are readily obtained. 33. Chart V shows, by isobaric lines, the mean pressure of the atmosphere in the northern hemisphere for January, in millimeters, reduced to the gravity of the parallel of 45°, and, by arrows, the prevailing directions, or rather the directions of all the resultants for the month. These latter are inserted, as in the case of Charts I and II, from theoretical considerations of the relations between the winds and the isobars, and not froin results deduced from actual observations. Chart VI shows the same for the month of July. The epochs of maxima and minima on the earth’s surface generally, from what has been stated, are nearly the 1st of January and July respectively, and not at the times of the least and greatest temperatures. . On the first of these charts, there are two areas of great barometrical depressions over the northern parts of the great oceans, and two areas of high barometer over each of the continents, and consequently having the isobars mostly crowded closely together, with corresponding strong prevailing winds. On the last of these charts, for July, in consequence of the reversal of the annual inequality, it is seen that these areas of low and high barometers are very much smoothed off, and consequently the isobars are much separated, with corresponding small velocities of the prevailing or resultant winds; for the winds in the same latitudes are very nearly inversely as the distances between the isobars, as will be explained in the second part of this work. . As the barometric pressures given on the chart are the observed pressures reduced to those of the gravity of the parallel of 45°, in comparing observed pressures in any part of the world with those given by the charts, the same reduction must first be made. The part of gravity depending upon latitude is expressed by the last term in the expression of g in (13). The effect upou the height of the mercurial coluinn is inversely as that upon gravity, and hence the observed column is too low toward the poles and too high toward the equator. The correction, therefore, to be applied to the observed height for the pressure of 0.76™ is 0.76™ x 0.00284 cos 2 0, which can be used for all parts of the earth’s surface without material error. This reduction is given for each fifth degree of latitude in the following— TABLE Latitude. | Reduction. || Latitude. Reduction. Latitude. | Reduction. g mm. 9 mm. 2 mm. 0 — 2.16 30 — 1.08 60 + 1.08 5 2.12 35 0. 74 65 1.39 ; 10 2. 03 40 — 0.37 10 1. 65 15 1. 87 45 0. 00 vis) 1. 87 20 1.65 50 + 0.37 60 2.03 25 — 1.39 55 + 0.74 65 | +212 39 CHAPTER III. THE GENERAL MOTIONS OF THE ATMOSPHERE. 34. Under the head of “The general motions of the atmosphere” are included all those motions which extend as a system over the whole globe, aud depend upon differences of temperature be- tween the equatorial and the polar regions at all seasons, and hence they comprise not only the mean motions of the atmosphere, but likewise the changes in these motions depending upon the seasons; but they do not include those motions or disturbances depending upon permanent differences, for the time, of temperature in different longitudes, upon local disturbances of temperature or of density from any cause, or upon the irregularities of the earth’s surface. The conditions to be satisfied in this case are those of equations (15), in which, since differences of temperature in Jongitude are not considered, D, log a’ vanishes, and consequently the last term of the secondequation. The complete solution of these equations is impossible, both on account of their complexity and the uncertain element of friction enterivg into them, the laws and the amount of which are unknown. Many important results, however, may. be deduced from their consideration and solution in special cases, from which approximate results may be obtained by neglecting the effects of friction, and the latter, with the aid of observation, may be shown in most cases to be very small. If the temperature and amount of aqueous vapor upon which a depends were the same over all parts of the earth’s surface, D, a! aud D, a! in (15) would vanish, and it is readily seen that the con- ditions of (15) in this case are satisfied with D,u = 0 and D,v = 0, and consequently with a state of rest and of uniform pressure over the whole globe. And if theatmosphere were set in motion by any external impulse, this motion, in the case of friction, would be speedily destroyed, and a state of rest ensue. There can be no winds, then, without a disturbance of the static equilibrinm by means of a difference of temperature or of aqueous vapor in different parts of the atmosphere. 35. In the case of no friction, where « is indepeudent of longitude, it is evident that P’ is like- wise independent of longitude, and the first member of the second of (15) must vanish, as well as the last two terms of the second member, and the equation is reduced to— 0= D2 +2 cos 0(n+ D,w) D,u Since we have u = r D,¢@ and v =r sin 6 D, ¢, this equation may be expressed in the following form :— 2 sin 0 cos 0 (n + D,g) D,¢ + sin? D? 9 = 0 The integral of this equation is— sin? 6 (n+ D,¢) =e in which cis a constant depending upon the initial east or west velocity, or value of D,¢, of the particle supposed to be not influenced in its motions by contiguous parts, as implied by putting F, = 0 in the original equation in (15). If we put 0 and v for the initial values of 6 and D,¢, we have— = c = sin’ 0 (n+ v) We shall therefore have, if we suppose the particles to have such an action upon each other as to reduce, in time, the motions of all the particles of the atmosphere upon the same parallel of lati- tude to the same, and that there is no resistance between the earth’s surfice and the atmosphere, Git G58 Boe wee sin? on + Die) = fo=s(ntonm in which m is the mass of the atmosphere, and— v= =f. sin? ov MST m 40 If the initial state of the atmosphere is that of rest relative to the earth’s surface, we have v, and consequently v’, = 0. . The first member of (37) expresses in terms of the earth’s radius the sum of all the areas de- scribed in a unit of time by a line drawn from the earth’s axis to each particle of the atmosphere, and this sum must always remain the same since no mutual actions of the particles upon each other can change it, and the velocity of each particle at the same distance from the earth’s axis, that is, upon the same latitude, must always be the same after they have been brought to this state by their mutual actions upon each other. We shall then have ; 2n (n+ v’) Osa aun. EDS ante : The first member of this equation represents the angular velocity of a particle of atmosphere around the earth’s axis depending upon the velocity of the earth’s rotation x and the angular velocity D, ¢ relative to the earth’s surface, and this velocity, it is seen, as the particle moves toward or from the pole, must be inversely as sin?0, and consequently inversely as the square of the distance from the axis of rotation, and this is independent of any law governing the motion toward or from the pole, just as in the case of the planets, or the motions of any free body controlled by a central force, whatever may be the law of that force. “Multiplying both members of (38) by r sin 4, it becomes the expression of the linear velocity. From (38) we get, in the case of a state of initial rest relative to the earth’s surface, in which case v’ = 0,— oy... 9 Dee (gaia *)* D,v = Deg= rsin0D,g= rn sag sind ) ‘The first of these expresses the. angular and the second the liuear velocity of eastw ard motion relative to the earth’s surface. If we put D, g = 0, the first of (39) gives— CC sin d= 5 from which we get 6 = 54° 44’, corresponding to the parallel of 35° 16’, where there is no east or west motion of the air. All velocities between this parallel and the pole are positive, and those between this parallel and the equator negative. If we substitute the preceding value of D,¢ in the first of (15), and neglect the effect upon P’ of a difference of density, and of the inertia of the fluid represented by D?2u, we get, since r D, log P’ is equal to De log P’, and F, vanishes in the case of no friction,— 1 1 — 72 y2 Gj 4 — 1