A METEOROLOGICAL TREATISE 
 
 ON THE 
 
 o n . _ / c 
 
 Circulation and Radiation 
 
 IN THE ATMOSPHERES OF 
 
 THE EARTH AND 
 
 OF THE SUN 
 
 BY 
 
 FRANK H. BIGELOW, M.A., L.H.D. 
 
 Professor of Meteorology in the U. S. Weather Bureau, 1891-1910, and in 
 the Argentine Meteorological Office since 1910 
 
 FIRST EDITION 
 FIRST THOUSAND 
 
 NEW YORK 
 
 JOHN WILEY & SONS, INC. 
 
 LONDON: CHAPMAN & HALL, LIMITED 
 1915 
 
Copyright, 1915, by 
 FRANK H. BIGELOW 
 
 PUBLISHERS PRINTING COMPANY 
 207-217 West Twenty-fifth Street, New York 
 
INTRODUCTION 
 
 METEOROLOGY as a science has failed to make progress 
 toward definite results for one fundamental reason. In a non- 
 adiabatic atmosphere the terms of the general equations of 
 motion, as computed from the ordinary prescribed formulas of 
 thermodynamics, do not balance as required. There are two 
 errors in the discussion: (1) There is a mixture of the non- 
 adiabatic and the adiabatic systems, and (2) the important 
 radiation terms have been omitted from the general equations. 
 More specifically, for the Boyle-Gay Lussac Law, P = p T R y 
 to be satisfied, at every point, it has been customary to borrow 
 R = gas constant from the adiabatic system, and apply it in the 
 non-adiabatic atmosphere. For example, three well known 
 treatments follow: 
 
 Bigelow. v. Bjerknes. Margules. 
 
 W*T Pl=(Tl\T^ fi./ 
 
 Tj P, \TJ P Q ~ \T 
 
 P* \T PQ \TJ Po \T U 
 
 ~ 
 
 = Constant RI = R = Con. 
 
 Margules' system is adiabatic, v. Bjerknes' is partly adiabatic 
 and partly non-adiabatic, Bigelow's is strictly non-adiabatic. 
 
 Ta-T ,_Cp. 
 
 TX-TO " Cu 
 
 Now it is true that each system satisfies P = p T R, but the 
 individual values of PI, 7\, pi, RI, are very different in the three 
 systems for the same initial values P , T , P O , RO, and on applying 
 them to practical observations the systems that are not strictly 
 non-adiabatic break down as regards computed and observed 
 values which should be in agreement. We can easily see the 
 
 iii 
 
 489536 
 
IV INTRODUCTION 
 
 separate consequences by the following equations that are readily 
 demonstrated: 
 
 Adiabatic: g fa - Zo ) = - Cp a (T a - To) = - Pa ~ - 
 
 g fa ZQ) = ^1 Cp a (Ti TO) = - * 
 
 Pad 
 
 Non-adiabatic: g fa - z ) = - HI Cp lQ (7\ - T ) - HI (Cp a - 
 
 g ( Zl -Z Q ) = - -^ - K (?i 2 - <?o 2 ) - (Qi - Co). 
 
 Gravity = Pressure -f- Circulation + Radiation 
 term. term. term. term. 
 
 In the adiabatic system by definition R a and Cp a are constant, 
 and there are no circulation and no radiation. In the non- 
 adiabatic system there are both circulation and radiation, but 
 these depend upon the departure of the specific heat from the 
 adiabatic value 
 
 ni (Cpa ~ Cpio) (7\ - To) = K (? 2 i - ? 2 o) + (Qi - <2o) 
 
 Hence, if R is constant it is a contradiction in terms to discuss 
 problems of circulation and radiation in the adiabatic or partially 
 adiabatic systems, as has been universally the method. 
 The facts of observation, furthermore, conform to 
 
 r> p 
 
 - ni Cpio (Ti - To) = -- > 
 
 and this implies that circulation and radiation are required to 
 make up the deficiency in respect to the gravity term between 
 two strata z\ and Z Q . Finally, the radiation term (Qi Q ) is 
 usually very much larger than the kinetic energy term ^ (qi 2 
 qf) y but it has never been incorporated in the meteorological 
 equations. 
 
 For these reasons the author has spent much time while in 
 the United States Weather Bureau, and especially while in the 
 
INTRODUCTION V 
 
 Argentine Meteorological Office, 1910 to date, in devising a 
 simple adjustment of the thermodynamic adiabatic equations, 
 found in all treatises, to an exact and practical form of computa- 
 tion which will adapt them to the non-adiabatic system prevail- 
 ing in the atmospheres of the earth and of the sun. The following 
 Treatise sets forth this new method of discussing the meteorologi- 
 cal problems, with sufficient detail to enable the reader to utilize 
 the formulas in practical computations. It contains the solution 
 of a number of problems that have heretofore been intractable 
 along the old lines of procedure: 
 
 1. The diurnal convection and the semi-diurnal barometric 
 waves, with the radiation. 
 
 2. The pressures and temperatures in cyclones and anti- 
 cyclones, with the circulation and radiation. 
 
 3. The thermodynamics of the atmosphere from balloon 
 ascensions to great altitudes. 
 
 4. The thermodynamics of the general circulation. 
 
 5. The distribution of the radiation in all latitudes and alti- 
 tudes to 20,000 meters. 
 
 6. The "solar constant" of radiation and the conflicting 
 results from pyrheliometers and bolometers. 
 
 7. The discrepancy in the absolute coefficient of electrical 
 conduction as derived from the several apparatus for dissipation, 
 and for the number and velocity of the ions. 
 
 8. The diurnal magnetic variations in the lower strata of the 
 atmosphere. 
 
 9. The non-periodic magnetic variations in their relation to 
 the solar radiation. 
 
 10. The magnetization and electrical terms in the sun at 
 very high temperatures. 
 
 I wish to express my appreciation and gratitude to my friend 
 and colleague, Dr. Walter G. Davis, Director of the Argentine 
 Meteorological Office, for his courteous co-operation in this work, 
 and in memory of our good old Cordoba days. 
 
 FRANK H. BIGELOW. 
 
 CORDOBA, ARGENTINA. 
 September, 1915. 
 
TABLE OF CONTENTS 
 
 CHAPTER I 
 
 PAGE 
 
 METEOROLOGICAL CONSTANTS AND ELEMENTARY FORMULAS .... 1 
 
 The Status of Meteorology 1 
 
 The Constants and Formulas of Static Meteorology 9 
 
 Three Series of Constants in Three Systems of Units 12 
 
 Variations from the Standard P , Po, RO, T 0t for the Same Point and 
 
 for Different Points on the Same Vertical Line 16 
 
 The Acceleration of Gravitation 18 
 
 The Density of the Atmosphere as a Mixture of Several Constituent 
 
 Gases .20 
 
 The General Formulas for the Mixture of Gases 23 
 
 The Inner Kinetic Energies, Work and Heat 26 
 
 The Fundamental Laws of Physics 28 
 
 The Kinetic Theory of Gases for the Atmosphere 30 
 
 The Temperatures and Temperature Gradients Observed at Different 
 
 Elevations in the Free Air 32 
 
 The Temperature Gradient in a Plateau 34 
 
 The Integral Mean Temperature and Gradient 36 
 
 The Virtual Temperature 37 
 
 /d T 
 
 -jr 37 
 
 The General Barometric Formula 39 
 
 Corrections to the Barometer 41 
 
 Examples of the Barometric Reduction Tables . 46 
 
 CHAPTER II 
 
 THERMODYNAMIC METEOROLOGY 50 
 
 General Formulas for the Computation of P, p, R, from the Ob- 
 served Temperatures T in a Free Non-Adiabatic Atmosphere . 50 
 
 The Adiabatic Equations 52 
 
 The Working Non-Adiabatic Equations 53 
 
 The Variable Values of n = ^ 57 
 
 The Differentiation of (172) 58 
 
 The Two Laws of Thermodynamics 64 
 
 The First Law of Thermodynamics 65 
 
 Fundamental Equations and Definitions 68 
 
 The Second Law of Thermodynamics 70 
 
 The Reversible Process 71 
 
 The Irreversible Process 71 
 
 Carnot's Cyclic Process 72 
 
 vii 
 
Vlll CONTENTS 
 
 PAGE 
 THERMODYNAMIC METEOROLOGY Continued 
 
 Cyclic Process for Vapors at Maximum Pressure 73 
 
 The Second Form of the Equations for Latent Heat . 75 
 
 Specific Heats 76 
 
 Examples of the Thermodynamic Data 77 
 
 Application of the Thermodynamic Formulas to the Non-Adiabatic 
 
 Atmosphere 80 
 
 Application of the Thermodynamic Formulas to Various Meteoro- 
 logical Problems 90 
 
 The Isothermal Region 91 
 
 The Diurnal Convection and the Semi-Diurnal Waves in the Lower 
 
 Strata 98 
 
 The Thermodynamic Structure of Cyclones and Anticyclones . . . 104 
 
 The Planetary Circulation and Radiation 113 
 
 The Thermodynamic Tables of the Planetary Circulation and 
 
 Radiation 118 
 
 CHAPTER III 
 
 THE THERMODYNAMICS OF THE ATMOSPHERE 135 
 
 The Co-ordinate Axes 135 
 
 The Co-ordinate Velocities and Accelerations 136 
 
 The Constituents of the Force in any Direction 136 
 
 The Forces of Inertia, Expansion, and Contraction 137 
 
 The Forces of Rotation 139 
 
 The Pressure Gradients 143 
 
 The Potential Gradient 144 
 
 The Equations of Continuity 145 
 
 The Operator V 2 , and the Total Differential j- { 146 
 
 Summary of the Equations of Motion in Rectangular, Cylindrical, 
 
 and Polar Co-ordinates 147 
 
 The Equations of Motion for the Rotating Earth in Cylindrical Co- 
 ordinates 148 
 
 in Polar Co-ordinates 149 
 
 Connection between the peneral Equations of Motion and the Ther- 
 mal Equations of Energy 151 
 
 The Equations for the Work of Circulation 153 
 
 CdP 
 The Evaluation of the Term / 154 
 
 Numerical Check on the Two Systems of Formulas 157 
 
 Numerical Evaluations of the Pressure Gradient 157 
 
 Evaluation of the Ratios -j-^ and -j 159 
 
 dB dx 
 
 To Find the Difference of Pressure (Bi B) at the Distance DI that 
 
 will just Balance the Eastward Velocity v 159 
 
 The Angular Velocity of the Earth's Rotation, a> 3 160 
 
 The Linear Absolute and Relative Velocities 161 
 
CONTENTS ix 
 
 PAGE 
 
 THE THERMODYNAMICS OF THE ATMOSPHERE Continued 
 
 Evaluation of the Barometric Gradients 162 
 
 Application of the General Equations of Motion to the Local Circula- 
 tions in the Earth's Atmosphere 163 
 
 Discussion of the Cylindrical Equations of Motion 164 
 
 Ferrel's Local Cyclone 164 
 
 The German Local Cyclone 166 
 
 The General Equation of Cylindrical Vortices 168 
 
 The Angular Velocity 171 
 
 The Total Pressure 171 
 
 Application of the Vortex Formulas to the Funnel-Shaped Tube . 172 
 
 to the Dumb-bell-Shaped Tube 172 
 
 The Total Pressure 173 
 
 The Relation Between Successive Vortex Tubes 173 
 
 The Second Form of the Cylindrical Equations of Motion in Terms 
 
 of the Current Function ^ 175 
 
 The Funnel-Shaped Vortex 175 
 
 The Dumb-bell-Shaped Vortex 177 
 
 The Deflecting Force 178 
 
 The Force of Friction 179 
 
 The Transformation of Energy and the Circulation of the Atmosphere 180 
 
 Case I. The Change of Position of the Layers 181 
 
 The Evaluation of / T dm in Linear Vertical Temperature Changes 183 
 
 Case II. Effect of an Adiabatic Expansion or Contraction in a Non- 
 
 Adiabatic Temperature Gradient 184 
 
 Case III. The Overturn of Deep Strata in the Column 185 
 
 Case IV. The Transformation of Two Masses of Different Tempera- 
 tures on the Same Level into a State of Equilibrium .... 187 
 Case V. Local Changes between Two Strata of Different Tempera- 
 tures 188 
 
 The General Circulation on a Hemisphere of the Earth's Atmosphere 189 
 Three Cases of the Slope of the Temperature Gradients and the Re- 
 sulting Velocity of the East and West Circulations 191 
 
 CHAPTER IV 
 
 EXAMPLES OF THE CONSTRUCTION OF VORTICES IN THE EARTH'S 
 
 ATMOSPHERE 196 
 
 The Cottage City Water Spout, August 19, 1896 199 
 
 The St. Louis Tornado, May 27, 1896 202 
 
 The De Witte Typhoon, August 1-3, 1901 205 
 
 The Ocean and Land Cyclones 207 
 
 The Ocean Cyclone, October 11, 1905 209 
 
 The Composition of Vortices 212 
 
 The Reversed Dumb-bell Vortex 213 
 
 Historical Review of the Three Leading Theories Regarding the 
 Physical Causes of Cyclones and Anticyclones in the Earth's 
 Atmosphere 216 
 
X CONTENTS 
 
 PAGE 
 
 EXAMPLES OF THE CONSTRUCTION OF VORTICES IN THE EARTH'S 
 ATMOSPHERE Continued 
 
 (1) Ferrel's Warm- and Cold-Center Cyclones 216 
 
 (2) Hann's 'Dynamic Cyclonic Whirls . . 216 
 
 (3) Bigelow's Asymmetric Cyclones and Anticyclones 216 
 
 The General and the Local Components 220 
 
 The Normal and the Local Velocities in Storms 221 
 
 The Normal and the Local Isobars in Cyclones and in Anticyclones . 229 
 
 The Graphic Construction of Resultants 232 
 
 The Normal and Local Isotherms in Cyclones and in Anticyclones . . 235 
 The Normal and Local Velocity Vectors in Cyclones and Anti- 
 cyclones 240 
 
 The Land Cyclone 243 
 
 Recapitulation of the Formulas for the Dumb-bell-Shaped Vortex . 250 
 
 The Meaning of the Tangential Angle i 251 
 
 Example of the Evaluation of the Terms in the Equations of Motion 
 
 for a Cyclone 254 
 
 CHAPTER V 
 
 RADIATION, IONIZATION, AND MAGNETIC VECTORS IN THE EARTH'S 
 
 ATMOSPHERE 261 
 
 The Determination of the Intensity of the Solar Radiation by Obser- 
 vations with the Pyrheliometer and the Bolometer 262 
 
 The Pyrheliometer 263 
 
 Example of the Practical Observations with the Pyrheliometer at 
 
 La Quiaca, Argentina, Sept. 22, 1912 264 
 
 The Bouguer Formula of Depletion 267 
 
 The Depletion of the Incoming Radiation from a Maximum Value 
 on the Cirrus Levels, as Determined by Observations at Different 
 
 Heights 274 
 
 The Relative Efficiency of 1 Gram of Aqueous Vapor .per Cubic 
 
 Meter in Absorbing the Incoming Radiation 276 
 
 The Bolometer and Its Energy Spectrum of Radiation 277 
 
 Evaluation of the Wien-Planck Formula of Radiation 278 
 
 The Values of -r4f in the Radiation Formula 290 
 
 dz 
 
 The Measures of the lonization of the Atmosphere 292 
 
 Notation and Elementary Relations 293 
 
 Electrostatic Relations per Unit Length 295 
 
 Conduction lonization Currents 296 
 
 Coefficient of Electrical Dissipation of the Atmosphere ^ . . . . 297 
 
 Elster and Geitel Dissipation Apparatus 297 
 
 Ebert Ion Counter 298 
 
 The Formulas for the Velocity of the Ions 299 
 
 Ebert Velocity Apparatus 300 
 
 Gerdian Apparatus for the Number and Velocity of the Ions . . . 302 
 
CONTENTS XI 
 
 PAGE 
 
 RADIATION, IONIZATION, AND MAGNETIC VECTORS IN THE EARTH'S 
 
 ATMOSPHERE Continued 
 The Cause of the Discrepancy in the Values of the Conductivity of the 
 
 Atmosphere as Determined by Two Methods 304 
 
 The Atmospheric Electric Potential 309 
 
 CHAPTER VI 
 
 TERRESTRIAL AND SOLAR RELATIONS 312 
 
 The Five Types of the Diurnal Convection in the Earth's Atmosphere 312 
 The Diurnal Variations of the Meteorological, Electrical, and 
 
 Magnetic Elements 317 
 
 The Diurnal Variations of the Terrestrial Magnetic Fields . . . 323 
 
 The Aperiodic Magnetic Vectors along the Meridians 329 
 
 The Synchronous Annual Variations of the Solar and the Terres- 
 trial Elements . . 335 
 
 The Aqueous Vapor in the Atmosphere 342 
 
 The Laws of the Evaporation of Water from Lakes, Pans, and Soils 
 
 with Plants 343 
 
 The Polarization of Sunlight in the Atmosphere 348 
 
 Solar Physics 352 
 
 The Spherical Astronomy of the Sun 356 
 
 The Magnetic Fields of the Earth and the Sun 360 
 
 Conclusion 363 
 
 CHAPTER VII 
 
 EXTENSION OF THE THERMODYNAMIC COMPUTATIONS TO THE TOP OF THE 
 
 ATMOSPHERE 364 
 
 Remarks on the Bouguer Formula 364 
 
 I. First Distribution of Temperature 365 
 
 Illustrations of the Use of Erroneous Densities . 372 
 
 The Thermodynamic Terms 374 
 
 The Constituents of the Solar and the Terrestrial Radiations in the 
 
 Earth's Atmosphere 375 
 
 The Total, Transmitted and Absorbed Amounts of the Solar and 
 
 the Terrestrial Radiations . . 377 
 
 II. Second Distribution of Temperature 380 
 
 Summary of the Computations for Twenty-one Balloon Ascensions . 384 
 
 The Effective Energy of Radiation and the Solar Constant .... 388 
 
 The Second Method of Discussing Pyrheliometric Data .... 395 
 
 The Annual Mean Variations 401 
 
 The General Summary 403 
 
 General Remarks 404 
 
 The Constants and Coefficients of Dry Air in the Kinetic Theory of 
 
 Gases throughout the Atmosphere 406 
 
 Summary of the Dimensions with Special Reference to the Equiv- 
 alents 414 
 
 General Problems in Atmospheric and Solar Physics 417 
 
A METEOROLOGICAL TREATISE ON THE 
 
 CIRCULATION AND RADIATION IN 
 
 THE ATMOSPHERES OF THE 
 
 EARTH AND OF THE SUN 
 
 CHAPTER I 
 Meteorological Constants and Elementary Formulae 
 
 The Status of Meteorology 
 
 MODERN meteorology may be said to have begun its "scientific 
 development about the year 1870. Since that time an enormous 
 mass of observations has been made, covering every branch of the 
 subject, but the classification of the data in any systematic form 
 has been singularly inadequate. This defective progress may be 
 attributed chiefly to two causes, the first, the practical units in 
 which the instrumental readings are commonly made, and the 
 second, the usual misapplication of the Boyle-Gay Lussac Law 
 in the atmosphere, whereby it is assumed that the gas coefficient 
 is constant and the conditions strictly adiabatic. Thus, for 
 the pressure P , thejiensity p Q , the temperature T , and the gas 
 coefficient ^ , the law is, 
 
 (1) Po = PO #o To, 
 
 and it must be satisfied at every point, but in passing from one 
 point to another, whereby PI p\ TI change values, the observa- 
 tions made in balloon and kite ascensions are not verified without 
 making R change with the height above the sea-level. This 
 variation of R carries with it a variability of the specific heat by 
 the formula, 
 
 (2) Cp = -L- Ri 
 
 where the ratio of the specific heats is k = Cp/Cv, and thence 
 the changes from one level to another follow, for the free heat 
 02 1~ Co), the entropy (Si - 5 ), the work (Wi - W ), the inner 
 [1] 
 
2 * ' ' -'KLETEORdtbGlCAL CONSTANTS AND FORMULAS 
 
 energy (Ui Uo), the radiation energy (Ki K ), and the 
 coefficient c and exponent a in the radiation function, 
 
 (3) K = c T. 
 
 It can be proved, furthermore, that unless Cp varies from the 
 adiabatic specific heat Cp a for R a constant, there can be no 
 circulation and no radiation of heat in the atmosphere, such as 
 actually exists. It is the purpose of this Treatise to develop 
 the working formulae on a systematic plan, so that they shall be 
 adapted to the computation of the necessary data throughout 
 the atmosphere, as in the diurnal convection, the local and 
 cyclonic circulation, and in the general planetary circulation, by 
 means of which numerous problems may be studied without 
 an undue amount of speculation. 
 
 The difficulty regarding the type of the observations available 
 is that the records are not suitable for use in the formulae without 
 special reductions, and that a number of different sets of units 
 are employed. There are three systems of units in more or less 
 consistent use, (l) the meter-kilogram-second-Centigrade degree 
 system (M. K. S. C), (2) the centimeter-gram-second-Centi- 
 grade degree system (C. G. S. C), (3) the foot-pound-second- 
 Fahrenheit degree system (F. P. S. F). The pressure, B n , 
 for the fundamental formulae is recorded in barometric milli- 
 meters of mercury, or in inches of mercury, instead of in force 
 units, employing the acceleration of gravity, 
 
 (4) g ^= Q.8060 (1 - 0.00260 cos 20) (l - y), 
 
 where 9.8060 in meters is the acceleration per second in latitude 
 y = 45 ? and at sea-level z = 0, for R = 6370191 meters, the 
 radius of the earth. The pressure of one atmosphere in units 
 of force is, 
 
 (5) P = g OPm B , 
 
 where p m = 13595.8 kilograms per cubic meter, the density of 
 mercury. Similarly, 
 
 (6) Po = go PO Jo, 
 
 where p = 1.29305 density of air in the same units, and / = 
 
STATUS OF METEOROLOGY 3 
 
 7991.04 the height of the homogeneous atmosphere. For B n = 
 0.760 meter, the pressure P = 101323.5 kilograms per square 
 meter. Again, in units of mass, 
 
 (7) po = = p m Bo = Po h= 10332.8. 
 
 Finally, in heat units, 
 
 P n 
 
 = 24.2106 
 
 where the mechanical equivalent of heat is 
 
 (9) A = 426.837 kilogram meters, one large calorie. 
 The pressure of the atmosphere is, therefore, 
 
 Po = 101323.5 in units of force, 
 
 po = 10332.8 in units of mass, 
 
 p A = 24.2106 in units of heat, 
 BQ = 0.760 in barometric units, 
 
 so that the common barometer readings must be wholly trans- 
 formed for practical computations in dynamic and thermody- 
 namic problems which involve circulation and radiation. 
 
 Similarly the (C. G. S. C) system and the (F. P. S. F) 
 system each involves four other values for one atmospheric 
 pressure, and it often happens that there occur crosses between 
 these three systems, so that meteorological literature is confused 
 and difficult to comprehend in any clear manner. 
 
 The temperature is usually recorded in Centigrade degrees, 
 C, or in Fahrenheit degrees, F, but the computations must be 
 executed in degrees of absolute temperature. 
 
 (10) T = 273 + / (C) Centigrade, 
 
 (11) T = 459.4 + / (F) Fahrenheit, 
 
 so that another series of transformations is required. It would 
 improve matters greatly to mark barometers in terms of PO and 
 thermometers in degrees T, and reconstruct the entire series of 
 working tables. The relations between the barometric and the 
 force pressures are as follows : 
 
4 
 
 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 TABLE 1 
 1 Scale division = 0.75 mm. A 5 = 100. AP 
 
 A5 
 
 in millimeters 
 
 AP 
 units of force 
 
 Unit Equivalents 
 
 100. 
 
 13332. 
 
 
 10. 
 
 1333.2 
 
 1.0 mm. A B = 133.32 A P 
 
 1. 
 
 133.32 
 
 
 0.1 
 
 13.332 
 
 0.75 mm. A B = 100. A P 
 
 75. 
 
 10000. 
 
 
 7.5 
 
 1000. 
 
 0.0075mm. A B = 1.0 A P 
 
 0.75 
 
 100. 
 
 
 0.075 
 
 10. 
 
 
 If the divisions on the barometer scale are made 0.75 millimeter, 
 the scale distance is 100 units of force in the (M. K. S. C) system, 
 so that the readings of the barometric pressure are immediately 
 available for all forms of dynamic and thermodynamic meteor- 
 ology. This scale is equally valuable for public purposes and 
 synoptic chart construction. 
 
 If the absolute temperatures (J^, 7\) are observed on two 
 levels of the atmosphere at the heights (z , 1), respectively, the 
 formulas permit the corresponding values to be computed for 
 (Po, Pi), (PO, PI), CRo, Ri), and (Cpo, Cpi). It can be shown that 
 
 T, fl, 
 
 FIG. 1. The temperature, velocity, and heat in the stratum (z l 2 ) 
 
 if Cp a is the specific heat for the adiabatic R a , and Cpw is the 
 mean specific heat for the stratum (z , Zi), the circulation and 
 the radiation are given by the formula, 
 
 (12) (Cp a - Cp lt ) (T a - 
 
 - Qo), 
 
STATUS OF METEOROLOGY 5 
 
 where the terms are related to the points as indicated in Fig. 
 1 for the stratum (z , Zi). 
 
 If (T a To) is the adiabatic temperature fall in the vertical 
 distance (si z ), and (Ti T ) the actual observed tempera- 
 ture fall, the ratio between them is, 
 
 , - (T a ~ TQ) dp Qi - Zp) _0o 
 
 = -(T,- To) "" a( Zl -z ) - a' 
 
 Again, if (q , <?i) are the corresponding velocities without re- 
 gard to direction, and (Qi - Q ) the loss of free heat in the 
 stratum, these quantities are all connected together by the 
 equation (12). This states that the kinetic energy of the circula- 
 tion for the unit mass ^ (<?i 2 q 2 ), together with the heat 
 exchange of the radiation energy of the free heat (Qi Q ), 
 depends upon the divergence of the actual mean specific heat 
 Cpio from the adiabatic value, 
 
 (14) . Cp a = 993.5787 at constant pressure. 
 The corresponding values for Cv a amd R a are 
 
 (15) Cv a = 706.5453 at constant volume. 
 Cp a - Cv a = R a = 287.0334 the gas coefficient. 
 
 Hence. -~^ = k = 1.4062486 the ratio of the specific heats. 
 
 Cv a 
 
 (16) Cp a - Cv a = Cv a (k - 1) 
 
 fto\ fl a 
 
 (18) 777 - 7; = 5- = r -- 7 = 2.461545. 
 
 Cp a - Cv a R a k - I 
 
 The corresponding values in a non-adiabatic atmosphere become, 
 Cp Cp k 
 
 (19) 
 (20) 
 
 Cp -Cv ' R 
 Cv Cv 
 
 cp - Cv R k - r 
 
 f T \n l 
 
 where R = R a l^r/ as will be proved. 
 
6 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 These two systems will be more fully developed, it being the 
 purpose here merely to point out the leading line of the construc- 
 tion of this work. By the definition of the specific heat, 
 
 (21) + Cp a (T. - T.) = - go (i - z), 
 for the unit mass, so that 
 
 = -~ = 0.0098695 C/meter, 
 
 (23) a = - * 1 - jt = ^ =-12- = ^- C/meter. 
 Zi Z Q tpio nCpa n 
 
 From (17), (19), (22), and (23) 
 
 k 1 R a R a . do R a a R a T\ T Q 
 
 It will be shown from the data of observation that 
 
 (25) C#,, (T a - To) = 2Z. 
 
 PlO 
 
 Subtracting (25) from (21), 
 
 (26) (Cp a - Qio) (T a - To) = - ^ (21 - z ) ^^ 
 
 Pio 
 
 Equating (12) and (26), 
 
 (27) - go ( Zl -**) = + - + Y 2 fe 2 - go 2 ) + (Qi - Co), 
 
 Pio 
 
 and this is the general equation of condition in the atmosphere, 
 showing that for the unit mass the force of gravitation is balanced 
 by the change of pressure, the kinetic energy of the circulation, 
 and the radiating heat. 
 
 Now, returning to the registration of the fundamental quan- 
 tities, in addition to the temperature T, and the pressure P, there 
 is also the velocity q. While the sum of the energy of the circula- 
 tion and the radiation can be computed through the difference 
 of the specific heats (Cp a Cpi Q ), there is no way to separate 
 the circulation from the radiation except through the direct 
 observation of the velocity. The radiation must be computed 
 
STATUS OF METEOROLOGY 
 
 indirectly through the gravitation, pressure, and circulation 
 terms taken together in an inequality. As these three terms 
 seldom balance in the free atmosphere, which is continually ex- 
 changing heat at every point, it is evident that the adiabatic 
 conditions, in connection with the general equation of motion, 
 are not capable of giving a complete solution of any of the im- 
 portant thermodynamic problems of the atmosphere. The 
 literature of meteorology is defective in this respect. It should 
 be noted that the point of departure for this treatment of the 
 problem consists in making R and Cp variable, as previously 
 stated. 
 
 The velocity vector q (, z>, w, a, 0) requires special con- 
 sideration as to the axes of co-ordinates and the angular direc- 
 
 Radius of the Earth 
 
 + Z direction outward 
 +W velocity upward 
 
 (X.Y.Z.) 
 (u.v.w.) 
 
 West 
 
 Parallel of latitude 
 f y = direction East 
 
 +V +V = velocity East 
 
 FIG. 2. The rectangular co-ordinate axes with component velocities and 
 
 angles 
 
 tions. There is great confusion in meteorology in the manner of 
 recording the motions of the atmosphere. The popular use of 
 the compass points giving the directions from which the wind 
 blows in the azimuth rotation N, E, S, W, is entirely inapplicable 
 
8 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 in meteorological computations. This system should be reversed 
 in two respects, (1) The vector direction is that toward which 
 the air moves, instead of that from which it blows, making an 
 azimuth difference of 180; (2) the proper co-ordinate axes make 
 the azimuth rotation (S, E, N, W). The result of these two 
 changes is effected by the formula. 
 
 (28) = 360 - A, 
 
 where A is the azimuth in degrees from the north through the 
 east, and the azimuth from the 'south through the east, the vector 
 being changed to record the direction "towards" instead of the 
 direction "from" which the wind moves. The rectangular co- 
 ordinate axes are fixed by the common convention of a right- 
 handed rotation about a radius of the earth with positive trans- 
 lation outward. 
 
 (Meridian) + x = axis positive southward; + u = velocity 
 south. 
 
 (Parallel) + y = axis positive eastward; + v = velocity 
 east. 
 
 (Radius) + z = axis positive outward; -f w = vertical 
 velocity. 
 
 (29) Horizontal Plane, s = (x 2 + y 2 )*. = (u 2 + *)*. 
 
 (30) Vertical Plane. r = (x 2 + y* + zrf. q = (u 2 + v* + w 2 ) 1 . 
 
 (31) Azimuth angle (S, E, N, W), tan = -- = . 
 
 z w 
 
 (32) Vertical angle (above horizon), tana = -- = . 
 
 The same co-ordinate relations should be employed in 
 terrestrial magnetism, atmospheric electricity, and vector physics 
 generally. 
 
 Besides recording the direction of motion very inconveniently, 
 the velocities themselves usually require a series of transforma- 
 tions to reduce than to practical dynamics. The anemometers 
 are commonly graduated in kilometers per hour, or in miles per 
 hour, but they should be graduated in meters per second for 
 
CONSTANTS AND FORMULAS 9 
 
 the (M. K. S.) system, and in feet per second for the (F. P. S.) 
 system in order to conform with the other terms of equation 
 (27). In the electrical self-registration of the wind direction it is 
 common to limit the compass points to eight in number, N, NE, 
 E, SE, S, SW, W, NW, but in all problems requiring accurate 
 wind deflecting components, as in studies of the diurnal convec- 
 tion, it is necessary to use at least 16 points of 22.5 each in order 
 to compute the observed, resultant, and deflecting vectors. 
 Finally, it would be much better to record the azimuth /3 (S, E, 
 N, W) in degrees of arc, as can be readily done by a good me- 
 chanical device. The vertical angle a must be computed instead 
 of observed, because it is small except in tornadoes and cannot 
 be ordinarily measured mechanically. It is very important 
 to record the wind vector (q, 0, a) in the system thus described, 
 in order to facilitate all studies in the higher problems. It 
 may be noted that .there is evidence to show that the wind 
 velocity recorded by the anemometer is on a scale about 20 
 per cent, greater in the United States than in Europe. This 
 subject should be fully tested as soon as possible. It is also 
 known that the ordinary anemometer registers excessive velocities 
 as compared with a force dynamometer, such that the recorded 
 value 40 means 33, 60 means 48, 80 means 62, thus introducing 
 great errors in the dynamic equations unless corrected. 
 
 The Constants and Formulas of Static Meteorology 
 
 Meteorology distributes itself into three parts in accordance 
 with the requirements of equation (27). Static meteorology 
 develops, 
 
 (33) - g(* - 2o ) = 
 
 and it is that which is generally used in the construction of 
 synoptic weather charts and the other elementary problems. 
 Dynamic meteorology develops, 
 
 (34) - g ( Zl - Zo ) = + ^^ + y, ( ?1 - <?'), 
 
 Pio 
 
10 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 and is concerned with the several general equations of motion 
 connecting circulation and pressure. 
 
 Thermodynamic meteorology develops, 
 
 (27) - g (ft - so) = + ^-=-^2 + 1 A (<?i 2 - <?o 2 ) + (Qi - Co), 
 
 Pio 
 
 and unites the radiation with the circulation and the pressure 
 through the functions of work and inner energy. It follows that 
 the term g (z\ z ) may be broken up into three parts : 
 
 (36) - g (z - Z Q ) = - g (zi - Z Q ) - g (zz - zi) - g (* - 22) 
 where we have, respectively, 
 
 r> 7> 
 
 (33) g (zi s ) = H L the pressure effect, 
 
 Pio 
 
 (37) - g (z 2 - zi) = + y 2 (qi 2 - g 2 ) the circulation effect, 
 
 (38) - g fa - a) = + (Qi - Qo) the radiation effect. 
 
 Each of these terms is effective in disturbing the normal pressure, 
 temperature, and density levels, which would assume fixed 
 positions when uninfluenced by the absorption and the emission 
 of solar and terrestrial radiation, the entire process being the 
 means of continually returning to normal equilibrium. 
 
 In order to derive the constants and the formulas for static 
 meteorology, the formulas (25) and (33) are united to form, 
 
 (39) - Pl ~J = fto fe - z ) = - Qio (T, - To), 
 
 where the mean gravity and the mean specific heat between 
 the two vertical points, z\ and z 0l are to be used. Then', 
 
 (40) - (Pi.- Po) = 10 Pio fe -Zo) = - pioQio (Ti - To). 
 
 Since for a column on a base of unit square area pio (z\ Z Q ) = 
 M, the mass that produces the pressure (Pi P ) when acted 
 upon by the force of gravity gi , in the differential equation, is, 
 
 (41) - J dP = J gdm = j gpdz = - JpCp d T. 
 
 If the upper limit is at the top of the atmosphere, and the 
 lower limit at the bottom of it, on the sea level in latitude 45, 
 
CONSTANTS AND FORMULAS 
 
 11 
 
 and for the temperature T = 273, (40) reduces to, 
 
 (42) Po = go po Zi = PO Cpo (273 - 7\). 
 When the temperature of reduction is 
 
 (43) 3Ti - 273 
 
 the last form in (42) disappears. The constants of static meteor- 
 ology conform to (42) for any substance whatsoever: water, 
 mercury, dry air, aqueous vapor, or mixtures. If P^ is the 
 pressure of one standard atmosphere the density must change 
 Tn an inverse proportion with the height. In the following 
 notation, 
 
 Substance 
 
 Density 
 
 Height 
 
 Column 
 
 Water 
 
 Pw 
 
 HUJ 
 
 Water column 
 
 Mercury 
 
 Pm 
 
 Bo 
 
 Barometer 
 
 Dry air 
 
 Po 
 
 /o 
 
 Homogeneous 
 
 Aqueous vapor 
 
 'Pz 
 
 h 
 
 Vapor column 
 
 (42) becomes, specifically, 
 
 (44) Po = go P w h w = go P m BQ = go Po /O = go pZ /2- 
 
 and, (water) (mercury) (dry air) (aqueous vapor) 
 
 p 
 
 (45) = p w h w = p m B Q = pp/p_ P2/2. 
 
 Before evaluating equations (44) and (45), it is necessary 
 to adopt the standard constants * of transformation between the 
 three fundamental systems (M. K. S.), (C. G. S.), (F. P. S.). 
 
 The equivalent units of length and volume are, 
 
 (46) 1 meter = 100 centimeters = 3.2809 feet. 
 
 1 meter 3 = 1000000 cm 3 = 35.3166 cu. ft. 
 The standard relation between volume and mass is, 
 1 cubic centimeter of water = 1 gram at the temperature 276.9. 
 
 (47) 1 kilogram = 1000 grams = 2.20462 pounds. 
 1000 kilograms = 1000000 grams = 2204.62 pounds. 
 
 *The subject of units and physical constants can be studied in Everett's 
 " Units and Physical Constants," Gray's " Smithsonian Physical Tables," 
 and in the text-books on physics generally. 
 
12 
 
 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 Hence, by division, the equivalents become, 
 kilograms 
 
 (48)1000 
 
 , 
 
 (meter) 3 (centimeter) 3 
 
 = 62.4237 
 
 16 . 0198 ki!2SI^? = 0.0160198^= 1 
 m 3 cm 3 
 
 (foot) 3 ' 
 pound 
 
 Three Series of Constants in Three Systems of Units 
 
 In Static Meteorology there are three series of constants for 
 force units, mass units, and heat units in the three systems of 
 units (M. K. S. C), (C. G. S. C), (F. P. S. F). These de- 
 velop from fundamental principles or definitions. Thus, to 
 illustrate by pressure: 
 
 1. Force pressure = mass X acceleration. 
 
 (49) 
 
 P = p m B go = po/ogo = M go = pogo = PA~- 
 
 2. Mass pressure = heat pressure X mechanical equivalent 
 of heat. 
 
 (50) 
 
 u A PA 
 M = = -r. 
 go A 
 
 3. Heat pressure = force pressure X heat equivalent of 
 gravity work. 
 
 (51) P A = AM = Po- = Ap Q . 
 
 o 
 
 These transformations apply to the heat terms R, Cp, Cv, in 
 the several systems. These factors become in the several unit 
 
 systems : 
 
 TABLE 2 
 GRAVITY AND MECHANICAL EQUIVALENTS OF HEAT 
 
 Work and Heat Equivalents 
 
 (M. K. S.) 
 
 (C. G. S.) 
 
 (F. P. S.) 
 
 go Acceleration of gravity 
 - Work equivalent of heat 
 
 Gravity-work of heat. 
 
 9.8060 
 426.837 
 
 4185 1 
 
 980.60 
 42683.7 
 
 41851000 
 
 32.173 
 777.93 
 
 25028 2 
 
 A 
 
 A Heat equivalent of work . . . 
 A_ J Heat equivalent of gravity 
 go ' work 
 
 0.002343 
 0.00023894 
 
 0.00002343 
 0.000000023894 
 
 0.0012855 
 0.000039954 
 
 A X (Heat in mechanical units) = Heat units of heat = calories. 
 
THREE SYSTEMS OF UNITS 
 
 13 
 
 In Tables 3, 4, and 5 have been collected together the con- 
 stants in the three unit systems. They illustrate practically 
 the formulas (1), (17), (18), (22), and (44) in Table 3, (50) in 
 Table 4, and (51) in Table 5. By combining these constants and 
 formulas a very large amount of static meteorology is derived. 
 
 TABLE 3 
 THREE SERIES OF CONSTANTS IN THREE SYSTEMS OF UNITS 
 
 (1) Gravitational Force Units 
 
 Formulas 
 
 S 
 
 (M. K. S. C) 
 Meter-Kg-Second 
 
 (C. G. S. C) 
 Cm.-Gram-Second* 
 
 (F. P. S. F) 
 Foot-Pound-Second 
 
 
 go 
 pw 
 hw 
 Po 
 
 go 
 pm 
 Bo 
 Po 
 
 go 
 
 PO 
 
 It 
 
 Po 
 
 go 
 
 P2 
 
 h 
 
 Po 
 
 ft 
 
 k -1 
 ft 
 
 k-l 
 1 
 -l 
 
 Ro 
 To 
 po 
 Po 
 
 lo 
 go 
 
 a 
 Ro 
 
 Cp a 
 Cv a 
 
 ao 
 
 Log. 
 9.8060 0.99149 
 1000 .0 3 . 00000 
 10.3329 1.01422 
 101323.5 5.00571 
 
 9 . 8060 . 99149 
 13595.8 4.13340 
 . 760 9 . 88081 
 101323 .5 5 . 00571 
 
 9.8060 0.99149 
 1.29305 0.11162 
 7991.04 3.90260 
 101323.5 5.00571 
 
 9.8060 0.99149 
 0.80427 9.90540 
 12847.6 4.10882 
 101323.5 5.00571 
 
 1.4062486 0.14806 
 0.4062486 9.60879 
 3.461545 0.53927 
 
 2.461545 0.39121 
 
 287.0334 2.45793 
 273. 2.43616 
 1.29305 0.11162 
 101323.5 5.00571 
 
 7991.04 3.90260 
 9.8060 0.99149 
 
 0.003663 7.56384 
 287.0334 2.45793 
 
 993.5787 2.99720 
 706.5453 2.84914 
 
 0.0098695 7.99429 
 
 Log. 
 980.60 2.99149 
 1.0000 0.00000 
 10332.9 3.01422 
 1013235. 6.00571 
 
 980.60 2.99149 
 13 . 5958 1 . 13340 
 76.0 1.88081 
 1013235. 6.00571 
 
 980.60 2.99149 
 .00129305 7.11162 
 799104. 5.90260 
 1013235. 6.00571 
 
 980.60 2.99149 
 .00080427 6.90540 
 1284760. 6.10882 
 1013235. 6.00571 
 
 1.4062486 0.14806 
 0.4062486 9.60879 
 3.461545 0.53927 
 
 2.461545 0.39121 
 
 2870334. 6.45793 
 273. 2.43616 
 0.00129305 7.11162 
 1013235. 6.00571 
 
 799104. 5.90260 
 980.60 2.99149 
 
 0.003663 7.56384 
 2870334 6.45793 
 
 9935787. 6.99720 
 7065453. 6.84914 
 
 .000098695 5.99429 
 
 Log. 
 32.173 1.50749 
 62.4237 1.79535 
 33.901 1.53021 
 68085 4.83305 
 
 32 .173 1 . 50749 
 848.70 2.92875 
 2.4935 0.39681 
 68085 4 . 83305 
 
 32.173 1.50749 
 0.080717 8.90696 
 26218.1 4.41860 
 68085. 4.83305 
 
 32 .173 1 . 50749 
 0.056009 8.69974 
 42249.1 4.62582 
 68085. 4.83305 
 
 1.4062486 0.14806 
 0.4062486 9.60879 
 3.461545 0.53927 
 
 2.461545 0.39121 
 
 1716.52 3.23465 
 491.4 2.69144 
 0.080717 8.90696 
 68085 4.83305 
 
 26218.1 4.41860 
 32.173 1.50749 
 
 0.002035 7.30856 
 1716.52 3.23465 
 
 5941.86 3.77392 
 4225.34 3.62586 
 
 0.0054146 7.73357 
 
 Density of water 
 
 Height 
 
 (44) Po = gopwhw.... 
 Gravity 
 
 Density of mercury . . . 
 Barometer height 
 (44) Po = gopmBo. . . . 
 
 
 Height 
 
 (44) Po - go polo 
 
 Gravity 
 
 Density aqueous vapor. 
 Height 
 
 (44) Po = gop 2 /2 
 
 Cp 
 Specific heats -^ 
 
 Ratio ^ - 1 . . 
 C v 
 
 (17) Const, press 
 (18) Const, vol 
 
 The Boyle-Gay Lussac 
 Law 
 
 (1) Po = P oRoTo 
 lo go = Ro To 
 
 (1) and (44) 
 1 
 
 a, = "Tfr 
 
 To 
 Ro ~ lo So a. 
 
 (17) CP =Ko jp-j.... 
 (18) Cv = Ro jri 
 
 (22)- d /-^- 
 
 ' d z Cpa 
 
14 
 
 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 (2) 
 
 TABLE 4 
 
 Mass or Units of Weight 
 
 Formulas 
 
 5 
 
 (M. K. S. C) 
 Meter-Kilgm.-Sec. 
 
 (C. G. S. C) 
 Cm.-Gm.-Sec. 
 
 (F. P. S. F) 
 Ft.-Pound-Sec. 
 
 (50) p = Pa/go 
 
 R = p/poTo.... 
 
 P 
 R 
 
 CP 
 
 Cv 
 
 Log. 
 10332.8 4.01422 
 29.2713 1.46644 
 
 101.3235 2.00571 
 72.0522 1.85765 
 
 Log. 
 1033.28 3.01422 
 2927.13 3.46644 
 
 10132.35 4.00571 
 7205.22 3.85765 
 
 Log. 
 2116.20 3.32556 
 53.353 1.72716 
 
 184.683 2.26643 
 131.330 2.11837 
 
 Ci>-R-^ 
 
 Cv=Cp - R 
 
 TABLE 5 
 
 (3) Heat Units 
 
 Formulas 
 
 S 
 
 (M. K. S. C) 
 
 (C. G. S. C) 
 
 (F. P. S. F) 
 
 Work equiv. heat 
 
 1 
 
 A 
 
 A 
 
 PA 
 
 R A 
 
 CP A 
 CA 
 
 426.837 2.63022 
 0.002343 7.36978 
 24.2106 1.38400 
 0.068583 8.83622 
 0.237406 9.37549 
 0.168823 9.22743 
 
 42683.7 4.63022 
 0.00002343 5.36978 
 0.024106 8.38400 
 0.068583 8.83622 
 0.237406 9.37549 
 0.168823 9.22743 
 
 777.93 2.89094 
 0.0012855 7.10906 
 2.72025 0.43461 
 0.068583 8.83622 
 0.237406 9.37549 
 0.168823 9.22743 
 
 Heat equiv. work 
 
 (5D P A =AP 
 R A = AR 
 
 C *A = ACp 
 Cv A = ACv... . 
 
 Work and Heat Units 
 
 One large calorie is the heat required to raise 1 kilogram of 
 water from to 1 C. 
 
 One small calorie (therm.) is the heat required to raise 1 
 gram of water from to 1 C. 
 
 One British thermal unit is the heat required to raise 1 pound 
 of water from 32 to 33 F. 
 
 One calorie = 1000 therms = 3.968 Br. th. u. = 426.837 
 kilogram meters. 
 
 One therm = 0.003968 Br. th. u. (3.968 = 2.2046 X 1.8.) 
 
 One dyne is the force which acting upon a gram for one second 
 generates a velocity of one centimeter per second; it produces 
 the C. G. S. unit of acceleration on one gram; it produces the 
 C. G. S. unit of momentum on any mass per second. 
 
 One erg is the amount of work done by one dyne working 
 
THREE SYSTEMS OF UNITS 15 
 
 through the distance of one centimeter; it is the C. G. S. unit of 
 energy. 
 
 One erg = 1 centimeter dyne = ^TTT;; = 0.0010198 gram cm. 
 
 - 980.60 xUx 100 = 0-000000010198 kilogram 
 
 meter. 
 
 One large calorie = 1 kilogram-degree C water = 426.837 
 
 kilogram meters 
 
 = 426.837 X 980.60 X 1000 X 100 = 
 4.1851 X 10 10 ergs, C. G. S. 
 
 One small calorie = 1 gram-degree C water = 426.837 
 
 gram meters 
 
 = 426.837 X 980.60 X 100 = 4.1851 X 
 
 10 7 ergs, C. G. S. 
 One British thermal unit = 1 pound-degree F water 
 
 o 280Q 
 = 426.837 X - -= 777.93 foot-pounds. 
 
 l.o 
 
 Work to Heat 
 
 T. The mechanical equivalent of heat is the work required 
 by work-friction to produce the given heat. 
 
 Log. 
 
 (52) -j = 426.837 2.63022 kilogram meters (M. K. S.) 
 
 = 42683.7 4.63022 gram centimeters (C. G. S.) 
 = 777.93 2.89094 foot-pounds (F. P. S.) 
 
 (53) il= 4185.1 3.62171 joules (M. K. S.) 
 
 = 41851000. 7.62171 ergs (C. G. S.) 
 
 = 25028.2 4.39843 absolute units (F. P. S.) 
 
 Heat to Work 
 
 A. The heat equivalent of work is the heat that is required 
 to do a given amount of work. 
 
16 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 Log. 
 
 (54) A = 0.002343 7.36978 kilogram calorie (M. K. S.) 
 
 = 0.00002343 5.36978 gram therm (C. G. S.) 
 
 = 0.0012855 7.10906 Br. th. units (F. P. S.) 
 
 (55) = 0.00023894 6.37829 (M. K. S.) 
 
 = 0.000000023894 2.37829 (C. G. S.) 
 
 = 0.000039954 5.60157 (F. P. S.) 
 
 I. FOR THE SAME POINT OR STATION 
 Variations from the Standard P , p , R Q , T Q 
 
 The several formulas derived from P = PO ^o TO apply 
 only to the sea level on latitude 45, but the variations, PI, pi, RI, 
 TI, are incessant in the earth's atmosphere, and the formulas 
 must be derived for passing from one condition to another. 
 Meteorology divides itself into two main branches according as 
 R is taken constant or variable, and it is a principal part of this 
 work to discuss the formulas when R is variable from one stratum 
 to another. When the point of observation is on the sea level or 
 on the lower land areas, it is proper to assume RI = RQ constant, 
 which greatly simplifies the computations. If the variations oc- 
 cur at the same place or station, go is also constant. For two 
 variable conditions of dry air at a given place we may write the 
 two in a ratio in several forms, using (44), (49), (50), (51) : 
 
 Hence, 
 
 . 
 PAOJ 
 
 ,__N P_ j>_ pA _ P T RQ _ Pm B go _ p I go 
 
 PO pQ pAQ PoToRo Pm^ogo PO IQ go 
 
 We shall confine our attention chiefly to the force pressures 
 P, PO in the (M. K. S.) system, and to the first series of constants 
 in Table 3. There are numerous equivalents which are easily 
 derived from the formulas. 
 
VARIATIONS IN THE VERTICAL LINE 17 
 
 P T o I B 
 
 (58) The pressure ratio, -^- = ^ = - = . 
 
 LQ PO J- PO*0 -OQ 
 
 The temperature ratio becomes, 
 
 The pressure ratio may have several forms, from which are 
 derived the pressure-density ratio, 
 
 (60) = ~ = (1 + a t). Auxiliaries. 
 
 p po 1 o po 
 
 ^ Pm go / i A D D 
 
 - (1 + a/). Po = #o P m go. 
 po 
 
 = PO ^o (1 + a 0- = VQ. 
 
 Po 
 
 = Ro T Q (1 + a t). P V = R T . 
 
 T P D 
 
 (61) The temperature ratio, -=.. 
 
 1 o " <y P 
 
 (62) The density ratio, = ^ . ^ 
 
 Po -TO ^ 
 
 ^ -^ 
 
 The temperature varies as the pressure and inversely as the 
 density; the density varies as the pressure and inversely as the 
 temperature. 
 
 II. FOR DIFFERENT POINTS ON ANY VERTICAL LINE z 
 The Variations of Gravity, Density, Temperature, and Pressure 
 
 The practical problems in static meteorology consist to a 
 considerable extent of the reduction of barometric pressures 
 from one elevation to another along a radius of the earth extended, 
 or inversely the determination of the difference of elevation be- 
 tween two measured barometric pressures along the same vertical. ' 
 The former process is applied in forming the synchronous charts 
 of pressure reduced to the sea level, or to any other adopted 
 plane which are used in public forecast charts of storm and 
 weather conditions, and the latter to preliminary surveys in 
 
18 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 mountain and plateau regions. The entire process is, however, 
 very complex in its application to particular cases, and it re- 
 quires much experience in managing the details of the computa- 
 tions. It will be necessary to describe somewhat fully the 
 several terms that are to be integrated from one level to another, 
 which enter the final barometric and hypsometric formulas. 
 Those here developed are such as are found in the Standard 
 ^Treatises, but another method of reduction will be introduced 
 at a later section. The first problem to describe is the gravity 
 value in any latitude,^, and at any height above the sea level, z; 
 the second considers the density of the air as a mixture of gases 
 and aqueous vapor in varying proportions; the third is to de- 
 termine the temperature gradients in a vertical direction in the 
 free air and within the land masses; and the fouxth is the use of 
 the barometer as an instrument of precision, together with the 
 discussion of the observed heights of the mercury column. All 
 the details easily found in good works on meteorology will be 
 very briefly mentioned. 
 
 7. The Acceleration of Gravitation J g$ z d z. 
 
 From formula (4), which is of geodetic origin, 
 (63) g$ = go (1 - 0.00260 cos 2 0), the latitude variation. 
 
 *" (64) g z = gj ^1 -~rj, the elevation variation. Hence, 
 (65) f^ d z = 9.8060 (1 - 0.00260 cos 
 
 The force of gravity varies inversely as the square of the 
 distance from the center of the earth. 
 
 ( 66 >>l- Rz 
 (bb) go " (R + *) - 
 
 -Z ). 
 
 (67) 
 
 - dz = (z - * ) - 
 
ACCELERATION OF GRAVITATION 
 
 19 
 
 The radius of the earth may be taken in the mean from 
 Bessel's spheroid, 
 
 (68) JRp = 6370191 meters, 6.8041525 log. 
 Rt = 20899600 feet, 7.3201380 log. 
 
 The computed values of formula (4), without any integration 
 in latitude and altitude, are given for a few selected points in 
 Table 6. 
 
 TABLE 6 
 EVALUATION OF FORMULA (4) 
 
 gfa = 9.8060 (1 - 0.00260 cos 2 0) ( 1 - ^) 
 
 z 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 45 
 
 40 
 
 30 
 
 20 
 
 10 
 
 
 
 Meters. 
 20000 
 15000 
 10000 
 5000 
 000 
 
 m. 
 9 . 7703 
 9.7857 
 9.8011 
 9.8165 
 
 m. 
 9.7688 
 9.7842 
 9.7996 
 9.8150 
 
 m. 
 9.7643 
 9.7797 
 9.7951 
 9.8105 
 
 m. 
 9.7574 
 9.7728 
 9.7882 
 9.8036 
 
 m. 
 
 9 . 7489 
 9 . 7643 
 9.7797 
 9.7951 
 
 m. 
 9 . 7444 
 9.7598 
 9.7752 
 9.7906 
 
 m. 
 9.7399 
 9.7553 
 9.7707 
 9.7861 
 
 m. 
 9.7314 
 9.7468 
 9.7622 
 9.7776 
 
 m. 
 9.7245 
 9.7399 
 9.7553 
 9.7707 
 
 m. 
 9.7200 
 9 . 7354 
 9.7508 
 9 . 7662 
 
 m. 
 9.7185 
 9.7339 
 9.7493 
 9 . 7647 
 
 9.8319 
 
 9 . 8304 
 
 9.8259 
 
 9.81909.8105 
 
 9.8060 
 
 9.8015 
 
 9 . 7930 
 
 9.7861 
 
 9.7816 
 
 9.7801 
 
 
 
 
 
 
 
 
 
 
 
 2z 
 The variation in height g -^- = 0.00308 meters per 1,000 
 
 meters, and 0.00308 feet per 1000 feet for g = 32.173 feefT 
 
 There has been a discussion as to the effect of the land masses 
 upon the action of gravity, whether the coefficient in the formula 
 should be 2.00, as developed in (66), or be modified. Ferrel 
 claims that it should remain 2.00; the Smithsonian Meteorologi- 
 cal Tables have adopted 1.96; and the International Meteoro- 
 logical Tables have taken 1.25, which latter value is here 
 adopted. The plateau regions of North and South America, Asia, 
 
 rr 
 
 and Africa will be best represented by 1.25 -~-, where z has a con- 
 siderable value, reckoned T rom the sea level. In balloon ascen- 
 sions from the ocean or from the low plains it may be better to 
 increase the value to 2.00, but this can be determined from obser- 
 vations by means of mercurial and aneroid barometers. In fact, 
 the aneroid barometer, when perfectly adjusted as a mechanism, 
 
20 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 measures the local hydrostatic pressure without any gravity 
 factor. Among the computations to be introduced in a later 
 section there will be examples of this action. Admitting the 
 coefficient 1.25, we have 
 
 (69) g z = g (l 1-25^) = 9.806 (1 - 0.000000196s) metric. 
 
 = 32.173 (1 - .0000000598 z) English. 
 = 9.806 - 0.00000192 z metric. 
 = 32.173 - 0.00000192 z English. 
 
 Similarly, the correction for height can be applied to any other 
 value of #0 , as found on the lower line of Table 6. 
 
 2. The Density of the Atmosphere as a Mixture of Several Con- 
 
 stituent Gases 
 
 For the practice of barometry it is sufficient to take account 
 of the atmosphere as a mixture of dry air, aqueous vapor and 
 carbon dioxide, commonly called carbonic acid.* In physical 
 problems there are the gases oxygen, nitrogen, hydrogen, carbonic 
 oxide, and traces of argon, helium, neon, krypton. We shall 
 summarize the treatment of several gases in a mixture and 
 the data of the kinetic theory of gases in this connection. 
 
 Adopt the notation for standard conditions as expressed by 
 Po = PO #o To, or Po v = Ro TV 
 
 Mixture. Dry Air. A J ueous Car A b ic 
 
 Vapor. Acid. 
 
 Density Pm0 p^ ^ P30 
 
 Volume v m0 v w v 20 v 30 
 
 The general equation for mixture is, 
 
 (70) p mQ V m(> = pio fllO + P20 ^20 + P30 ^30- 
 
 The values to be assigned to the terms are: 
 
 Dry air Plo = 1.29278. v 10 = v mo - v 2Q - v 30 . 
 
 * In view of the possible existence of the real carbonic acid (H 2 C O 3 ) at 
 the low temperature of the isothermal layer, the use of the word " acid " instead 
 of anhydride can not be commended. 
 
DENSITY OF THE ATMOSPHERE 21 
 
 Aqueous vapor p2 = 0.622 pi . v 2Q = variable amounts. 
 Carbonic acid pso = 1.529 pi . 30 = 0.0004 v m0 . 
 Mixture p m0 to be computed. v m0 = v w + v 20 + v 30 . 
 
 Introducing these values in (70) and dividing by v mQ 
 
 (71) ^ = 1.29278(1-^ - ^ + 0.622^ + 1.529^), 
 
 \ Z>mO VmO V m0 V m O/ 
 
 = 1.29278 fl - 0.378 + 0.529 }. 
 \ v m0 v m0 J 
 
 Since the ratio of the volume of the constituent to the volume 
 of the mixture is the same as that of the partial pressure of the 
 constituent to the total pressure of the mixture, we have generally 
 
 VnO Pn0 pn_ e_ 
 
 v m0 " p m0 " p m " B' 
 
 where p n0 and p m0 are for the normal data (P TO), 
 p n and p m are for any (P T) , 
 e and B are the barometric pressures. 
 
 Hence, = , and ^ = 0.0004, so that, 
 
 Z>mO >Q V m O 
 
 (73) p m0 = 1.29278 (l - 0.378^ + 0.00021 V 
 
 It is customary to unite the terms for the dry air and the 
 carbonic acid in the normal density, 
 
 (74) po = 1.29278 + 0.00027 = 1.29305 per cubic meter. 
 
 In (73) e^is to be taken in meters of mercury in the (M. K. S.) 
 system, in millimeters in the (C. G. S.) system, and in feet in the 
 (F. P. S.) system. Since e varies incessantly in the atmosphere 
 no fixed value can be assigned to it on any level. The reduction 
 from the normal (P Q p m0 T ) to any other condition (P p m T) on 
 the same level z is given by (60), substituting p m for /o, and 
 Pmo for /o, 
 
 p 'T' p 'T 1 / p 
 
 (75) p m = - PmQ = ~ P Q 1 ~ 0.378 - 
 
 p 
 
22 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 B 1 1 
 
 (76) p m = -5- p m0 
 
 6/) (l + 0.378) 
 
 Since we retain the density p = 1.29305 for the value of 
 p m0 , with dry air at normal pressure in this equation, the 
 corresponding barometric pressure in the fraction expressing 
 the partial pressure of aqueous vapor must be B -, butj^may 
 be any pressure whatever, according to the dryness of the air. 
 
 C z e 
 
 The Integral Mean / 0.378 -=-. 
 
 'so >o 
 
 The pressure of the aqueous vapor decreases from the ground 
 upward in a geometric ratio, which is expressed approximately 
 on the average by the formula, 
 
 (77) e = e 10 6517 in meters, e = e 21381 in feet. 
 
 It will be shown that the barometric pressure diminishes by 
 a similar law, 
 
 (78) B = Bo 10 1840 in meters, B = B 10 60367 in feet. 
 
 p 
 
 By combining these in the ratio 5- it becomes, 
 
 (79) 4- = -^ 1Q~ 10091 in meters, ~- = -~ lO' 33 ^ in feet. 
 
 > >Q & JL>O 
 
 These can be reduced from the common base 10 to the 
 Naperian base e by the modulus M = 0.43429. 
 
 10091 X M = 4383, 33108 X M = 14378. 
 The expression for the integral mean from^Zpjto z is 
 
 (80) j8 = - 0.378 ~ / V ^ssa j z f or t he metric system. 
 
 Z ZQ >Q ^ZQ 
 
 I en C z - 
 
 (81) j8 = - 0.378 - / e 14378 d z for the English system. 
 
 Z ZQ >Q 'ZQ 
 
 These can be developed in a series, as shown in the "Report 
 on the International Cloud Observations," UJ3.JSLJ3., 198 Z _ 
 page 491,^" 
 
FORMULAS FOR MIXTURE OF GASES 23 
 
 metnc - 
 
 i / z \ z ~] 
 
 ~T\ \4383 / ^ * ' ' J 
 (83) ft = 0.378-1 [l - 4l yb* ) + 4t (74^) 2 - 
 
 JJQ ^i ^J-TcO/O / ol ^XTtOiO' 
 
 1 / V IT 
 
 -4!Vl4378)+-"J En S llsh - 
 
 That is to say, having the vapor pressure j^ and the barometric 
 
 pressure B Q at the surface one can compute the average value 
 
 & 
 of the integral of the term 0.378 ^ up to the height z. It is 
 
 commonly impractical to measure the values of e and B at 
 several points in the atmosphere, and for many computations 
 this method of mean integration upward from the surface is 
 quite sufficientjpr_pjractical work. Also, it is a very expeditious 
 method when using the humidity table 92, page 548, for metric 
 measures and Table 19 of the "Barometry Report/' U. S. W. B., 
 
 o 
 
 1900, page 108, for English measures. The mean values of -g 
 
 for the air column is often taken as the arithmetical mean of 
 the observed values at an upper statiqn_z and a lower station J;Q.____ 
 In the case of balloon and kite ascension, the registered relative 
 humidity, temperature, and pressure can be computed to the 
 integral mean value required. When only temperature and 
 pressure are registered, this correction to the density of the 
 atmosphere is not available in the hypsometric formula. 
 
 The General Formulas for the Mixture of Gases 
 
 The general principles controlling the mixture of gases are 
 so often useful in meteorology that it will be convenient to collect 
 together the common formulas expressing the several processes. 
 It will now be proper to pass from the system of M. K. S. units 
 to the system of C. G. S. units, and to bring forward the terms 
 
24 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 applicable to the thermodynamics of the atmosphere and the 
 kinetic theory of gases. We adopt the following notation: 
 
 a = atomic weight. 
 
 K 
 
 m = molecular weight = -5-. 
 
 K. f 
 
 (84) K = the absolute gas constant = m R = ~ . 
 
 * 
 
 Compute K in the (C. G. S.) system, using the gas hydrogen, 
 m H = 2, p H = 0.000089996. Log = 5.95422 - 10. 
 
 82482000 
 
 = 1-9708 small calories or therms. 
 
 Using the values for air m = 28.736, p = 0.00129305, the 
 same result is obtained. This formula applies to the three 
 systems and to all gases. The density and molecular weight of 
 hydrogen are related to those of other gases so that, 
 
 (85) m p H = m H p, 
 
 and for this reason hydrogen is the standard. 
 
 n = the number of molecules in a unit volume. 
 N = the number of molecules in a V- volume. 
 
 M = N m = Mass. 
 
 (86) Number. N = = V n = ^~ 
 
 m K 1 
 
 P V m P V 
 
 (87) Mass. M = N m = V nm = T = TT^ = 
 
 P M_ PM 
 
 R^T ' nm ~ RT p' 
 
 M M MKT 
 
 (88) Volume. V = M v = - -=- = 
 
 p nm m P 
 
FORMULAS FOR MIXTURE OF GASES 25 
 
 (89) Pressure. jP^ = - -= = nK T = =- = ~RT = 
 
 nmRT = p R T. 
 
 (90) Density. p = = WW = TT= yr = n~^. 
 
 , m v . 1 1 F A" T RT 
 
 (91) Volume of unit mass, v = = - -= = = . 
 
 p nm M m P P 
 
 (92) Constant. R = = - = = - = -^-. 
 
 m M T T nm T p 
 
 Referring to the standard gas hydrogen there are some special 
 values for the gas constant. 
 
 Logs. 
 n p ~v H ~ m gram cm. 2 
 
 (94) 
 
 (95) K = ^^* = ^^ = 82482000 gram X 
 PH PH J- o 
 
 sec. 2 temp. 
 
 
 If 7? r is the gas constant for the heat energy at T, 
 R m is the gas constant for the molecular energy, 
 R a is the gas constant for the atomic energy. Then, 
 
 3 r the mean kinetic energy 
 ( Ub ) KI = K T + K m -\- K n = K = =- = - 
 
 2 T absolute temperature . 
 
 (97) R z = ^ Ri R = Cv the specific heat at constant volume. 
 
 o 
 
 (98) R 3 = - RI = Cp the specific heat at constant pressure. 
 
 o 
 
 These are related to the potential and kinetic energies in the 
 following relations, and thence to the specific heats at constant 
 pressure and constant volume. 
 
26 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 Inner Potential Energies 
 
 (99) / = J m -f J a * The inner potential energy of molecules 
 
 and atoms. 
 
 o o r> o 
 
 (100) J m =-Pv=- = - RT. Inner molecular potential 
 
 w & p & 
 
 energy. 
 / = Inner atomic potential energy. 
 
 2 
 /o = ^ 7^ (inner viriol). Initial inner potential energy. 
 
 o 
 
 (101) V = - I S (X x + Y y + Z.) the mean viriol for the 
 
 force (X. Y. Z.). 
 
 Inner Kinetic Energies 
 
 (102) H = H m -f H a . The inner kinetic energy of molecules 
 
 and atoms. 
 
 H m = Inner molecular kinetic energy. 
 
 H a = Inner atomic kinetic energy. 
 
 Total Heat and Work Energies 
 
 Q = The heat or total inner energy = Cv T. 
 W = the work or total external potential energy. 
 
 (103) Q = J + H + W = (| 1 - R) T + / . Total inner 
 
 energy. 
 p 2 
 
 (104) W = Pv = -- = RT = -V e (outer viriol). External 
 
 p o 
 
 potential energy. 
 
 *J m and To relate to the trifling rearrangements of parts which are the only 
 changes that can occur in ordinary chemical and physical reactions. We can 
 not attack the enormous stores of energy shut up within the atoms. 
 
INNER KINETIC ENERGIES, WORK, AND HEAT 27 
 
 (105) U = J - /o + W = J m + J a - /o + W. Potential 
 
 energy. 
 
 (106) U W = J JQ. Accession of inner potential energy. 
 
 (107) V = H = - l - S (X x + Y y + ZJ = | U = mean viriol 
 
 or work done. 
 
 The gas constants are again denned thus: 
 
 TJ 
 
 (108) R T = 7p (heat) = the ratio of the inner kinetic energy to T. 
 
 (109) R m = j^-^mq 2 where q 2 = the mean square velocity. 
 
 (110) R a = -^ -n m^i 2 , n = the number of atoms in a molecule. 
 
 1 2> 
 
 The Specific Heats of Monatomic Gases 
 
 5 55 
 
 (111) Cp= -R l -R + R==R 2 + R = -R l = -R. 
 
 7? 2 7? 2 7? 
 = 3* 1 = 3 R * 
 
 / ~* - <-\\ /~*at 3? 7? Z? 3? . J2 - 3? 1? 
 
 3 
 
 T> T) T> 
 
 2 ^ _ Kl _ y^ 2 
 
 , ON Cp R Z + R 5 
 
 (113) pr- = ^ = r- = 1.67 = k for monatomic gases 
 
 (mercury). 
 
 c-c k - 
 
 ' (20) ' 
 
 (116) Cp - Cv = R = ^. Compare (15), (16). 
 
 O 
 
 2 CD 
 
28 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 , 11ft v Ja Jm 3 Cp-Cv 5Cv-3Cp 
 
 (118) -Q- -Q- - 2 oT ~2C~ V 
 
 The above formulas are in mechanical units, and they can be 
 transposed into heat units on multiplying them by A = the heat 
 equivalent of work which is A = 0.00002343 in ,the C. G. S. 
 system by Table 2. 
 
 The Fundamental Laws of Physics 
 
 (119) Boyle's (Mariotte) Law. P v Q = Pv= (-} = R T = 
 
 \p/ T 
 
 constant for constant T. 
 
 CT) \ ~D 
 
 =rj = = constant for constant v. 
 
 Pv 
 
 (121) Boyle-Gay Lussac Law. -=r = R = constant in per- 
 
 fect gases. 
 
 (122) Avogadro's Law. N = = T^ = constant for (P. 
 
 m A i 
 
 V. T.) constant. 
 
 (123) Clausius' Law. J mi <?i 2 = f m z qf constant kinetic 
 
 energy. 
 
 (124) Dalton's Law. P = PI + P 2 + . . . = f (J mi qS + 
 
 J mz qz* + ...). Pressures. 
 
 In ordinary gases as distinguished from perfect and ideal 
 gases, all these formulas are more or less defective on account of 
 the internal action of the atoms and the molecules upon each 
 other under the stresses of electrical and other mechanical forces. 
 Many formulas, with constant coefficients and exponents, have 
 been devised to take account of these physical variations, but 
 they will not be further mentioned in this place. 
 
 The Formulas for the Mixture of Several Gases 
 
 (125) Pressure. P = P RT = ~- = nmR T = n K T = 
 
 K . RTM 
 
FORMULAS FOR THE MIXTURE OF SEVERAL GASES 29 
 
 (126) P = P, + P 2 + P 3 + . . . = (! + m + 
 
 (127) Pi = . *' - P. 
 
 n n 
 
 (128) Pi : P 2 : P 3 . . . = V l : V 2 : V 3 . . . = 
 
 Rt-Mi:RtM 9 :KiM^ 
 
 (129) P V = (M l R l + M 2 R 2 + . . .) T = 
 
 (130) Mass. = , , > ^^ 
 
 m mi nh m$ 
 
 ... M 1 +M 2 +M 3 +. 
 
 n Mi M* Ma 
 
 (1 32) n^,n,...,,... = P,,P,,P,... 
 
 (133) Density. n m = HI mi + n 2 nh + n s m 3 + . . . = 
 
 PI + P2 + PS + . . . = P = -. 
 
 (134) 
 
 11 
 
 (135) Gas Constant. R - 
 
 . . . K 
 
 n 2 m 2 
 
 * RI R 2 R 3) etc., in these equations refer to the various gas coefficients of 
 the several gases. 
 
30 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 (136) Inner Energy. M U = M l U l + M 2 / 2 + M 3 U 3 + . . . . 
 
 (137) Entropy. M S = M^ Si -f M 2 S 2 + M z S 3 + . . . . 
 
 (138) Concentration, c = ~ '- LL -. 
 
 (139) c R = 
 
 The Kinetic Theory of Gases for the Atmosphere 
 
 The various formulas involving specific heat can all be de- 
 duced from the kinetic theory of gases, and it is therefore desir- 
 able to have at least approximate values of the constants of the 
 principal gases which are the constituents of the atmosphere. 
 These are arranged in Table 7, so that the formulas from which 
 they are derived suggest by definition the exact meaning of the 
 several terms. It is much better to depend upon formulas for 
 defining constants than upon any extended verbal description 
 for the sake of accuracy and brevity. On the other hand, it is 
 not possible to study any advanced research problem in atmos- 
 pheric physics without depending upon the several terms in 
 the kinetic theory of gases. In the present status of physics, 
 research is attempting to make out the connection between the 
 theory of mechanical collisions in the molecules of a gas and the 
 corresponding dynamic electric and magnetic forces, but this 
 investigation is incomplete. 
 
 It would be very desirable that some international commission 
 should adopt a series of consistent constants for the terms of 
 Table 7, in order that all computations may be made on the 
 same basis. At present there are small variations in the values 
 in consequence of adopting slightly different fundamental con- 
 stants from which the others are derived. It is probable that 
 sufficient agreement exists among chemists and physicists as to 
 these elementary constants, in order to make this a practical 
 proposition. 
 
KINETIC THEORY OF GASES 
 
 31 
 
 .a o> 
 
 118 
 
 .; 
 
 O O 
 
 5.$ 
 
 ir 
 
 *H "O 
 
 a=a 
 
 H ':!TOiii4i ii 
 
 : II 5^.y?- t SSSS-eo i S tec 
 
 I I 
 
 IS C 
 10 
 
 fl B 
 .2 S 3 
 
 Hi 
 
 n v P 
 
32 
 
 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 This table was computed for the constants, 
 K = 82481110. Absolute gas constant, dynes/cm. 2 
 *- P = 1013235. One atmosphere in dynes/cm. 2 
 
 -j- = 41852800. Mech. equivalent heat in ergs. 
 
 It is very desirable that the International Meteorological 
 Committee should fix standard values throughout the table. 
 
 The Temperature and the Temperature Gradients Observed at 
 Different Elevations in the Free Air 
 
 The actual temperature of the atmosphere at any point is 
 the resultant of the force of gravitation as balanced by the 
 
 TABLE 8 
 
 EXAMPLES OF TEMPERATURE AND TEMPERATURE GRADIENTS AT 
 DIFFERENT ELEVATIONS 
 
 
 Lindenburg 
 
 Lindenburg 
 
 Atlantic Ocean 
 
 Victoria Nyanza 
 
 Station 
 
 Apr. 27, 1909 
 
 May 5, 1909 
 
 Sept. 25, 1907 Lat. 
 
 Summer, 1908 
 
 
 Lat. + 52 
 
 Lat. + 52 3 
 
 + 35Long. + 36 J 
 
 Lat. J 
 
 Height z 
 
 
 AT 
 
 
 AT 
 
 
 AT 
 
 
 AT 
 
 in meters 
 
 T 
 
 1000 
 
 
 1000 
 
 
 1000 
 
 
 1000 
 
 18000 
 
 
 
 
 
 
 
 
 
 
 
 226 5 
 
 4-30 
 
 
 
 190 5 
 
 6 5 
 
 17000 
 
 
 
 223.5 
 
 + 1.0 
 
 222.9 
 
 + 2.7 
 
 197.1 
 
 -5.5 
 
 16000 
 
 
 
 222.5 
 
 4-18 
 
 220.2 
 
 + 0.1 
 
 202.6 
 
 - 4 2 
 
 15000 
 
 
 
 220.7 
 
 4-10 
 
 220.1 
 
 + 0.2 
 
 206 8 
 
 4 
 
 14000 
 
 
 
 219.7 
 
 + 28 
 
 219.9 
 
 - 0.2 
 
 210.8 
 
 -52 
 
 / 13000 
 
 202.7 
 
 - 0.8 
 
 216.9 
 
 + 5.6 
 
 220.1 
 
 - 0.2 
 
 216.0 
 
 -6.6 
 
 12000 
 
 203.5 
 
 - 6.4 
 
 211.3 
 
 - 1.7 
 
 220.3 
 
 - 7.8 
 
 222.6 
 
 -8.8 
 
 11000 
 
 209.9 
 
 - 8.8 
 
 213.0 
 
 - 8.4 
 
 228.1 
 
 - 7.9 
 
 231.4 
 
 -7.5 
 
 10000 
 
 218.7 
 
 - 9.9 
 
 221.4 
 
 - 9.1 
 
 236.0 
 
 -10.0 
 
 238.9 
 
 -7.2 
 
 9000 
 
 228.6 
 
 -11.9 
 
 230.5 
 
 - 8.8 
 
 246.0 
 
 - 7.7 
 
 246.1 
 
 -4.6 
 
 8000 
 
 240.5 
 
 - 8.8 
 
 239.3 
 
 - 8.4 
 
 253.7 
 
 - 7.4 
 
 250.7 
 
 -7.3 
 
 7000 
 
 249.3 
 
 - 7.0 
 
 247.7 
 
 - 7.9 
 
 261.1 
 
 - 6.3 
 
 258.0 
 
 -5.4 
 
 6000 
 
 256.3 
 
 - 6.6 
 
 255.6 
 
 - 7.3 
 
 267.4 
 
 - 6.6 
 
 263.4 
 
 -5.8 
 
 5000 
 
 262.9 
 
 - 6.2 
 
 262.9 
 
 - 4.2 
 
 274.0 
 
 - 4.5 
 
 269.2 
 
 -6.5 
 
 4000 
 
 269.1 
 
 - 6.6 
 
 267.1 
 
 - 5.8 
 
 278.5 
 
 - 3.4 
 
 274.7 
 
 -6.1 
 
 3000 
 
 , 275.7 
 
 - 5.3 
 
 272.9 
 
 - 3.9 
 
 281.9 
 
 - 5.2 
 
 280.8 
 
 -7.6 
 
 2000 
 
 281.0 
 
 - 5.5 
 
 276.8 
 
 - 2.9 
 
 287.1 
 
 - 3.4 
 
 288.4 
 
 -7.8 
 
 1000 
 
 286 5 
 
 7 9 
 
 279 7 
 
 1 6 
 
 290 5 
 
 6 
 
 296 2 
 
 
 000 
 
 294.4 
 
 
 281.3 
 
 
 296.5 
 
 
 
 
 
 
 
 
 
 hydrostatic pressure, the circulation, and the radiation. It is 
 the most important element to be observed, and from it all the 
 
TEMPERATURE GRADIENTS IN FREE AIR 
 
 33 
 
 other terms can be computed, provided the velocity and the 
 vapor pressure are also given by the observations. In order to 
 have the data in concrete form so that the formulas may become 
 practical, four examples are taken from the observations, at 
 Lindenburg, Germany, in the Tropic North Atlantic Ocean, and 
 at Victoria Nyanza, Africa. Table 8 records the height in 
 metersj^ the absolute temperature T, the vertical temperature 
 
 ^ T 
 
 gradient per 1000 meters , and Table 9 the relative humidity 
 
 -LUUU 
 
 R. H., and the vapor pressure in millimeters e. The latter is 
 computed by taking from the Tables of vapor pressure in saturat- 
 ed air at given temperatures the dew-point vapor pressure, and 
 multiplying by the relative humidity. The Smithsonian tables 
 have been extended to include approximately the vapor pressure 
 
 TABLE 9 
 
 EXAMPLES OF THE CORRESPONDING RELATIVE HUMIDITY AND VAPOR 
 PRESSURE AT DIFFERENT ELEVATIONS 
 
 Height z 
 
 R.H. 
 
 e 
 
 R.H. 
 
 e 
 
 R.H. 
 
 e 
 
 R.H. 
 
 e 
 
 19000 
 
 per cent. 
 
 mm. 
 
 per cent. 
 
 mm. 
 
 per cent. 
 
 mm. 
 
 per cent. 
 
 mm* 
 
 18000 
 
 
 
 43 
 
 0.020 
 
 
 
 33 
 
 0.000 
 
 17000 
 
 
 
 43 
 
 .004 
 
 47 
 
 0.014 
 
 33 
 
 .000 
 
 16000 
 
 . 
 
 
 
 43 
 
 .012 
 
 47 
 
 .012 
 
 33 
 
 .001 
 
 15000 
 
 
 
 
 
 44 
 
 .011 
 
 47 
 
 .012 
 
 34 
 
 .002 
 
 14000 
 
 
 
 44 
 
 .010 
 
 47 
 
 .012 
 
 34 
 
 .003 
 
 13000 
 
 61 
 
 '6.002 
 
 44 
 
 .007 
 
 47 
 
 .012 
 
 34 
 
 .004 
 
 12000 
 
 61 
 
 .002 
 
 43 
 
 .004 
 
 47 
 
 .012 
 
 34 
 
 .010 
 
 11000 
 
 61 
 
 .043 
 
 43 
 
 .004 
 
 47 
 
 .027 
 
 34 
 
 .029 
 
 10000 
 
 61 
 
 .012 
 
 43 
 
 .012 
 
 47 
 
 .063 
 
 34 
 
 .058 
 
 9000 
 
 61 
 
 .039 
 
 43 
 
 .033 
 
 47 
 
 .183 
 
 34 
 
 .134 
 
 8000 
 
 62 
 
 .135 
 
 44 
 
 .084 
 
 48 
 
 .402 
 
 35 
 
 .218 
 
 7000 
 
 64 
 
 .284 
 
 45 
 
 .209 
 
 51 
 
 .843 
 
 36 
 
 .449 
 
 6000 
 
 69 
 
 .741 
 
 47 
 
 .473 
 
 54 
 
 1.544 
 
 41 
 
 .831 
 
 5000 
 
 72 
 
 1.400 
 
 50 
 
 .972 
 
 56 
 
 2.750 
 
 50 
 
 1.667 
 
 4000 
 
 67 
 
 2.370 
 
 51 
 
 1.420 
 
 58 
 
 3.910 
 
 73 
 
 3.770 
 
 3000 
 
 67 
 
 3.710 
 
 42 
 
 1.910 
 
 66 
 
 5.600 
 
 69 
 
 5.440 
 
 2000 
 
 70 
 
 5.590 
 
 41 
 
 2.460 
 
 65 
 
 7.770 
 
 60 
 
 7.800 
 
 1000 
 
 63 
 
 7.250 
 
 60 
 
 4.390 
 
 72 
 
 10.700 
 
 57 
 
 12.030 
 
 000 
 
 54 
 
 15.620 
 
 58 
 
 4.730 
 
 78 
 
 16.770 
 
 
 
 at very low temperatures. These values of T and e will be used 
 in illustrating the barometric reduction formulas. By formula 
 
34 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 (22) the adiabatic temperature gradient per 1000 meters is 
 -9.869 C., or per 1000 feet -5.415 F. In Table 8 it is seen 
 how widely the actual temperature gradients differ from this value, 
 and it is this circumstance that compels us to reconstruct the 
 entire range of standard thermodynamic formulas, in order to 
 adapt them to practical work in the earth's atmosphere. The 
 slow progress of meteorological physics is due to this difference 
 of gradient more than to any other cause. The temperature 
 gradients are incessantly varying in the atmosphere from large 
 temperature falls, A jT, to considerable temperature gains, 
 -f A T. This change of sign gives rise to the subject of the 
 inversion of temperature of which examples occur often at night 
 near the ground, and usually in the isothermal region. 
 
 The values of the vapor pressure below T = 273 are those 
 of the Smithsonian Tables extended. 
 
 The Temperature Gradient in a Plateau from the Sea Level to the 
 Surface of the Ground 
 
 An important part of barometry is the determination of the 
 temperature gradient within the land mass forming a plateau 
 region, as in the Rocky Mountain district of the United States, 
 by means of which the pressure observed at a station on the 
 surface may be reduced to the sea level, in order to be combined 
 with those stations having low elevations, so as to make a 
 synchronous map of storm conditions for the entire country. 
 This problem is one of considerable difficulty, and it must be 
 solved in accordance with the prevailing local conditions, so 
 that no fixed rules can be given for its treatment. A very 
 extensive reduction for the United States is found in the "Report 
 on Barometry," already mentioned, but its leading principles 
 can be briefly summarized. Having low-level stations in the 
 eastern, central, and Pacific districts, the problem is to connect 
 up the stations on the plateau at different elevations above the 
 sea level, by means of the average temperature gradient within 
 the land mass, which is very different from the gradient above 
 the plateau in the free air. The first step is to construct from 
 the available data approximate temperature gradients, which 
 
TEMPERATURE GRADIENT IN A PLATEAU 35 
 
 can be used for short distance reductions, in longitude, in latitude, 
 and on the vertical. Then certain reference vertical lines are 
 chosen, as the intersection of the parallels and meridians for each 
 5-degree interval, and for the planes 1000 feet apart. To these 
 points are reduced the stations by numerous combinations, so that 
 the same station is reduced to several selected reference points. 
 These points now lie on vertical lines, and each line of tempera- 
 tures may by plotting be extended downward to sea level. The 
 second step is to draw the sea-level isotherms between the 
 central and the Pacific districts, joining across the plateau 
 region by the most probable curves. The third is to compare 
 by interpolation these horizontal temperatures with those 
 obtained from vertical extension, and by mutual adjustments 
 the two sets may be made to agree harmoniously. This inter- 
 locking of a horizontal system with a vertical system is able to 
 produce by mutual checking a very exact agreement between 
 the two sets. In this way the sea-level temperature was found 
 beneath the plateau, and thence the temperature gradients in 
 a vertical direction were computed. These gradients differ 
 greatly from one another in different parts of the plateau, they 
 differ from month to month at the same place, and there is no 
 fixed gradient which can be used at any given station. The 
 station gradients fall into distinct classes in the several parts of 
 the plateau in respect to the yearly variations, and diagrams 
 were constructed to serve for the individual stations. The 
 monthly gradient may be roughly summarized for comparison 
 with the free-air gradients. 
 
 A T C. AT F. 
 
 The reduction from 7^7- - to - _ , -- is effected by 
 100 meters 100 feet 
 
 the factor 1 = 0.55. This is from 1 X ^~ = p^ ' 55 ' 
 
 This plateau gradient is only 37 per cent, of the free air adiabatic 
 gradient. Extensive reduction barometric tables were con- 
 structed for many stations in the United States, which are used 
 in compiling the weather forecast charts, of which further mention 
 will be made. 
 
36 
 
 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 TABLE 10 
 
 THE MEAN MONTHLY TEMPERATURE GRADIENTS IN THE ROCKY MOUNTAIN 
 PLATEAU OF THE UNITED STATES 
 
 A 27100 
 
 Jan. 
 
 Feb. 
 
 March 
 
 Apr. 
 
 May 
 
 June 
 
 
 Per 
 100 ft. 
 Per 
 100m. 
 
 -0.191 
 -0.348 
 
 -0.174 
 -0.317 
 
 -0.145 
 -0.264 
 
 -0.202 
 -0.368 
 
 -0.194 
 -0.353 
 
 -0.215 
 -0.391 
 
 A 77100 
 
 July 
 
 Aug. 
 
 Sept. 
 
 Oct. 
 
 Nov. 
 
 Dec. 
 
 Year 
 
 Per 
 100 ft. 
 Per 
 100m. 
 
 -0.175 
 -0.319 
 
 -0.179 
 -0.326 
 
 F 
 -0.168 
 
 -0.306 
 
 -0.191 
 -0.348 
 
 -0.181 
 -0.329 
 
 -0.187 
 -0.340 
 
 op 
 -0.202 
 
 -0.368 
 
 The Integral Mean Temperature and Gradient 
 
 In reducing pressures from one level to another it is necessary 
 to know the mean temperature of the actual air column in the 
 free air, or in the hypothetical air column within a plateau, or 
 between a mountain summit and the sea level, or other plane of 
 reference. These are found from the summation of the tem- 
 peratures at several levels as in Table 8, and dividing the sum by 
 the number of the strata taken. 
 
 (140) 
 
 from (23). 
 
 For two strata it is the arithmetical mean, 
 (141) 7\ = J (7 1 ! + To). 
 
 For example, in the balloon ascension, April 27, 1909, sum the 
 T z through the several strata; also, only the top and bottom of 
 the same thick layer, as indicated. 
 
INTEGRAL MEAN TEMPERATURE AND GRADIENT 37 
 
 Layer in meters 
 
 -s r 
 
 z 
 
 i (TJ. + r ) 
 
 000 to 4000 
 
 281.34 
 
 281.75 
 
 000 to 9000 
 
 264.43 
 
 260.15 
 
 000 to 13000 
 
 248.51 
 
 243.55 
 
 It is not usually sufficient to take the mean value of the top 
 and bottom temperature of a thick layer for the mean tempera- 
 ture of the column, and the error of reduction is dependent upon 
 the discrepancy between T m and T iQ . If the gradient is uniform 
 between two strata T m = T w , and the difference vanishes. The 
 length of vertical distance that permits 7\ to be used depends 
 upon local temperature distributions, and each case must be 
 carefully examined. The same rule applies to the vapor pressure 
 e, and any other meteorological element. The determination of 
 the integral mean with accuracy is one of the hardships of practi- 
 cal meteorology, upon which a large amount of labor is necessarily 
 expended. 
 
 The Virtual Temperature T r . 
 
 It is sometimes convenient to combine the actual mean 
 temperature T m with the expression for the vapor pressure 
 
 p 
 
 term 0.378-^ to form the so-called virtual temperature T r , by 
 the formula, 
 
 (142) 2T- = T w (l + 0.378 J) 
 
 The barometric reduction can then be carried forward as if 
 the dry air and the aqueous vapor were compounded in one gas 
 whose equivalent temperature is T r . 
 
 /I rrt 
 ^jT. 
 
 The ratio of the change of temperature d T to the prevailing 
 temperature T is related to the logarithm in the follov/ing useful 
 auxiliary formulas, which are often needed in substitutions. 
 
38 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 They are applicable through such strata, thin or thick, as have 
 uniform temperature gradients, whether the temperature in- 
 creases or diminishes in a vertical direction. They are given 
 for the adiabatic and the non-adiabatic temperature variations. 
 
 /< ^o\ I a a 1 a /I T 1 1 T> \ 
 
 (143) / -=- = T = M lo S r" = M ( lo S r ~ lo S r o). 
 
 J 1 a J- aO M- 1 o M 
 
 (144) f^ = r '~ r = i log p = ^ (log Tx - log To). 
 J 1 lio M 1 Q M 
 
 Ti-T 1 
 
 10 
 
 1 
 
 (n- 1) M g R ~ (n- 1) M 
 
 The Temperature Variations and the Specific Heat 
 
 It is convenient to make the transfer from the non-adiabatic 
 temperature loss to the adiabatic temperature loss, in connection 
 with the specific heat, by using, as in (13), 
 
 T a -T 
 
 (146) i Cp a (T, - To) = Cp a (T a -To) = - go(zi-z ). 
 
 (147) m Cp 10 (T, - To) = Cp 10 (T a - To) = Pl ~ P 
 
 Pi0 
 
 These formulas will be fully illustrated in a later chapter. 
 It is evident that many combinations can be made by employing 
 the formulas (143) to (147), and they are very practical in de- 
 veloping the formulas. 
 
 T f j- r^ Bo I 0r Zi Z \ 
 
 Transformation of -5- = -^ f 1 + 1.25 ^ J. 
 
 The introduction of the plateau effect upon gravity in (69) 
 has its parallel in the effect upon the barometric pressure, which 
 is similarly modified. We have for both cases, 
 
 (i> . -i. (i + 
 
GENERAL BAROMETRIC FORMULA 39 
 
 Since 1 + 1.25 ~^~ is a small variation from unity, the 
 K 
 
 general formula is applicable, 
 
 (150) Com. log (1 + x) = M (x - % x* + % x* - . . .). 
 
 Passing to common logarithms (149) becomes on neglecting 
 the powers above x, 
 
 (151) log J = log | + log (l + 1.25 *-^) , 
 
 . Bo , M 1.25 , 
 
 = log -g + -- (2i - Z ). 
 
 It will be found in the barometric formula that, 
 
 P ' B 
 
 (152) Zi z = K log -5- (approx.) = K log -_- (approx.). Hence, 
 
 /ieo\ 1 ^ 1 -^ i 1 nt-M K BQ 
 
 (153) log - = log - + 1.25 -- log - , 
 
 logj (1 + 0.00157) = log ~ (1 + y). 
 
 The General Barometric Formula 
 
 The several auxiliary formulas now deduced make it very 
 simple to derive the barometric reduction formula connecting 
 together the heights (z\ . z ) and the mercurial pressures (5i, B ). 
 From (41) the differential pressure is, 
 
 (154) - dP = p.gdz. Divide by P, 
 
 (155) - d -j-=^gdz. Divide by |, 
 
 d P P 
 
 (156) - -5- . - = g . d z. Substitute from (75), 
 
 * P 
 
 (157) - -- . - l + 0.378 = g d z. 
 
 po 1 o 
 
40 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 Substitute P from (49) and g+ f from (63), (64), 
 _ dP Bo Pm go d i Q 370 __\ = 
 
 X Po J- \ -t>Q/ 
 
 go (1 - 0.0026 cos2(j>) (l - -jj{} d z. 
 Pass to common logarithms by the factor -^. and integrate, 
 
 /IKON i -^ -^Q Pm T m / e\ 
 
 (159) log -p- . -vp- . ( 1 + 0.378 -=- ) = 
 
 r M po 1 o \ JDQ/ m 
 
 e 
 =- 
 
 o DQ m 
 
 - 0.0026 cos 
 
 The last gravity form is from (67). The constant 
 
 (160) K = 2~ = 18400 (Metric), and JRTi = 60367.7 (English). 
 M po 
 
 T m is the mean temperature of the column (140), and the 
 integral mean of ( 1 + 0.378-^-) is accomplished by the observa- 
 
 \ JJQ/ 
 
 tions along the column, or by integrating from the surface by 
 (82) and (83) . Substituting (153) 
 
 (161) log ~ (1 + 0.00157) K(l+ 0.00367 9) (l + 0.378 } 
 
 & \ -L>0/ m 
 
 Barometer. " Const. Temperature. pSsure. 
 
 (1 + 0.0026 cos 20) l + l = zi - Z . 
 
 Gravity in Gravity in 
 
 Latitude. Height. 
 
 Substituting the numerical values and combining, 
 (162) log So = log B + 18429+67 Y0 2 C.+ o.003 < 
 
 - 0.378^-) (1 - 0.0026 cos 20). Metric. 
 (163) log = logB ~ 
 
 - 0.378^-) (1 - 0.0026 cos 20). English. 
 
CORRECTIONS TO THE BAROMETER 41 
 
 These can be expressed in the general form, 
 
 (164) log B, = log B + m (1 - 0) (1 ~r) 
 
 = log B + m m ft m ?. 
 
 In view of the uncertainty attaching to our knowledge at 
 any time of the distribution of the vapor pressure in the air 
 column, it is desirable to keep the term m ft separate as a correc- 
 tion to the difference between the logarithms. Similarly the 
 gravity term is retained by itself because in many computations 
 it is small and can be neglected. Complete reduction tables are 
 given in the " Report on the International Cloud Observa- 
 tions" for the metric system, Tables 91, 92, 93, and in the 
 " Report on the Barometry of the United States, Canada, and 
 the West Indies" for the English system, Tables 13 to 21. 
 From these logarithm tables many forms of numerical tables 
 without logarithms can be constructed for special purposes. 
 
 Corrections to the Barometer 
 
 The mercurial barometer requires several corrections before 
 the pressure can be used in practice. 
 
 1. Correction to the Standard Temperature. The instrument 
 is constructed of parts whose coefficients of expansion with 
 changes of temperature are not the same, as for the mercury 
 and the brass scale. Adopting the notation, 
 
 t = the temperature of the attached thermometer. 
 t m = the standard temperature of mercury, C., 32 F. 
 t s = the standard temperature of the brass scale, C., 62 F. 
 m = the coefficient of expansion of mercury, 0.0001818 per 
 degree Centigrade, 0.0001010 per degree Fahrenheit. 
 
 n = the coefficient of expansion of brass, 0.0000184 per degree 
 Centigrade, 0.0000102 per degree Fahrenheit. 
 The accepted formulas are as follows : 
 
 (165) B n - B = - B ( m ~ n \ t f or B n and B in millimeters. 
 
 1 + m t 
 
 inches. 
 
42 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 The English form reduces to, 
 
 (167) B n - B -B 10978 + 1.112 /' 
 
 The necessary reduction tables are found in nearly all 
 compilations of Meteorological Tables. 
 
 2. Correction to the Standard Gravity, g&. This is the gravity 
 variation in latitude as given in (63), from which is obtained, 
 
 (168) 45 - B+ = - B+ (l - ~ } = - B+ 0.00260 cos 2<. 
 
 V 45 / 
 
 The temperature and gravity corrections are applied as 
 instrumental corrections to the actual barometric reading at a 
 given hour. 
 
 3. Correction to a Standard Barometer or Patron. Each 
 barometer as an instrument has certain minor deficiencies which 
 cannot be readily analyzed, and in order to make a number of 
 barometers homogeneous, so as to give strictly comparable 
 pressures, it is necessary that they be severally standardized 
 by comparison with an adopted normal or patron barometer. 
 The Kew barometer is used for many standard comparisons, 
 and each weather service keeps its own standard which has been 
 carefully compared with the Kew instrument. Within each 
 national service the barometers are compared, and an instru- 
 mental correction is given for each barometer before sending to 
 a station. Sometimes these corrections hold steadily for long 
 intervals, and sometimes they change suddenly and erratically. 
 Whenever there are local removals, or whenever a barometer is 
 cleaned, its correction must again be determined. Frequent 
 inspections and comparisons with a portable secondary standard 
 are necessary if a homogeneous series of pressures is to be secured . 
 It is not possible to be too painstaking in respect of the inter- 
 barometric corrections. 
 
 4. The Station or Removal Correction. If it happens that at a 
 given station there are any removals of the barometer from one 
 office to another, as so frequently happens in large cities, and the 
 elevation is thereby changed from time to time, it is necessary 
 to adopt a standard elevation for the station and reduce the 
 
CORRECTIONS TO THE BAROMETER 43 
 
 series of readings taken at any other height to this level, which 
 will persist as the adopted station elevation from the beginning 
 to the end of the service. When the change in height is con- 
 siderable these corrections depend upon the temperature in the 
 course of the year. A correction card for instrumental and 
 station removal errors should accompany each barometer, and 
 be carefully recorded as part of the history of the instrument. 
 In preparing homogeneous tables of pressure for use in solar 
 physics and other cosmical problems, it is indispensable that 
 all barometer readings should be carefully treated in this manner. 
 The homogeneous system for the United States has thus been 
 prepared by the author to cover the barometric pressure, the 
 temperature, the vapor pressure, and the precipitation from the 
 year 1871 to date, and the published data of the Weather Bureau 
 are all on that basis. Similar homogeneous data are being pre- 
 pared for Argentina and for other countries. 
 
 5. The Correction from the Surface Temperature (t) to the Mean 
 Temperature of the Air Column (8) in Barometric Reductions. It 
 is obviously so difficult to determine the relation of the surface 
 temperature / to the mean air column temperature within a 
 land mass 0, as from a station on a plateau to the sea level, that 
 a special study was made of this subject in order to facilitate a 
 prompt reduction of the observed pressure to the corresponding 
 sea-level pressure. These are needed for transmission by 
 telegraph to a central office where the daily weather forecast 
 charts are constructed. Unfortunately there is no simple rule 
 connecting / and 0, and in many cases the difference 6 t is 
 very variable. Reduction tables are first computed with the 
 adopted elevation H, and a series of assumed values of 6 for 
 several barometric pressures in steps of 0.10 inch. Then the 
 relation between / and having been found, the surface tem- 
 perature is used as the argument for the table in place of 6. The 
 practical value of / taken in the United States, where the observa- 
 tions are made at 8 A.M. and 8 P.M. daily, is the mean of the cur- 
 rent dry-bulb temperature and that taken twelve hours before. 
 This gives a fair temperature average for the day, and it tends 
 to eliminate some of the local effects of passing storms. It has 
 
44 
 
 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 been found to work well in the practice of ten years. In order 
 to illustrate the differences between / and 6 in the course of the 
 year, as the temperatures change from summer to winter, a 
 few examples are extracted from Table 53 of the Barometry 
 Report, where the heights are in feet, and temperatures are 
 Fahrenheit. 
 
 TABLE 11 
 RELATION BETWEEN THE SURFACE TEMPERATURE / AND THE MEAN 
 
 Boise, 
 
 Salt Lake 
 
 Independence, 
 
 Helena, 
 
 Pike's Peak, 
 
 Battleford, 
 
 Idaho 
 
 City, Utah 
 
 Cal. 
 
 Mont. 
 
 Colo. 
 
 Canada 
 
 2739 
 
 4366 
 
 3910 
 
 4110 
 
 14111 
 
 1608 
 
 1 
 
 e 
 
 t 
 
 e 
 
 t 
 
 9 
 
 / 
 
 e 
 
 t 
 
 e 
 
 t 
 
 e 
 
 -42 
 
 -40 
 
 
 
 
 
 -48 
 
 -40 
 
 -61 
 
 -40 
 
 
 
 -32 
 
 -30 
 
 
 
 
 
 -37 
 
 -30 
 
 -49 
 
 -30 
 
 -40 
 
 -30 
 
 -23 
 
 -20 
 
 -24 
 
 -20 
 
 
 
 -26 
 
 -20 
 
 -35 
 
 -20 
 
 -29 
 
 -20 
 
 -13 
 
 -10 
 
 -14 
 
 -10 
 
 
 
 -15 
 
 -10 
 
 -23 
 
 -10 
 
 -17 
 
 -10 
 
 - 3 
 
 - 
 
 - 5 
 
 
 
 - 3 
 
 '6 
 
 - 4 
 
 
 
 -12 
 
 
 
 - 6 
 
 
 
 7 
 
 10 
 
 5 
 
 10 
 
 6 
 
 10 
 
 7 
 
 10 
 
 - 2 
 
 10 
 
 5 
 
 10 
 
 17 
 
 20 
 
 14 
 
 20 
 
 15 
 
 20 
 
 20 
 
 20 
 
 5 
 
 20 
 
 16 
 
 20 
 
 27 
 
 30 
 
 24 
 
 30 
 
 24 
 
 30 
 
 29 
 
 30 
 
 10 
 
 30 
 
 27 
 
 30 
 
 37 
 
 40 
 
 34 
 
 40 
 
 35 
 
 40 
 
 38 
 
 40 
 
 16 
 
 40 
 
 38 
 
 40 
 
 47 
 
 50 
 
 44 
 
 50 
 
 48 
 
 50 
 
 46 
 
 50 
 
 22 
 
 50 
 
 49 
 
 50 
 
 57 
 
 60 
 
 55 
 
 60 
 
 60 
 
 60 
 
 55 
 
 60 
 
 30 
 
 60 
 
 59 
 
 60 
 
 67 
 
 70 
 
 66 
 
 70 
 
 71 
 
 70 
 
 66 
 
 70 
 
 38 
 
 70 
 
 69 
 
 70 
 
 77 
 
 80 
 
 76 
 
 80 
 
 81 
 
 80 
 
 76 
 
 80 
 
 47 
 
 80 
 
 79 
 
 80 
 
 87 
 
 90 
 
 86 
 
 90 
 
 92 
 
 90 
 
 86 
 
 90 
 
 56 
 
 90 
 
 89 
 
 90 
 
 97 
 
 100 
 
 96 
 
 100 
 
 102 
 
 100 
 
 96 
 
 100 
 
 
 
 99 
 
 100 
 
 Similarly, the relations between the surface / and the mean 
 free-air temperature, or the mean plateau temperature, 6, have 
 been prepared for reductions to the sea-level plane, the 3,500- 
 foot level, and the 10,000-foot planes for over 200 stations, so 
 that synchronous charts can be constructed on each of these 
 three planes simultaneously. Such charts were prepared for 
 one year, in part by telegram and in part by card reports, so 
 that the pressure charts could be studied on the sea level, on 
 the mean-plateau level, and in the two-mile level. These com- 
 parisons were so suggestive and instructive in respect of the 
 progress of storms and the areas of precipitation as to make 
 them of great value in practical forecasts of weather conditions. 
 The trend of the upper-level isobars shows clearly the course of 
 
CORRECTIONS TO THE BAROMETER 45 
 
 the storm track for 24 to 36 hours, whereas the sea-level isobars 
 have very little evidence of this kind. This is because the closed 
 isobars on the sea level have generally opened up into sweeping 
 curves on the two-mile level. Similarly, the rain areas are 
 indicated by the region of most oblique crossing of the lower 
 with the upper isobars. There is a great future for meteorology 
 in the use of these upper level charts. 
 
 6. The Plateau Correction C A0 H. An extensive discussion 
 of the reduced pressures on the sea-level plane showed a series of 
 plateau differences depending upon a station constant C = 0.001 
 usually, A = the departure of the monthly from the annual 6, 
 and H = the height of the station in feet in units of 1,000 feet, 
 so that A B = C. A 6. H. This plateau correction was computed 
 and applied to all the plateau stations of the United States. It 
 seems to take account of the effect of the land mass in the course 
 of the year upon the temperature distribution, which is very 
 complex in its action. 
 
 7. The Local Correction A A. After the corrections above 
 mentioned have been applied, there are still a few stations which 
 require a small correction A A to make them harmonize with the 
 pressure system surrounding them. The cause is still obscure 
 and is very local, possibly due to the wind action near the office. 
 
 8. The Local Vapor Pressure Correction. The prevailing 
 relative humidity and the corresponding vapor pressure are 
 approximate functions of the temperature in each locality, so 
 that an approximate value of the correction to the barometer 
 due to the presence of the aqueous vapor can be found for each 
 station and applied along with the other corrections. 
 
 9. The Station Pressure Reduction Charts. It should be noted 
 that all the barometric corrections have been made in terms of 
 the surface temperature so that this t and the barometer reading 
 B. when corrected for the several instrumental errors, become the 
 arguments for the reduction to any plane. In practical work, 
 instead of corrections from the station to the sea level or other 
 plane of reference being furnished to the several stations, there 
 have been prepared, for the arguments (/, B), the reduced sea- 
 level pressure at once in a sufficiently expanded form of tables to 
 
46 
 
 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 permit of quick and accurate interpolation; similar tables were 
 provided for the 3,500-foot and the 10,000-foot planes. With 
 these auxiliary station tables the reduced pressures are promptly 
 obtained, and transformed into the telegraph cipher code for 
 transmission to other offices. In this way a large number of 
 stations in the United States receive in several telegraph circuits, 
 by interchange of messages, the necessary data for all map con- 
 struction. Making the observations at 8 A.M., 75th meridian 
 time in all districts, for example, the data are received, recorded, 
 interpreted as forecasts, and usually retransmitted to all parts 
 of the country within two hours, or by 10 o'clock. 
 
 Examples of the Barometric Reduction Tables 
 
 "Barometry Report," U. S. W. B., 1900-01, Tables 13-21. 
 " International Cloud Report/' U. S. W. B., 1898-99, Tables 
 91-93. 
 
 In order to illustrate the barometric formulas in practice, the 
 example of Santa Fe, N. M., is here given. 
 
 TABLE 12 
 
 REDUCTION TO THE SEA LEVEL BY THE W-TABLES IN LOGARITHMS 
 SANTA FE, NEW MEXICO 
 
 Height = 7013 feet. Longitude, 105 57' Latitude, 35 41' 
 
 Arguments 
 
 Jan. 
 
 April 
 
 July 
 
 Oct. 
 
 Year 
 
 Station B 
 Temperature . . .6 
 Vapor Pressure CQ 
 
 23.180 in. 
 32.0F. 
 0.145 in. 
 
 23.177 in. 
 54. 3 F. 
 0.199 in. 
 
 23.362 in. 
 79.0F. 
 0.574 in. 
 
 23.294 in. 
 58.0F. 
 0.294 in. 
 
 23. 248 in. 
 56.0F. 
 0.274 in. 
 
 Logarithm log B. 
 Table 17 m 
 Table 19C, - $m 
 Table 20 ym 
 
 Sum. log BO 
 
 1.36511 
 + .11594 
 -.00016 
 -.00011 
 
 1 48078 
 
 1.36506 
 + .11090 
 -.00022 
 -.00010 
 
 1.47564 
 
 1.36851 
 + .10581 
 -.00061 
 -.00010 
 
 1 . 47361 
 
 1.36725 
 + .11011 
 -.00033 
 -.00010 
 
 1 . 47693 
 
 1.36638 
 + .11054 
 -.00031 
 -.00010 
 
 1 47651 
 
 Sea level .... B 
 
 30.254 
 
 29.898 
 
 29.759 
 
 29.987 
 
 29.958 
 
 Table 19A Arg I . 
 Tablel9BArg.II 
 
 .0018 
 .0014 
 
 .0025 
 .0020 
 
 .0073 
 .0058 
 
 .0037 
 .0030 
 
 .0035 
 .0028 
 
 Reduction B B 
 
 7.074 
 
 6.721 
 
 6.397 
 
 6.693 
 
 6.710 
 
EXAMPLES OF BAROMETRIC REDUCTION 
 SECOND FORM OF TABLE, NUMERICAL 
 
 47 
 
 Table 21, Sec. I. . 
 
 7.028 
 
 6.762 
 
 6.487 
 
 6.719 
 
 6.742 
 
 Table 21, Sec. II. 
 
 -f. 064 
 
 -.018 
 
 -.041 
 
 + .004 
 
 -.003 
 
 /?w Ym 
 
 -.018 
 
 --.022 . 
 
 -.049 
 
 -.030 
 
 -.028 
 
 Reduction 
 
 7.074 
 
 6.722 
 
 6.397 
 
 6 693 
 
 6.711 
 
 
 
 
 
 
 
 The station pressure B is already corrected for the tempera- 
 ture, gravity, instrumental and removal errors. 
 
 For the argument ^ (t Sp + O or y 2 (t 8a + t 8p ), take 0. 
 
 For the arguments (B. 0), in Table 17, take m. 
 
 For the arguments (B . e ) in Table 19 A, take Arg. I (below). 
 
 For the arguments (Arg. I, H), in Table 19 J5, take Arg. II 
 (below) . 
 
 For the arguments (Arg. II, m), Table 19 C, take - m. 
 For the arguments ($, m), in Table 20, take f m. 
 
 Numerical Form 
 
 For the arguments (H, B = 30 inches), in Table 21, 1, take 
 first reduction. 
 
 For the arguments H, B^ (approx.), in Table 21, II, proceed 
 by trials. 
 
 Interpolate the correction for humidity from Table 21, III. 
 
 Interpolate the correction for gravity from Table 21, IV. 
 
 These two methods work very rapidly after a little practice, 
 and the reductions are valid to the 0.001 inch of pressure. In 
 order to illustrate the method of reduction for the plateau the 
 following example is given in Table 13. 
 
 The assumed station pressure B has the four station correc- 
 tions applied. The body of the reduction table was computed 
 for assumed values of 6, which correspond with certain surface 
 temperatures /, computed from the .consecutive 8 o'clock pairs 
 as observed. This t becomes then the practical argument for 
 station reductions. The mean annual 6 for Santa Fe was taken 
 63 F., and A B at any time is the variation (0 63). The 
 
48 
 
 METEOROLOGICAL CONSTANTS AND FORMULAS 
 
 TABLE 13 
 
 SANTA FE, NEW MEXICO 
 
 REDUCTION OF PRESSURE TO THE SEA LEVEL, THE 3,500- AND 
 10,000-FooT PLANES 
 
 I. Reduction to sea level 
 Elevation, 7,013 feet. Longitude, 105 57'. Latitude, 35 41' 
 
 Temp. 
 
 Correction for 
 
 22.40 
 
 22.60 
 
 22.80 
 
 23.00 
 
 23.20 
 
 23.40 
 
 23.60 
 
 23.80 
 
 24.00 24.20 
 
 I 
 
 e 
 
 C.A 
 
 0.H 
 
 e 
 
 LA 
 
 Reduction Bo B from w-Table. 
 
 -27 
 -16 
 - 5 
 5 
 16 
 26 
 35 
 43 
 50 
 58 
 67 
 77 
 88 
 
 -20 
 -10 
 
 10 
 20 
 30 
 40 
 50 
 60 
 70 
 80 
 90 
 100 
 
 -.50 
 -.44 
 -.38 
 -.32 
 -.26 
 20 
 
 
 
 +7.79 
 .59 
 .40 
 .22 
 7.05 
 6 88 
 
 7.86 
 . 66 
 .46 
 .28 
 7.11 
 6 94 
 
 7.93 
 
 .73 
 .53 
 .34 
 .17 
 7 00 
 
 8.00 
 7.80 
 .60 
 
 .41 
 .24 
 7 07 
 
 8.07 
 7.86 
 .66 
 .47 
 .30 
 7 13 
 
 8.14 
 7.93 
 .73 
 .54 
 .36 
 19 
 
 8.21 
 7.99 
 .79 
 .60 
 
 .42 
 25 
 
 8.28 
 8.06 
 7.86 
 .67 
 .49 
 31 
 
 8.35 
 8.3 
 7.93 
 .73 
 .55 
 37 
 
 8.42 
 .20 
 8.00 
 7.80 
 .61 
 43 
 
 
 
 
 
 .00 
 -.01 
 01 
 
 .00 
 .00 
 00 
 
 -.14 
 
 -.01 
 
 00 
 
 .72 
 
 .78 
 
 6.84 
 
 6.90 
 
 6.96 
 
 7.02 
 
 7.08 
 
 7.14 
 
 .20 
 
 .26 
 
 -.08 
 -.02 
 + 04 
 
 -.01 
 -.01 
 - 02 
 
 .00 
 .00 
 00 
 
 .57 
 .43 
 .29 
 
 .63 
 .49 
 35 
 
 .69 
 .55 
 41 
 
 .75 
 .60 
 46 
 
 .81 
 .66 
 52 
 
 6.87 
 .72 
 58 
 
 6.93 
 
 .77 
 63 
 
 6.99 
 .83 
 68 
 
 7.05 
 6.89 
 74 
 
 7.11 
 6.95 
 ,80 
 
 + .10 
 + .16 
 
 + .22 
 
 -.03 
 -.03 
 -.03 
 
 .00 
 .00 
 .00 
 
 .16 
 6.04 
 5.91 
 
 .22 
 6.09 
 5.97 
 
 .28 
 .15 
 6.02 
 
 .33 
 .20 
 6.07 
 
 .39 
 .26 
 6.13 
 
 .44 
 .31 
 6.18 
 
 .49 
 .36 
 6.23 
 
 .54 
 .41 
 6.28 
 
 .60 
 .47 
 6.34 
 
 .66 
 .52 
 6.39 
 
 Date 
 
 Jan. 
 
 Feb. 
 
 Mch. 
 
 Apr. 
 
 May 
 
 June 
 
 July 
 
 Aug. 
 
 Sept. 
 
 Oct. 
 
 Nov. 
 
 Dec. 
 
 5 
 
 28.9 
 
 30.6 
 
 37.0 
 
 44.2 
 
 52.7 
 
 62.0 
 
 67.4 
 
 67.0 
 
 62.0 
 
 53.2 
 
 41.7 
 
 33.1 
 
 15 
 
 27.9 
 
 32.0 
 
 39.5 
 
 46.6 
 
 55.7 
 
 65.1 
 
 68.5 
 
 66.3 
 
 59.9 
 
 49.8 
 
 37.6 
 
 30.8 
 
 25 
 
 29.3 
 
 34.5 
 
 41.9 
 
 49.6 
 
 58.8 
 
 66.2 
 
 67.8 
 
 64.2 
 
 56.5 
 
 45.7 
 
 35.3 
 
 29.8 
 
 Note. A A and C.&O.H have been united. 
 
 II. Reduction to the 3,500-foot plane 
 
 Temp. 
 
 Correction for 
 
 22.40 
 
 22.60 
 
 22.80 
 
 23.00 
 
 23.20 
 
 23.40 
 
 23.60 
 
 23.80 
 
 24.00 
 
 24.20 
 
 I 
 
 0i 
 
 C.A 
 0.H 
 
 e 
 
 AA 
 
 Reduction Bi B from w-Table 
 
 -24 
 
 -20 
 
 -.27 
 
 
 
 +3.61 
 
 3.65 
 
 3.68 
 
 3.71 
 
 3.75 
 
 3.78 
 
 3.81 
 
 3.84 
 
 3.87 
 
 3.90 
 
 14 
 
 10 
 
 24 
 
 
 
 .52 
 
 56 
 
 59 
 
 62 
 
 66 
 
 69 
 
 72 
 
 75 
 
 78 
 
 81 
 
 - 4 
 
 
 
 -.21 
 
 
 
 .44 
 
 .47 
 
 .50 
 
 .53 
 
 .57 
 
 .60 
 
 .63 
 
 .66 
 
 .69 
 
 .72 
 
 fi 
 
 10 
 
 18 
 
 
 
 36 
 
 39 
 
 42 
 
 45 
 
 48 
 
 51 
 
 54 
 
 57 
 
 60 
 
 63 
 
 16 
 
 ?0 
 
 -.15 
 
 
 
 .29 
 
 .32 
 
 35 
 
 38 
 
 41 
 
 44 
 
 46 
 
 49 
 
 52 
 
 55 
 
 ?6 
 
 30 
 
 -.12 
 
 
 
 22 
 
 25 
 
 28 
 
 30 
 
 33 
 
 36 
 
 39 
 
 42 
 
 45 
 
 47 
 
 36 
 
 40 
 
 -.09 
 
 . . . 
 
 
 .15 
 
 .18 
 
 .21 
 
 .23 
 
 .26 
 
 .29 
 
 .32 
 
 .35 
 
 .38 
 
 .40 
 
 45 
 
 50 
 
 -.06 
 
 
 
 .08 
 
 .11 
 
 .14 
 
 .16 
 
 .19 
 
 .22 
 
 .25 
 
 .28 
 
 .31 
 
 .33 
 
 55 
 
 60 
 
 -.03 
 
 .66 
 
 .00 
 
 3.02 
 
 3.05 
 
 .08 
 
 .10 
 
 .13 
 
 .16 
 
 .18 
 
 .21 
 
 .24 
 
 .26 
 
 64 
 
 70 
 
 .00 
 
 -.01 
 
 .00 
 
 2.96 
 
 2.99 
 
 3.02 
 
 3.04 
 
 .07 
 
 .10 
 
 .12 
 
 .15 
 
 .17 
 
 .19 
 
 74 
 
 80 
 
 + .03 
 
 -.01 
 
 .00 
 
 .90 
 
 .93 
 
 2.96 
 
 2.98 
 
 3.01 
 
 3.04 
 
 .06 
 
 .09 
 
 .11 
 
 .13 
 
 84 
 
 90 
 
 + .06 
 
 -.02 
 
 .00 
 
 .84 
 
 .87 
 
 .90 
 
 .92 
 
 2.95 
 
 2.98 
 
 3.00 
 
 3.03 
 
 3.05 
 
 .07 
 
 94 
 
 100 
 
 + .09 
 
 -.02 
 
 .00 
 
 2.79 
 
 2.82 
 
 2.84 
 
 2.86 
 
 2.89 
 
 2.92 
 
 2.94 
 
 2.97 
 
 2.99 
 
 3.01 
 
EXAMPLES OF BAROMETRIC REDUCTION 
 
 III. Reduction to the 10,000-foot plane 
 
 49 
 
 Temp. 
 
 Correction for 
 
 22.40 
 
 22.60 
 
 22.80 
 
 23.00 
 
 23.20! 23. 40 
 
 23.60 
 
 23.80 
 
 24.00 
 
 24.20 
 
 t 
 
 02 
 
 C.A 
 
 e.H 
 
 e 
 
 AA 
 
 ' Reduction Bt - B from m-Table. 
 
 -20 
 
 -?0 
 
 -.21 
 
 
 
 -2.67 
 
 2.70 
 
 2.73 
 
 2.75 
 
 2.78 
 
 2.80 
 
 2.82 
 
 2.85 
 
 2.87 
 
 2.89 
 
 -10 
 
 -10 
 
 -.18 
 
 
 
 - .61 
 
 .64 
 
 .67 
 
 .69 
 
 .72 
 
 .74 
 
 .76 
 
 .79 
 
 .81 
 
 .83 
 
 1 
 
 
 
 -.15 ... 
 
 
 - .56 
 
 .59 
 
 .61 
 
 .63 
 
 .66 
 
 .68 
 
 .70 
 
 .73 
 
 .75 
 
 .77 
 
 12 
 
 10 
 
 - 12! ... 
 
 
 .51 
 
 .54 
 
 .56 
 
 .58 
 
 .61 
 
 .63 
 
 .65 
 
 .67 
 
 .69 
 
 .71 
 
 23 
 
 20 
 
 09 
 
 
 
 46 
 
 .49 
 
 .51 
 
 .53 
 
 .56 
 
 .58 
 
 .60 
 
 .62 
 
 .64 
 
 .66 
 
 34 
 
 30 
 
 -.06 
 
 
 
 - .41 
 
 .44 
 
 .46 
 
 .48 
 
 .51 
 
 .53 
 
 .55 
 
 .57 
 
 .59 
 
 .61 
 
 45 
 
 40 
 
 -.03 
 
 .00 
 
 .66 
 
 - .37 
 
 .39 
 
 .41 
 
 .43 
 
 .46 
 
 .48 
 
 .50 
 
 .52 
 
 .54 
 
 .56 
 
 56 
 
 50 
 
 -.01 
 
 + .01 
 
 .00 
 
 - .32 
 
 .34 
 
 .36 
 
 .38 
 
 .41 
 
 .43 
 
 .45 
 
 .47 
 
 .49 
 
 .51 
 
 67 
 
 60 
 
 + .01 
 
 + .01 
 
 .00 
 
 - .28 
 
 .30 
 
 .32 
 
 .34 
 
 .37 
 
 .39 
 
 .41 
 
 .43 
 
 .45 
 
 .47 
 
 78 
 89 
 
 70 +.03 
 80 +.06 
 
 + .01 
 + .02 
 
 .00 
 .00 
 
 - .24 
 - .20 
 
 .26 
 .22 
 
 .28 
 .24 
 
 .30 
 ' .26 
 
 .32 
 .28 
 
 .34 
 .30 
 
 .36 
 .32 
 
 .38 
 .34 
 
 .40 
 .36 
 
 .42 
 .38 
 
 100 
 
 90 
 
 + .09 
 
 + .02 
 
 .00 
 
 -2.16 
 
 2.18 
 
 2.20 
 
 2.22 
 
 2.24 
 
 2.26 
 
 2.28 
 
 2.30 
 
 2.32 
 
 2.34 
 
 value of C for Santa Fe happens to be 0.00086 and H is taken in 
 units of 1,000 feet, 7.01. Hence, for the sea level, 
 
 for = - 20, C.A0.H = 0.00086 X (-83) X 7.01 
 
 (-63) 
 
 
 20 
 40 
 60 
 80 
 100 
 
 (-43) 
 (-23) 
 (- 3) 
 (+17) 
 (+37) 
 
 - 0.50 
 -0.38 
 -0.26 
 
 - 0.14 
 -0.02 
 + 0.10 
 + 0.22 
 
 The plateau stations always seem to require such a correction 
 in order to make a harmonious network of pressures with the 
 surrounding low-level stations. It is easier to make this correc- 
 tion in the form given above, rather than attempt to trace out 
 its effect upon the mean temperature 6, as related to the surface 
 temperature t. The entire subject needs a fuller theoretical 
 discussion if possible. The vapor pressure correction e is the 
 mean value as for the argument surface /, and suffices for these 
 station reduction tables up to the 0.01 inch. The final station 
 reductions to the sea level were made for the arguments (t. B}, 
 and applied to the assumed values of B, so that for the same 
 arguments (t. B), the value of B Q is immediately read by an 
 easy interpolation. 
 
 Similar reductions were made for the 3,500-foot plane, and 
 the 10,000-foot plane. They were checked by reduction from the 
 station to 3,500 feet, to sea level, from sea level to 10,000 feet, and 
 thence in a circuit back to the station pressure. This was done 
 for all the numerous plateau stations in the United States. 
 
CHAPTER II 
 Thermodynamic Meteorology 
 
 General Formulas for the Computation of P, p, R,from the Observed 
 Temperatures T in a Free Non-Adiabatic Atmosphere 
 
 IT is easily seen from the discussion of the barometer how 
 many complexities this instrument introduces in practice, on 
 account of the series of corrections, and by reason of the system 
 of units employed, which separates the data from all other 
 thermodynamic terms occurring in meteorology. There is 
 need, then, of developing another system of reduction, by which 
 it may be possible to pass from the temperatures T t observed 
 in the free air up to great heights, to the corresponding pressures 
 P, densities p, and gas coefficients R, so that the general law 
 P = p R T shall continuously be satisfied throughout the 
 atmosphere. If the mercurial barometer is needed on the surface 
 to give a base for vertical reductions, it is not practical to carry 
 it to heights on kites and balloons. The aneroid may be used 
 to check the resulting computed pressures, but not to give the 
 actual pressure for the dependent formulas. Fortunately, there 
 is a simple and comprehensive set of formulas for this purpose, 
 which will now be developed. 
 
 For any temperature vertical gradient a, the temperature 
 T at the height z above T is, 
 
 J rr* 
 
 (169) T = T Q - a z, so that, d T = -adz, and d z = --- . 
 
 The differential equation for pressure variations with the 
 height is from (41), 
 
 (170) dP = pg dz= -- p godT, by substituting d z. 
 From the Boyle-Gay Lussac Law, P = p R Q T, by division, 
 
 we obtain, since by (24) ^- 
 
 , 
 a /CQ K 1 
 
 dT nk dT 
 
 aR T ~ k - 1 T 
 50 
 
GENERAL FORMULAS FOR NON-ADIABATIC ATMOSPHERE 51 
 
 Passing to logarithms and to limits, this gives, 
 - 1 - (^\ 
 
 PO \T O ) 
 
 nk 
 
 Having observed 7\ and T on two levels, at the vertical 
 distance apart z\ z , the pressure PI can be computed from 
 the pressure P on the lower level. We proceed to determine 
 the density pi, and gas coefficient RI, which correspond with 
 PI = piRiTi on the Zi-level, when P = p R Q To is given on 
 the_2g-leyel. By successive stages from the surface the same 
 formulas will arrive at any altitude where the temperature TV- 
 is known. From two successive levels, we have the ratio, 
 
 (173) 75- = 5~^F , and by transforming, /^ 
 
 r'o p KQ 1 o 
 
 ( . 
 * ; 
 
 At this point the entire treatment of thermodynamic 
 meteorology diverges. If the gas coefficient is taken constant, 
 RI = RQ, and 
 
 (175) fl. 
 
 For example, V. Bjerknejs in equation J jageJ>l, " Dynamic 
 Meteorology and Hydrography," Carnegie Institution of Wash- 
 ington, 1910, uses this form for the ratio pi/p , since in his 
 system of units go = 1. This is the common way of treating 
 the matter, but it is easy to see that this derivation of the non- 
 adiabatic densities from the well-known adiabatic equation is 
 inconsistent with the analogue of the pressures in (172), which 
 
 k 
 simply multiplies the exponent T - - by n, so that for the 
 
 K 1 
 
 densities the exponent should be 
 
 , _ . 
 
 Proceeding in the second way it is obvious that, preserving 
 the same treatment for density as for pressure, we should take, 
 
52 THERMODYNAMIC METEOROLOGY 
 
 / 17A v 
 
 (176) = ( jT 
 po VI o 
 
 Ri /TV*- 1 ' 
 Ro =: \T ) 
 
 In order to check these results by (172) and (173), 
 
 - ^ +(n-l) + l 
 
 > 
 
 which is correct. This process makes R a variable in the existing 
 non-adiabatic atmosphere, so that the air is not distributed by 
 gravitation like an adiabatically expanding gas, in which there 
 is no circulation and no change of heat contents by radiation and 
 absorption from level to level. On the contrary, the observations 
 prove that usually there is circulation and radiation going on 
 to preserve the gravitation equilibrium with the existing pressure 
 variations or gradients. As stated, the entire system of ther- 
 modynamics takes on a new form through the fact that the 
 specific heat must also be a variable along with the gas coefficient. 
 
 (2) Cp-^R. 
 
 We shall return to explain the consequences of this funda- 
 mental property of the atmosphere, which is in reality a gaseous 
 mixture of rapidly varying thermodynamic capacities, in con- 
 sequence of the effect of the absorption of solar radiation and 
 the emission of atmospheric radiation in various ways. 
 
 The Adiabatic Equations 
 
 The correlative adiabatic equations follow at once by putting 
 n = 1, and a = a , 
 
 (179) Pressure = 
 
 -TO \J- o 
 
 (180) Density^ = (^ 
 
 Po W o> 
 
 (181) Gas constant RI = R . 
 
WORKING NON-ADIABATIC EQUATIONS 53 
 
 The Working N on- Adiabatic Equations 
 
 (182) Pressure, log P, - log P = ^ (log 7\ - log r ). 
 
 (183) Density, log Pl - log Po = ^~ (log 7\ - log T ). 
 
 (184) Gas coefficient, log RI - log R Q = (-!) (log Z\ - log T ). 
 
 These equations were published in the Monthly Weather 
 Review, March, 1906, (38), (39), (40), and they have been 
 illustrated by numerous applications to balloon and kite ascen- 
 sions with excellent results, up to great altitudes, as 20,000 
 meters. The following example shows the method of arranging 
 the computation so as to proceed from level to level, the com- 
 puted PI, pi, RI of one becoming P , p , R for that next above 
 it. The example is taken at random from our computations, 
 some of the results being compiled in Bulletin No. 3, Argentine 
 Meteorological Office, 1913. The constants are taken from 
 Table 3 in the (M. K. S.) system. The surface values of.P , PO, 
 R , TQ are assumed to conform to the adiabatic system, while PI, 
 pi, RI, TI above the surface are computed by the non-adiabatic 
 system. 
 
 At the height z = 116 meters the density is computed from 
 the adiabatic formula, 
 
 P 
 
 Also, P = go p m Bo, where B Q is in meters. It will be noted 
 that the check is complete. It may be stated that the ob- 
 served values of B at Lindenburg are usually about 1 mm. 
 higher than the computed values B c . This constitutes a cri- 
 terion upon the adjustment of the aneroid, which, in ascending, 
 lags in registration- and records a pressure corresponding to 
 a lower level than that assumed for the temperature T at the 
 height z. 
 
54 
 
 THERMODYNAMIC METEOROLOGY 
 
 88-" 
 
 22 
 
 <N 
 
 H 
 
 M 
 
 H 
 
 CO 
 
 S J 
 
 O <M O 
 
 t^ TjH |> t^ 
 
 T i O CO 
 
 00 O CO <N 
 
 t>- Tt< Q *O 
 
 <M rt< O iO 
 
 f^ 
 
 l> T-H 
 
 3 
 
 3 8 
 
 ? r 
 
 g^g^ 
 
 "8 
 
 co i> oo 
 
 t- TJH T^ 
 
 00 1> b- 
 
 o o 
 
 <N TjH 
 
 T-H I> 
 
 sg 
 
 I o 
 
 
 
 =>S 
 
 00 Ttl O "* i-H 
 CO TtH O 10 Tt^ 
 
 8 
 
 05 CO 
 
 O i I T-H O 
 
 i> co co i> 
 
 Tf IO 
 
 
 
 N I *f 
 
 o o 
 
 -^ r* 
 
 ^E 
 
 00 <N 
 
 % 2g 
 
 OS 00 O CO O 
 
 TjH CO CO CO TfH 
 
 " Tt< i- 1 CO CO 
 CO CO 
 
 o o 
 
 CO CO 
 
 ^ I 
 
 i 
 
 rH CO CO 
 
 O T-H CD 
 
 iO <N CO O 
 
 T-H rt< 00 
 
 GO IO O CO 
 
 CO 
 
 O 
 
 iO O5 C^l 
 
 OS GO 00 
 
 ^^ ^* ^^ 
 CO <N CO 
 t^* t^- l>- 
 
 CO CO 
 
 
 00 
 
 I I 
 
 cc 
 
 v~ 
 
 
 Tf 
 
 
 
 
 
 
 
 Tj 
 
 T 
 g 
 
 '3 
 
 ^ O( 
 
 ^ 
 O 
 
 H 
 
 D 
 ) 
 
 
 
 
 iO (M CO (M 
 
 !> OO 00 CO 
 
 O5 Oi t^* t^ 
 
 O5 O5 OO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 " 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 b-H 
 
 
 
 -t 
 
 
 t 
 
 i 
 
 
 
 
 r^ 
 
 ^ 
 
 
 
 
 
 fcj 
 
 
 
 
 c 
 
 8 1 
 
 M 
 
 4- 
 
 '5 
 
 1 
 t" 
 J 
 
 
 
 rE 
 C 
 
 *. 
 
 ; i 
 1 
 1 
 
 si 
 
 ; c 
 
 I 
 
 i; 
 
 S 
 i 
 
 t 
 
 ^ 
 
 ^_ 
 
 ' J 
 
 v X 
 
 1 
 
 3 ^ 
 
 X ^ 
 
 C| 
 
 ? 1 
 
 
 5 
 j 
 
 1 
 
 -H 
 
 , 8 
 
 ^ 
 
 | 
 
 H 
 1 
 
 Ot 
 
 t. 
 
 j 
 
 1 
 1 
 
 bj 
 
 ^ 
 
 o 1 
 
 
 
 1 
 
 ^_- 
 
 D J 
 
 j 
 
 D *8 
 ' j 
 
 
 7^ 
 
 b/ 
 
 ^C 
 
 ,1 "^ 
 
 ) ] PQ == '. bfl 
 CO . O 
 
WORKING NON-ADIABATIC EQUATIONS 
 
 55 
 
 *o 
 
 CO 
 
 7 i 
 
 t^ CM iO CMTt<O5 CO i I 
 
 d '+ ^ ^ ^ ^ CM 
 
 O5 b- i CM 
 
 t>- *-* 
 
 00 i-H 
 
 Tt< TjH 
 
 CM 00 
 
 CO Tf< t>- 
 
 O O5 i-H 
 
 cO l> 
 
 O5* t^. 
 I 
 
 O5 CM 
 
 CO CM 
 
 ^ CO 
 CO O5 
 
 CO CO 
 
 w S 
 
 
 
 rH 00 CO O5 i-H CO 
 
 O5 00 O O 00 O 
 
 ^ 
 CM 
 
 II I 
 
 d d 
 
 _,. ^o 
 
 O5 CO 
 
 p CO p 
 
 07 V 
 
 s 
 
 CM 
 
 i-H 00 i-H 
 
 o 
 
 i-H 
 
 
 
 i-H 00 b- 
 
 CM 
 
 OO 
 
 co O ^ 
 
 t- , 
 
 CM 
 
 s 
 
 l> O 
 
 
 C^ CO CO ^D CO CO CO 
 
 05CO + 
 
 co 
 
 >O 
 
 
 
 
 
 
 
 
 
 
 f 
 
 
 
 
 
 
 
 (X, 
 
 1 
 
 =- a 
 w> b 
 
 
 
 % ' 
 
 bo 
 n 
 
 bo 5 
 
 44 5 
 
 
 bO 
 I 
 
 S 
 bo bO bo bo 
 
 j? 
 
 s 
 
 7 1 
 
 tf ^ 
 
 ^ ^ . J 
 
 o 6 ^^ S I 
 
 * I * 8 
 
 I -H+ Q - 
 
 L^ K - II *O 
 
 bo -= ^ 
 
 o 
 
 
 bo & I 
 
 o 
 
 u 
 
 'Q, 
 
 . cyo 
 
 ^ 
 
 I I 
 
 KJTJ bO bO 
 
 .2 ^ -2 
 
 ^ co" ^co" bf) 
 
 00 oo 00 t- ^ 
 
 11 i-H i-H i-H 
 
 
 00 
 (* 
 
 
 
 
 05 00 
 
 d d 
 
 CO 
 Oi 
 
 d d 
 
 CO O5 
 
 ^ d 
 
 s 
 
 CO 
 
 s 
 
 8 
 
 II o 
 
 v2 
 
 a; cs 
 
 e 
 3 ff 
 
 S"S 
 
 I! 
 
 "rt * 
 (U O 
 
 (51 
 
 * 
 
56 
 
 THERMODYNAMIC METEOROLOGY 
 
 TABLE 15 
 
 COMPARISON OF THE PRESSURES COMPUTED BY THE NON-ADIABATIC 
 FORMULA AND THOSE OBSERVED AT GREAT HEIGHTS 
 
 Heights 
 
 Lindenburg 
 
 Lindenburg 
 
 Atlantic Ocean 
 
 Atlantic Ocean 
 
 in 
 
 May 5, 1909 
 
 July 27, 1908 
 
 Sept. 9, 1907 
 
 June 19, 1906 
 
 Meters 
 
 Lat. +52 
 
 Lat. +52 
 
 Lat. +26 
 
 Lat. -2 
 
 z 
 
 Be 
 
 Bo 
 
 Bc-B, 
 
 Be 
 
 Bo 
 
 Bc-Bo 
 
 Be 
 
 B 
 
 Bc-Bo 
 
 Be 
 
 Bo 
 
 Bc-Bt 
 
 19000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 18000 
 
 
 
 
 .0589 
 
 .0600 
 
 -.0011 
 
 
 
 
 
 
 
 17000 
 
 .0679 
 
 .0670 
 
 + .0009 
 
 .0692 
 
 .0690 
 
 + 2 
 
 .0742 
 
 .0740 
 
 + !6662 
 
 
 
 
 16000 
 
 .0794 
 
 .0790 
 
 + 4 
 
 .0808 
 
 .0790 
 
 + 18 
 
 .0861 
 
 .0860 
 
 + 1 
 
 
 
 
 15000 
 
 .0927 
 
 .0930 
 
 3 
 
 .0945 
 
 .0950 
 
 - 5 
 
 .1004 
 
 .1010 
 
 - 6 
 
 .1021 
 
 .1020 
 
 + .0001 
 
 14000 
 
 .1082 
 
 .1090 
 
 - 8 
 
 .1101 
 
 .1110 
 
 - 9 
 
 .1177 
 
 .1170 
 
 + 7 
 
 .1203 
 
 .1200 
 
 + 3 
 
 13000 
 
 .1266 
 
 .1270 
 
 4 
 
 .1289 
 
 .1300 
 
 - 11 
 
 .1381 
 
 .1380 
 
 + 1 
 
 .1409 
 
 .1410 
 
 - 1 
 
 12000 
 
 .1484 
 
 .1490 
 
 6 
 
 .1512 
 
 .1520 
 
 - 8 
 
 .1615 
 
 .1610 
 
 + 5 
 
 .1641 
 
 .1640 
 
 + 1 
 
 11000 
 
 .1744 
 
 .1750 
 
 6 
 
 .1772 
 
 .1780 
 
 8 
 
 .1881 
 
 .1880 
 
 -1- 1 
 
 .1902 
 
 .1900 
 
 + 2 
 
 10000 
 
 .2041 
 
 .2050 
 
 - 9 
 
 .2067 
 
 .2080 
 
 - 13 
 
 .2179 
 
 .2180 
 
 - 1 
 
 .2191 
 
 .2190 
 
 + 1 
 
 9000 
 
 .2374 
 
 .2390 
 
 - 16 
 
 .2399 
 
 .2410 
 
 - 11 
 
 .2509 
 
 .2510 
 
 1 
 
 .2512 
 
 .2510 
 
 + 2 
 
 8000 
 
 .2746 
 
 .2760 
 
 - 14 
 
 .2769 
 
 .2780 
 
 - 11 
 
 .2875 
 
 .2880 
 
 5 
 
 .2870 
 
 .2870 
 
 
 
 7000 
 
 .3159 
 
 .3180 
 
 - 21 
 
 .3181 
 
 .3190 
 
 - 9 
 
 .3281 
 
 .3290 
 
 9 
 
 .3268 
 
 .3260 
 
 + 8 
 
 6000 
 
 .3619 
 
 .3630 
 
 - 11 
 
 .3639 
 
 .3650 
 
 - 11 
 
 .3732 
 
 .3740 
 
 8 
 
 .3709 
 
 .3710 
 
 - 1 
 
 5000 
 
 .4128 
 
 .4130 
 
 - 2 
 
 .4149 
 
 .4160 
 
 - 11 
 
 .4233 
 
 .4240 
 
 - 7 
 
 .4200 
 
 .4200 
 
 
 
 4000 
 
 .4696 
 
 .4700 
 
 4 
 
 .4716 
 
 .4720 
 
 - 4 
 
 .4789 
 
 .4800 
 
 - 11 
 
 .4747 
 
 .4750 
 
 3 
 
 3000 
 
 .5330 
 
 .5340 
 
 - 10 
 
 .5344 
 
 .5340 
 
 + 4 
 
 .5402 
 
 .5410 
 
 - 8 
 
 .5356 
 
 .5350 
 
 + 6 
 
 2000 
 
 .6035 
 
 .6040 
 
 - 5 
 
 .6040 
 
 .6040 
 
 
 
 .6077 
 
 .6060 
 
 + 17 
 
 .6031 
 
 .6030 
 
 + 1 
 
 1000 
 
 .6823 
 
 .6840 
 
 - 17 
 
 .6806 
 
 .6810 
 
 - 4 
 
 .6821 
 
 .6820 
 
 + 1 
 
 .6783 
 
 .6780 
 
 + 3 
 
 Surface 
 
 .7599 
 
 .7599 
 
 
 
 .7551 
 
 .7551 
 
 
 
 .7640 
 
 .7640 
 
 
 
 .7610 
 
 .7610 
 
 
 
 The differences between B c and B Q are probably due to an 
 assignment of the temperature to a slightly erroneous height,* 
 owing to the movement of the balloon ahead of the record of 
 the barograph and thermograph, which requires a correction 
 for lag. The pressure recorded by the aneroid is for the mixture 
 of dry air and aqueous vapor, so that by (75) for the same height, 
 where T = T and p = p , 
 
 (185) P = P (l - 0.378 J; 
 
 Bo = B (l - 0.378 ^ 
 
 where P or B is the dry air pressure, and P or B Q the 
 pressure in the mixture, e being the vapor pressure in the 
 
 * The heights have been read from the aneroid record and are not corrected 
 for the supposed lag; but the error is less for T than for P because T changes 
 more slowly than P. 
 
THE VARIABLE VALUES OF H = -~ 
 
 57 
 
 same system of units. The connection between (182) and 
 (159) is such that they can easily be shown to be identical, 
 after the action of the mercurial barometer has been made to 
 equal that of an aneroid. 
 
 The Variable Values of n = - 
 
 The introduction of n into the adiabatic formulas converts 
 them into the non-adiabatic formulas, and at the same time adds 
 circulation and radiation to a static atmosphere. Hence, by (13), 
 
 = _ _T l -T _ -(T a - To) 
 
 a 
 
 (186) 
 
 n = = 
 
 marks the natural transition from static to dynamic and ther- 
 modynamic meteorology. It is important, therefore, to under- 
 stand the full significance of the ratio between the adiabatic 
 and the non-adiabatic temperature gradients. Since (T a T ) 
 
 TABLE 16 
 EXAMPLES OF THE VALUE OF n 
 
 Height z 
 
 Lindenburg 
 May 5, 1909 
 
 Lindenburg 
 July 27, 1908 
 
 Atlantic Ocean 
 Sept. 9, 1907 
 
 18000 
 
 
 - 3.6553 
 
 
 17000 
 
 +1.0966 
 
 -16.4489 
 
 - 2.5972 
 
 16000 
 
 -5.4830 
 
 - 6.1684 
 
 - 0.9399 
 
 15000 
 
 -9.8694 
 
 -49.3467 
 
 - 2.9027 
 
 14000 
 
 -3.5248 
 
 - 6.1684 
 
 + 4.9347 
 
 13000 
 
 -1.7624 
 
 - 2.5972 
 
 + 1.9739 
 
 12000 
 
 +5.8055 
 
 + 2.0999 
 
 + 1.3337 
 
 11000 
 
 + 1.1749 
 
 + 1.1611 
 
 + 0.9676 
 
 10000 
 
 +1.0846 
 
 + 1.2986 
 
 + .1611 
 
 9000 
 
 +1.1215 
 
 + 1.1344 
 
 + .1611 
 
 8000 
 
 +1.1749 
 
 + 1.1749 
 
 + .3901 
 
 6000 
 
 +1.3520 
 
 + 1.6179 
 
 + .3901 
 
 4000 
 
 + 1.7016 
 
 + 1.8980 
 
 + .2653 
 
 2000 
 
 +3.5248 
 
 + 1.2986 
 
 + 2.5972 
 
 1000 
 
 -4.9347 
 
 + 12.3366 
 
 -12.3367 
 
 and (Ti TO) are usually each negative with an increase in 
 elevation, n is generally a positive quantity, but it becomes 
 
58 THERMODYNAMIC METEOROLOGY 
 
 negative whenever there is an inversion of temperature, or 
 temperature increase with the height, as near the surface of the 
 ground in the early morning, or in the isothermal layer at great 
 heights. If TI = Ta, n = 1, and the gradient is adiabatic; 
 if TI = T Q , n = and there is no temperature change with 
 the height; if TI > T Q , n is negative, and if TI is only a little 
 greater than T Q for the change in elevation z\ z , n will be 
 a large negative quantity. Table 16 gives a few examples of the 
 values of n. 
 
 Table 16 indicates the wide range through which n passes 
 in practical reductions, and it is easily seen how valueless the 
 formulas become for meteorological discussions where n is 
 assumed to be unity, as is commonly the procedure. Further- 
 more, since the value of n must always be carried to the fourth 
 decimal it has not seemed worth while to construct general 
 reduction tables, because they would be very extensive or re- 
 quire complex interpolation. 
 
 The Differentiation of (172) 
 
 Since n is a variable in equation (172), we proceed to differ- 
 entiate it for P, T, n, variables. 
 
 (187) log = log . Differentiate, 
 
 nk TT T k dn 
 
 4-1 
 
 r. + log r 
 
 k 
 Substitute P = p R T and Cp = R 
 
 , 
 
 rv J. 
 
 (190) = n Cp d T + CpTlQgTdn. 
 
 p M. 
 
 Take the integral between limits for the mean p, 
 
 (191) Pl ~ P = % Cfco (3Ti - To) + ^ CA, r 10 log p 
 
 PlO M 2 o 
 
DIFFERENTIATION OF (172) 59 
 
 By (144), (146), (147), using the mean values Cp w , 
 
 (192) Pl ~ o P = * Cp w (7\ - To) + (m - n ) Cp a 
 
 (Tt - To). 
 
 = ! c# 10 (T\ - r ) + i c/> a (ri - r ) - 
 
 WoC> (Ti-To) 
 
 = cfco (r a - r ) + cp a (T a - TO) - cp a 
 
 (T a - To) 
 
 = Cp a (T a - To) - (Cp a - Cfco) (T a - T ). 
 
 Since p and Cp are variable in the stratum (zi z ), the 
 mean values are pio and Cp W) while HI continues constant 
 within the stratum, which must not be taken too thick to allow 
 this approximation to hold true. The result is twofold. First, 
 
 (193) Pl ~ P = C# w (T a - To) = i C#i (r x - To). 
 
 Pio 
 
 The adiabatic system, on the other hand, gives, 
 
 (194) Pra ~ P = C# B (r fl - To) = - g fe - so). 
 
 PaO 
 
 Hence, the difference between the two systems is, 
 
 (195) - - = (Cp a - Cp) (T a - To). 
 PaO PIO 
 
 From the common dynamic equation for pressure and 
 velocity, which will be proved in a later section, and adding a 
 term for the dynamic energy of radiation heat, we obtain by 
 substitution in (192) the working and fundamental equation, as 
 the second result, 
 
 (196) go fe - a.) = - fl ~ P - J (?, 2 - <?o 2 ) - 2i - Co). 
 
 Pio 
 
 These have already been quoted in (21), (25), (26), (27). 
 From (192), using the last form, we find, 
 
 (197) go (z, - Sb) * - ^ ^ - (Cp a - Cp 10 ) (T a - To), 
 
 Pio 
 
 and by comparison of the last terms (196), (197), 
 
 (198) - (ft - Co) - i (?i 2 - <?o>) - - (Cp a -Cp 1Q ) (T a -T Q ). 
 
60 THERMODYNAMIC METEOROLOGY 
 
 (199) - (<2i - <2o) = - (Cpa ~ C# w ) (T a - To) + i (?i 2 -?o 2 ). 
 
 That is to say, the variation from the true adiabatic system 
 is due to the kinetic energy of heat (<2i Q ), and circulation 
 J (qi z g 2 ) for the unit mass, which is equivalent to the 
 .variation of the mean specific heat of the stratum from the 
 adiabatic specific heat, Cp a = 993.5787 in Table 3, multiplied 
 by the change in temperature between the bottom T and top 
 Ta of the layer z\ ZQ. Unfortunately, there seems to be no 
 way to separate (Qi Q Q ) from \ (qj <? 2 ) through the 
 specific heat, except by using the direct observations of the 
 velocity, and then computing (Qi Q Q ) by means of (199). 
 Those observatories which record P, T, the pressure and the 
 temperature, but not the R. H. and <?, the relative humidity and 
 velocity of motion of the stratum, cannot enter upon any problem 
 in circulation or radiation in the atmosphere. 
 
 From equation (196) we obtain, 
 
 (200) - = gdz + qdq + dQ, and, 
 
 P 
 
 (201) -dP = gpdz+pqdq+pdQ. 
 From the equation (190) is derived, 
 
 (202) - d P = p n Cp d T - p -^ Cp T log T. d n, 
 
 and by means of (146), (144), (21), this becomes, 
 
 (203) - d P = g p d z - p j Cp T log T. d n. 
 
 The usual adiabatic pressure variation dP = gpdz, as 
 in (170), is converted into the non-adiabatic form with circulation 
 
 and radiation by subtracting p T> Cp T log T d n, which includes 
 
 the differentials first for heat, and second for circulation. 
 
 If equation (203) is treated in the same manner as (170) we 
 shall obtain by substitutions, 
 
 foriA\ ( T \ ( T \ \ Zj-Zp l dp Zi- 
 
 (204) f^) -(~r) =(a -a) T = 
 
 
 
 which is the difference between the temperature ratios in the 
 adiabatic and the non-adiabatic systems. 
 
DIFFERENTIATION OF (172) 
 
 61 
 
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62 
 
 THERMODYNAMIC METEOROLOGY 
 
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DIFFERENTIATION OF (172) 63 
 
 There are a few critical remarks that can now be made to 
 advantage, in view of the principles that have been assumed in 
 many important meteorological papers. 
 
 (1) By (201) and (203) it is not proper to assume that the 
 variation of the pressure in a vertical direction is proportional 
 to the mass variation, d P = g d m, because this excludes 
 the circulation and the radiation. It is a contradiction in terms 
 to seek for solutions of radiation problems under this limitation. 
 
 (2) By (198), (199), if it is assumed that R and Cp are 
 constants and the change of temperature is adiabatic, there can 
 be no circulation and no radiation. It is a contradiction in terms 
 to seek for solutions of radiation problems under this assumption. 
 
 i*d P 
 
 (3) By (193) the integral / depends upon a varying 
 
 Cp. It follows that if P. T. are observed by the instruments, 
 if RQ is taken constant in the Boyle-Gay Lussac Law, and if p 
 is then computed directly from P = p R Q T, this value in the 
 
 /dP 
 will always lead to fictitious results, because the 
 p 
 
 circulation and the radiation are excluded from the dis- 
 cussion. 
 
 (4) In the integration of (190), in strata where n is constant, 
 as they can be made by taking the layer thin enough, it is proper 
 
 to use the mean values of the several terms, pio, Cpio, TIQ, 
 
 jp p _ _ p 
 
 1 i / "\ 
 
 PlO = 2 \Pl + POJ. 
 
 /* 
 J P 
 
 f 
 
 Pio 
 nCpdT = n Cp 1Q (^ - T ), Cp w = J (Cpi + Cp ). 
 
 I T I TI 
 
 M TQ M TQ 
 
 since Cp = Cp a and n = 1 in this term. 
 
 (5) It has been customary to evaluate the equations of 
 motion in the atmosphere while omitting the heat term (Qi Qo) 
 in (196). The result is that it is impossible to balance the other 
 three terms, so that the great problem of the relation of the 
 circulation to the observed pressures in the general and the local 
 circulations has been insoluble. In order to exhibit more fully 
 
64 
 
 THERMODYNAMIC METEOROLOGY 
 
 the relation of these four terms, two examples of balloon ascen- 
 sions to great heights are added. 
 
 TABLE 18 
 THE EVALUATION OF (Q l Q ) IN BALLOON ASCENSIONS (196) 
 
 Station 
 
 Lindenburg, May 5, 1909 
 
 Atlantic Ocean, Sept. 9, 1907 
 
 jj 
 
 * 
 _/_ ._ \ 
 
 Pi-Po 
 
 -J 
 
 -(Qi- 
 
 
 Pt-Po 
 
 -J 
 
 _ (Ql _ 
 
 
 
 
 pio 
 
 (v-^ 
 
 Qo) 
 
 
 pio- 
 
 (V-tf 
 
 Qo) 
 
 
 17000 
 
 9806.0 
 
 6437.6 
 
 + 3.6 
 
 3358.6 
 
 + 6.2 
 
 6755.8 
 
 -26.8 
 
 3065.9 
 
 +11.1 
 
 16000 
 
 9806.0 
 
 6629.2 
 
 + 5.6 
 
 3169.2 
 
 + 2.0 
 
 7301.2 
 
 + 6.4 
 
 2496.1 
 
 +2.3 
 
 15000 
 
 9806.0 
 
 6963.7 
 
 +11.2 
 
 2817.1 
 
 +14.0 
 
 7886.4 
 
 +38.9 
 
 1885.6 
 
 - 4.9 
 
 14000 
 
 9806.0 
 
 7357.0 
 
 + 9.2 
 
 2433 . 5 
 
 + 6.3 
 
 8288.6 
 
 +104.8 
 
 1424.2 
 
 -11.6 
 
 13000 
 
 9806.0 
 
 7851.0 
 
 + 5.6 
 
 1939.7 
 
 + 9.7 
 
 8508.6 
 
 +53.0 
 
 1228.5 
 
 +15.9 
 
 12000 
 
 9806.0 
 
 8300 . 4 
 
 + 1.7 
 
 1499.8 
 
 + 4.1 
 
 8666.2 
 
 -89.3 
 
 1227 . 9 
 
 + 1.2 
 
 11000 
 
 9806.0 
 
 8489 . 
 
 - 0.9 
 
 1313.9 
 
 + 4.0 
 
 8706.6 
 
 -33.2 
 
 1130.2 
 
 + 2.4 
 
 10000 
 
 9806.0 
 
 8530.6 
 
 + 0.9 
 
 1268.8 
 
 + 5.7 
 
 8721.0 
 
 - 1.4 
 
 1079.9 
 
 + 6.5 
 
 9000 
 
 9806.0 
 
 8565.0 
 
 + 8.1 
 
 1227.5 
 
 + 5.4 
 
 8769.4 
 
 + 4.2 
 
 1025.7 
 
 + 6.7 
 
 8000 
 
 9806.0 
 
 8615.8 
 
 +13.2 
 
 1176.9 
 
 + 0.1 
 
 8845.2 
 
 +30.4 
 
 928.1 
 
 + 2.3 
 
 7000 
 
 9806.0 
 
 8668.8 
 
 +16.1 
 
 1114.1 
 
 + 7.0 
 
 8958.0 
 
 -54.3 
 
 895.0 
 
 + 7.3 
 
 6000 
 
 9806.0 
 
 8750.2 
 
 + 8.8 
 
 1044.0 
 
 + 3.0 
 
 9074.0 
 
 -53.8 
 
 777.6 
 
 + 8.2 
 
 5000 
 
 9806.0 
 
 8885.0 
 
 + 5.2 
 
 909.2 
 
 + 6.6 
 
 9222 . 
 
 +18.3 
 
 568.1 
 
 - 2.4 
 
 4000 
 
 9806.0 
 
 9051 . 5 
 
 + 5.4 
 
 745.6 
 
 + 3.5 
 
 9340.5 
 
 -24.6 
 
 485.9 
 
 + 4.2 
 
 3000 
 
 4903.0 
 
 4589.3 
 
 + 7.0 
 
 312.3 
 
 - 5.6 
 
 4706.0 
 
 - 2.2 
 
 204.1 
 
 - 4.9 
 
 2500 
 
 4903.0 
 
 4636.4 
 
 +11.7 
 
 257.5 
 
 - 2.6 
 
 4725.8 
 
 - 6.2 
 
 179.2 
 
 + 4.2 
 
 2000 
 
 4903.0 
 
 4689.1 
 
 - 5.9 
 
 217.3 
 
 + 2.5 
 
 4764 . 9 
 
 - 4.5 
 
 135.7 
 
 + 6.9 
 
 1500 
 
 4903.0 
 
 4738.9 
 
 -22.1 
 
 174.4 
 
 +11.8 
 
 4815.6 
 
 - 0.8 
 
 87.6 
 
 + 0.6 
 
 1000 
 
 4903.0 
 
 4813.8 
 
 -32.5 
 
 104.4 
 
 +17.3 
 
 4846.6 
 
 +13.7 
 
 9.9 
 
 +32.8 
 
 500 
 
 3765.5 
 
 3753.8 
 
 -11.5 
 
 19.5 
 
 + 3.7 
 
 4865.7 
 
 -10.7 
 
 0.9 
 
 - 1.9 
 
 * More exactly g should diminish with height. 
 
 It is easy to see that in the higher levels (Qi Qo) is a domi- 
 nant term, and that there is no possibility of a balance between 
 the other three taken by themselves.. There is a delicate 
 interaction of the four terms, such that the circulation seeks to 
 adjust the pressure and the radiation to the demands of gravity. 
 The minor errors A are to be chiefly ascribed to the fact that 
 the observed temperatures are not quite correct at the height z. 
 In many cases there are pair values of -f A and A easily 
 adjusted on this hypothesis. 
 
 The Two Laws of Thermodynamics 
 
 Before proceeding further it is necessary to summarize the 
 two fundamental laws of thermodynamics, the first being that 
 of the conservation of energy, and the second being that of the 
 
THE TWO LAWS OF THERMODYNAMICS 65 
 
 decrease of entropy or increase of expenditure in a non-conserva- 
 tive system. The first law is defined as follows: In a conservative 
 system, from which no heat escapes and into which no heat is re- 
 ceived, the sum of all the changes in the energy, whether large or 
 small, remains constant. In a system in communication with the 
 outside world the amount of the energy gained or lost by it is 
 equal to that delivered to or received from the outside world. 
 The second law may be described in several ways. Heat cannot 
 by itself pass out of a colder into a warmer body (Clausius). 
 It is impossible to construct a periodically acting machine which 
 does nothing else than raise a weight in expending work and 
 cooling a reservoir of heat (Planck). There exists in nature a 
 quantity which always changes itself only in the same direction 
 in all the variations which take place in nature; this is the 
 entropy. It is in nowise possible to diminish the entropy of a 
 system of bodies without there being left changes in other bodies. 
 If such changes do not remain, then the entropy of a system can 
 continue the same. Every physical or chemical change takes 
 place in such a way that entropy either decreases or remains 
 the same, but the outside world tends continually toward 
 maximum entropy. It is necessary and sufficient for the 
 equilibrium of a separate structure that for all possible changes 
 in the state of the structure, the changes of entropy be zero or 
 negative (Weinstein). 
 
 The First Law of Thermodynamics 
 
 The following series of formulas are available in the adi- 
 abatic and non-adiabatic systems using the proper values of 
 P. p. R. T. 
 
 P 
 p ~'~ R f 
 
 RT 
 
 i) = . 
 
 dT dP 
 
 (205) 
 (206) 
 (207) 
 (208) 
 
 P 
 Pv 
 
 dT 
 
 d v 
 
 = P RT 
 
 = RT. 
 Pdv 
 
 + vd 
 
 P 
 
 dP 
 
 = P 
 
 
 R 
 
 R 
 
 T 
 
 dT P ' P 
 
66 THERMODYNAMIC METEOROLOGY 
 
 dv dT dP_ 
 
 (209) : ~Y p 
 
 Referring to the series of equations 84-118 for the definition 
 of the terms, we have generally, 
 
 (210) Inner Energy. U = H + J = Q-W. 
 
 (211) U = Cv fT.pdv = - fp.dv. 
 
 (212) dU 
 (213) 
 
 7? 7" 1 
 
 (2H) d U = Cv d T + -- dv - Pdv. 
 
 (215) dU = CpdT - Pdv. 
 
 (216) d U = Cvd T = (Cp - R) d T. 
 
 (217) External Work. W = (K) + V = J + H + V. 
 
 (218) dW = dQ-dU. 
 
 (219) dW = TdS -dU. 
 
 (220) dW = d(TS- U} -SdT. 
 
 (221) dW =* -dF -SdT. 
 
 (222) d W = P d v. 
 
 dP 
 
 (223) dW = RdT - . 
 
 (224) Heat Energy. Q = W + U. 
 
 (225) () = TF + # + / = # + #- 
 
 (226) Q = [(K) + V] external + [H + J] inter- 
 
 nal + (R) friction. 
 
 (227) 
 
 (228) 
 
 (229 dQ = CvdT-\-RdT v dP = Cp d T 
 
 (228) JQ = Cz;jr+ ~ - dv = Cvd T + Pdv. 
 
THE FIRST LAW OF THERMODYNAMICS 67 
 
 (230) dQ-C, 
 
 K 
 
 (231) dQ = TdS. 
 From (218) in heat units, we obtain, 
 
 (232) First Law. A d W = d Q - A d U, and by (222), 
 
 AdU 
 
 (233 ) A dPd ..(AQ\ d 
 
 [ dTdv ~ dT\dv) ~J~v 
 
 dP dAP dCv 
 
 (234) A IT = ^T ~ ^ 
 
 *(->) 
 
 V * 
 
 <** 
 
 (236) - Qi = - Ui - Wi, Heat expended outward (negative). 
 + ft = + #2 + P^>, Heat received inward (positive). 
 
 (237) A Q = Q 2 - Qi = (U 2 - Ui) + (W> - Wi), Resultant 
 
 heat energy. 
 
 Among the definitions we have, 
 
 V = the external gravity potential acting inwards, to- 
 gether with the centrifugal force of the earth's 
 
 rotation at the angular velocity w = , at 
 the perpendicular distance tcr from the axis, 
 
 (238) V = - fg d z + y*f (vutfdm. 
 
 (239) (K) = the kinetic energy of motion of the mass m 
 
 with the velocity q. (K) = | m (f = H + /. 
 
 (240) F = the free energy or the thermodynamic potential 
 
 at a constant volume. F = U T S. 
 
 (241) U = The bound energy. U - F = T S. 
 
 (242) = the thermodynamic potential at a constant 
 
 pressure, F + Pv. <f> = U - TS + Pv. 
 
 = U-TS + RT. 
 
68 THERMODYNAMIC METEOROLOGY 
 
 Fundamental Equations and Definitions 
 
 It is convenient to have for ready reference the fundamental 
 equations and several definitions of thermodynamic processes, 
 though they cannot be further developed in this connection. 
 It is the purpose of this treatise to prepare such data for meteor- 
 ology as can be admitted into the large group of well-known 
 equations which have been heretofore inapplicable in the atmos- 
 phere for lack of the necessary correct values of P. />. R. T. 
 There are several variables : P.v.T. Q.W.U .S.R. 
 
 (243) dU = A l dQ- dW = A l TdS - Pdv. 
 = Pdv. 
 
 (244) 
 
 dF = - A^SdT - Pdv. 
 
 fA*\ JT dF + Pdv dF 
 
 (245) dT== ~~~A^S ' dv== ~~ 
 
 d<f> = - 
 
 Adidbatic Processes, d Q = and d S = 0. 
 Adiabatic. d Q = signifies no gain of heat from the outside 
 and no loss of heat to the outside from the 
 system. 
 Isentropic. dS = signifies that the entropy remains con- 
 
 stant. 
 
 Isodynamic Processes, d U = and d T = 0. 
 Isodynamic. d U = 0. The inner energy remains constant. 
 Isothermal, d T = 0. The temperature remains constant. 
 
 Isoenergetic Processes. dW = 0, d P = Q, dv = 0, d p = Q. 
 Isometric, d W = 0. The expenditure of external work is the 
 
 same. 
 
 Isobaric. d P = 0. The pressure remains unchanged. 
 Isochoric. dv = 0. The volume is constant during the 
 
 process. 
 
 Isopyknic. d p = 0. The density is constant during the 
 
 process. 
 
FUNDAMENTAL EQUATIONS AND DEFINITIONS 69 
 
 Isopiestic Processes, d P = 0. 
 
 Isopiestic. dP = 0. The pressure is constant while the 
 
 other variables change. 
 
 Isoelastic Processes. dR = 0. 
 
 Isoelastic. dR = 0. In ideal gases the gas coefficient is 
 
 constant. 
 
 Evaluation of dQ in Terms of P. v. T. through the Entropy S. 
 Taking the following three pairs of variables, they lead to 
 the definition of the specific, latent, and expansion heats. 
 
 (246) Variables (v.T). d Q = T (||) dT + T (~) d v = 
 
 \O JL ' v V O V ' T 
 
 + C T dv. 
 
 (247) Variables (P. T). dQ = T d T + TJ dP 
 
 (248) Variables (P.v). d Q = T (|) dT+T (|) d v = 
 
 Hence by comparison the definitions become, 
 Specific Heats. Latent Heats. Expansion Heats. 
 
 (249) C P = T ( . (250) C T = r( (251) r p = r(|f ) . 
 
 p . T p p 
 
 (252) c, = T . (253) rr = r. (254) r > r) 
 
 These occur in These occur in These occur in 
 
 radiation. evaporation. convection. 
 
 The subscript indicates the term which remains constant. 
 
 Evaluation ofdQin Terms of P. v. T. through the Inner 
 Energy U. 
 
 (255) Variables 
 
 (r. T). dQ =A (|^) d T + A [ (|^) T +P~]d v. 
 
70 
 
 THERM ODYNAMIC METEOROLOGY 
 
 (256) Variables (P. T). d Q =A [ (|~) + P |~]d T + 
 
 (257) Variables (P. v) . d Q = A (|^) d P + f (|^) + P~| 
 
 * o Jr ' p UAO v * i 
 
 Specific Heat 
 
 d v. 
 
 p 
 
 Latent Heat 
 
 Expansion Heat 
 
 ^ - () 
 
 Latent Heat 
 
 Specific Heat 
 
 - (H). 
 
 -dfi 
 
 Expansion Heat 
 
 (263) r,= 
 
 By inter comparisons and substitutions very numerous equa- 
 tions can be constructed. Compare Weinstein's "Thermody- 
 namik." Those for entropy are, 
 
 (264) S vT = S + Cv log T + A R log v. 
 
 (265) S TP = S + Cp log T - A R log P. 
 
 The Second Law of Thermodynamics 
 This is derived from equation (231). 
 
 (266) Second Law. dS =-~. 
 
 This gives rise to two processes in nature, the reversible, in 
 which after a series of transformations the original state is 
 reached, and the irreversible process, in which the original state 
 is permanently lost. 
 
THE REVERSIBLE AND IRREVERSIBLE PROCESSES 71 
 
 For the reversible process, d S = -=r = 0. 
 
 From (225) for d Ui = d U 2 = d W 2 = 0, we have, 
 
 (267) Q 2 - Q l + W 1 = 0. 
 
 (268) From the first law. d Q 2 - d Qi + d Wi = 0. 
 
 (269) From the second law. d S 2 - d Si = -r - - = 0. 
 
 L2 1\ 
 
 T 
 
 (270) Solving these equations, d Qi = ^ d Wl ' 
 
 r,' a - Q 2 " TI - 2Y d - g 2 " TX - r 2 * 
 
 (272) ^ = (Tl ~ ^ =1 - ^- 2 . The efficiency of the engine. 
 ^i ^ i i 
 
 ^ of 1 M ^ - n ( 1 -} 
 - ft - r. -- ft - - 
 
 (274) Carnot's Function = -^ in mechanical units for 7\ > T 2 
 
 1 1 
 
 = -T-TFT in heat units. 
 A T! 
 
 T! = the temperature of the source of heat energy. 
 T 2 = the temperature of the sink of the energy. 
 The energy runs down from the source to the sink. 
 
 For the irreversible process, d S = -TJT > 0. 
 
 (275) From the first law. d Q 2 - d Q l + d Wi = 0. 
 
 (276) From the second law. d S 2 - d Si = -J^- - -Jr* > 0. 
 
 lz J-i 
 
 Solving these equations. 
 
 (277) dW l 
 
72 
 
 THERMODYNAMIC METEOROLOGY 
 
 (278) Ford IF, = 0, 
 
 (279) d Qi >~r 
 
 . 
 
 1 1 
 
 d Q 2 
 
 Carnot's Cyclic Process 
 
 As an example of a reversible process we may describe the 
 Carnot Cycle, in which a unit mass^with the initial condition 
 (Pi. i. jfi) passes by an isothermal change to a second condition 
 
 O Volume v 
 
 FIG. 3. Carnot's cycle 
 
 (P 2 - 2. ^i), then by an adiabatic change to the third condition 
 (Ps. z; 3 . T 2 ), thence reversing from this extreme point by another 
 isothermal change to (P 4 . z> 4 . T 2 ), and finally by another adiabatic 
 change to the initial (Pi.Vi. TI). This is illustrated in Fig. 3. 
 
 Ti = temperature of the source RI. 
 Tz temperature of the sink R 2 . 
 
 1 and 3 = Isothermal processes. 
 
 2 and 4 = Adiabatic processes. 
 1 and 2 = Work of expansion. 
 
 3 and 4 = Work of compression. 
 + Q = Heat received. 
 
 Q = Heat expended. 
 
 + W = Work of expansion. 
 
 W = Work of compression. 
 
CARNOT'S CYCLIC PROCESS 73 
 
 Summarizing by the first law. 
 
 Isotherm. Adiab. Isotherm. Adiab. 
 
 (280) Q = Q l -Q 2 = W 
 
 = Wi + W 2 -W 3 -Wt 
 
 nTi V3 T* vi Ts vtTi 
 
 (281) = IPdv +/Pdv -IPdv -IPdv, 
 
 Jv\ Ti Jv2 Ti Jv* T* Jvi Ti 
 
 (282) Since P = ^ = - Cv j^- by (228) iordQ = 0. 
 
 /Ti r T * r V4 T 2 r Tl 
 
 = R f -dv - fCvdT +R/ ~dv - fCv d T, 
 
 J VI v JTI Jvz v JT-L 
 
 (283) Since R log v + Cv log T = Const, by (228) for d Q = 0. 
 
 (284) = R (T, - r.) log ^ = R (T, - T 2 ) log = 
 
 It follows that log = log , or = . 
 
 (285) (k - 1) log Pl = log TV By (180). 
 
 (286) +(k-i) log vi + log Tt = (k- 1) log v, + log TV 
 
 (287) + (k - 1) log v 2 .+ log T! = (k - 1) log fl 3 + log T 2 . 
 
 From the Second Law by (271) and (180), 
 k- i , k- i 
 
 LATENT HEAT 
 Cyclic Process for Vapors at Maximum Pressure 
 
 A second example of the reversible process is found in the 
 cycle through which vapors pass in changing to liquids, by the 
 latent heat which is required in effecting this transformation. 
 
74 
 
 THERMODYNAMIC METEOROLOGY 
 
 Vapor Liquid Solid Total 
 
 Mass Mi M 2 M 3 M = Mi + M 2 = Constant. 
 
 Specific Heat Ci C 2 C 3 
 Latent Heat ... r\ r 2 
 
 Volume Vi v 2 v 3 v = Vi + v%. 
 
 Liquid 
 
 \ 
 
 Vapor 
 
 -Q, 
 
 \ 
 
 O e h f V 
 
 FIG. 4. Cyclic process for maximum vapor pressure 
 
 (289) Product. M (vi + v 2 ) = Mi Vi+M 2 v 2 = Mi Vi + (M MI) v 2 . 
 
 (291) dMv = M dv = (vi v 2 ) dMi. 
 
 This is the mass which evaporates in the expansion while 
 (z> 3 z; 4 ) d M 2 condenses during compression. Hence 
 
 (292) ^ = -^- 
 
 d v Vi v 2 
 
 The general equation of condition is, 
 
 (293) A P d v = r 2 - , d v = r 2 - - d v in heat units. 
 
 d v Vi v 2 
 
 For d v = we have by (209), P = T -r^. Hence, 
 
 dP M 
 
 (294) APdv = AT T ^dv = r 2 -dv,s 
 
 (295) 
 (296) 
 
 r 2 
 
 A Ti ~ T latent heat of vaporization 
 
 of liquid to vapor (^i v 2 ) in heat units. 
 
 r 3 = A T 2 
 
 ^- 2 - 7-=-, latent heat of melting of 
 i a 1 2 
 
 solid to liquid (v 2 v 3 ) in heat units. 
 
SECOND FORM OF THE EQUATIONS FOR LATENT HEAT 75 
 
 The Second Form of the Equations for Latent Heat 
 
 For external equilibrium where there is no exchange with the 
 surrounding medium, the conditions are : 
 
 Total Vapor Liquid Solid 
 
 (297) Masses. M = M i + M 2 + M s . 
 
 (298) Specific volumes, v = v\ + % + Vs- 
 
 (299) Volume. Mv = M i Vi + M 2 v 2 + M 3 %. 
 
 (300) Energy. M Z7 = Mi C/i + Jlf 2 U 2 + M s U 3 . 
 
 (301) Entropy. MS = M l Si + M 2 S 2 + M 3 5*3. 
 
 For internal equilibrium the general equation is, 
 
 (302) yl 1 M 5 5 = A 1 S M a 5 5 a + ^l 1 S S a 8 M a , 
 Where a takes the values 1. 2. 3. in succession. 
 
 Since U = Q - W, and TS = U + W = U + Pdv = Q, 
 we have by differentiation and substitution, 
 
 (303) SS = ^ + L^ 
 
 (304) A*M SS = S ^-^ + 2 Ma ^ 3ga + A*S a SM a = 0. 
 
 -^ a -'a 
 
 The three independent conditions for interpreting this 
 equation are, 
 
 (305) For the masses, S 6 M a = 0. 
 
 (306) For the volumes, S M a 5 v a + 2 fl 5 Af a = 0. 
 
 (307) For the energies, S Af a 6 / a + S Z7 fl 6 M a = 0. 
 After eliminating from the equations 6 Jl/2 . 5 V2 . 6 C/2- 
 
 (308) A^ d S = (4- - 7^-} M, d Ui - (^ - ^-} M* d U 3 
 
 \1 1 1 2/ \^ 2 ^ 3/ 
 
 = 0. 
 
76 THERMODYNAMIC METEOROLOGY 
 
 The conditions of equilibrium for the maximum entropy are 
 uniform temperature, T = T\ = T 2 = jT 3 , and uniform pressure, 
 
 Pi = P* = Pi. 
 Hence we have by selection, 
 
 (309) r 2 = Ti (Si - 5 2 ) = A (Ui - U 2 ) + PI (vi - v 2 ) for vapor 
 
 and liquid. 
 
 (310) r, = T 2 (S 2 - S t ) = A (U 2 - Z7 8 ) + P* (v* - %) for liquid 
 
 and solid. 
 
 SPECIFIC HEATS 
 
 A third example of a reversible cyclic process is given in the 
 specific heats, C\ for vapor, C 2 for fluid. From the first law by 
 (234), we have, 
 
 , dP d.AP d.Cv 
 (234) A jf = -^ ~j^. 
 
 Develop these terms successively by substitutions. 
 Differentiate the last form of (293), dividing by d v, 
 
 d ' AP M 
 
 r 2 d (DI v 2 ) -i M 
 Ldf ~ (vi - v 2 ) d T] vi-v* 
 
 The specific heat of the mixture is, by (136), 
 (312) Cv = C 2 (M-M 1 )+CiMi-r 2 j~. Hence, by (289), 
 
 'fQiA\ f^\^L - C*dMi CidMi r 2 \~dvi-d ^ 2 "1 
 
 L4j (2) dv " dv dv " (v! + v 2 ) I dT J 
 
 ^ By (292), 
 dCv 
 
 Subtract (1) - (2). 
 
SPECIFIC HEATS 77 
 
 /i^ i ^ /- . . , 
 
 (317) -7-=; -\- C 2 Ci = A TT -p= in heat units. 
 # i M. u, L 
 
 The Specific Heats in Terms of the Latent Heats 
 
 It can be proved by differentiation of the first forms of (309) 
 and (310) and the necessary substitutions that, 
 
 (318) (Cfc - (Q) 2 - f* - + _*_ [ ( f| )p _ f ?) J, 
 
 vapor-liquid. 
 
 <> *> - > -I? - ? + ^[.(t?),- (I?) ,) 
 
 liquid-solid. 
 Compare Planck's "Thermodynamik." 
 
 Examples of the Thermodynamic Data 
 
 1. Carnot's Cycle. In Fig. 3 the area enclosed between the 
 isotherms (1.3) and the adiabats (2.4) represents the work 
 done, W, in the cyclic process, and the figure is called the indicator 
 diagram. This is used in studying the efficiency of engines, 
 whether the process is natural or mechanical, and there is a very 
 large literature on the subject. No applications have as yet 
 been made in the atmosphere, for two reasons, the first because 
 of the difficulty of tracing out the history of a given mass, and 
 the second because the values of Cv in the formula are not 
 constant, and the true values of it have not heretofore been 
 computed. However, it is possible to take a standard mass, 
 as one kilogram of air at an initial point, and trace out the 
 conditions through which it must have passed in rising from the 
 surface through changing P. v. T. till it arrives at the surface 
 again in the original state, even though the path during its 
 circulation may not be known. This work will be reserved for 
 further studies. 
 
 2. Cyclic Process for Vapors. In Fig. 4, we have, 
 
 (320) (z> 2 Vi) d Mi the mass which evaporates in expansion. 
 
 (321) (z> 3 z> 4 ) d M 2 the mass which condenses in compression. 
 
78 THERMODYNAMIC METEOROLOGY 
 
 (322) b k = -TJ, d T the increase in the pressure. 
 
 (323) W = (vi v 2 ) -T-, .d Mid T the work done in the area 
 
 a, b, c, d. 
 
 (324) Qi = r 2 dMi = C 2 d MidT the heat received in expansion. 
 
 (325) Q 2 
 
 (326) Q 2 = CidMidT --.d Mi d T the heat expended in 
 
 compression. 
 
 (327) Q = Qi-Q 2 = (j^ + C 2 - 
 Since A = ^, this becomes as in (317), 
 
 (328) -~ + C 2 - Ci = A (vi - v 2 ) jj, in heat units. 
 
 Values of the Latent Heat and Specific Heats 
 
 (329) r 2 = 606.5 - 0.708 t for water to aqueous vapor. 
 n = 80.066 for ice to water. 
 
 Cpi = 0.4810 Cp 2 = 1.0000 Cp* = 0.5020. 
 
 Example 3. Water to Vapor at 100 C., by Vaporization 
 
 Ti = 273 + 100 = 373. 
 
 z>i = 1658 the volume of 1 gram of aqueous vapor at 100 C. 
 
 ^2 = 1 the volume of 1 gram of water at 0. 
 
 vi-v 2 = 1658 - 1 = 1657. 
 
 -r~ = 27.2 millimeters of mercury. 
 
 - 2 - 3894 x 10 ~ 8 = H85TOOO- ^ = 2 ' 37829 - 10) 
 
 - X = 3.1856 X 10 ~ 5 (log = 5.50319 - 10) 
 
 go 7oU 
 
 reduction from work units and mm. to C. 
 G. S. units heat. 
 
VALUES OF LATENT HEAT AND SPECIFIC HEATS 79 
 
 (295) Latent Heat. r 2 = T : fa - %) ~ = 373 X 1657 X 
 
 go a 1 
 
 27.2 X 3.1856 X 10 " 5 = 535.5. 
 
 Example 4. Ice to Water at by Melting 
 T 2 = 273 + = 273. 
 v z = 1.00 volume of 1 gram of water at 0. 
 v 3 = 1.09 volume of 1 gram of ice at 0. 
 % - v 3 = - 0.09. 
 
 d P 
 -r= = 134.6 atmospheres. 
 
 (296) Latent Heat. r 8 = . T 2 fa - v s ) ~ = 273 X (- 0.09) 
 
 (-134.6) X 3.1856 X 10 ~ 5 X 760 = 80.066. 
 Example 5. Aqueous Vapor in Contact with Water at 100. 
 
 &-- 0.708 (||) = 4.931 (||) = 0.001 
 dT \oTJ p \o 1 J P 
 
 Cp 2 = 1.0300 P = 760mm. T = 373 
 
 <3, 8 , , 
 
 0.001] = - 0.5504. 
 Specific Heat. C#i = 1.0300 - 0.5490 = 0.4810. 
 
 Example 6. Water in Contact with Ice at 
 
 ^ = 0.6400 (||)=- 0.00006 (||) = + 0.00011. 
 dT \oTJ P \ol / p 
 
 Cp 2 = 1.0000 water at 0. 
 
 80 80 
 
 = 0.6400 -273 + ^^^t- 0.00006- 
 
 0.00011] 
 
 = 0.6400 - 0.2930 + 0.1510 = 0.4980. 
 Specific Heat. Cp z = 1.0000 - 0.4980 = 0.5020. 
 
THERMODYNAMIC METEOROLOGY 
 
 Example 7. Pressure of Vapor in Contact with Water and Ice 
 
 P 12 = 4.57 vapor-water. Vi = 205000. 
 
 p 13 = 760 vapor-ice. v 2 = 1.00 
 
 p 23 = water-ice. v 3 = 1.09 
 
 T = 0.0074 C., the fundamental temperature. 
 
 r 12 41851000 760 
 (295) Vapor on water, -j^ -- T ^_ ~ - 1013235- 
 
 606.5 3.1391 X 10 4 
 
 204999 = 0.3402mm. 
 
 Vapor on ice. 
 
 (606.5 + 80) 3.1391 X 10 4 
 
 273 " 204999 -0.3851mm. 
 
 ^0 
 
 (296) Water on ice. - - 
 
 80.066 3.1391 X 10 4 
 
 ' 102294 ' mm - 
 
 (-0.09) 
 
 Compare Planck's "Thermodynamik." 
 
 Application of the Thermodynamic Formulas to the Non-Adiabatic 
 
 Atmosphere 
 
 The foregoing formulas would apply to an adiabatic atmos- 
 phere, using the constants of Table 3, wherein Cp, Cv, R are 
 constants, but they do not apply to the existing non-adiabatic 
 atmosphere, because it is not an ideal gas, rather a mixture of 
 gases which are undergoing rapid changes of condition through 
 variations in the heat contents by insolation and radiation. 
 They can, however, be adapted to the earth's atmosphere by 
 suitable modifications, which depend upon the formulas de- 
 veloped under static meteorology. The following summary is 
 sufficient for working purposes. 
 
APPLICATION TO NON-ADIABATIC ATMOSPHERE 81 
 
 Entropy 
 
 T 
 jr - R 
 
 / QQ1 \ e e ft ~ Qo r>. TI T Q 1 PI 
 
 (331) 61 - 6 = rp ~ = HI Cp a ^ -- TT Rio log -5- 
 
 ^ 
 
 dQ dT dP dT v 
 
 (330) dS = -jr '= Cp-jr - R~p~ = Cp^r - j,dP. 
 
 10 JL 10 
 
 M 
 
 Work Against External Forces 
 
 (332) dW = Pdv = RdT - = Cp ^-^ d T - . 
 
 p k p 
 
 (333) Wi - Wo = Pio (vi - VQ) = R w (T a - T ) L 
 
 Pio 
 
 k - 1 , p, 
 
 Pio 
 
 (334) 
 
 Pio 
 Energy 
 
 k-ldP 
 
 = CpdT r -- . 
 k p 
 
 (335) A - U = Cvu (T a - To) = (Cp a - RIO) (T a - T ) = 
 
 Cp a (T a ~ To) ~ Cp w ~^ (T a - T ) 
 = (Qi - Go) - (W l - Wo) = Cp a (T a ~ To) ~ 
 
 pio 
 Heat Energy 
 
 (336) dQ = CvdT + RT~ = Cv d T + Cp ^ d T - 
 
 k - 
 
82 THERMODYNAMIC METEOROLOGY 
 
 (337) Q 1 -Q = T w (S, -So) = (Cp a - Cp w ] (T a - T ) = 
 
 Cv w (T a - T ) + Pio (vi - vo). 
 
 = (Ui - Uo) + (Wi - Wo) = Cpa (Ta - To) - 
 
 Pi- Po 
 
 Pio 
 Radiation Function 
 
 P - TdS p- P* 
 ~ ~~ ~ f - 
 
 (Krt r >- '- P "'- p 
 
 (339) * = 3^ . = -^^ - P 10 = VI _ VQ - Pm 
 
 (341) log K! - log ^o = A (log r, - log T ). 
 
 (342) , _ log *i - log go 
 
 (3 . 42) - log r, - log r.- 
 
 The Radiation Coefficients and Exponents 
 
 (343) #10 = C Ti Q A . log #10 = log C + ,4 log r 10 . 
 
 (344) KM = c T 10 a . log #10 = log c +a log T 10 . 
 
 (345) log c = log Co + (-4 - 4) log B. 
 
 C = 9.12 X 10 ~ 5 . B = 1.66 X 10 ~ 2 . 
 
 (346) logc = - 5.906 - (2.220) (A - 4). 
 
 (347) c =C Q B A ~ 4 = 9.12 X 10~ 5 (1.66 X lQ-y~ 4 . 
 
 These formulas will be fully explained and illustrated in the 
 examples that follow (pages 8485). 
 
 Working Equations 
 -Co 
 
 (331) Si - So 
 
 (333) Wi - W = R'w (T a - To) - 
 
 Pi - Po 
 
 Pio 
 
WORKING EQUATIONS 83 
 
 (335) Ui-U = (Qi-Qo)-(Wi-W ). 
 (339) K 10 = Ul ~ Uo . 
 
 (340) =() 
 
 Ao Wo/ 
 
 log Zi log 
 
 In order to illustrate the formulas of computation (331) 
 to (342) the data of Table 17 are continued in Table 19. 
 It must be especially noted that - (Cp a - Cpio) (T a - T ), 
 which by (198) includes the kinetic energy of circulation and the 
 kinetic energy of radiation - J (q? - q 2 ) - (Q l - Q ), is not 
 carried forward, but only (Qi Q ), the energy of radiation. 
 If the former were taken for the computations beyond this point 
 the circulation would be treated as true radiation, which is 
 improper. The sign - (Q l - Q ) in Table 17 is changed to 
 -f (Qi Qo) in Table 19. In computing the mean entropy 
 from one level to another, the mean temperature T iQ = f (T[ -f 
 To) is taken from Table 14 in successive pairs. The entropy 
 generally increases with the height, and always does so unless 
 there is an inversion of temperature, or an excess of wind varia- 
 tion in velocity between the levels, such as occurred in this case 
 between 500 and 1,000 meters. 
 
 In applying (333) for computing the work (Wi Wo) we 
 must now compute R\o corresponding with (Qi Q Q ), which 
 differs from R 1Q taken by pairs in Table 14, since R w implies 
 the circulation as well as the radiation. The formulas now be- 
 come 
 
 (348) (Cp a - Cp' lo ) (T a - To) = (Qi - Qo). 
 
 (349) Cp\ Q = Cpa - %-^ = 993.58 - ^=-|. 
 
 
 
 a - 
 
 (350) 
 
84 
 
 THERMODYNAMIC METEOROLOGY 
 
 M 
 
 H 05 
 
 U O5 
 
 s ^ 
 
 g s 
 
 1 1 
 
 <J 
 w 
 
 fl I 
 
 M 
 
 B 
 
 H 
 D 
 
 CM 
 
 ! 
 
 l> 
 
 CM CM 
 
 CO 
 
 I I 
 
 CM rH 
 
 s 
 
 1 I 
 
 rH 
 1 1 
 
 O5 O GO O t>- CO O5 
 
 O5 CO J> CO O5 TH O 
 
 O rH TH 00 CM <M TH 
 TH TH !> 
 
 T-H 10 
 
 1 1 
 
 1 1 
 
 O5 
 1 I 
 
 sis s 
 
 CO.H 
 i i 
 
 00 O 
 
 1 1 
 
 CO >O 
 (N CO 
 
 CO O5 
 O5 1^ 
 CO CO 
 
 O O 
 | 
 
 + 
 
 O5 >O IO 
 O5 TH CO 
 
 05 CO 
 TH 
 
 S CO-H 
 
 CO 
 
 1 I 
 
 O 
 
 (M 
 
 s 
 
 CM 00 
 
 CM O 
 
 00 O 
 
 (M 
 
 co 
 
 05 
 
 O5 CO 
 CO O 
 
 !> O5 >O O 
 CO O5 TH CO 
 
 O O 
 
 II 
 
 CM (M l> 
 
 
 
 10 
 
 rH 00 
 T( 
 
 CO TH 
 
 i 7 
 
 1 1 
 
 o>o> 
 
 I I 
 
 
 
 Co Co 
 1 I 
 
 Co Co 
 
 bfl 
 
 1 1 
 
 .<N 
 
 
 + I 
 
 CO 
 
 + 
 
 + + 
 
 Tt< CO CO 
 
 + + I 
 
 + + I 
 
00 
 
 l> 10 CO 
 CO 
 
 WORKING EQUATIONS 
 
 oo o oo b- 
 
 00 CM ^H O 
 
 rH CO" i-H IO Ob- T}^COO(MTft 
 
 i ( CO T-H O OOCO O CO O CO CM 
 
 OS O T-H OS 
 
 + 
 
 I I 
 
 00 00 O "# 
 
 I I ! 
 
 t^ CO 
 
 GO T-H OS 1O 
 
 Tt< 1C 00 rH 
 
 (N l> 00 O 
 
 OS O rH OS 
 
 8 
 
 < 
 
 I I 
 
 * 
 
 ' 00 00 O CO 
 
 I I I 
 
 00 CO b- (M 
 
 O5 CO CO r^ t^ CO 
 
 ^* CO CO t^* CO OS 
 
 *O CO CO GO Tfrl CO 
 
 iO OS O O O CO 
 
 CO C5 O r-t '06 
 
 I + 
 
 CO O O O 
 
 CO CO CO t I 
 
 00 CO 
 00 iO 
 
 I 7 
 
 D CO CO CO 
 ' 00 l> O 
 
 I I I 
 
 OO CO TJH CM CO CO OS b- CM 
 
 CO CO CO b- O3 -^f OS CM CM CO O 
 
 10 oo i i ^H ^ ^H oc^ T i co 10 oo co 
 
 IO OS CO CO CO CO OS CO CO C^ *O CO OO 
 
 CO OSOT-i'o6 TJH'O 'o6l>Or}J 
 
 i + i T i i i 
 
 ^^ T^ OS 1( ^ *O OS OO CO CO CO 
 
 ^ lO^JiiOCl T-I COO1O(N 
 
 OO T-HOOSC'MCO O COCMOSCO 
 
 TJH OOs2?T^(N C<lcO T-t(NT^l>.ii 
 
 >O O OS w . O CO O5I> OC^OCOI> 
 
 CO OOs'oo' TJH' TJ< *o6l>OT}< 
 
 i + i T i i i 
 
 T-H OS 
 
 ^ OS rf* 
 
 TJH Olt^iOcOOS r^C<l 
 
 ^o oososoo osi^ 
 
 CO CO OS O ' 00 rji b- 
 
 00 
 
 I +11 
 
 Tf lO 1C OS 
 
 O b- (M 
 
 b- 00 00 CM 
 
 ' 00 t>^ O rji 
 
 I I I 
 
 1>* CO l>* !> CO CO 
 
 co co T- i co t^ co o 
 
 ^^ Oi Oi ^^ *-O O5 CO 
 
 O OS O '06 Tj<' CN 
 
 + I \ 
 
 CO i- 1 O T-H 
 
 00 t O CO 
 
 I I I 
 
 OS S 
 
 CO O OS O 00 
 
 I + 
 
 OS lO 
 
 OS IO 
 
 ^ OS 
 
 I \ 
 
 ^ 2 
 
 "^r 
 00 
 
 O OS O 
 
 i 
 
 I I 
 
 85 
 
86 
 
 THERMODYNAMIC METEOROLOGY 
 
 COMPARISON OF Rio WITH R\ 
 
 Height z 
 
 116 
 
 500 
 
 1000 
 
 1500 
 
 2000 
 
 2500 
 
 3000 
 
 4000 
 
 5000 
 
 RIO 
 R'IO 
 
 
 287.28 
 282.91 
 
 286.87 
 287.35 
 
 285.34 
 285.65 
 
 283.14 
 281.68 
 
 280.50 
 277.95 
 
 278.21 
 275.46 
 
 275.59 
 274.09 
 
 272.06 
 268.89 
 
 R'IO is generally smaller than R w in these observations. 
 
 P P 
 The term is taken directly from Table 17, and 
 
 PlO 
 
 Wo is easily computed. Then Ui U Q follows from (335). 
 
 In computing the radiation, 
 
 Ui-Ui 
 
 the values of 
 
 v are the reciprocal of the density p in Table 14. Had p been 
 computed by formula (175), which takes R constant, instead of 
 by (176), with R variable, it is seen at once how erroneous 
 would have been the derived radiations, because the values of 
 KIQ depend upon the small differences fyi v ) in succession. 
 These radiations are mean values for the strata concerned. It 
 is important to study the relation of the radiation to the tem- 
 perature, and to compare the exponents of formula (340) with 
 the exponent of a full radiation in the Stephan Law, which is 4. 
 This subject is complex in the earth's atmosphere as will be in- 
 dicated. The problem is as follows: The values of K in re- 
 lation to T by (340) are in the form of ratios, whereas in the 
 Stephan Law (344) they stand related through a coefficient. 
 If the constituent of the ratio is in the form (343), it is quite 
 certain that the coefficients (C . c} are not equal, nor are the expo- 
 nents (^4 .a). We proceed to develop the relations between 
 C and c, A and a. The equation (343) gives three terms, K w , 
 A, TIG, from which to compute C, and it is necessary to indicate 
 what are the relative values of log C and A . With the data of 
 Table 19 in the first section of Table 20, compute A log T 10 
 and subtract this from log K iQ to obtain log C. The negative 
 sign before the logarithm affects only the characteristic. Thus, 
 logarithm 11.944 gives the number 8.79 X 10~~". In this 
 
WORKING EQUATIONS 
 
 87 
 
 way the values of log C and A were computed for twelve balloon 
 ascensions, of which two examples are given in Table 21. It 
 is readily seen that log C is negative, as the temperature T de- 
 creases with the height z, and positive in regions of inversion of 
 temperature. The magnitude of log C depends upon the ratio 
 
 rri 
 
 If the temperature changes slowly with the height the 
 
 TABLE 20 
 COMPUTATION OF LOG C AND ( a . LOG c} 
 
 Section I. log C = log Kio - A log Tio (343) 
 Lindenburg, April 27, 1909 
 
 z 
 
 TlQ 
 
 log Tio 
 
 116 
 
 500 
 292.25 
 
 1000 
 
 288.30 
 2 45984 
 
 1500 
 
 284.90 
 2 45469 
 
 2000 
 
 282 . 15 
 2 45048 
 
 2500 
 
 279.85 
 2 44693 
 
 3000 
 
 277.20 
 2 44279 
 
 4000 
 
 272 .40 
 2 43521 
 
 5000 
 
 266.00 
 2 42488 
 
 log.log Tio 
 
 
 
 39091 
 
 39000 
 
 38925 
 
 38862 
 
 38789 
 
 38654 
 
 38469 
 
 log A 
 
 
 
 0.78581 
 
 68298 
 
 67320 
 
 . 68362 
 
 0.67620 
 
 0.55160 
 
 62707 
 
 A 
 
 
 
 6 11 
 
 4 82 
 
 4 71 
 
 4 83 
 
 4 74 
 
 3 56 
 
 4 24 
 
 log (A log 
 Tio) 
 
 
 
 1.17672* 
 
 1.07298 
 
 1.06245 
 
 1.07224 
 
 1.06409 
 
 0.93814 
 
 1.01176 
 
 A log Tio 
 
 
 
 15.021 
 
 11.830 
 
 11 547 
 
 11.810 
 
 11 . 590 
 
 8.672 
 
 10 275 
 
 log Kio 
 
 
 
 4 96534 
 
 4 94187 
 
 4 92519 
 
 4 90796 
 
 4 83566 
 
 4 84816 
 
 4 80528 
 
 Kio 
 
 
 
 92330 
 
 87472 
 
 84176 
 
 80902 
 
 76853 
 
 70495 
 
 63867 
 
 loeC 
 
 
 
 11 944 
 
 7 112 
 
 -7 378 
 
 7 098 
 
 7 246 
 
 4 176 
 
 6 530 
 
 C 
 
 
 
 ll 
 8.79X10 
 
 7 
 
 1.29X10 
 
 7 
 
 2.39X10 
 
 7 
 
 1.25X10 
 
 7 
 
 1.76X10 
 
 1.50X10 
 
 3.39X10 
 
 Section II. log c = log Kio - a log Tio (344) 
 
 Assumed 
 
 
 
 3 82 
 
 3 82 
 
 3 82 
 
 3 82 
 
 3.82 
 
 3.81 
 
 3.81 
 
 log ai 
 
 
 
 0.5821 
 
 0.5821 
 
 0.5821 
 
 0.5821 
 
 0.5821 
 
 0.5809 
 
 0.5809 
 
 log.log Tio 
 
 
 
 0.3909 
 
 0.3900 
 
 0.3893 
 
 0.3886 
 
 0.3879 
 
 0.3865 
 
 0.3847 
 
 log ai log 
 Tio 
 
 
 
 9730 
 
 . 9721 
 
 0.9714 
 
 0.9707 
 
 0.9700 
 
 0.9674 
 
 0.9656 
 
 
 
 
 9 397 
 
 9 378 
 
 9 363 
 
 9.348 
 
 9.333 
 
 9.277 
 
 9.238 
 
 log Kio 
 
 
 
 4.965 
 
 4 942 
 
 4.925 
 
 4.908 
 
 4.836 
 
 4.848 
 
 4.805 
 
 log c 
 
 
 
 5 568 
 
 5 564 
 
 5 562 
 
 -5.560 
 
 -5.503 
 
 -5.571 
 
 -5.567 
 
 
 
 
 3 824 
 
 3 822 
 
 3 821 
 
 3.820 
 
 3.794 
 
 3.825 
 
 3.823 
 
 Second 
 Assumed 
 
 
 
 
 
 
 
 3.807 
 
 3.817 
 
 3.817 
 
 
 
 
 
 
 
 
 0.5806 
 
 0.5817 
 
 0.5817 
 
 log log 7\o 
 
 
 
 
 
 
 
 0.3879 
 
 0.3865 
 
 0.3847 
 
 log . 02 log 
 
 Tio 
 
 
 
 
 
 
 
 0.9685 
 
 0.9682 
 
 0.9664 
 
 
 
 
 
 
 
 
 9.300 
 
 9.294 
 
 9.256 
 
 log Kio 
 
 
 
 
 
 
 
 4.836 
 
 4.848 
 
 4.805 
 
 
 
 
 
 
 
 
 -5.536 
 
 -5.554 
 
 -5.549 
 
 a 
 
 
 
 
 
 
 
 3.809 
 
 3.817 
 
 3.815 
 
88 
 
 THERMODYNAMIC METEOROLOGY 
 
 Working Formulas 
 
 (343) K 1Q = C7V 
 
 log C = log KM A log TIG 
 (344) K 10 = c 2V 
 
 log KIQ = log c + a log JIG 
 
 log C = log KIQ a log TIQ. 
 
 Assume trial #1 and adjust by Table 22. The final pair 
 log c and a should fall on the same line. 
 
 TABLE 21 - 
 
 EXAMPLES OF LOG C . A AND LOG c . a THE COEFFICIENTS AND 
 EXPONENTS IN THE RADIATION FORMULAS (343), (344) 
 
 z 
 
 Lindenburg, April 27, 1909 
 
 Lindenburg, May 5, 1909 
 
 logC 
 
 A 
 
 logc 
 
 a 
 
 logC 
 
 A 
 
 logc 
 
 a 
 
 17000 
 
 
 
 
 
 
 
 
 
 16000 
 
 
 
 
 
 +28 . 034 
 
 10 . 34 
 
 5.435 
 
 3.764 
 
 15000 
 
 
 
 
 
 +52.871 
 
 -21.11 
 
 -5.461 
 
 3.775 
 
 14000 
 
 
 
 
 
 + 16.828 
 
 5.78 
 
 -5.489 
 
 3.788 
 
 13000 
 
 - 4.356 
 
 3.45 
 
 -5.531 
 
 3.807 
 
 + 10.437 
 
 - 3.41 
 
 -5.537 
 
 3.810 
 
 12000 
 
 - 7.140 
 
 4.86 
 
 -5.563 
 
 3.822 
 
 -27.533 
 
 13.29 
 
 -5.563 
 
 3.821 
 
 11000 
 
 - 5.962 
 
 3.64 
 
 -5.571 
 
 3.825 
 
 - 6.181 
 
 4.40 
 
 -5.548 
 
 3.814 
 
 10000 
 
 - 1.410 
 
 3.44 
 
 -5.560 
 
 3.820 
 
 - 3.288 
 
 3.09 
 
 -5.558 
 
 3.819 
 
 9000 
 
 - 2.114 
 
 2.73 
 
 -5.549 
 
 3.815 
 
 - 4.021 
 
 3.62 
 
 -5.558 
 
 3.819 
 
 8000 
 
 - 3.211 
 
 3.11 
 
 -5.539 
 
 3.810 
 
 - 6.506 
 
 4.26 
 
 -5.560 
 
 3.820 
 
 7000 
 
 - 7.178 
 
 4.79 
 
 -5.538 
 
 3.810 
 
 - 5.249 
 
 3.95 
 
 -5.557 
 
 3.818 
 
 6000 
 
 - 9.719 
 
 5.40 
 
 -5.548 
 
 3.814 
 
 - 5.401 
 
 3.89 
 
 -5.560 
 
 3.820 
 
 5000 
 
 - 6.530 
 
 4.24 
 
 -5.549 
 
 3.815 
 
 -11.132 
 
 6.48 
 
 -5.547 
 
 3.814 
 
 4000 
 
 - 4.176 
 
 3.56 
 
 -5.554 
 
 3.817 
 
 - 4.050 
 
 3.63 
 
 -5.560 
 
 3.820 
 
 3000 
 
 - 7.246 
 
 4.74 
 
 -5.536 
 
 3.809 
 
 - 8.115 
 
 5.25 
 
 -5.567 
 
 3.823 
 
 2500 
 
 - 7.098 
 
 4.83 
 
 -5.560 
 
 3.820 
 
 -18.387 
 
 9.23 
 
 -5.577 
 
 3.828 
 
 2000 
 
 - 7.378 
 
 4.71 
 
 -5.562 
 
 3.821 
 
 -20.982 
 
 9.81 
 
 -5.590 
 
 3.833 
 
 1500 
 
 - 7.112 
 
 4.82 
 
 O . OvVx 
 
 3.822 
 
 -14.180 
 
 7.64 
 
 -5.601 
 
 3.838 
 
 1000 
 
 -11.944 
 
 6.11 
 
 -5.568 
 
 3.824 
 
 +24,708 
 
 -8.29 
 
 -5.593 
 
 3.834 
 
 500 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ratio is small. At the same time the ratio h~ J , which does 
 
 \Ar,/ 
 
 not have a coefficient or exponent, registers a change in the 
 
WORKING FORMULAS 
 
 89 
 
 atmosphere which is closely connected with the variation of 
 the pressure P. Hence there are large changes in log C and A 
 which are opposite in sign, but both increasing or diminishing 
 together. Under nearly normal conditions it is seen that A is 
 approximately 4.00, which would be the value for a full radiating 
 black body. The entire series of values A, log C, were collected 
 in groups, and the mean values when plotted fall on a straight 
 line, of which the equation was found to be, 
 
 (347) C = C B A ~\ 
 
 (345) log C = log Co + (A - 4) log B. 
 
 (346) logC = - 5.960 + (A - 4) (- 2.220). 
 
 (347) C = 9.12 X 10~ 5 (1.66 X 10 ~ 2 ) A ^ 4 . 
 
 The development of a portion of this formula from A = 4.00 
 to A = 3.50 is given in Table 22. A = 4.00 corresponds 
 with a full black radiator and A = 3.50 corresponds with the 
 theoretical value for the atmospheric air. The mean value was 
 found to be A = 3.82 near the surface. The notation (log c . 
 a) indicates the values in the constituent formula, while (log C . 
 A) are used for the ratio formula. 
 
 TABLE 22 
 EVALUATION OF THE FORMULA (346) 
 
 a logc 
 
 a log c 
 
 a log c 
 
 a logc 
 
 4.00 -5.960 
 3.99 -5.938 
 3.98 -5.916 
 3.97 -5.893 
 3.96 -5.871 
 
 3.95 -5.849 
 3 94 5 827 
 
 3.85 -5.627 
 3.84 -5.605 
 3.83 -5.583 
 3.82 -5.560 
 3.81 -5.538 
 
 3.80 -5.516 
 3 79 _5 494 
 
 3.70 -5.294 
 3.69 -5.272 
 3.68 -5.250 
 3.67 -5.227 
 3.66 -5.205 
 
 3.65 -5.183 
 3 64 -5 161 
 
 3.55 -6.961 
 3.54 -6.939 
 3.53 -6.917 
 3.52 -6.894 
 3.51 -6.872 
 
 3.50 -6.850 
 
 3.93 5.805 
 
 3.78 -5.472 
 
 3.63 -5.139 
 
 
 3.92 -5.782 
 
 3.77 -5 449 
 
 3.62 5.116 
 
 
 3 91 5 760 
 
 3 76 5 427 
 
 3 61 5 094 
 
 
 
 
 
 
 3.90 -5.738 
 
 3 75 5 405 
 
 3 60 - 5 072 
 
 
 3.89 -5.716 
 3.88 -5 694 
 
 3.74 -5.383 
 3 73 5 361 
 
 3.59 -5.050 
 3 58 -5 028 
 
 
 
 3.87 -5.671 
 
 3.72 -5.338 
 
 3.57 5.005 
 
 
 3.86 -5 649 
 
 3 71 -5 316 
 
 3 56 -6 983 
 
 
 
 
 
 
90 THERMODYNAMIC METEOROLOGY 
 
 The negative sign applies only to the characteristic 
 log Co = - 5.960 Co = 9.12 X 10~ 5 
 logB = - 2.220 B = 1.66 X 10~ 2 . 
 
 By means of Table 22 we proceed in Section II of Table 
 20 to compute log c and a from log C and A . Assume an 
 approximate value ai, as 3.82, and compute di log 7\ from the 
 value in Section I. Subtract from log K\ Q for log c, and in 
 Table 22 interpolate that value a , which is the pair value of 
 log c. If this value a Q agrees with the assumed a\ the check is 
 complete. If, on the other hand, these values of logc and a 
 do not quite agree, as in the examples under z = 3000, 4000, 5000, 
 take the mean value between the assumed a\ and computed 
 a Q , and proceed again with a 2 to compute logc and a. The 
 second trial is usually successful if a\ has been chosen with some 
 practice. The corresponding values of (log c . a) are found in 
 the examples of Table 21, and it is seen that the irregularities 
 of (log C . A) have disappeared. Log c and a usually decrease 
 slowly with the elevation and with the increase of latitude from 
 the equator. 
 
 These results check by log c + a log TIQ = log KIQ. 
 
 Application of the Thermodynamic Formulas to Various Meteor- 
 ological Problems 
 
 It has seemed necessary to give an extended example of the 
 method of computing the thermodynamic values in the non- 
 adiabatic atmosphere, on account of the complexity of the 
 computations, and because of the numerous valuable results 
 dependent upon them. In Bulletin No. 3 of the Argentine 
 Meteorological Office, 1913, will be found the results for many 
 types of data in considerable detail. We can here summarize 
 them briefly, depending upon diagrams to bring out the general 
 ideas, in particular respecting the isothermal region, the diurnal 
 convection, the circulation in cyclones and anti-cyclones, and 
 the general circulation of the atmosphere. 
 
THE ISOTHERMAL REGION 91 
 
 The Isothermal Region 
 
 It has been found by balloon ascensions to great elevations, 
 up to 20,000 meters or more, that the temperature of the atmo- 
 sphere diminishes at the rate of about 6.0 C. per 1,000 meters 
 up to an elevation of 12,000 meters in Europe, or 15,000 meters 
 in the tropics, or even to 20,000 meters over the equator, while 
 above these elevations the temperature is nearly constant or 
 increases a little to the highest levels explored. There have 
 been many conjectures as to the cause of the permanence of the 
 heat of this isothermal region, as overflow of the tropic heat 
 to mid-latitudes, conductional transportation of heat from the 
 lower to the higher levels, production of ozone by the incoming 
 solar radiation in the upper atmosphere and absorption of the 
 short waves of the solar radiation in the same region. There are 
 objections to each of these hypotheses so obvious that we pro- 
 ceed at once to examine the thermodynamic data for at least a 
 statement of the case, if not a complete explanation of the facts. 
 
 The computations were executed for the following balloon 
 ascensions, as reported in the volumes: 
 
 Europe Atlantic Tropics 
 
 Lindenburg, April 27, 1909 (52). Sept. 25, 1907 (35). 
 
 May 5, 1909 " Sept. 9, 1907 (25). 
 
 May 6, 1909 " Aug. 29, 1907 (13). 
 
 July 27, 1908 " July 29, 1907 (13). 
 
 Sept. 2, 1909 " June 19, 1906 (-2). 
 
 Mailand, Sept. 7, 1906 (45). Victoria Nyanza, 1908 (0). 
 
 The mean values are compiled in Table 23, and illus- 
 trated in Figs. 5 and 6. Since the data in the isothermal region 
 are not so complete as below it, these results are to be considered 
 as instructive rather than definitive. It will require the work 
 of many years to accumulate and compute the data necessary 
 for normal conditions. The temperatures show that there 
 are as wide local fluctuations in the isothermal region as below 
 it. Furthermore, the temperatures are lowest over the equator, 
 200, and gradually increase to 210 in the tropics, or 215 in 
 
92 
 
 THERMODYNAMIC METEOROLOGY 
 
 Europe. From these minimum values the temperatures in 
 the isothermal region increase about 10 degrees. These facts 
 appear clearly in the Table 23, and the diagrams, where 
 
 13000 
 
 10000 
 
 5000 
 
 000 
 
 P- K 
 
 Pressure 
 
 Radiation 
 
 Energy 
 
 Density 
 
 25000 
 
 50000 
 
 75000 
 
 100000 
 
 0.7500 
 
 1.0000 
 
 1.2500 
 
 1-5000 
 
 10000 
 
 5000 
 
 000 
 
 200 C 
 
 Temperatu e 
 
 \ 
 
 
 Free Heat 
 
 250 275 -3000 
 
 Full .Line for Europe 
 DottedJLine for Tropics 
 FIG. 5 
 
 -2000 
 
 -1000 
 
 the other data are presented. The data on Fig. 5 (P, K, p, 
 Qi QQ) may be considered primary, and those on Fig. 6 
 (Cp, Si - So, Wi - W Q , Ui - UQ) secondary, as being the 
 machinery of the thermal engine. It is noted that (P,K, p) 
 
THE ISOTHERMAL REGION 
 
 93 
 
 have one configuration, and (Q, Cp, S, W, U) another configura- 
 tion. The former is more immediately under the control of 
 gravitation acting downward, and the latter is the result of 
 
 15000 
 
 5000 
 
 000 
 
 C p Specific Heat 
 
 En 
 
 tropy 
 
 \ 
 
 600 700 
 
 800 900 1000 -10.00 -5.00 
 
 15000 
 
 10000 
 
 5000 
 
 Work 
 
 - U Inner Energy 
 
 000 
 4000 5000 6000 7000 -8000 -7500 -7000 -6500 
 
 Full Line for Europe 
 Dotted Line for Tropics 
 FIG. 6 
 
 radiation. The circulation q is the balancing governor to the 
 engine which keeps the other two parts in equilibrium. 
 
 (<2i - Qo). There is a persistent supply of heat from four 
 conditions: (l) That from the heated earth as the source; 
 (2) that in the lower strata, due to convection within 2,000 
 
94 
 
 THERMODYNAMIC METEOROLOGY 
 
 TABLE 23 
 
 MEAN THERMODYNAMIC VALUES FOR EUROPE 
 
 z 
 
 T 
 
 P 
 
 p 
 
 Cp 
 
 g 
 
 Qi-Qo 
 
 Si -So 
 
 Wi- 
 
 W 
 
 Ui-U 
 
 KlQ 
 
 20000 
 19000 
 18000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 17000 
 
 219.5 
 
 9141 
 
 0.2220 
 
 657.15 
 
 0.8 
 
 -3231 
 
 -14.747 
 
 4658 
 
 -7887 
 
 16704 
 
 16000 
 
 223.4 
 
 10983 
 
 .2476 
 
 689.35 
 
 2.5 
 
 -2791 
 
 -12.620 
 
 4962 
 
 -7753 
 
 18497 
 
 15000 
 
 221.2 
 
 12819 
 
 .2764 
 
 731.44 
 
 3.9 
 
 -2392 
 
 -11.158 
 
 5209 
 
 -7602 
 
 20083 
 
 14000 
 
 220.3 
 
 14972 
 
 .3086 
 
 769.99 
 
 8.6 
 
 -1955 
 
 - 8.641 
 
 5540 
 
 -7494 
 
 21850 
 
 13000 
 
 215.7 
 
 17017 
 
 .3422 
 
 799.60 
 
 14.3 
 
 -1702 
 
 - 7.487 
 
 5730 
 
 -743l!23883 
 
 12000 
 
 215.8 
 
 19942 
 
 .3830 
 
 836.65 
 
 16.4 
 
 -1381 
 
 - 6.326 
 
 5948 
 
 -7329J26548 
 
 11000 
 
 218.7 
 
 23341 
 
 .4283 
 
 863.20 
 
 17.8 
 
 -1190 
 
 - 5.345 
 
 6096 
 
 -7285 
 
 29953 
 
 10000 
 
 226.0 
 
 27198 
 
 .4780 
 
 873.16 
 
 19.3 
 
 -1089 
 
 - 4.722 
 
 6164 
 
 -7253 
 
 34502 
 
 9000 
 
 232.8 
 
 31606 
 
 .5315 
 
 884.82 
 
 20.7 
 
 -1131 
 
 - 4.768 
 
 6248 
 
 -737940630 
 
 8000 
 
 241.1 
 
 36509 
 
 .5890 
 
 890.49 
 
 18.4 
 
 -1034 
 
 - 4.229 
 
 6302 
 
 -733645275 
 
 7000 
 
 249.0 
 
 41972 
 
 .6504 
 
 897.75 
 
 16.3 
 
 - 920 
 
 - 3.647 
 
 6329 
 
 -7249 
 
 51345 
 
 6000 
 
 256.1 
 
 48053 
 
 .7162 
 
 907.35 
 
 12.6 
 
 - 793 
 
 - 3.083 
 
 6405 
 
 -7198 
 
 57729 
 
 5000 
 
 262.6 
 
 54819 
 
 .7865 
 
 919.62 
 
 14.1 
 
 - 650 
 
 - 2.578 
 
 6511 
 
 -7160 
 
 64352 
 
 4000 
 
 268.3 
 
 62353 
 
 .8620 
 
 934.08 
 
 13.0 
 
 - 530 
 
 - 1.957 
 
 6601 
 
 -7131 
 
 71739 
 
 3000 
 
 274.3 
 
 70722 
 
 .9429 
 
 947.41 
 
 12.6 
 
 - 380 
 
 - 1.375 
 
 6717 
 
 -7097 
 
 77430 
 
 2000 
 
 279.1 
 
 80030 
 
 1.0299 
 
 964.88 
 
 11.8 
 
 - 241 
 
 - 0.857 
 
 6825 
 
 -7066 
 
 86011 
 
 1000 
 
 284.8 
 
 90345 
 
 1.1224 
 
 979.50 
 
 9.2 
 
 - 76 
 
 - 0.241 
 
 6969 
 
 -7045 
 
 94582 
 
 100 
 
 289.4 
 
 100419 
 
 1.2101 
 
 993.58 
 
 6.1 
 
 
 
 
 
 
 
 
 
 
 
 meters of the ground, and involving a supply of latent heat by 
 the condensation of aqueous vapor into water in cloud formation 
 by (295); (3) that in the cirrus cloud region, 9,000 to 15,000 
 meters elevation, according to the latitude, due to ice formation 
 from frozen water or vapor by (296). In this cirrus region there 
 are other sources of heat supply, such as an accumulation of 
 heat from absorption of radiation producing the new rate of 
 loss of free heat per 1,000 meters. Take the differences in the 
 ((?i (?o) columns, and the mean values fall into two groups, 
 omitting those in the cirrus layers. 
 
 Europe, 12000 to 17000 A (Q l - Q ) = - 370 per 1000 meters. 
 
 Tropics, 14000 to 17000 = - 476 
 
 Europe, 1000 to 11000 =-133 
 
 Tropics, 1000 to 13000 " = - 140 " " 
 
THE ISOTHERMAL REGION 
 
 95 
 
 TABLE 23 
 MEAN THERMODYNAMIC VALUES FOR THE ATLANTIC TROPICS 
 
 a 
 
 20000 
 19000 
 18000 
 
 r 
 
 P 
 
 p 
 
 Cp 
 
 <? 
 
 Qi-Qo 
 
 Si -So 
 
 Wi- 
 Wo 
 
 Ui-Ut 
 
 Kio 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 17000 
 
 227.0 
 
 9894 
 
 0.2256 
 
 668.93 
 
 8.0 
 
 -3012 
 
 -13.359 
 
 4911 
 
 -7922 
 
 17488 
 
 16000 
 
 217.3 
 
 11213 
 
 .2513 
 
 709.00 
 
 7.4 
 
 -2570 
 
 -10.292 
 
 5203 
 
 -7774 
 
 18647 
 
 15000 
 
 209.9 
 
 13331 
 
 .2788 
 
 788.87 
 
 8.8 
 
 -1898 
 
 - 8.567 
 
 5609 
 
 -7506 
 
 19328 
 
 14000 
 
 212.8 
 
 15665 
 
 .3128 
 
 814.91 
 
 8.5 
 
 -1584 
 
 - 7.361 
 
 5750 
 
 -7334 
 
 21544 
 
 13000 
 
 215.9 
 
 18398 
 
 .3502 
 
 835.84 
 
 14.3 
 
 -1470 
 
 - 6.697 
 
 5911 
 
 -7381 
 
 24731 
 
 12000 
 
 221.6 
 
 21442 
 
 .3911 
 
 855.58 
 
 13.8 
 
 -1345 
 
 - 5.949 
 
 6024 
 
 -7369 
 
 28286 
 
 11000 
 
 230.3 
 
 24942 
 
 .4355 
 
 861.08 
 
 13.3 
 
 -1302 
 
 - 5.545 
 
 6060 
 
 -7362 
 
 32588 
 
 10000 
 
 238.9 
 
 28853 
 
 .4830 
 
 865.62 
 
 12.4 
 
 -1250 
 
 - 5.135 
 
 6086 
 
 -7336 
 
 37288 
 
 9000 
 
 247.7 
 
 33204 
 
 .5338 
 
 869.26 
 
 12.0 
 
 -1182 
 
 - 4.699 
 
 6127 
 
 -7309 
 
 42938 
 
 8000 
 
 254.9 
 
 38038 
 
 .5879 
 
 878.51 
 
 11.8 
 
 -1100 
 
 - 4.252 
 
 6199 
 
 -7298 
 
 47822 
 
 7000 
 
 260.1 
 
 43413 
 
 .6459 
 
 887.63 
 
 10.7 
 
 -1000 
 
 - 3.769 
 
 6271 
 
 -7271 
 
 53935 
 
 6000 
 
 268.3 
 
 49379 
 
 .7079 
 
 899.98 
 
 9.9 
 
 - 881 
 
 - 3.247 
 
 6368 
 
 -7248 
 
 59582 
 
 5000 
 
 274.1 
 
 56008 
 
 .7742 
 
 913.55 
 
 6.6 
 
 - 700 
 
 - 2.534 
 
 6473 
 
 -7173 
 
 65980 
 
 4000 
 
 278.5 
 
 63380 
 
 .8453 
 
 931.88 
 
 6.5 
 
 - 532 
 
 - 1.907 
 
 6591 
 
 -7123 
 
 72304 
 
 3000 
 
 283.9 
 
 71569 
 
 .9217 
 
 946.73 
 
 6.5 
 
 - 397 
 
 - 1.388 
 
 6694 
 
 -7091 
 
 78606 
 
 2000 
 
 288.8 
 
 80625 
 
 1.0031 
 
 963.44 
 
 5.8 
 
 - 187 
 
 - 0.641 
 
 6846 
 
 -7033 
 
 86055 
 
 1000 
 
 292.4 
 
 90699 
 
 1.0908 
 
 984.35 
 
 4.4 
 
 - 24 
 
 - 0.081 
 
 6911 
 
 -6935 
 
 93656 
 
 000 
 
 299.5 
 
 101753 
 
 1.1837 
 
 993.58 
 
 5.8 
 
 
 
 
 
 
 
 
 
 
 
 In the Europe group 9,000 to 10,000 is omitted, and in the 
 tropic group 10,000 to 12,000 is omitted, as being regions of 
 special local supply. It appears that heat is lost at about three 
 times greater rate in the isothermal region than in the lower 
 levels. This occurs at the same time the temperature is rising 
 in the isothermal region but falling in the lower levels. The 
 Victoria Nyanza ascensions give A (Qi Q ) = 144 through- 
 out the region 3,000 to 18,000, but in this case no inversion 
 of T was found. (4) The principal fact to be explained is the 
 slow rate of loss of heat in the convectional region, 140 per 
 1,000 meters, as compared with that in the isothermal region, 
 about 400. This is easily accounted for by the following facts : 
 The incoming solar radiation is of short wave lengths, and pen- 
 etrates to the earth's surface, having only a small amount of 
 
96 THERMODYNAMIC METEOROLOGY 
 
 selective absorption of radiant energy. This is transformed at 
 the earth into long waves, in changing the temperature energy 
 from 7,000 to 300, and this heat escapes to space partly by radi- 
 ation and partly by vertical convection, the latter extending to 
 the isothermal layer, whose height varies with the latitude, and 
 the heat contents of the air at the surface. In the general ver- 
 tical convection of the atmosphere, as the temperature of a 
 unit mass changes from T to 7\, in the vertical distance z\ z , 
 there is an evolution of heat Qi Qo = Cp 1Q (T\ jT ), which 
 is added to the atmosphere throughout the convectional region. 
 If (Qi Qo) is the natural loss of heat by radiation without 
 convection, and ((V QQ} the amount evolved by convective 
 cooling of the temperature of the rising mass, then we have, 
 400 = (Qi Qo) the heat loss in the isothermal region, and 140 = 
 (Qi ~~ Qo) ~~ (Qi ~ Qo) that in the convection region. Hence 
 260 = (Qi Qo) the heat evolved by vertical motion. This 
 subject will require a fuller development than is at present 
 available, and it is complicated by the fact that the vertical 
 distance through which the mass moves is not well known. The 
 air that has risen by convection with cooling and evolution of 
 heat in one place falls again in other places with heating and ab- 
 sorption of heat. Such places of descending air are, during the 
 night, in the permanent high-pressure belts and in the wander- 
 ing anti-cyclones. This subject will be sufficiently illustrated 
 in the following sections. 
 
 P, - K and p. 
 Pressure, Radiation-Potential, and Density. 
 
 The pressure is found to change continuously in a smooth 
 curve from the surface upward, that in the tropics being some- 
 what higher in value than in Europe. An entirely similar 
 curve is developed by the potential radiation K, the values 
 being always higher than P. The density is also given in a 
 smooth curve, the tropics and Europe crossing at about 7,000 
 meters. It would seem, then, that the ultimate purpose of 
 T. Q and the other dependent terms is to so regulate the pressure 
 
PRESSURE, RADIATION-POTENTIAL, AND DENSITY 97 
 
 and radiation that they shall change steadily from level to level, 
 under the attraction of the earth's gravitation, and that all the 
 other thermodynamic values mutually adjust themselves to 
 produce this simple result. Hence, the problem depends upon 
 the rate of loss of A (Qi Q ) in the lower and the isothermal 
 regions, which is distinctly a physical question. A reason has 
 been already indicated why the rate of transfer of heat should 
 be greater in one region than in the other, and why the iso- 
 thermal separates from the convectional region. 
 By (339) we have the following equation: 
 
 A<n K . 
 
 (339) K 10 - 
 
 This equation is immediately derived from (259), where 
 
 C\ /^ 
 
 C T = ( J is the latent heat while the temperature is constant; 
 
 it also comes from (309), where the latent heat r 2 of vaporization 
 is derived by primary analysis; or it may be taken from (338), 
 (339); finally it is found by computation that the last form 
 through the gas coefficients R a , RIO is satisfied, as can easily be 
 verified by the data of Tables 14 and 19. The small dis- 
 crepancies are due entirely to the velocity term which was 
 eliminated by means of Table 17. KIQ is negative in sign 
 because (U\ U ) is negative while fyi v ) is positive. It 
 seems, then, that the latent heat, 
 
 Hence, the escape of heat or radiation in the earth's atmosphere 
 depends entirely upon the divergence of the gas coefficient ^10 
 from the adiabatic value R a , as was stated. The divergence of 
 the lines P 10 and K^ in Fig. 5 measures this term. If the 
 velocity is also considered it will be a line near the KIQ line, 
 slightly adjusting it to make the pressure transitions gradual. 
 This is the function of the horizontal cloud motions of flowing 
 
98 THERMODYNAMIC METEOROLOGY 
 
 strata so generally seen in the atmosphere. This confirms the 
 principle of equations (36) to (38), which indicate the relations 
 of pressure, circulation, and radiation to gravitation. 
 
 It is easy to see that such data are capable of making all the 
 general thermodynamic formulas (205) to (328), and many others, 
 applicable to the earth's non-adiabatic atmosphere. It should 
 be carefully noted that the density p, and the gas coefficient R, 
 must be computed by (176), (177), and not by (175) for R a con- 
 stant; that the effective specific heat Cp is variable, and that 
 radiation depends upon this fact. The principal quantities to 
 obtain by observation are the temperature T, and the velocity 
 of circulation q at the height z, and hence the observations for 
 temperature alone, omitting q, are not capable of giving correct 
 radiation data. Finally, the variation of pressure d P is not 
 proportional to the mass gpdz = gdm, but by (201) the terms 
 p q d q + p d Q must be added for circulation and radiation, or 
 else P = K, which is to exclude them from the problem, and 
 reduce it to the unusual adiabatic case. One can now perceive 
 that there is no possibility of solving the general equations of 
 motion in cyclones and anti-cyclones, and in all the other types 
 of circulation, without first eliminating the heat term d Q. Nearly 
 all attempts of meteorologists to solve the circulation problems 
 have been futile chiefly on this account, because of the assumed 
 necessity of ascribing to friction, and to the deflecting force of 
 the earth's rotation on a moving mass, values which they do not 
 actually possess. We shall be able to explain this more fully 
 in the chapter on Dynamic Meteorology, but now proceed to 
 illustrate more at length the thermodynamic terms in other 
 typical conditions of the atmosphere. 
 
 The Diurnal Convection and the Semi-diurnal Waves in the Lower 
 
 Strata 
 
 There is a series of problems relating to the semi-diurnal 
 waves observed at the surface, which have been much discussed 
 without satisfactory results, as the semi-diurnal barometric 
 waves and the several electrical and magnetic waves which are 
 
CONVECTION AND SEMI-DIURNAL WAVES 99 
 
 associated with the diurnal convection. At the surface the 
 temperature has only a single diurnal wave, and it has not been 
 possible to match these two series of data in a definite relation 
 of cause and effect. The difficulty in studying the general 
 problem has been the lack of observations in the free air above 
 the surface, especially during the night. The only exception 
 to this defect is the series of kite ascensions at the Blue Hill 
 Observatory, 1897-1902, which were discussed in my papers, 
 Monthly Weather Review, February to August, 1905. In these it 
 was shown that the single diurnal temperature wave changes 
 into a semi-diurnal wave at about 400 meters above the surface, 
 and that the semi-diurnal waves die away within two or three 
 thousand meters of the ground. It was also indicated that the 
 other data, namely, vapor pressure, atmospheric electric potential, 
 ionization, and magnetic fields, have diurnal variations closely 
 matching the diurnal circulation. This section gives the result 
 of another discussion of the subject, using the data of the Cordoba 
 and Pilar stations, Argentina. 
 
 There are two hypotheses regarding the origin of the semi- 
 diurnal barometric waves: (1) The forced oscillation of the 
 entire atmosphere, as proposed by Lord Kelvin, and developed 
 by Margules, Hann, Jaerisch, Gold, and others; (2) the effect 
 of the diurnal convection, proposed in general terms by Espy, 
 and studied by Ferrel, Koppen, Sprung, Bigelow, and others. 
 The important objections to the former hypothesis are, that 
 these waves do not embrace the entire atmosphere and are 
 usually confined within 2,000 meters of the surface; that the 
 analytical equations and harmonic analyses merely represent, 
 in other forms, the data assumed for their coefficients, and do 
 not reach the origin of the physical causes; and that these 
 equations have not the radiation and heat terms, which are 
 more important than the friction and the deflecting force of 
 rotation. 
 
 The method of treating the Cordoba-Pilar data was to assume 
 a uniform temperature T, and pressure P, on the 3,500-meter 
 level, and then, by studying the observed temperatures and wind 
 velocities on several levels, 000, 200, 400 ... 3,500 meters, at 
 
100 
 
 THERMODYNAMIC METEOROLOGY 
 
 Scale 
 AT 
 
 p.m. 
 
 2a.m. 
 
 6a.m. 
 
 10 a.m. 
 
 2p.m. 
 
 6p.m. 
 
 10 p.m. 
 
 2a.m. 
 
 2500 
 
 2000 
 
 1500 
 
 1000 
 
 400 
 200 
 
 000 
 
 + 3.0 
 
 + 2.0 
 
 + 1.0 
 
 
 
 -1.0 
 -2.0 
 -3.0 
 -40 
 
 280.9 
 
 284.7 
 
 NN. 
 
 291.5 
 293.3 
 295.0 
 296.3 
 296.4 
 295.8 
 
 FIG. 7. A T Temperature of the semi-diurnal waves above 400 meters and 
 
 the diurnal wave near the surface 
 T is the mean temperature on the given level. 
 
 AP 
 
 .10 p.mj 2 a.m. 
 
 6a.m. 
 
 10 a.m. 
 
 2p.m. 
 
 6p.m. 
 
 10 p.m. 
 
 2a.m. 
 
 2500 
 
 2000 
 
 1500 
 
 1000 
 800 
 600 
 400 
 200 
 000 
 
 75463 
 
 Units 
 +160 
 -1-120 
 f 80 
 + 40 
 
 
 40 
 -80 
 -120 
 -160 
 
 \X 
 
 85087 
 
 90253 
 92387 
 94555 
 96771 
 99027 
 101338 
 
 FIG. 8. A P Pressure of the semi-diurnal waves in all these strata vanish- 
 ing at about the level 2500 meters 
 
CONVECTION AND SEMI-DYURNAi; ^WAVES' J 
 
 101 
 
 the hours (2, 6, 10) A.M., (2, 6, 10) P.M., proceed by computations 
 entirely similar to those of Tables 14, 17, 19, to derive a 
 pressure P at the surface which would exactly match the mean 
 
 AB 
 
 10p.m. 2a.m. 6a.m. 10a.m. 2p.m. 6p.m. 10p.m. 2a.m. 
 
 + 1.00 
 0.00 
 -1.00 
 
 
 
 
 _~n 
 
 
 
 
 
 
 ^^ 
 
 ^ 
 
 
 \ 
 
 
 fr 
 
 '^ 
 
 
 
 
 
 
 \ 
 
 \ 
 
 / 
 
 
 
 
 
 wn T. 
 
 ne=Otse 
 Line = Cc 
 
 rved Data 
 mputed D 
 
 ^VJ 
 
 ita 
 
 / 
 
 
 
 Dotted 
 
 FIG. 9. Computed and observed pressure waves at the surface 
 
 10 p.m. 
 
 2 a.m. 
 
 6 a.m. 
 
 10 a.m. 
 
 2 p.m. 
 
 6 p.m. 
 
 10 p.m. 
 
 2a.m. AQ 
 
 -70 
 
 -70 
 
 2500 
 
 2000 
 
 1500 
 
 -50 
 
 1000 
 
 800 
 
 -50 
 
 -40 
 
 600 
 
 400 
 200 
 000 
 
 -30 
 
 -20 
 
 -10 
 
 
 
 + 10 
 + 20 
 
 y/ 
 
 0+20 
 
 FIG. 10. The loss of heat for every 200 meters A Q = (Q l Q ) 
 
 pressure as observed at Cordoba. It required several trial 
 computations to accomplish this result, especially during the 
 night hours where direct observations were lacking above the 
 
102 TIIERVOD ^NAMIC METEOROLOGY 
 
 surface, and the outcome is indicated in Tables 24 and 25, 
 and in Figs. 7, 8, 9, 10. All the other data, as computed, will 
 be found in Bulletin No. 3, O. M. A. Transferring the data of 
 Table 24 to the diagrams, it is seen that the single diurnal 
 temperature wave at the surface transforms into semi-diurnal 
 waves at 400 meters, and that these die away at 2,500 meters. 
 This result confirms the discussion of the Blue Hill data in all 
 respects. The corresponding pressure waves are semi-diurnal 
 throughout these strata from the surface to 2,500 meters, and 
 there they vanish. The night wave is weaker than the day wave 
 in consequence of the temperature inversion near the surface, 
 both waves being nearly equal at 1,000 meters. 
 
 Since the barometric variations are alone of interest, the 
 base line has been taken as that near sea level. The discrepancy 
 at 2 P.M. is due to an imperfect temperature distribution at that 
 hour, which should be made cooler by a few tenths of a degree. 
 
 By adjusting the vertical temperatures a little it would be 
 possible to reproduce the observed curve with precision through 
 the non-adiabatic computations. There are no other tempera- 
 tures which could reproduce this pressure wave at the surface, 
 and this fact is proof of the cause of the observed pressure system. 
 In order to understand more fully the origin of the temperature 
 system, the values of A (Qi Q Q ), the variations on the daily 
 mean are plotted on Fig. 10 and the curves of equal heat losses 
 are drawn. The data are somewhat imperfect, but the general 
 result is not doubtful. It shows that there are two principal axes 
 of heat exchange, that in the afternoon, 2 P.M., at the surface, to 
 8 P.M., at 2,500 meters, and that in the night from 1 A.M at 2,500, 
 to 4 A.M., at the surface. That is to say, the air rises obliquely in 
 the afternoon to the right and falls in the night, also to the right. 
 The air rises and falls in such a zigzag path as gives a turning- 
 point at about 10 P.M. above and 10 A.M. below, judging by the 
 crests. The rising air cools by expansion, and the falling air 
 heats by compression, the former producing the afternoon 
 wave and the latter the night wave to within 400 meters of the 
 surface on Fig. 7. At this level the more rapid cooling of the 
 ground during the night makes itself felt, and there is radiation 
 
CONVECTION AND SEMI-DIURNAL WAVES 103 
 
 TABLE 24 
 
 THE SEMI-DIURNAL TEMPERATURE WAVES IN THE STRATA 400 TO 2,500 
 METERS, TOGETHER WITH THE SINGLE DIURNAL TEMPERATURE WAVE 
 AT THE SURFACE 
 
 z 
 
 r 
 
 P 
 
 2 
 
 -02H2o) 
 
 T 
 
 P 
 
 2 
 
 -02H2o) 
 
 
 
 2 A. 
 
 VI. 
 
 
 
 6 A 
 
 .M. 
 
 
 2500 
 
 281. 
 
 75464 
 
 3.0 
 
 + 151.8 
 
 280. 8 
 
 75466 
 
 6.0 
 
 + 140.2 
 
 2000 
 
 285. 
 
 80153 
 
 4.0 
 
 +131.9 
 
 284 .2 
 
 80162 
 
 7.0 
 
 +120.1 
 
 1500 
 
 288 .6 
 
 85080 
 
 5.0 
 
 + 111.9 
 
 287 .5 
 
 85104 
 
 7.0 
 
 + 97.6 
 
 1000 
 
 291 .9 
 
 90230 
 
 5.0 
 
 + 38.5 
 
 290 .6 
 
 90288 
 
 6.0 
 
 + 34.9 
 
 800 
 
 293 .6 
 
 92368 
 
 5.0 
 
 + 40.2 
 
 292 .3 
 
 92428 
 
 5.0 
 
 + 33.1 
 
 600 
 
 295 .0 
 
 94530 
 
 4.0 
 
 + 30.0 
 
 294 .3 
 
 94604 
 
 4.0 
 
 + 25.8 
 
 400 
 
 295 .8 
 
 96743 
 
 4.0 
 
 + 16.1 
 
 295 .4 
 
 96820 
 
 4.0 
 
 + 13.2 
 
 200 
 
 293 .7 
 
 99018 
 
 3.0 
 
 - 11.7 
 
 293 .4 
 
 99095 
 
 3.0 
 
 - 15.8 
 
 000 
 
 231 .7 
 
 101349 
 
 2.0 
 
 
 291 .0 
 
 101440 
 
 2.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 A. 
 
 M. 
 
 
 
 2p 
 
 M. 
 
 
 2500 
 
 280. 7 
 
 75462 
 
 9.4 
 
 + 144.2 
 
 280. 9 
 
 75462 
 
 9.7 
 
 + 146.7 
 
 2000 
 
 284 .0 
 
 80173 
 
 9.4 
 
 + 114.2 
 
 284 .8 
 
 80162 
 
 10.3 
 
 +133.0 
 
 1500 
 
 287 .0 
 
 85112 
 
 9.4 
 
 + 87.4 
 
 288 .8 
 
 85084 
 
 10.6 
 
 + 133.0 
 
 1000 
 
 289 .5 
 
 90313 
 
 8.3 
 
 + 23.0 
 
 292 .6 
 
 90228 
 
 9.2 
 
 + 49.2 
 
 800 
 
 291 .0 
 
 92460 
 
 8.1 
 
 + 19.7 
 
 294 .8 
 
 92355 
 
 8.7 
 
 + 49.2 
 
 600 
 
 292 .5 
 
 94656 
 
 7.9 
 
 + 18.9 
 
 296 .9 
 
 94510 
 
 8.3 
 
 + 49.9 
 
 400 
 
 294 .5 
 
 96886 
 
 7.6 
 
 + 16.2 
 
 299 .0 
 
 96703 
 
 7.9 
 
 + 52.2 
 
 200 
 
 296 .2 
 
 99154 
 
 6.0 
 
 + 23.4 
 
 300 .5 
 
 98930 
 
 7.0 
 
 + 52.2 
 
 000 
 
 297 .7 
 
 101459 
 
 4.0 
 
 
 301 .3 
 
 101214 
 
 5.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6 P.I 
 
 I. 
 
 
 
 10 F 
 
 .M. 
 
 
 2500 
 
 281. 5 
 
 75458 
 
 8.0 
 
 +159.3 
 
 280. 8 
 
 75466 
 
 5.0 
 
 + 142.0 
 
 2000 
 
 286 .0 
 
 80142 
 
 9.0 
 
 + 156.5 
 
 284 .3 
 
 80170 
 
 6.0 
 
 +117.6 
 
 1500 
 
 290 .0 
 
 85038 
 
 9.0 
 
 + 145.6 
 
 287 .1 
 
 85102 
 
 6.0 
 
 + 88.5 
 
 1000 
 
 293 .5 
 
 90166 
 
 8.0 
 
 + 51.9 
 
 290 .7 
 
 90294 
 
 6.0 
 
 + 34.9 
 
 800 
 
 296 .0 
 
 92278 
 
 8.0 
 
 + 58.9 
 
 292 .2 
 
 92432 
 
 5.0 
 
 + 28.0 
 
 600 
 
 297 .3 
 
 94426 
 
 7.0 
 
 + 51.6 
 
 294 .2 
 
 94606 
 
 5.0 
 
 + 29.0 
 
 400 
 
 298 .0 
 
 96636 
 
 6.0 
 
 + 42.1 
 
 295 .1 
 
 96835 
 
 4.0 
 
 + 19.9 
 
 200 
 
 298 .6 
 
 98867 
 
 5.0 
 
 + 31.6 
 
 295 .7 
 
 99095 
 
 3.0 
 
 + 3.2 
 
 000 
 
 299 .0 
 
 101149 
 
 4.0 
 
 
 294 .3 
 
 101414 
 
 2.0 
 
 
 
 
 
 
 
 
 
 
 
 from the descending air to the ground, and an inversion of tem- 
 perature under ordinary circumstances. The friction and the 
 earth's deflection have very little influence on the temperature 
 and pressure conditions, and the circulation cannot be studied 
 by itself until the radiation or heating terms have been eliminated. 
 
104 
 
 THERMODYNAMIC METEOROLOGY 
 
 TABLE 25 
 
 THE CORRESPONDING VALUES OF B c =P/g Pm ON THE LEVEL 2 = 000 
 METERS ARE Now GIVEN, AND COMPARED WITH THE OBSERVED VALUES 
 OF # BY MEANS OF THE DIFFERENCES 
 
 Surface 
 
 2A.M. 
 
 6A.M. 
 
 10 A.M. 
 
 2 P.M. 
 
 6P.M. 
 
 10 P.M. 
 
 Mean 
 
 B c 
 Bo 
 A BC 
 
 760.20 
 760.16 
 +0 08 
 
 760.88 
 760.75 
 +0 76 
 
 761.03 
 760.98 
 +0 91 
 
 759.18 
 759.52 
 -0 94 
 
 758.70 
 758.71 
 1 42 
 
 760.70 
 760.59 
 
 +0 58 
 
 760.12 
 760.12 
 
 A BO 
 
 +0 04 
 
 +0 63 
 
 +0 86 
 
 -0 60 
 
 1 41 
 
 +0 47 
 
 
 
 
 
 
 
 
 
 
 These curves can be reduced to the harmonics if desired. The 
 value of the radiation exponent is a = 3.82 throughout the 
 twenty-four hours. 
 
 The Thermodynamic Structure of Cyclones and Anticyclones 
 
 There has been much speculation regarding the forces that 
 generate the powerful circulations in storms known by the name 
 of cyclones and anticyclones, or low-pressure and high-pressure 
 areas respectively. These will be more fully mentioned in the 
 chapter on Dynamic Meteorology, but here we proceed to apply 
 the principles just illustrated in the diurnal convection. From 
 numerous kite and balloon ascensions in all parts of these local 
 circulations, it has been learned what is the usual distribution 
 of the temperature, and from the cloud observations what is the 
 direction and velocity of the wind motion or " vector" in all 
 areas, and all altitudes up to at least 10,000 meters. Compare 
 the International Cloud Report, 1898, the Monthly Weather 
 Review, January to July, 1902, April to June, 1904, January to 
 August, 1906, October, 1907, to February, 1909, also the daily 
 Weather Synoptic Charts, for numerous studies and details. 
 From these data we have selected the temperatures T, and veloc- 
 ities q, given in Tables 26, 25, and Figs. 11, 12. The computed 
 values of the pressure P, and the free heat (Qi - Qo) are given 
 in the same tables and figures, while the other thermodynamic 
 data are summarized in Bulletin No. 3, 0. M. A. Only three 
 diagrams are extracted from the tables, but these are enough to 
 
6000 meterp High Area 
 
 Low Area 
 
 266 
 
 2000 meters 
 
 260 
 
 284 288 292 
 
 Full lines = Temperatures T 
 Dotted lines = Variations of Heat (Q, Q ) 
 FIG. ii. The temperatures and heat variations in high and low areas 
 
High Area 
 
 5000 meter? 
 
 FIG. 12. Pressures and wind velocities in high and low areas 
 
THERMODYNAMIC STRUCTURE OF CYCLONES 
 
 TABLE 26 
 SOME VALUES OF T. P. q(Q l - Q ) IN CYCLONES 
 
 TEMPERATURE T 
 
 107 
 
 
 000 
 
 500 
 
 1000 
 
 1500 
 
 2000 
 
 2500 
 
 3000 
 
 4000 
 
 5000 
 
 5. 760 
 S. 750 
 5. 740 
 C. 735 
 
 290.0 
 
 287.5 
 284.0 
 280.0 
 
 289.0 
 286.4 
 282.6 
 279.0 
 
 288.0 
 284.5 
 280.8 
 277.5 
 
 286.3 
 282.7 
 279.0 
 275.5 
 
 284.3 
 280.4 
 276.8 
 273.2 
 
 282.1 
 278.0 
 274.4 
 271.0 
 
 279.8 
 276.0 
 271.6 
 269.0 
 
 274.5 
 270.5 
 266.0 
 263.0 
 
 268.0 
 264.0 
 260.0 
 257.0 
 
 E. 760 
 E. 750 
 E. 740 
 
 278.0 
 281.0 
 283.5 
 
 276.6 
 279.3 
 281.4 
 
 274.8 
 277.0 
 279.1 
 
 272.4 
 274.6 
 276.6 
 
 269.7 
 272.0 
 274.0 
 
 267.0 
 269.5 
 271.0 
 
 264.5 
 267.0 
 268.0 
 
 259.5 
 261.7 
 263.0 
 
 255.0 
 256.0 
 257.0 
 
 N. 750 I 
 N. 750 II 
 
 N. 740 
 
 266.0 
 272.5 
 277.0 
 
 265.8 
 271.2 
 276.0 
 
 264.5 
 269.7 
 274.8 
 
 263.0 
 268.0 
 273.0 
 
 261.5 
 266.0 
 270.8 
 
 259.5 
 263.9 
 268.5 
 
 257.0 
 261.7 
 266.0 
 
 250.0 
 256.0 
 260.5 
 
 243.0 
 249.0 
 254.0 
 
 W. 760 
 W. 750 
 W. 740 
 
 268.0 
 273.0 
 277.0 
 
 267.5 
 272.6 
 276.0 
 
 266.8 
 272.0 
 275.0 
 
 266.0 
 271.0 
 273.7 
 
 265.0 
 269.8 
 272.0 
 
 264.0 
 268.0 
 270.0 
 
 262.0 
 266.0 
 268.0 
 
 258.0 
 261.0 
 263.0 
 
 253.0 
 255.0 
 256.5 
 
 PRESSURE P 
 
 S. 760 
 
 101322 
 
 95518 
 
 90010 
 
 84814 
 
 79893 
 
 75208 
 
 70778 
 
 62566 
 
 55161 
 
 5. 750 
 
 99988 
 
 94194 
 
 88724 
 
 83548 
 
 78628 
 
 73953 
 
 69540 
 
 61367 
 
 54000 
 
 S. 740 
 
 98655 
 
 92873 
 
 87418 
 
 82230 
 
 77340 
 
 72683 
 
 68278 
 
 60127 
 
 52801 
 
 C. 735 
 
 97990 
 
 92158 
 
 86672 
 
 81482 
 
 76565 
 
 71908 
 
 67498 
 
 59366 
 
 52051 
 
 E. 760 
 
 101322 
 
 95275 
 
 89562 
 
 84140 
 
 78997 
 
 74124 
 
 69515 
 
 61016 
 
 53424 
 
 E. 750 
 
 99988 
 
 94064 
 
 88462 
 
 83150 
 
 78114 
 
 73338 
 
 68812 
 
 60467 
 
 52993 
 
 E. 740 
 
 98655 
 
 92864 
 
 87380 
 
 82166 
 
 77223 
 
 72532 
 
 68072 
 
 59861 
 
 52486 
 
 N. 750 I 
 
 99988 
 
 93706 
 
 87884 
 
 82372 
 
 77165 
 
 72273 
 
 67641 
 
 59114 
 
 51465 
 
 2V. 750 II 
 
 99988 
 
 93890 
 
 88142 
 
 82708 
 
 77587 
 
 72747 
 
 68163 
 
 59737 
 
 52179 
 
 N. 740 
 
 98655 
 
 92747 
 
 87176 
 
 81896 
 
 76915 
 
 72187 
 
 67720 
 
 59481 
 
 52080 
 
 W. 760 
 
 101322 
 
 95067 
 
 89220 
 
 83654 
 
 78460 
 
 73532 
 
 68911 
 
 60404 
 
 52860 
 
 W. 750 
 
 99988 93983 
 
 88252 
 
 82868 
 
 77790 
 
 72997 
 
 68477 
 
 60150 
 
 52690 
 
 W. 740 
 
 98655 
 
 92745 
 
 87156 
 
 81890 
 
 76930 
 
 72223 
 
 67779 
 
 59603 
 
 52256 
 
 VELOCITY q 
 
 
 
 
 
 
 
 
 
 
 
 S. 760 
 
 4.1 
 
 6.5 
 
 9.0 
 
 13.9 
 
 21.0 
 
 26.5 
 
 27.0 
 
 24.0 
 
 23.0 
 
 5. 750 
 
 4.9 
 
 9.0 
 
 12.0 
 
 15.4 
 
 21.2 
 
 27.0 
 
 26.0 
 
 24.0 
 
 26.0 
 
 S. 740 
 
 6.5 
 
 11.0 
 
 14.0 
 
 16.9 
 
 22.8 
 
 30.0 
 
 34.0 
 
 34.0 
 
 33.0 
 
 C. 735 
 
 7.0 
 
 11.0 
 
 15.0 
 
 19.0 
 
 30.0 
 
 40.0 
 
 42.0 
 
 40.0 
 
 39.0 
 
 E. 760 
 
 5.2 
 
 8.6 
 
 12.4 
 
 16.0 
 
 19.8 
 
 23.0 
 
 24.0 
 
 25.0 
 
 26.0 
 
 E. 750 
 
 5.0 
 
 9.0 
 
 13.0 
 
 17.0 
 
 22.0 
 
 26.0 
 
 27.0 
 
 27.0 
 
 27.0 
 
 E. 740 
 
 5.0 
 
 7.5 
 
 11.0 
 
 15.0 
 
 22.0 
 
 29.0 
 
 29.0 
 
 29.0 
 
 28.0 
 
 TV. 750 I 
 
 3.5 
 
 5.0 
 
 5.8 
 
 6.0 
 
 7.0 
 
 7.5 
 
 8.0 
 
 8.0 
 
 7.5 
 
 N. 750 II 
 
 4.5 
 
 6.0 
 
 8.0 
 
 11.0 
 
 14.5 
 
 18.0 
 
 20.5 
 
 24.5 
 
 25.0 
 
 TV. 740 
 
 6.0 
 
 8.0 
 
 11.0 
 
 15.0 
 
 18.0 
 
 22.0 
 
 26.0 
 
 31.0 
 
 33.0 
 
 W. 760 
 
 5.0 
 
 8.0 
 
 12.0 
 
 17.0 
 
 22.0 
 
 26.0 
 
 27.0 
 
 28.0 
 
 31.0 
 
 W. 750 
 
 6.0 
 
 9.0 
 
 13.0 19.0 
 
 24.0 
 
 27.0 
 
 28.0 
 
 30.0 
 
 33.0 
 
 W. 740 
 
 6.0 
 
 10.0 
 
 14.0 20.0 
 
 25.0 
 
 29.0 
 
 32.0 
 
 34.0 
 
 37.0 
 
 1 
 
 
 
 
 
 
 
 
 
108 
 
 THERMODYNAMIC METEOROLOGY 
 
 FREE HEAT (Qi - Qo) 
 
 5. 760 
 
 
 -45.9 
 
 -118.4 
 
 -209.7 
 
 -333.0 
 
 -386.3 
 
 -313.4 
 
 -789.6 
 
 -872.3 
 
 5. 750 
 
 
 -61.2 
 
 -124.5 
 
 -189.3 
 
 -297.2 
 
 -374.2 
 
 -253.8 
 
 -631.9 
 
 -861.6 
 
 5. 740 
 
 
 -69.2 
 
 -125.2 
 
 -186.0 
 
 -316.2 
 
 -425.8 
 
 -403.3 
 
 -659.0 
 
 -764.7 
 
 C. 735 
 
 ... 
 
 -70.5 
 
 -150.6 
 
 -221.4 
 
 -465.6 
 
 -597.1 
 
 -377.6 
 
 -625.6 
 
 -810.9 
 
 E. 760 
 
 
 -54.5 
 
 -129.3 
 
 -189.8 
 
 -248.3 
 
 -282.7 
 
 -283.5 
 
 -671.9 
 
 -851.9 
 
 E. 750 
 
 
 -55.7 
 
 -122.8 
 
 -184.1 
 
 -263.8 
 
 -303.7 
 
 -276.4 
 
 -621.4 
 
 -773.4 
 
 E. 740 
 
 
 -55.0 
 
 -104.1 
 
 -167.4 
 
 -286.0 
 
 -377.2 
 
 -232.2 
 
 -571.8 
 
 -695.6 
 
 N. 750 I 
 
 
 -50.1 
 
 -124.3 
 
 -184.7 
 
 -251.3 
 
 -304.7 
 
 -352.3 
 
 -791.3 
 
 -890.2 
 
 N. 750 II 
 
 
 -40.6 
 
 -109.8 
 
 -183.5 
 
 -254.0 
 
 -316.8 
 
 -356.8 
 
 -828.4 
 
 -874.3 
 
 2V. 740 
 
 
 -48.6 
 
 -130.3 
 
 -213.5 
 
 -261.7 
 
 -338.5 
 
 -398.1 
 
 -864.0 
 
 -919 6 
 
 W. 760 
 
 
 -59.7 
 
 -158.1 
 
 -260.1 
 
 -360.9 
 
 -427.8 
 
 -417.6 
 
 -960.4 
 
 -1-204 . 
 
 W. 750 
 
 
 -62.5 
 
 -167.1 
 
 -286.6 
 
 -364.4 
 
 -392.4 
 
 -394.8 
 
 -925.4 
 
 -1110.3 
 
 JF. 740 
 
 
 -66.7 
 
 -151.7 
 
 -271.5 
 
 -341.0 
 
 -389.2 
 
 -417.7 
 
 -860.9 
 
 -1041.4 
 
 TABLE 27 
 SOME VALUES OF T. P. q (Q l <2 ) IN ANTICYCLONES 
 
 TEMPERATURE T 
 
 A 
 
 C B 
 
 000 
 
 500 
 
 1000 
 
 1500 
 
 2000 
 
 2500 
 
 3000 
 
 4000 
 
 5000 
 
 14 
 15 
 16 
 17 
 
 S. 770 I 
 S. 770 II 
 S. 775 
 C. 777 
 
 278.0 
 273.5 
 272.0 
 272.0 
 
 277.4 
 273.0 
 271.0 
 271.0 
 
 276.0 
 272.0 
 269.5 
 269.2 
 
 274.6 
 270.4 
 267.8 
 267.0 
 
 273.0 
 268.7 
 266.0 
 264.5 
 
 271.0 
 267.0 
 264.0 
 262.0 
 
 268.6 
 265.0 
 262.0 
 259.5 
 
 264.0 
 260.0 
 256.5 
 254 . 5 
 
 259.0 
 254.0 
 250.0 
 248.0 
 
 11 = 18 
 19 
 20 
 
 E. 760 
 E. 770 
 
 E. 775 
 
 268.0 
 264.0 
 266.0 
 
 267.5 
 263.3 
 265.0 
 
 266.8 
 262.7 
 264.0 
 
 266.0 
 262.0 
 262.9 
 
 265.0 
 261.2 
 261.5 
 
 264.0 
 259.7 
 259.8 
 
 262.0 
 258.0 
 258.0 
 
 258.0 
 254.0 
 254.0 
 
 253.0 
 249.5 
 249.0 
 
 21 
 22 
 23 
 
 N. 760 
 N. 770 
 
 N. 775 
 
 271.0 
 272.0 
 272.0 
 
 269.8 
 271.4 
 271.0 
 
 268.2 
 270.3 
 269.5 
 
 266.3 
 269.1 
 267.6 
 
 264.2 
 267.6 
 265.5 
 
 262.0 
 265.4 
 263.1 
 
 259.4 
 262.6 
 260.4 
 
 254.0 
 257.0 
 255.0 
 
 248.0 
 251.0 
 249.5 
 
 24 
 25 
 26 
 
 W. 760 
 W. 770 
 
 W. 775 
 
 282.0 
 278.0 
 274.0 
 
 281.2 
 277.1 
 273.3 
 
 280.3 
 276.0 
 271.8 
 
 279.1 
 
 274.7 
 270.0 
 
 277.7 
 273.0 
 268.0 
 
 276.0 
 271.0 
 265.9 
 
 273.8 
 268.5 
 263.7 
 
 268.0 
 263.0 
 258.0 
 
 262.0 
 257.0 
 252.0 
 
 PRESSURE P 
 
 14 
 
 5. 770 I 
 
 102655 
 
 96583 
 
 90794 
 
 85330 
 
 80146 
 
 75292 
 
 70677 
 
 62160 
 
 54551 
 
 15 
 
 5. 770 II 
 
 102655 
 
 96594 
 
 90550 
 
 85028 
 
 79805 74865 
 
 70220 
 
 61644 
 
 53970 
 
 16 
 
 S. 775 
 
 103322 
 
 97020 
 
 91080 
 
 85466 
 
 80164 i 75157 
 
 70435 
 
 61741 
 
 53949 
 
 17 
 
 C. 777 
 
 103588 
 
 97270 
 
 91316 
 
 85670 
 
 80343 
 
 75289 
 
 70520 
 
 61740 
 
 53888 
 
 18 
 
 E. 760 
 
 101322 
 
 95067 
 
 89220 
 
 83654 
 
 78460 
 
 73532 
 
 68911 
 
 60404 
 
 52860 
 
 19 
 
 E. 770 
 
 102655 
 
 96232 
 
 90184 
 
 84492 
 
 79144 
 
 74122 
 
 69392 
 
 60711 
 
 53016 
 
 20 
 
 E. 775 
 
 103322 
 
 96906 
 
 90818 
 
 85126 
 
 79753 
 
 74698 
 
 69928 
 
 61187 
 
 53419 
 
 21 
 
 N. 760 
 
 101322 
 
 95110 
 
 89266 
 
 83736 
 
 78512 
 
 73577 
 
 68913 
 
 60321 
 
 52646 
 
 22 
 
 N. 770 
 
 102655 
 
 96394 
 
 90676 
 
 85088 
 
 79857 
 
 74893 
 
 70197 
 
 61549 
 
 53803 
 
 23 
 
 N. 775 
 
 103322 
 
 97020 
 
 91080 
 
 85472 
 
 80153 75142 
 
 70395 
 
 61656 
 
 53845 
 
 24 
 
 W. 760 
 
 101322 
 
 95376 
 
 89716 
 
 84410 
 
 79394 
 
 74638 
 
 70136 
 
 61824 
 
 54349 
 
 25 
 
 W. 770 
 
 102655 
 
 96563 
 
 90772 
 
 85312 
 
 80146 ! 75273 
 
 70648 
 
 62111 
 
 54476 
 
 26 
 
 W. 775 
 
 103322 
 
 97073 
 
 91176 
 
 85596 
 
 80327 
 
 75357 
 
 70647 
 
 61974 
 
 54203 
 
THERMODYNAMIC STRUCTURE OF CYCLONES 
 VELOCITY q 
 
 109 
 
 14 
 
 5. 770 I 
 
 5.0 
 
 7.0 
 
 10.0 
 
 12.0 
 
 16.0 
 
 19.0 
 
 20.0 
 
 21.0 
 
 22.0 
 
 15 
 
 5. 770 II 
 
 5.0 
 
 7.0 
 
 8.0 
 
 9.0 
 
 12.0 
 
 15.0 
 
 16.0 
 
 17.0 
 
 18.0 
 
 16 
 
 5. 775 
 
 3.0 
 
 5.0 
 
 7.0 
 
 8.0 
 
 9.0 
 
 11.0 
 
 12.0 
 
 14.0 
 
 16.0 
 
 17 
 
 C. 777 
 
 4.0 
 
 5.0 
 
 6.0 
 
 8.0 
 
 9.0 
 
 10.0 
 
 11.0 
 
 12.0 
 
 14.0 
 
 18 
 
 E. 760 
 
 5.0 
 
 8.0 
 
 12.0 
 
 17.0 
 
 22.0 
 
 26.0 
 
 27.0 
 
 28.0 
 
 31.0 
 
 19 
 
 E. 770 
 
 5.0 
 
 7.0 
 
 10.0 
 
 14.0 
 
 19.0 
 
 23.0 
 
 25.0 
 
 26.0 
 
 28.0 
 
 20 
 
 E. 775 
 
 4.0 
 
 6.0 
 
 9.0 
 
 12.0 
 
 16.0 
 
 19.0 
 
 20.0 
 
 22.0 
 
 24.0 
 
 21 
 
 N. 760 
 
 5.0 
 
 8.0 
 
 11.0 
 
 14.0 
 
 18.0 
 
 22.0 
 
 24.0 
 
 26.0 
 
 27.0 
 
 22 
 
 2V. 770 
 
 6.0 
 
 9.0 
 
 13.0 
 
 16.0 
 
 20 
 
 23.0 
 
 26.0 
 
 28.0 
 
 30.0 
 
 23 
 
 N. 775 
 
 4.0 
 
 7.0 
 
 10.0 
 
 13.0 
 
 17.0 
 
 21.0 
 
 23.0 
 
 24.0 
 
 26.0 
 
 24 
 
 W r . 760 
 
 5.0 
 
 8.0 
 
 10.0 
 
 13.0 
 
 14.0 
 
 15.0 
 
 16.0 
 
 17.0 
 
 18.0 
 
 25 
 
 PF. 770 
 
 5.0 
 
 7.0 
 
 9.0 
 
 11.0 
 
 12.0 
 
 13.0 
 
 14.0 
 
 15.0 
 
 16.0 
 
 26 
 
 W. 775 
 
 4.0 
 
 6.0 
 
 8.0 
 
 10.0 
 
 12.0 
 
 12.5 
 
 13.0 
 
 14.0 
 
 15.0 
 
 FREE HEAT (Qi - Qo) 
 
 14 
 
 5. 770 I 
 
 
 49 7 
 
 131 4 
 
 -189 
 
 282 
 
 -337 1 
 
 350 6 
 
 804 4 
 
 977 1 
 
 15 
 
 5. 770 II 
 
 
 51 8 
 
 121 5 
 
 -186 1 
 
 266 1 
 
 331 1 
 
 359 
 
 837 1 
 
 988 
 
 16 
 
 5. 775 
 
 
 -43.3 
 
 -113.2 
 
 -167.7 
 
 -224.8 
 
 -289.5 
 
 -332.2 
 
 -794.6 
 
 -934.0 
 
 17 
 
 C. 777 
 
 
 -39.8 
 
 -104.0 
 
 -164.6 
 
 -204.8 
 
 -248.9 
 
 -293.1 
 
 -706.1 
 
 -868.1 
 
 18 
 19 
 
 E. 760 
 E. 770 
 
 .**.* 
 
 -59.7 
 51 
 
 -158.1 
 142 8 
 
 -260.1 
 
 -242 8 
 
 -360.9 
 352 
 
 -427.8 
 420 2 
 
 -417.6 
 442 4 
 
 -960.4 
 972 2 
 
 -1204.5 
 -1194 7 
 
 20 
 
 E. 775 
 
 
 -45.9 
 
 -130.2 
 
 -209.3 
 
 -299.6 
 
 -356.1 
 
 -379.1 
 
 -918.7 
 
 -1109.7 
 
 21 
 22 
 23 
 
 TV. 760 
 TV. 770 
 N. 775 
 
 
 -53.2 
 -61.3 
 51 8 
 
 -125.7 
 -154.6 
 126.7 
 
 -191.5 
 -219.9 
 192 9 
 
 -270.1 
 -311.0 
 270 2 
 
 -335.3 
 -356.9 
 333 5 
 
 -346.0 
 -408.2 
 343 4 
 
 -770.4 
 -834.3 
 740 4 
 
 -895.7 
 -980.2 
 923 1 
 
 24 
 
 W. 760 
 
 
 55 2 
 
 123 9 
 
 206 7 
 
 246 8 
 
 304 4 
 
 355 1 
 
 -808 8 
 
 -942 4 
 
 25 
 
 W. 770 
 
 
 -47.2 
 
 -119.8 
 
 -188.3 
 
 -238.6 
 
 -292.2 
 
 -338.8 
 
 -780.6 
 
 -921.9 
 
 26 
 
 W. 775 
 
 
 -47.7 
 
 -119.5 
 
 181.1 
 
 -238.1 
 
 -272.5 
 
 -321.0 
 
 -762.3 
 
 -903.0 
 
 
 
 
 
 
 
 
 
 
 
 
 indicate clearly the principles that are involved in their structure. 
 Continuing the computations to higher levels, it is found that the 
 temperature lines or isotherms, the pressure lines or isobars, and 
 the velocity lines or vectors, coincide at every point in direction. 
 In the lower levels these lines cross each other at various angles 
 in the areas marked 1 to 26 on the sea level of Fig. 12, which 
 shows the order of the computations: (1) The upper undisturbed 
 circulation, where T. P. q coincide, belongs to the general 
 circulation, and (2) the lower disturbed circulation, where 
 T. P. q do not coincide, to the combined general and local circula- 
 tion; (3) the purely local circulation, the cyclone and anti- 
 cyclone proper, can be separated from the second by vector 
 composition, since (2) is the resultant of (l) and (3). It is 
 found that the disturbing circulation (3) is similar in configuration 
 
110 THERMODYNAMIC METEOROLOGY 
 
 to that at sea level up to 10,000 meters, if it penetrates the general 
 eastward drift so high, that it usually increases in intensity to 
 the 3,000-meter level, and then gradually dies out as the head is 
 stripped away in the rapidly flowing upper currents. There 
 is no evidence of a change in the type of the circulation, and 
 therefore the physical origin of the structure of a cyclone is the 
 same throughout. In the International Cloud Report it was 
 shown that a series of warm currents from the south interlock 
 alternately with another series of cold currents from the north, 
 in the United States and adjacent regions, and that these local 
 circulations are the mixing regions where the interchange of 
 the temperature goes on toward a thermodynamic equilibrium 
 under the force of gravitation. Fig. 11 shows the temperature 
 distribution T and the free-heat distribution (Qi Qo). The 
 temperature is deflected to the south on the east side of the 
 high area, and to the north on the east side of the low area. The 
 deflections of the isotherms diminish with the altitude and 
 finally disappear as these disturbing currents diminish in strength. 
 There are no cold-center anticyclones, and no warm-center 
 cyclones, as has been assumed in many theoretical discussions. 
 The distribution of (Qi Q ) is in elliptical figures whose centers 
 are on the border of the high and low areas, and they show where 
 the exchange of heat is going on most vigorously. The radiation 
 heat increases with the height in consequence of the general 
 radiation of the atmosphere increasing upward. It would be 
 well to separate the purely local (Qi Q ) from the general as 
 can be done by computation. The pressures of Fig. 12 depend 
 entirely upon the temperature assigned to the several areas, and 
 not upon the circulation, the deflecting force, the centrifugal 
 force, or the friction, or any other minor condition. The air 
 column, though temporary in position at a given instant, presses 
 upon the level of computation, in consequence of the air masses 
 which are determined only by the density, since this in turn is 
 a function of the temperature. The given temperature structure 
 must be continuously renewed by circulation of air from the warm 
 and cold regions, or else the gravitation would soon flatten 
 down all the disturbed temperature and pressure levels. The 
 
WARM, COLD, AND LEAKAGE CURRENTS 111 
 
 ultimate source of heat is the sun's insolation, chiefly on the 
 tropics, and radiation in general from the atmosphere. The 
 tropics are the boiler and the polar regions the condenser of 
 the thermal engine, and the cyclones and anticyclones are the 
 working machinery of motion. The general circulation depends 
 upon the heat of the tropics, with westward drift to the south 
 of 33 latitude, and eastward drift in middle latitudes from 33 
 to 66 latitude, that in the polar zone being irregular. There 
 are, however, centers of general action along the high-pressure 
 belt separating the westward from the eastward drifts, such 
 that there are leakage currents from one of these zones to the 
 other. There are such centers of action over the tropic north 
 and south Atlantic and Pacific Oceans, those in the same hemi- 
 sphere being broken through by the western and eastern conti- 
 nents. Such leakage currents flow northward from the Gulf of 
 Mexico over the United States, and from the north Pacific upon 
 the northwestern States in a southward direction. The inter- 
 flowing of these two series of warm and cold currents upon the 
 United States and Canada is the immediate cause of the num- 
 erous cyclones and anticyclones that wander eastward over 
 this region. Forecasts are made of the probable detailed action 
 of the weather conditions in all areas, as learned by experience 
 with the types that these local circulations assume. The 
 operation of the several elements, temperature, pressure, wind, 
 and precipitation, is very complex and irregular, so that practical 
 forecasts are difficult and uncertain, except in the cases of 
 vigorously developed storms, which move along paths quite 
 well determined by the pressure and temperature distributions. 
 It should be noted in Fig. 12 that there is a saddle of higher 
 pressure to the north of the center in low areas, and to the south 
 of high areas there is a saddle of low pressure. These gradually 
 diminish with the height and usually disappear above the 3,000- 
 meter level. It can be seen that the winds on the sea level 
 generally blow out of a high area into a low area, by curves 
 having reversed spiral forms crossing the isobars at angles 
 varying with the place. The winds flow more closely along 
 the isobars at higher levels, as stated, and from 5,000 meters to 
 
112 THERMODYNAMIC METEOROLOGY 
 
 10,000 meters it would be safe to draw the isobars and isotherms 
 from the wind directions as observed in the high clouds. It 
 follows that high-level pressure and temperature charts are 
 the true indicators of the general movement of storms across 
 the continent, because they show the direction of the eastward 
 drift when the isobars on the sea level do not clearly indicate it. 
 Such charts were prepared by the author for the sea level, the 
 3,500-foot plane, and the 10,000-foot plane for the United States, 
 and they proved to be most instructive for the public forecast 
 service. 
 
 Attention is called to the fact that the isobars are all marked 
 in the notation of Table 1, units of force B = P/100, and the 
 isotherms in absolute temperatures T. Thus, we have for the 
 pressure : 
 
 D Tt Mercury 
 
 f D mm. 
 
 103322 1033.2 775.0 
 
 102655 1026.5 770.0 
 
 101322 1013.2 760.0 
 
 99988 999.9 750.0 
 
 98655 986.6 740.0 
 
 97990 979.9 735.0 
 
 P is the dynamic pressure in the M. K. S. System; B is 
 this pressure divided by 100 for practical use; mm. is the milli- 
 meters of mercury of a barometer. B and mm. are related 
 very closely in the ratio 4 to 3. Hence, by making the scale of 
 a barometer in divisions each three-fourths of a millimeter, it 
 would be only necessary to multiply the reading by 100 to obtain 
 the dynamic pressures useful in all computations on the dynamics 
 and thermodynamics of the atmosphere. This change in units 
 is so simple, and so far-reaching in its beneficial results, that it 
 is strongly recommended to meteorologists. Generally, a full 
 set of Tables should be constructed to supersede the mercurial 
 British and the Metric systems now in use. 
 
 Further attention is called to the fact that these computa- 
 tions fully satisfy equation (196) in a non-adiabatic atmosphere, 
 and that, therefore, the author's theory of the non-asymmetric 
 cyclone and anticyclone, due to interflowing currents at different 
 temperatures subject to the attraction of gravitation at every 
 
PLANETARY CIRCULATION AND RADIATION 113 
 
 point, is fully verified, since the computed and the observed 
 values are in agreement. 
 
 The Planetary Circulation and Radiation. The Observations of 
 Temperature and Velocity 
 
 The greatest difficulty in discussing the problems of the 
 planetary circulation and radiation consists in determining the 
 proper temperatures and velocities of the circulation in all 
 latitudes from the equator to the pole, and at all altitudes from 
 the surface up to the practical limit of the balloon ascensions, 
 as 30,000 meters. The number of available observations is very 
 limited throughout the tropics, they are lacking entirely in 
 the arctic zone, and above 14,000 meters in the isothermal 
 region they are insufficient for our purposes. In spite of these 
 difficulties it has been thought proper to execute the extensive 
 computations, for the sake of the general instruction regarding 
 various unsolved problems of meteorology, which depend upon 
 such data. There are several accessible reports and compilations 
 on the results of balloon ascensions, and we utilize them without 
 further references: Rykachef for Russia, Dines for England, 
 Teisserenc de Bort for France, Wegener for Germany, Rotch for 
 St- Louis, Teisserenc de Bort and Rotch for the Atlantic Ocean, 
 Berson for Victoria Nyanza and East Africa. Table 27 
 contains a summary of the original mean observations arranged 
 according to the latitude, and Table 30 contains the adopted 
 temperature system, which fairly represents this type of distribu- 
 tion. An inspection of these original temperatures presents a 
 great difficulty when they are compared with the wind velocities 
 and directions in the tropics. It is seen that there is a decrease 
 of temperature in the convectional region from the equator to 
 the pole, except in the low levels of the tropics, as indicated 
 in Fig. 13, Case II. When the temperature rises towards the 
 pole there is westward wind, as in the trades of the tropics; 
 when the temperature falls toward the pole there is eastward 
 drift, as in the temperate zones. This was first developed by 
 Bigelow, 1904, and confirmed by De Bort and Rotch in their 
 report, 1909, thus establishing a fundamental property of all 
 
114 
 
 THERMODYNAMIC METEOROLOGY 
 
 atmospheric motions as indicated by Helmholtz. The trades 
 blow steadily westward at a moderate velocity, while the east- 
 ward drift reaches a mean velocity of about 35 meters per second 
 
 z 
 
 Height 
 
 in 
 Meters 
 
 18000 
 
 17000 
 
 16000 
 
 15000 
 
 14000 
 
 13000 
 
 12000 
 
 11000 
 
 10000 
 
 9000 
 
 8000 
 
 7000 
 
 6000 
 
 5000 
 
 4000 
 
 3000 
 
 2000 
 
 1000 
 
 000 
 
 Case II High Temperatures 
 in the Isothermal Region 
 
 90 80 70 60 50 40 30 20 10 
 
 Case I Low Temperatures 
 in the Isothermal Region 
 
 70 60 50 40 30 20 10 O c 
 
 FIG. 13. Two typical cases of the observed temperatures in the earth's 
 atmosphere up to 19,000 meters 
 
 in middle latitudes, where the temperature falls rapidly toward 
 the pole at high elevations. 
 
 These cases illustrate the movement of a temperature maxi- 
 mum from the tropics into the temperate zones of the isothermal 
 region. 
 
 The general questions of temperature are greatly complicated 
 by the necessity of adapting them to the observed velocities, 
 and for those the observations are too limited in number in the 
 isothermal region and in the arctic zone to be decisive. Table 
 33 contains one system of velocities, which conform to the 
 adopted temperatures of Table 30. The velocities are 
 directed westward in the convection region of the tropics, with 
 maximum on the 4,000-meter level, minimum at the 13,000- 
 meter level, and a region of alternately westward and eastward 
 velocities above that level, except immediately over the equator, 
 where the wind is steadily westward. In the latitudes on the 
 poleward side of the high-pressure belt, which is in latitude 30, 
 
PLANETARY CIRCULATION AND RADIATION 
 
 115 
 
 TABLE 27 
 
 THE MEAN OBSERVED TEMPERATURES IN THE ATMOSPHERE FROM BALLOON 
 ASCENSIONS ARRANGED IN THE ORDER OF LATITUDE 
 
 
 Russia 
 
 Eng- 
 land 
 
 Ger- 
 
 many 
 
 France 
 
 St. 
 Louis 
 
 Atlantic Ocean Tropics 
 
 Viet. 
 Nyanza 
 
 Lati- 
 
 
 
 
 
 
 
 
 
 
 tude 
 
 56 
 
 53 
 
 51 
 
 49 
 
 39 
 
 35 
 
 25 15 
 
 5 
 
 
 
 Num- 
 
 
 
 
 
 
 
 
 
 ber 
 
 143 
 
 200 
 
 380 
 
 581 
 
 23 
 
 12 
 
 6 
 
 8 
 
 6 
 
 12 
 
 19000 
 
 
 
 
 
 
 
 
 
 
 
 18000 
 
 
 
 
 
 
 
 
 
 
 190 5 
 
 17000 
 
 
 
 
 
 
 
 
 
 
 197 1 
 
 16000 
 
 
 
 
 
 
 
 
 
 
 202 6 
 
 15000 
 
 
 
 
 
 213 3 
 
 
 
 
 
 206 8 
 
 14000 
 
 
 219.1 
 
 218.6 
 
 218.9 
 
 212.2 
 
 211.2 
 
 210.7 
 
 210.0 
 
 214.1 
 
 210.8 
 
 13000 
 
 .... 
 
 219.3 
 
 218.5 
 
 218.6 
 
 214.2 
 
 214.6 
 
 212.6 
 
 216.2 
 
 218.1 
 
 216.0 
 
 12000 
 
 218.3 
 
 219.6 
 
 218.8 
 
 217.8 
 
 216.7 
 
 219.4 
 
 217.4 
 
 223.6 
 
 224.7 
 
 222.6 
 
 11000 
 
 217.7 
 
 219.4 
 
 220.2 
 
 219.0 
 
 221.0 
 
 225.8 
 
 224.7 
 
 231.8 
 
 231.8 
 
 231.4 
 
 10000 
 
 218.9 
 
 223.1 
 
 223.4 
 
 223.7 
 
 226.2 
 
 233.7 
 
 233.2 
 
 240.2 
 
 239.3 
 
 238.9 
 
 
 
 
 
 
 
 
 
 
 
 9000 
 
 224.3 
 
 228.4 
 
 228.6 
 
 229 .5 232 . 9 
 
 242.0 
 
 241.8 
 
 248.2 
 
 247.2 
 
 246.1 
 
 8000 
 
 231.4 
 
 235.2 
 
 235.0 
 
 236.8 239.8 
 
 250.0 
 
 249.6 
 
 255.5 
 
 254.6 
 
 250.7 
 
 7000 
 
 239.1 
 
 241.8 
 
 242.2 
 
 244.0 248.5 
 
 257.6 
 
 256.6 
 
 261.8 
 
 261.4 
 
 258.0 
 
 6000 
 
 246.4 
 
 248.8 
 
 249.3 
 
 251.1 
 
 256.1 
 
 264.4 
 
 263.0 
 
 267.6 
 
 267.3 
 
 263.4 
 
 5000 
 
 253.1 
 
 255.4 
 
 256.1 
 
 257.6 
 
 262.7 
 
 270.5 
 
 268.8 
 
 272.8 
 
 272.1 
 
 269.2 
 
 4000 
 
 259.3 
 
 261.7 
 
 262.3 
 
 263.6 
 
 268.6 
 
 275.8 
 
 274.6 
 
 277.9 
 
 277.2 
 
 274.7 
 
 3000 
 
 265.3 
 
 268.8 
 
 268.0 
 
 269.0 
 
 273.5 
 
 280.9 
 
 279.3 
 
 282.8 
 
 282.0 
 
 280.8 
 
 2000 
 
 269.9 
 
 272.5 
 
 273.1 
 
 273.7 
 
 277.8 
 
 285.4 
 
 283.9 
 
 287.5 
 
 286.7 
 
 288.4 
 
 1000 
 
 274.3 
 
 2,77.1 
 
 277.6 
 
 278.3 
 
 281.1 
 
 290.9 
 
 288.7 
 
 292.2 
 
 292.3 
 
 296.2 
 
 000 
 
 277.1 
 
 281.3 
 
 282.9 
 
 282.5 
 
 285.9 
 
 298.9 
 
 296.2 
 
 298.4 
 
 300.8 
 
 
 
 as given in Table 29 for all elevations, the wind is eastward, 
 increasing from the surface to a maximum on the 9,000-10,000- 
 meter level, where it suddenly falls in velocity, and prevails 
 eastward or variable throughout the isothermal region. These 
 velocity conditions conform to Bigelow's observations at Wash- 
 ington, D. C., 1896-97, International Cloud Report, Charts 
 11, 14, and Monthly Weather Review, April, May, June, 1904. 
 The reports of the International Committee show that for the 
 hemisphere at large, these westward and eastward circulations 
 tend to concentrate about "centers of action," wherein the 
 continuity of the high-pressure belt around the globe in longitude 
 is broken up into sections, one over the Atlantic Ocean in each 
 hemisphere, and another over the Pacific Ocean in each hemi- 
 sphere. This subdivision is due to the mutual influence of 
 
116 
 
 THERMODYNAMIC METEOROLOGY 
 
 oceans and continents through the induced temperatures, and 
 the configurations of the great currents of the general circulation 
 depending upon them. The high-pressure belt is itself produced, 
 in that latitude, by the downflow of air which has originally 
 risen in the tropics; the segregation is accompanied by low-level 
 " leakage currents " from the tropics to the temperate zones, which 
 form the warm parts of cyclones and anticyclones. The cold 
 streams from the polar zones meet these warm currents in mid- 
 latitudes, and their interaction produces the local circulations of 
 storms, under the force of gravitation acting on warm and cold 
 masses in contact with each other. 
 
 For our special purpose in this connection, there has been 
 much discussion regarding the existence of the " antitrade" 
 winds blowing eastward in the upper levels, as above 12,000 
 meters. The observational data are themselves conflicting, but 
 this points to a very important feature in the theory of the 
 planetary circulation. Collecting some of the data for very 
 high altitudes, 9,000 to 17,000 meters, we have the following 
 typical exhibit: 
 
 TABLE 28 
 
 THE NUMBER OF WINDS FROM EIGHT COMPASS POINTS, IN DIFFERENT 
 LATITUDES, AT 9,000 TO 17,000 METERS ALTITUDE 
 
 Station 
 
 N. 
 
 N.E. 
 
 E. 
 
 S.E. 
 
 [s. 
 
 S.W. 
 
 w. 
 
 N.W. 
 
 To'l 
 
 Lat. 
 
 Lindenburg 
 
 18 
 
 17 
 
 14 
 
 4 
 
 1 
 
 16 
 
 2 
 
 q 
 
 81 
 
 52 
 
 Ponta Delgada . 
 
 6 
 
 16 
 
 8 
 
 4 
 
 1 
 
 10 
 
 1? 
 
 q 
 
 66 
 
 37.7 
 
 and Madeira. 
 
 
 
 
 
 
 
 
 
 
 32.6 
 
 Teneriffe 
 
 3 
 
 
 
 ?, 
 
 ?, 
 
 3 
 
 ?,?, 
 
 1 
 
 g 
 
 38 
 
 28.5 
 
 St. Vincent 
 
 
 
 ?, 
 
 13 
 
 4 
 
 5 
 
 2 
 
 
 
 
 
 26 
 
 17 
 
 Victoria Nyanza 
 Ascension 
 
 5 
 1 
 
 10 
 
 o 
 
 36 
 5 
 
 10 
 
 
 2 
 
 
 4 
 
 
 4 
 
 
 10 
 
 8 
 
 81 
 14 
 
 
 
 -8 
 
 
 
 
 
 
 
 
 
 
 
 
 It is evident that there is an alternation of wind direction 
 between N.E. and S.W. in all latitudes, but that the westward 
 wind prevails over the equator, and to some extent predominates 
 in all latitudes to 50 or 60. This can only mean that in the 
 isothermal region the temperatures increase from the equator 
 
PLANETARY CIRCULATION AND RADIATION 117 
 
 to that latitude, as on Fig. 13, Case II, "High Temperatures 
 in the Isothermal Region"; for all the wind directions blowing 
 eastward, the opposite temperature gradient must occur, of 
 fall from the equator toward the pole, as in Case I, "Low 
 Temperatures in the Isothermal Region." We infer that in 
 the general circulation there are heat maxima, or warm crests, 
 which form near the equator, Case I, and move toward the pole 
 to middle latitudes, Case II. The temperature gradients in the 
 isothermal region are, therefore, very unsteady at any place, and 
 there is a continuous mixture of the air currents, along with a 
 vigorous radiation of heat from below, as will be further indicated. 
 The computed data of Case II are here produced in Tables 29-42. 
 While there are instances of "antitrade" winds at high elevations, 
 the "trade" winds apparently penetrate to very high altitudes 
 at other times. These observations were all made on the eastern 
 side of the north Atlantic Tropic Ocean, and on the eastern edge 
 of the high-pressure section of that region, where westward winds 
 from N.E.- to S.W.-ward prevail as part of the forced circula- 
 tion. On the western side of the ocean and western edge of this 
 same section, it is probable that the " antitrade" wind will pre- 
 dominate much more vigorously, but this is again a localized 
 effect of temperature and pressure distribution. An inspection 
 of the Victoria Nyanza temperatures, Table 27, indicate a 
 very pronounced fall of temperature on the levels from 16,000 
 to 18,000 meters, such as would produce a violent westward 
 circulation, which does not seem to exist. It would be proper 
 to confirm these valuable observations at other points over the 
 equator, whenever practical. Similarly the temperatures of 
 St. Louis relative to Europe would demand a violent westward 
 wind at the high levels, which likewise does not regularly exist. 
 These facts show how difficult it is to construct a satisfactory 
 system of temperatures and velocities for the planetary circula- 
 tion. There is great need for high-level balloon ascensions 
 recording temperature, humidity, and wind velocity and direc- 
 tion, upon which to base the computations for the other terms 
 in the problems of the atmosphere. 
 
118 THERMODYNAMIC METEOROLOGY 
 
 The Thermodynamic Tables of the Planetary Circulation and 
 
 Radiation 
 
 After the preceding explanations regarding the observational 
 data and formulas, the reader can easily study the results of 
 the computations for Case II, in Tables 29-42, so that only 
 special points of interest will be indicated. 
 
 Table 29. The pressure maximum is near latitude 30 at all 
 elevations; the minimum near the pole is much lower than that 
 at the equator. 
 
 Table 30. The temperature maximum is in the high-press- 
 ure maximum throughout the convectional region, and in the 
 isothermal region it moves from near the equator toward the pole; 
 there is a sharp drop in the temperature in passing from the 
 convectional to the isothermal region; this boundary is located 
 at 9,000-12,000 meters on the poleward side of 40 and it lies 
 between 12,000 and 16,000 in the tropics; when the temperatures 
 in the isothermal region are relatively cold the boundary is 
 at high elevations, and when warm at low elevations respectively; 
 over anticyclones the isothermal region is at high altitudes, 
 and over cyclones at low latitudes; it is high in winter and low 
 in summer; its elevation depends upon the temperature and 
 gravitation conditions in the convection region and not on any 
 inherent forces of its own; it is distinguished in its physical 
 properties from the convectional by some properties which will 
 be indicated under the topic of radiation. 
 
 Table 31. The density has nearly the same value on 
 the same level of the tropics as a broad minimum, and it in- 
 creases toward the pole, much more in the convectional than in 
 the isothermal region. 
 
 Table 32. The gas coefficient and the dependent specific 
 heat are variables, though assumed to be constant at the surface 
 before radiation changes it with the elevation, but decreas- 
 ing upward generally, much more near the pole than over 
 the equator; there is an irregularity in passing to the isothermal 
 region, accompanied by the change of temperature and velocity 
 of circulation. The check P = p T R is confirmed at every 
 point. 
 
THERMODYNAMIC TABLES 
 
 119 
 
 TABLE 29 
 
 THE PLANETARY CIRCULATION AND RADIATION 
 The pressure P in the units of force (M. K. S.) 
 
 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 
 
 19000 
 
 5795 
 
 5854 
 
 5964 
 
 6184 
 
 6356 
 
 6587 j 6945 
 
 6832 
 
 6637 
 
 6445 
 
 18000 
 
 6768 
 
 6838 
 
 6980 
 
 7224 
 
 7409 
 
 7709 8128 
 
 8038 
 
 7848 
 
 7683 
 
 17000 
 
 7922 
 
 8003 
 
 8148 
 
 8433 
 
 8672 
 
 9023 i 9550 
 
 9457 
 
 9273 
 
 9132 
 
 16000 
 
 9247 
 
 9367 
 
 9536 
 
 9843 
 
 10110 
 
 10519 
 
 11177 
 
 11127 
 
 10955 
 
 10829 
 
 15000 
 
 10823 
 
 10935 
 
 11136 
 
 11521 
 
 11833 
 
 12311 
 
 13133 
 
 13084 
 
 12915 
 
 12807 
 
 14000 
 
 12667 
 
 12797 
 
 13028 
 
 13448 
 
 13794 
 
 14408 
 
 15407 
 
 15374 
 
 15215 
 
 15108 
 
 13000 
 
 14825 
 
 14978 
 
 15248 
 
 15740 
 
 16144 
 
 16837 
 
 18041 
 
 18028 17869 
 
 17754 
 
 12000 
 
 17351 
 
 17530 
 
 17846 
 
 18422 
 
 18821 
 
 19674 
 
 21076 
 
 21069 20899 
 
 20782 
 
 11000 
 
 20308 
 
 20513 
 
 20887 
 
 21560 
 
 22028 
 
 22972 
 
 24555 
 
 24523 
 
 24345 
 
 24224 
 
 10000 
 
 23768 
 
 24013 
 
 24405 
 
 25135 
 
 25781 
 
 26808 
 
 28481 
 
 28403 
 
 28220 
 
 28098 
 
 9000 
 
 27818 
 
 28105 
 
 28596 
 
 29367 
 
 30099 
 
 31177 
 
 32860 
 
 32726 
 
 32542 
 
 32418 
 
 8000 32509 
 
 32838 
 
 33367 
 
 34213 
 
 35012 
 
 36068 
 
 37721 
 
 37534 
 
 37344 
 
 37222 
 
 7000 i 37859 
 
 38226 
 
 38772 
 
 39666 
 
 40517 
 
 41530 
 
 43113 
 
 42877 
 
 42678 
 
 42558 
 
 6000 
 
 43930 
 
 44344 
 
 44876 
 
 45781 
 
 46666 
 
 47620 
 
 49111 
 
 48812 
 
 48605 
 
 48495 
 
 5000 
 
 50811 
 
 51251 
 
 51742 
 
 52631 
 
 53516 
 
 54414 
 
 55772 
 
 55427 
 
 55219 
 
 55110 
 
 4000 
 
 58606 
 
 59038 
 
 59464 
 
 60297 
 
 61156 
 
 61967 
 
 63164 
 
 62791 
 
 62576 
 
 62491 
 
 3000 
 
 67437 
 
 67816 
 
 68143 
 
 68867 
 
 69678 
 
 70378 
 
 71355 
 
 70963 
 
 70772 
 
 70707 
 
 2000 i 77432 
 
 77703 
 
 77877 
 
 78455 
 
 79164 
 
 79722 
 
 80428 
 
 80046 
 
 79862 
 
 79838 
 
 1000 ! 88730 
 
 88838 
 
 88836 
 
 89168 
 
 89704 
 
 90104 
 
 90462 
 
 90126 
 
 89958 
 
 89950 
 
 000 
 
 101521 
 
 101388 
 
 101135 
 
 101149 
 
 101414 
 
 101588 
 
 101548 
 
 101216 
 
 101042 
 
 101058 
 
 Formula: log P = log Po 
 
 nk 
 
 (log r - log To). 
 
 (182) 
 
 TABLE 30 
 THE TEMPERATURE T IN ABSOLUTE DEGREES CENTIGRADE 
 
 z 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 
 
 19000 
 
 219.0 
 
 219.2 
 
 219.4 
 
 220 .4 
 
 220.8 
 
 218.6 
 
 214.8 
 
 210.9 
 
 203.0 
 
 193.0 
 
 18000 
 
 218.6 
 
 218.8 219.0 
 
 220.0 
 
 220.6 
 
 218.4 
 
 214.6 
 
 210.6 
 
 204.0 
 
 196.0 
 
 17000 218.3 
 
 218.5 218.7 
 
 219.7 
 
 220.4 
 
 218.2 
 
 214.4 
 
 210.3 
 
 205.0 
 
 199.0 
 
 16000 218.0 
 
 218.2 
 
 218.4 
 
 219.4 
 
 220.2 
 
 218.0 
 
 214.2 
 
 210.0 
 
 206.0 
 
 202.0 
 
 15000 
 
 217.7 
 
 217.9 
 
 218.1 
 
 219.1 
 
 220.0 
 
 217.8 
 
 214.0 
 
 211.0 
 
 208.0 
 
 205.0 
 
 14000 
 
 217.4 
 
 217.6 
 
 217.8 
 
 218.8 
 
 219.8 
 
 218 
 
 215.0 
 
 213.0 
 
 210.0 
 
 209.0 
 
 13000 
 
 217.2 
 
 217.4 
 
 217.6 
 
 218.6 
 
 219.6 
 
 219.0 
 
 218.0 
 
 216.2 
 
 215.0 
 
 214.0 
 
 12000 
 
 217.0 
 
 217.2 
 
 217.4 
 
 218.4 
 
 219.4 
 
 220.0 
 
 221.0 
 
 222.0 
 
 221.0 
 
 220.0 
 
 11000 
 
 216.8 
 
 217.0 
 
 217.2 
 
 218.2 
 
 219.2 
 
 221.0 
 
 226.4 
 
 228.3 
 
 227.0 
 
 226.0 
 
 10000 
 
 216.6 
 
 216.8 
 
 217.0 
 
 218.0 
 
 219.0 
 
 222.1 
 
 234.3 
 
 236.3 
 
 235.5 
 
 234.5 
 
 9000 
 
 216.4 
 
 217 
 
 218.5 
 
 220.3 
 
 222.2 
 
 230.4 
 
 243.6 
 
 245.4 
 
 244.2 
 
 243.2 
 
 8000 
 
 221.8 
 
 222.1 
 
 224.5 
 
 227.3 
 
 230.0 
 
 238.6 
 
 251.6 
 
 253.2 
 
 252.5 
 
 251.5 
 
 7000 227.0 
 
 227.5 
 
 230.5 
 
 234.8 
 
 238.0 
 
 246.2 
 
 259.2 
 
 260.5 
 
 259.5 
 
 258.5 
 
 6000 
 
 232.8 
 
 233.0 
 
 237.0 
 
 241.8 
 
 245.7 
 
 253.0 
 
 265.8 
 
 266.3 
 
 265.5 
 
 264.8 
 
 5000 
 
 237.3 
 
 239.0 
 
 243.0 
 
 248.3 
 
 253.0 
 
 259.5 
 
 271.7 
 
 271.5 
 
 270.5 
 
 269.5 
 
 4000 
 
 241.5 
 
 244.2 
 
 248.3 
 
 254.2 
 
 259.0 
 
 265.8 
 
 
 
 277.3 
 
 276 5 
 
 275.5 
 
 274.0 
 
 3000 
 
 245.3 
 
 248.7 
 
 253.3 
 
 259.7 
 
 264.8 
 
 271.3 
 
 282.9 
 
 281.4 
 
 280.0 
 
 279.0 
 
 2000 
 
 249.0 
 
 253.0 
 
 257.7 
 
 264.6 
 
 270.6 
 
 276.6 
 
 288.4 
 
 286.2 
 
 285.0 
 
 284.0 
 
 1000 
 
 252.5 
 
 257.0 
 
 261.9 
 
 269.0 
 
 276.0 
 
 281.6 
 
 292.0 
 
 290.0 
 
 289.5 
 
 289.0 
 
 000 
 
 255.0 
 
 260.0 
 
 265.0 
 
 273.0 
 
 281.0 
 
 288.0 
 
 299.5 
 
 298.5 
 
 298.7 
 
 298.0 
 
 The observations made in balloon and kite ascensions. 
 
120 THERMODYNAMIC METEOROLOGY 
 
 Table 33. The adopted wind velocity indicates an east- 
 ward movement with the positive sign (+), westward with 
 the negative sign ( ), and alternating with the (=F) signs; it 
 is quite likely that further observations will enable us to improve 
 this mean table of velocities; these must ultimately be so ad- 
 justed to the air masses associated with them that the sum of the 
 momenta of rotation about the earth's axis shall be equal to zero 
 in order that the period of the earth's rotation may be constant, 
 as indicated by astronomical observations, and this involves 
 the corresponding pressure, temperature, density, and radiation 
 of heat from point to point throughout the entire atmosphere. 
 
 Table 34. The kinetic energy of circulation from one 
 level to another acts as a balance in the action of gravitation 
 against the pressure and heat terms in the general equation. 
 This action is very pronounced in passing from the convectional 
 to the isothermal region; it is strong in some parts of cyclones 
 and anticyclones, and in tornadoes, being due to rapid changes in 
 the temperatures for short distances. When the pressure and 
 heat terms are deficient, the kinetic energy makes it up by an 
 increase in the velocity; when in excess, it balances the same by 
 decreasing the velocity; the numerous horizontal currents in 
 the atmosphere, as seen by the cloud motions in different direc- 
 tions, exhibit this process which is incessantly at work adjusting 
 these delicate differences between pressure and radiation to the 
 controlling force of gravitation ; these adjust themselves mutually 
 at every point in the atmosphere, and are not propagated at 
 long range from one distant point to another. 
 
 Table 35. The hydrostatic pressure per unit density is 
 computed from Tables 29, 31, for use in the general equation 
 (196), which must always be satisfied. It should be carefully 
 remembered that the density is to be computed by equation 
 (176), and not by equation (175). The term decreases upward, 
 and generally from the equator to the pole, though there is 
 a small maximum in the convectional region just north of the 
 high-pressure belt, in the levels 5,000 to 9,000 meters. 
 
 Table 36. The free heat on which radiation depends gives 
 the change in the heat contents of the unit mass per 1,000 
 
THERMODYNAMIC TABLES 
 
 121 
 
 TABLE 31 
 
 THE PLANETARY CIRCULATION AND RADIATION 
 The density p in kilograms per cubic meter 
 
 z 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 O 8 
 
 19000 
 
 0.1810 
 
 0.1788 
 
 0.1776 
 
 0.1769 
 
 0.1754 
 
 0.1756 
 
 0.1753 
 
 0.1737 
 
 0.1700 
 
 0.1670 
 
 18000 
 
 .2022 
 
 .1997 
 
 .1986 
 
 .1976 
 
 .1956 
 
 .1964 
 
 .1961 
 
 .1950 
 
 .1915 
 
 .1892 
 
 17000 
 
 .2261 
 
 .2233 
 
 .2217 
 
 .2206 
 
 .2188 
 
 .2197 
 
 .2199 
 
 .2189 
 
 .2156 
 
 .2140 
 
 16000 
 
 .2524 
 
 .2498 
 
 .2480 
 
 .2463 
 
 .2440 
 
 .2456 
 
 .2459 
 
 .2458 
 
 .2428 
 
 .2416 
 
 15000 
 
 .2823 
 
 .2788 
 
 .2768 
 
 .2754 
 
 .2729 
 
 .2740 
 
 .2758 
 
 .2758 
 
 .2730 
 
 .2722 
 
 14000 
 
 0.3157 
 
 0.3118 
 
 0.3096 
 
 0.3074 
 
 0.3043 
 
 0.3064 
 
 0.3090 
 
 0.3093 
 
 0.3066 
 
 0.3061 
 
 13000 
 
 .3531 
 
 .3487 
 
 .3463 
 
 .3437 
 
 .3403 
 
 .3423 
 
 .3457 
 
 .3464 
 
 .3438 
 
 .3433 
 
 12000 
 
 .3949 
 
 .3900 
 
 .3872 
 
 .3835 
 
 .3796 
 
 .3824 
 
 .3861 
 
 .3870 
 
 .3843 
 
 .3840 
 
 11000 
 
 .4417 
 
 .4362 
 
 .4331 
 
 .4300 
 
 .4245 
 
 .4270 
 
 .4304 
 
 .4311 
 
 .4284 
 
 .4282 
 
 10000 
 
 .4939 
 
 .4878 
 
 .4844 
 
 .4796 
 
 .4748 
 
 .4765 
 
 .4783 
 
 .4786 
 
 .4758 
 
 .4759 
 
 9000 
 
 0.5524 
 
 0.5456 
 
 0.5415 
 
 0.5357 
 
 0.5300 
 
 0.5306 
 
 0.5295 
 
 0.5293 
 
 . 5265 
 
 0.5269 
 
 8000 
 
 .6172 
 
 .6094 
 
 .6043 
 
 .5972 
 
 .5902 
 
 .5885 
 
 .5841 
 
 .5835 
 
 .5806 
 
 .5813 
 
 7000 
 
 .6878 
 
 .6790 
 
 .6724 
 
 .6634 
 
 .6548 
 
 .6505 
 
 .6424 
 
 .6414 
 
 .6385 
 
 .6394 
 
 6000 
 
 .7645 
 
 .7546 
 
 .7660 
 
 .7346 
 
 .7240 
 
 .7170 
 
 .7047 
 
 .7033 
 
 .7003 
 
 .7016 
 
 5000 
 
 .8479 
 
 .8364 
 
 .8255 
 
 .8112 
 
 .7981 
 
 .7884 
 
 .7714 
 
 .7698 
 
 .7669 
 
 .7684 
 
 4000 
 
 0.9384 
 
 0.9249 
 
 0.9114 
 
 0.8935 
 
 0.8775 
 
 0.8647 
 
 . 8428 
 
 0.8413 
 
 0.8382 
 
 0.8402 
 
 3000 
 
 1.0370 
 
 1.0207 
 
 1.0041 
 
 0.9821 
 
 0.9628 
 
 0.9466 
 
 0.9191 
 
 0.9177 
 
 0.9148 
 
 0.9173 
 
 2000 
 
 1 . 1440 
 
 1 . 1244 
 
 1.1071 
 
 1 . 0775 
 
 1.0543 
 
 1 . 0344 
 
 1 . 0008 
 
 0.9998 
 
 0.9969 
 
 1 . 0001 
 
 1000 
 
 1.2604 
 
 1.2367 
 
 1.2125 
 
 1.1802 
 
 1 . 1523 
 
 1.1284 
 
 1 . 0880 
 
 1.0878 
 
 1.0850 
 
 1.0886 
 
 000 
 
 1.3870 
 
 1.3586 
 
 1.3296 
 
 1.2908 
 
 1.2574 
 
 1.2289 
 
 1.1812 
 
 1.1814 
 
 1.1785 
 
 1 1825 
 
 Formula: log p 
 
 log Po + j- (log T - log To). 
 
 (183) 
 
 TABLE 32 
 THE GAS COEFFICIENT R, WHICH Is A VARIABLE 
 
 z 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 
 
 19000 
 
 146.14 
 
 149.37 
 
 153.03 
 
 158.60 
 
 164.12 
 
 171.56 
 
 184.39 
 
 186.48 
 
 192 . 33 
 
 199.92 
 
 18000 
 
 153 . 12 
 
 156.51 
 
 160.45 
 
 166.18 
 
 171.70 
 
 179.71 
 
 193.14 
 
 195.72 
 
 200.88 
 
 207.12 
 
 17000 
 
 160.47 
 
 164.01 
 
 168.01 
 
 174.01 
 
 179 . 86 
 
 188.24 
 
 202 . 54 
 
 205.43 
 
 209.77 
 
 214.42 
 
 16000 
 
 168.03 
 
 171.88 
 
 176.06 
 
 182.21 
 
 188.17 
 
 196.94 
 
 217 . 10 
 
 215.62 
 
 219.05 
 
 221.92 
 
 15000 
 
 176.09 
 
 179.98 
 
 184.36 
 
 190.95 
 
 197.10 
 
 206.29 
 
 222.48 
 
 224 . 88 
 
 227.53 
 
 229.53 
 
 14000 
 
 184.53 
 
 188.61 
 
 193.20 
 
 199.95 
 
 206.21 
 
 215.68 
 
 231.91 
 
 233.39 
 
 236.28 
 
 236.14 
 
 13000 
 
 193.29 
 
 197.56 
 
 202.37 
 
 209.44 
 
 216.00 
 
 224.58 
 
 239.38 
 
 240.73 
 
 241.75 
 
 241.63 
 
 12000 
 
 202.46 
 
 206.94 
 
 211.93 
 
 219.37 
 
 225.99 
 
 233.85 
 
 246.98 
 
 245.24 
 
 246 . 08 
 
 245.98 
 
 11000 
 
 212.07 
 
 216.76 
 
 222 . 04 
 
 229.79 
 
 236.72 
 
 243.45 
 
 251.97 
 
 249.17 
 
 250.38 
 
 250.28 
 
 10000 
 
 222.14 
 
 227.05 
 
 232.58 
 
 246.01 
 
 247.95 
 
 253.29 
 
 254.13 
 
 250.64 
 
 251.86 
 
 251.77 
 
 9000 
 
 232 . 68 
 
 237.39 
 
 241.68 
 
 248.85 
 
 255.56 
 
 255.06 
 
 254.74 
 
 251.96 
 
 253 . 09 
 
 253.01 
 
 8000 
 
 237.48 
 
 242.61 
 
 245.94 
 
 252 . 06 
 
 257.92 
 
 256.88 
 
 256.66 
 
 254.07 
 
 254.69 
 
 254.62 
 
 7000 
 
 242.48 
 
 247.47 
 
 250.16 
 
 254.65 
 
 259 . 99 
 
 259.30 
 
 258.96 
 
 256.62 
 
 257.57 
 
 257.51 
 
 6900 
 
 247.35 
 
 252.22 
 
 253.79 
 
 257.74 
 
 262.33 
 
 262.51 
 
 262.21 
 
 260 . 62 
 
 261.39 
 
 261.05 
 
 5000 
 
 252 . 54 
 
 256.39 
 
 257.92 
 
 261.31 
 
 265.05 
 
 265.98 
 
 266.11 
 
 265.19 
 
 266.19 
 
 266.15 
 
 4000 
 
 258.59 
 
 261.39 
 
 262.76 
 
 265.47 
 
 269.08 
 
 269 . 62 
 
 270.28 
 
 269.94 
 
 270.97 
 
 271.46 
 
 3000 
 
 265.12 
 
 267 . 15 
 
 267.91 
 
 270.02 
 
 273.29 
 
 274.04 
 
 274.43 
 
 274.72 
 
 276.27 
 
 276.28 
 
 2000 
 
 271.82 
 
 273.15 
 
 273.71 
 
 275.18 
 
 277.48 
 
 278.65 
 
 278.66 
 
 279.74 
 
 281.07 
 
 281.11 
 
 1000 
 
 278.81 
 
 279 . 51 
 
 279.75 
 
 280.88 
 
 282.05 
 
 283.56 
 
 284.74 
 
 285.70 
 
 286.38 
 
 285.93 
 
 000 
 
 287.03 
 
 287.03 
 
 287 . 03 
 
 287.03 
 
 287 . 03 
 
 287 . 03 
 
 287.03 
 
 287.03 
 
 287.03 
 
 287.03 
 
 Formula: log R = log Ro + (n-l) (log T - log To). . . . (184) 
 Check P = Tp Rat every point (173) 
 
122 TIIERMODYNAMIC METEOROLOGY 
 
 meters, and the table indicates how far the atmosphere has de- 
 parted from the adiabatic state. The term increases upward, 
 and generally from the equator to the pole, but there is a mini- 
 mum on entering the isothermal region in middle latitudes and 
 a region of marked irregular progression in the values. Un- 
 fortunately there is no way to compute this term directly, as it 
 depends upon the evaluation of the specific heat and the velocity 
 of the circulation, by equation (199). It is therefore a great loss 
 to science when an observatory measures the temperatures, but 
 not the humidity and wind velocity, at different levels, because 
 the entire subject of thermodynamic meteorology is thereby 
 excluded from further discussions. The magnitude of the term 
 (Qi Qo) can be seen in Table 36 to be very great in the upper 
 levels, and that in comparison with it the kinetic energy of the 
 circulation, as in Table 34, is very small. Hence, all those the- 
 ories of atmospheric circulation which depend upon gravitation, 
 pressure, and circulation alone, and omit heat changes through 
 radiation, either absorption or emission, have no permanent 
 value. The difficulty of determining the heat term has, no doubt, 
 been the cause of this defect which prevails in meteorological 
 literature, but it is none the less a fatal defect in this branch of 
 science. Table 43, p. 131, gives the second differences of the 
 heat contents, and it is the rate of change of the heat per 1,000 
 meters, or the radiation rate, which prevails on the average in 
 all parts of the atmosphere, and it is fundamental to all studies 
 of "solar constant " and solar insolation. This table will be dis- 
 cussed more fully under the subject of Bolometry and Pyrhe- 
 liometry, and at this place only its leading features will be noted. 
 Taking the hemisphere as a whole, the mean rate of radiation is 
 157.1 in the convectional region, and 283.1 in the isother- 
 mal region, almost twice as great in the latter. This changes 
 greatly in latitude, as can be seen in the means for the isothermal 
 region (/), and the convectional region (C), respectively. The 
 means on the table are taken outside the layers of transition, and 
 these give nearly a constant value in C from the equator to lati- 
 tude 60, but an increasing value to the pole; in the / region there 
 is a minimum in the tropics, maximum in the middle latitudes, 
 
THERMODYNAMIC TABLES 
 
 123 
 
 TABLE 33 
 
 THE PLANETARY CIRCULATION AND RADIATION 
 The velocity q in meters per second 
 
 z 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 
 
 19000 
 
 0.0 
 
 3.0 
 
 4.0 
 
 4.0 
 
 5.0 
 
 T5.0 
 
 T6.0 
 
 T7.0 
 
 + 9.0 
 
 -12.0 
 
 18000 
 
 0.0 
 
 3.0 
 
 4.0 
 
 5.0 
 
 6.0 
 
 6.0 
 
 T7.0 
 
 =F8.0 
 
 +~10.0 
 
 -12.0 
 
 17000 0.0 
 
 3.0 
 
 4.0 
 
 6.0 
 
 7.0 
 
 7.0 
 
 T8.0 
 
 +9.0 
 
 +10.0 
 
 -12.0 
 
 16000 
 
 0.0 
 
 4.0 
 
 5.0 
 
 7.0 
 
 8.0 
 
 9.0 
 
 T6.0 
 
 T6.0 
 
 T10.0 
 
 -11.0 
 
 15000 
 
 0.0 
 
 5.0 
 
 6.0 
 
 8.0 
 
 9.0 
 
 10.0 
 
 T4.0 
 
 +4.0 
 
 T 7.0 
 
 -10.0 
 
 14000 
 
 0.0 
 
 6.0 
 
 7.0 
 
 9.0 
 
 10.0 
 
 11.0 
 
 T2.0 
 
 T2.0 
 
 =F 4.0 
 
 - 9.0 
 
 13000 
 
 0.0 
 
 6.0 
 
 8.0 
 
 10.0 
 
 11.0 
 
 12.0 
 
 4.0 
 
 5.0 
 
 T 2.0 
 
 - 8.0 
 
 12000 
 
 0.0 
 
 7.0 
 
 9.0 
 
 11.0 
 
 12.0 
 
 13.0 
 
 9.0 
 
 5.0 
 
 - 3.0 
 
 - 7.0 
 
 11000 
 
 0.0 
 
 7.0 
 
 11.0 
 
 13.0 
 
 15.0 
 
 16.0 
 
 12.0 
 
 6.0 
 
 - 4.0 
 
 - 5.0 
 
 10000 
 
 0.0 
 
 8.0 
 
 15.0 
 
 17.0 
 
 20.0 
 
 20.0 
 
 15.0 
 
 6.0 
 
 - 5.0 
 
 - 3.0 
 
 9000 
 
 0.0 
 
 5.6 
 
 25.2 
 
 31.6 
 
 35.0 
 
 34.0 
 
 21.7 
 
 0.8 
 
 - 3.8 
 
 - 2.0 
 
 8000 
 
 0.0 
 
 4.8 
 
 22.8 
 
 28.9 
 
 31.5 
 
 29.0 
 
 16.0 
 
 .-2.6 
 
 - 5.0 
 
 - 2.5 
 
 7000 
 
 0.0 
 
 4.0 
 
 19.6 
 
 25.8 
 
 26.9 
 
 22.0 
 
 9.0 
 
 -4.9 
 
 - 6.2 
 
 - 2.6 
 
 6000 
 
 0.0 
 
 3.2 
 
 15.9 
 
 21.4 
 
 21.8 
 
 15.4 
 
 2.4 
 
 -6.8 
 
 - 7.0 
 
 - 2.7 
 
 5000 
 
 0.0 
 
 2.0 
 
 12.4 
 
 17.6 
 
 17.2 
 
 11.4 
 
 2.6 
 
 -8.6 
 
 - 7.4 
 
 - 2.8 
 
 4000 
 
 0.0 
 
 0.8 
 
 9.0 
 
 13.8 
 
 13.6 
 
 9.2 
 
 6.6 
 
 -9.4 
 
 - 7.6 
 
 - 2.8 
 
 3000 
 
 0.0 
 
 0.4 
 
 6.6 
 
 11.0 
 
 11.0 
 
 7.4 
 
 7.3 
 
 -9.4 
 
 - 7.2 
 
 - 2.8 
 
 2000 
 
 0.0 
 
 1.6 
 
 5.0 
 
 9.4 
 
 9.4 
 
 6.4 
 
 6.1 
 
 -8.4 
 
 - 6.2 
 
 - 2.4 
 
 1000 
 
 0.0 
 
 2.4 
 
 3.8 
 
 8.0 
 
 8.0 
 
 5.6 
 
 4.1 
 
 -6.7 
 
 - 4.6 
 
 - 1.4 
 
 000 
 
 0.0 
 
 3.4 
 
 3.4 
 
 7.4 
 
 7.1 
 
 5.1 
 
 2.0 
 
 -4.6 
 
 - 2.6 
 
 - 0.4 
 
 From the observations made in balloon and kite ascensions. 
 
 TABLE 34 
 THE KINETIC ENERGY OF CIRCULATION X (2 2 i <Z 2 o) 
 
 z 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 
 
 19000 
 
 0.0 
 
 0.0 
 
 0.0 
 
 + 4.5 
 
 + 5.5 
 
 + 5.5 
 
 + 6.5 
 
 + 7.5 
 
 + 9.5 
 
 0.0 
 
 18000 
 
 0.0 
 
 0.0 
 
 0.0 
 
 + 5.5 
 
 + 6.5 
 
 + 6.5 
 
 + 7.5 
 
 + 8.5 
 
 0.0 
 
 0.0 
 
 17000 
 
 0.0 
 
 + 3.5 
 
 + 8.5 
 
 + 6.5 
 
 + 7.5 
 
 + 16.0 
 
 - 14.0 
 
 -22.5 
 
 0.0 
 
 -11.5 
 
 16000 
 
 0.0 
 
 + 4.5 
 
 + 5.5 
 
 + 7.5 
 
 + 8.5 
 
 + 9.5 
 
 - 10.0 
 
 -10.0 
 
 -25.5 
 
 -10.5 
 
 15000 
 
 0.0 
 
 + 5.5 
 
 + 6.5 
 
 + 8.5 
 
 + 9.5 
 
 + 10.5 
 
 - 6.0 
 
 - 6.0 
 
 -16.5 
 
 - 9.5 
 
 14000 
 
 0.0 
 
 0.0 
 
 + 7.5 
 
 + 9.5 
 
 + 22.0 
 
 + 11.5 
 
 + 6.0 
 
 +10.5 
 
 - 6.0 
 
 - 8.5 
 
 13000 
 
 0.0 
 
 + 6.5 
 
 + 8.5 
 
 + 10.5 
 
 + 11.5 
 
 + 12.5 
 
 + 32.5 
 
 + 5.0 
 
 + 2.5 
 
 - 7.1 
 
 12000 
 
 0.0 
 
 0.0 
 
 +20.0 
 
 + 24.0 
 
 + 40.5 
 
 + 43.5 
 
 + 31.5 
 
 + 5.5 
 
 + 3.5 
 
 -12.5 
 
 1 1000 
 
 0.0 
 
 + 7.5 
 
 +52.0 
 
 + 60.0 
 
 + 87.5 
 
 + 72.0 
 
 + 40.5 
 
 - 0.0 
 
 + 4.5 
 
 - 8.0 
 
 10000 
 
 0.0 
 
 +16.3 
 
 -54.9 
 
 +354.8 
 
 +412.5 
 
 +378.0 
 
 +123.0 
 
 -17.7 
 
 - 5.3 
 
 - 2.5 
 
 9000 
 
 0.0 
 
 - 4.2 
 
 -57.6 
 
 -81.3 
 
 -116.4 
 
 -157.5 
 
 -107.4 
 
 + 3.1 
 
 + 5.3 
 
 + 1.1 
 
 8000 
 
 0.0 
 
 - 3.5 
 
 -67.8 
 
 - 84.8 
 
 -134.3 
 
 -178.5 
 
 - 87.5 
 
 + 8.6 
 
 + 6.7 
 
 + 0.3 
 
 7000 
 
 0.0 
 
 - 2.9 
 
 -65.7 
 
 -103.8 
 
 -124.2 
 
 -123.4 
 
 - 37.6 
 
 +11.1 
 
 + 5.3 
 
 + 0.2 
 
 6000 
 
 0.0 
 
 - 3.1 
 
 -49.5 
 
 - 74.1 
 
 - 89.7 
 
 - 53.6 
 
 + 0.5 
 
 +13.9 
 
 + 2.9 
 
 + 0.2 
 
 5000 
 
 0.0 
 
 - 1.7 
 
 -36.4 
 
 - 59.7 
 
 - 55.4 
 
 - 22.7 
 
 + 18.4 
 
 + 7.2 
 
 + 1.5 
 
 0.0 
 
 4000 
 
 0.0 
 
 - 0.2 
 
 -18.7 
 
 - 34.7 
 
 - 32.0 
 
 -14.9 
 
 + 4.8 
 
 0.0 
 
 - 3.0 
 
 0.0 
 
 3000 
 
 0.0 
 
 + 1.2 
 
 - 9.3 
 
 - 16.3 
 
 - 16.3 
 
 - 6.9 
 
 - 8.0 
 
 - 8.9 
 
 - 6.7 
 
 - 1.0 
 
 2000 
 
 0.0 
 
 + 1.6 
 
 - 5.3 
 
 - 12.2 
 
 - 12.2 
 
 - 4.8 
 
 - 10.2 
 
 -12.8 
 
 - 8.6 
 
 - 1.9 
 
 1000 
 
 0.0 
 
 + 2.9 
 
 - 1.4 
 
 - 4.6 
 
 - 6.8 
 
 - 2.7 
 
 - 6.4 
 
 -11.8 
 
 - 7.2 
 
 - 0.9 
 
 000 
 
 0.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 To obtain the kinetic energy these numbers must be multiplied by the appropriate values 
 of p:o. 
 
124 THERMODYNAMIC METEOROLOGY 
 
 and minimum in the polar zone. The broad fact is clear that 
 about twice as much heat radiates through the isothermal region 
 as through the convection region. At the strata of transition the 
 radiation is in some latitudes positive, as at the 10,000-meter 
 level from 30 to 60 latitude. These special sources of heat 
 will be further considered in discussing the value of the "solar- 
 constant" of radiation. The importance of such data in radia- 
 tion problems is very great, because it seems to explain several 
 processes in the atmosphere that have been very obscure. 
 
 Table 37. The entropy increases upward and from the 
 equator to the pole, though the same variation occurs in the 
 transition to the isothermal region as was noted for the heat 
 contents. 
 
 Table 38. The work against external forces , such as expansion 
 when a mass of air is raised from one level to another in the 
 circulation, is to be computed through a new value of R', which 
 differs from the R of (184) in consequence of the elimination 
 of the velocity of the circulation, as explained in (199), (333), 
 on Table 19. It is the circulation term which continu- 
 ally interferes with a perfect check to this series of equations, 
 and on that account, therefore, must be most carefully 
 observed. 
 
 Table 39. The inner energy is the heat contents remaining 
 after the work has been done externally, and it increases upward 
 and from the equator to the pole, with the same variation be- 
 tween the convectional and the isothermal regions. 
 
 Table 40. The radiation function is the rate of change of the 
 inner energy with the change of the volume, and it may be 
 computed by several formulas. The K\Q differs from Pi by the 
 rate of change of the heat contents with the change in volume, 
 or it may be seen to depend upon the variation of the gas coeffi- 
 cient R. The value of K i0 generally decreases with the altitude, 
 and there is a small maximum in the high-pressure belt of the 
 convectional region, but this shifts toward the pole in the iso- 
 thermal region. The immediate problem is to determine the 
 relation to the temperature through the approximate Stefan 
 formula, and this has been done in the same way as that followed 
 
THERMODYNAMIC TABLES 
 
 125 
 
 TABLE 35 
 
 THE PLANETARY CIRCULATION AND RADIATION 
 
 Pi - 
 
 The Hydrostatic Pressure per Unit Density 
 
 PlO 
 
 z 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 
 
 19000 
 
 5078.2 
 
 5198.1 
 
 5401 . 4 
 
 5552.5 
 
 5676.6 
 
 6032.3 
 
 6370.4 
 
 6540.2 
 
 6697.9 
 
 6651 . 2 
 
 18000 
 
 5387.5 
 
 5508.6 
 
 5556.6 
 
 5782.0 
 
 6095.6 
 
 6314.3 
 
 6836.6 
 
 6855.0 
 
 6998.9 
 
 7187.5 
 
 17000 
 
 5537.1 
 
 5765.0 
 
 5909.0 
 
 6038.5 
 
 6214.4 
 
 6428.8 
 
 6985.8 
 
 7185.8 
 
 7338.7 
 
 7449.5 
 
 16000 
 
 5893 . 9 
 
 5932.7 
 
 6082.3 
 
 6431.6 
 
 6665.4 
 
 6897.7 
 
 7497.2 
 
 7503 . 8 
 
 7600.0 
 
 7699.7 
 
 15000 
 
 6167.3 
 
 6305.5 
 
 6466.7 
 
 6532.2 
 
 6794.8 
 
 7226.0 
 
 7777.0 
 
 7826.5 
 
 7936.5 
 
 7956.5 
 
 14000 
 
 6453 . 3 
 
 6603 . 1 
 
 6766.3 
 
 6962.3 
 
 7291.3 
 
 7490 . 
 
 8045.2 
 
 8094.0 
 
 8161.2 
 
 8149.2 
 
 13000 
 
 6754.0 
 
 6908.4 
 
 7082 . 8 
 
 7366.2 
 
 7436.2 
 
 7828.3 
 
 8294 . 7 
 
 8293.0 
 
 8321.8 
 
 8325.7 
 
 12000 
 
 7069.0 
 
 7220.8 
 
 7413.5 
 
 7704.4 
 
 7975.7 
 
 8149.3 
 
 8520.6 
 
 8445.0 
 
 8479.5 
 
 8475.8 
 
 11000 
 
 7396.3 
 
 7575.8 
 
 7668.0 
 
 7860.7 
 
 8345.6 
 
 8490.4 
 
 8640 . 
 
 8531.2 
 
 8571.2 
 
 8569.0 
 
 10000 
 
 7740.8 
 
 7919.5 
 
 8169.8 
 
 8335.6 
 
 8594.6 
 
 8675.4 
 
 8690 . 2 
 
 8577.4 
 
 8623.2 
 
 8615.8 
 
 9000 
 
 8021.6 
 
 8195.8 
 
 8327.8 
 
 8555.8 
 
 8771.6 
 
 8740.2 
 
 8730.4 
 
 8641.2 
 
 8674.0 
 
 8669.8 
 
 8000 
 
 8199.2 
 
 8364.0 
 
 8466.6 
 
 8643 . 6 
 
 8843.4 
 
 8816.8 
 
 8799.4 
 
 8724 . 6 
 
 8750.0 
 
 8742.0 
 
 7000 
 
 8359.8 
 
 8595.0 
 
 8606.8 
 
 8748.2 
 
 8919.2 
 
 8906.2 
 
 8897.0 
 
 8826.6 
 
 8854 . 
 
 8854 . 6 
 
 6000 
 
 8535.2 
 
 8702.6 
 
 8737.6 
 
 8862.8 
 
 9001.4 
 
 9026.2 
 
 9025.6 
 
 8980.5 
 
 9015.7 
 
 9000 . 
 
 6000 
 
 8727.2 
 
 8842 . 8 
 
 8892.2 
 
 8993 . 6 
 
 9119.0 
 
 9137.4 
 
 9158.6 
 
 9141.0 
 
 9166.4 
 
 9177.0 
 
 4000 
 
 8940 . 8 
 
 9023.6 
 
 9061.5 
 
 9138.4 
 
 9261 . 
 
 9287.8 
 
 9297.4 
 
 9291.6 
 
 9350.8 
 
 9349.0 
 
 3000 
 
 9165.4 
 
 9217.6 
 
 9234.4 
 
 9310.4 
 
 9405.0 
 
 9433.6 
 
 9451.0 
 
 9473.2 
 
 9509.4 
 
 9524.5 
 
 2000 
 
 9397.8 
 
 9432 . 
 
 9461.2 
 
 9490 . 8 
 
 9553.0 
 
 9600.4 
 
 9607.2 
 
 9657.0 
 
 9698.4 
 
 9682.0 
 
 1000 
 
 9663.0 
 
 9671.6 
 
 9676.5 
 
 9697.5 
 
 9719.4 
 
 9743.6 
 
 9770 . 8 
 
 9776.8 
 
 9793.3 
 
 9781.4 
 
 000 
 
 
 
 
 
 
 
 
 
 
 
 
 .... 
 
 
 
 
 
 
 
 
 
 
 Computed from Tables 29, 31. 
 
 TABLE 36 
 THE HEAT WHICH GIVES RISE TO RADIATION (Q\ Q ) 
 
 z 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 
 
 19000 
 
 4694.2 
 
 4581.1 
 
 4451.2 
 
 4253.8 
 
 4064.2 
 
 3800.2 
 
 3350 . 6 
 
 3270.0 
 
 3079.6 
 
 2853.1 
 
 18000 
 
 4449.3 
 
 4331.0 
 
 4195.5 
 
 3982.7 
 
 3794.3 
 
 3514.3 
 
 2990.1 
 
 2944.8 
 
 2791.4 
 
 2605.0 
 
 17000 
 
 4204.4 
 
 4064.9 
 
 3920.2 
 
 3707.8 
 
 3512.0 
 
 3210.4 
 
 2736.1 
 
 2677 .4 
 
 2481.0 
 
 2363.8 
 
 16000 
 
 3927.8 
 
 3793.2 
 
 3643.8 
 
 3424 . 3 
 
 3216.4 
 
 2908.5 
 
 2391.6 
 
 2291.5 
 
 2203.3 
 
 2104.8 
 
 15000 
 
 3645.9 
 
 3504.4 
 
 3350.2 
 
 3140.1 
 
 2907.2 
 
 2587.5 
 
 2050.2 
 
 1934.7 
 
 1900.1 
 
 1861.0 
 
 14000 
 
 3352.2 
 
 3131.0 
 
 3041.4 
 
 2803.6 
 
 2571.7 
 
 2273.9 
 
 1749.7 
 
 1696.7 
 
 1646.3 
 
 1653.7 
 
 13000 
 
 3045.8 
 
 2811.2 
 
 2719.8 
 
 2470.7 
 
 2244 . 6 
 
 1962.7 
 
 1465.6 
 
 1499.7 
 
 1470.5 
 
 1483.8 
 
 12000 
 
 2724.1 
 
 2568.5 
 
 2372.2 
 
 2109.6 
 
 1866.1 
 
 1609.4 
 
 1233.1 
 
 1355.2 
 
 1322.5 
 
 1341.5 
 
 11000 
 
 2388.8 
 
 2217.5 
 
 1988.4 
 
 1714.0 
 
 1439.6 
 
 1248.7 
 
 1120.4 
 
 1268.5 
 
 1222.6 
 
 1238.1 
 
 10000 
 
 2036.9 
 
 1888.9 
 
 1759.7 
 
 1093.8 
 
 792.7 
 
 744.4 
 
 990.6 
 
 1238.5 
 
 1185.9 
 
 1185.9 
 
 9000 
 
 1774.7 
 
 1611.1 
 
 1534.0 
 
 1331.4 
 
 1151.3 
 
 1218.7 
 
 1177.8 
 
 1159.1 
 
 1126.8 
 
 1133.6 
 
 8000 
 
 1607.5 
 
 1438.2 
 
 1399.5 
 
 1235.4 
 
 1093.6 
 
 1167.3 
 
 1085.8 
 
 1074.0 
 
 1049.0 
 
 1057.6 
 
 7000 
 
 1439.1 
 
 1273.3 
 
 1263.3 
 
 1157.2 
 
 1008.1 
 
 1016.1 
 
 941.2 
 
 959.0 
 
 936.0 
 
 948.0 
 
 6000 
 
 1267.2 
 
 1121.2 
 
 1114.6 
 
 1013.9 
 
 887.3 
 
 832.2 
 
 781.0 
 
 810.4 
 
 791.2 
 
 800.4 
 
 5000 
 
 1075.2 
 
 958.1 
 
 948.3 
 
 867.6 
 
 736.2 
 
 697.8 
 
 625.3 
 
 658.0 
 
 628.9 
 
 622.9 
 
 4000 
 
 860.2 
 
 777.7 
 
 760.0 
 
 693.8 
 
 573.4 
 
 534.2 
 
 496.8 
 
 501.1 
 
 461.1 
 
 449.7 
 
 3000 
 
 634.2 
 
 575.6 
 
 563.7 
 
 559.4 
 
 414.2 
 
 372.3 
 
 366.3 
 
 342.5 
 
 292.4 
 
 285.9 
 
 2000 
 
 400.3 
 
 364.2 
 
 357.3 
 
 319.7 
 
 260.5 
 
 202 .4 
 
 192.4 
 
 160.1 
 
 121.7 
 
 122.1 
 
 1000 
 
 140.4 
 
 125.2 
 
 125.8 
 
 109.7 
 
 91.9 
 
 62.1 
 
 45.5 
 
 34.6 
 
 18.5 
 
 24.4 
 
 000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Formula: - (Qi - Qo) = - (Cp* - 
 Check q (zi - Zo) = l - 
 
 (T* - To) + \ (<Zi 2 - <Z0 2 ) 
 - Q ) 
 
 (199). 
 
 (196). 
 
 Comparing the numerical values of these terms, it is seen how impossible it is to solve 
 any problems in dynamic meteorology without the term (Qi Qo). 
 
126 THERMODYNAMIC METEOROLOGY 
 
 on Table 20, the intermediate steps of computing log C and A 
 for the ratios Ki/K Q and Ti/To being here omitted. 
 
 Table 41. The log c in the formula diminishes slowly from the 
 surface upward, it increases from the equator to the pole in 
 the convectional region, but decreases in the isothermal region, 
 the result being a nearly uniform value on the high levels of the 
 isothermal region. The minus sign affects only the characteristic 
 of the logarithm, and for a mean value log c = 5.500, c = 
 3.162X 10~ 5 . The Kurlbaum coefficient in the Stefan formula 
 for a perfect radiator is taken at 7.68 X 10~ n (C. G. min. C) 
 = 5.32 X 10~ 6 joules per square meter per sec., so that the 
 air radiates at six times the rate of a perfect radiator in the 
 ether, as when the sun transmits its radiant energy to the earth. 
 The quantity c relates to a summation of the successive radia- 
 tions from one air volume to another throughout a layer 1,000 
 m. deep. The number of repeated transfers in this depth can 
 only be approximately inferred from the knowledge that the 
 efficient radiating layer of the atmosphere, or that for which 
 any further increase of depth does not add to the radiant in- 
 tensity, amounts to several meters. The latter quantity is con- 
 siderably smaller than the superficial radiation from a black 
 solid. 
 
 Table 42. The exponent a diminishes from the surface up- 
 ward; it increases from the equator to the pole in the convec- 
 tional region, but diminishes in the isothermal region. Its 
 mean value is 3.82 instead of 4.00, as in the Stefan formula, 
 at the surface. Comparing the data we have for the 
 
 Perfect Radiator, K w =5.32 X 10~ 6 TV' 00 joule/sq. m. sec. 
 
 (K. M. S. system) and for the 
 Atmosphere, K 10 = 3.162 X 10~ 5 Tic 3 ' 82 per 1,000 m. 
 
 The consequences of these data in theories of atmospheric 
 radiation will require much careful investigation in several 
 directions. 
 
THERMODYNAMIC TABLES 
 
 127 
 
 TABLE 37 
 
 THE PLANETARY CIRCULATION AND RADIATION 
 The Entropy (5, - S ) 
 
 2 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 
 
 19000 
 
 -21.454 
 
 -20.919 
 
 -20.307 
 
 -19.318 
 
 -18.415 
 
 -17.392 
 
 -15.606 
 
 -15.516 
 
 -15.133 
 
 -14.669 
 
 18000 
 
 -20.363 
 
 -19.799 
 
 -19.166 
 
 -18.111 
 
 -17.208 
 
 -16.099 
 
 -13.938 
 
 -13.993 
 
 -13.650 
 
 -13.190 
 
 17000 
 
 -19.269 
 
 -18.612 
 
 -17.933 
 
 -16.885 
 
 -15.942 
 
 -14.720 
 
 -12.768 
 
 -12.740 
 
 -12.073 
 
 -11.790 
 
 16000 
 
 -18.026 
 
 -17.392 
 
 -16.692 
 
 -15.615 
 
 -14.613 
 
 -13.348 
 
 -11.171 
 
 -10.884 
 
 -10.644 
 
 -10.343 
 
 15000 
 
 -16.755 
 
 -16.090 
 
 -15.363 
 
 -14.338 
 
 -13.221 
 
 -11.875 
 
 - 9.558 
 
 - 9.126 
 
 - 9.091 
 
 - 8.990 
 
 14000 
 
 -15.427 
 
 -13.936 
 
 -13.971 
 
 -12.819 
 
 -11.705 
 
 -10.407 
 
 - 8.082 
 
 - 7.906 
 
 - 7.748 
 
 - 7.819 
 
 13000 
 
 -14.029 
 
 -12.937 
 
 -12.505 
 
 -11.308 
 
 -10.226 
 
 - 8.942 
 
 - 6.677 
 
 - 6.845 
 
 - 6.745 
 
 - 6.838 
 
 12000 
 
 -12.559 
 
 -11.831 
 
 -10.917 
 
 - 9.664 
 
 - 8.509 
 
 - 7.299 
 
 - 5.512 
 
 - 6.019 
 
 - 5.904 
 
 - 6.019 
 
 11000 
 
 -11.024 
 
 -10.224 
 
 - 9.159 
 
 - 7.859 
 
 - 6.570 
 
 - 5.635 
 
 - 4.864 
 
 - 5.330 
 
 - 5.287 
 
 - 5.377 
 
 10000 
 
 - 9.408 
 
 8.510 
 
 - 8.079 
 
 - 4.990 
 
 - 3.593 
 
 - 3.291 
 
 - 4.146 
 
 - 5.031 
 
 - 4.944 
 
 - 4.965 
 
 9000 
 
 - 8.100 
 
 - 7.337 
 
 - 6.926 
 
 - 5.949 
 
 - 5.092 
 
 - 5.197 
 
 - 4.757 
 
 - 4.649 
 
 - 4.537 
 
 - 4.583 
 
 8000 
 
 - 7.164 
 
 - 6.398 
 
 - 6.152 
 
 - 5.348 
 
 - 4.672 
 
 - 4.816 
 
 - 4.251 
 
 - 4.182 
 
 - 4.098 
 
 - 4.147 
 
 7000 
 
 - 6.268 
 
 - 5.531 
 
 - 5.403 
 
 - 4.836 
 
 - 4.169 
 
 - 4.071 
 
 - 3.728 
 
 - 3.643 
 
 - 3.566 
 
 - 3.623 
 
 6000 
 
 - 5.397 
 
 - 4.751 
 
 - 4.644 
 
 - 4.138 
 
 - 3.558 
 
 - 3.248 
 
 - 2.905 
 
 - 3.014 
 
 - 2.952 
 
 - 2.996 
 
 5000 
 
 - 4.491 
 
 - 3.966 
 
 - 3.861 
 
 - 3.454 
 
 - 2.876 
 
 - 2.589 
 
 - 2.278 
 
 - 2.402 
 
 - 2.304 
 
 - 2.992 
 
 4000 
 
 - 3.534 
 
 - 3.156 
 
 - 3.030 
 
 - 2.700 
 
 - 2.181 
 
 - 1.989 
 
 - 1.774 
 
 - 1.796 
 
 - 1.660 
 
 - 1.626 
 
 3000 
 
 - 2.566 
 
 - 2.295 
 
 - 2.206 
 
 - 2.134 
 
 - 1.547 
 
 - 1.359 
 
 - 1.283 
 
 - 1.207 
 
 - 1.035 
 
 - 1.016 
 
 2000 
 
 - 1.596 
 
 - 1.428 
 
 - 1.375 
 
 - 1.198 
 
 - 0.953 
 
 - 0.725 
 
 - 0.663 
 
 - 0.609 
 
 - 0.424 
 
 - 0.426 
 
 1000 
 
 - 0.553 
 
 - 0.484 
 
 - 0.478 
 
 - 0.405 
 
 - 0.330 
 
 - 0.218 
 
 - 0.154 
 
 - 0.118 
 
 - 0.063 
 
 - 0.083 
 
 000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Formula : (Si -So) 
 
 - . - (331) . 
 
 TABLE 38 THE EXTERNAL WORK (Wi - W ) 
 
 z 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 
 
 19000 
 
 3601.4 
 
 3688.7 
 
 3854.4 
 
 3948 . 5 
 
 4017.9 
 
 4256.9 
 
 4344.3 
 
 4652.0 
 
 4754.7 
 
 4942.6 
 
 18000 
 
 3849.6 
 
 3926.9 
 
 3935.8 
 
 4099.8 
 
 4358 . 9 
 
 4496.7 
 
 4867.6 
 
 4872.0 
 
 4972 . 4 
 
 5107.2 
 
 17000 
 
 3918.8 
 
 4109.2 
 
 4208.6 
 
 4276.8 
 
 4396.1 
 
 4523.4 
 
 4943.4 
 
 5126.4 
 
 5222.6 
 
 5299.5 
 
 16000 
 
 4195.8 
 
 4195.7 
 
 4302.1 
 
 4588.0 
 
 4761.7 
 
 4905.1 
 
 5355.2 
 
 5333 . 
 
 5403 . 6 
 
 5474.9 
 
 15000 
 
 4387.8 
 
 4485.0 
 
 4601.7 
 
 4606.5 
 
 4801.8 
 
 5140.7 
 
 5536.4 
 
 5552.7 
 
 5652.6 
 
 5661.2 
 
 14000 
 
 4588.9 
 
 4645.9 
 
 4812.1 
 
 4939.4 
 
 5201.4 
 
 5314.0 
 
 5717.8 
 
 5751.3 
 
 5804 . 
 
 5794.1 
 
 13000 
 
 4801.0 
 
 4887.7 
 
 5035.7 
 
 5247.1 
 
 5251.8 
 
 5562.5 
 
 5885.3 
 
 5893.3 
 
 5913.8 
 
 5921.5 
 
 12000 
 
 5023.1 
 
 5129.9 
 
 5266.0 
 
 5581.0 
 
 5681.9 
 
 5781.4 
 
 6044.0 
 
 6003.6 
 
 6028.7 
 
 6030.5 
 
 11000 
 
 5253 . 5 
 
 5383 . 6 
 
 5409.6 
 
 5523 . 
 
 5928.7 
 
 6018.3 
 
 6130.8 
 
 6065.7 
 
 6091.6 
 
 6093.9 
 
 10000 
 
 5496.4 
 
 6532.3 
 
 5845.3 
 
 5818.7 
 
 6019.3 
 
 6056.5 
 
 6143.5 
 
 6102.3 
 
 6133.0 
 
 6125.6 
 
 9000 
 
 5701.4 
 
 5828.4 
 
 5938.1 
 
 6107.6 
 
 6271.4 
 
 6259.4 
 
 6237.8 
 
 6143.2 
 
 6166.6 
 
 6164.4 
 
 000 
 
 5830.8 
 
 5945.7 
 
 6038.0 
 
 6167.7 
 
 6326.5 
 
 6321.2 
 
 6280.2 
 
 6202.0 
 
 6220.2 
 
 6214.7 
 
 7000 
 
 5942.7 
 
 6130.0 
 
 6138.9 
 
 6249.8 
 
 6377.6 
 
 6366.8 
 
 6336.1 
 
 6271.0 
 
 6291.2 
 
 6295.6 
 
 6000 
 
 6068.4 
 
 6193.7 
 
 6226.8 
 
 6322.9 
 
 6424.9 
 
 6433.7 
 
 6418.4 
 
 6381.8 
 
 6411.4 
 
 6398.4 
 
 5000 
 
 6205.0 
 
 6286.7 
 
 6333.3 
 
 6411.4 
 
 6498.8 
 
 6501.0 
 
 6506.4 
 
 6498.2 
 
 6515.2 
 
 6524.1 
 
 4000 
 
 6356.4 
 
 6415.4 
 
 6448.2 
 
 6506.0 
 
 6593.8 
 
 6609.2 
 
 6608.1 
 
 6603.5 
 
 6651.2 
 
 6646.0 
 
 3000 
 
 6515.8 
 
 6551.0 
 
 6564.4 
 
 6639 . 2 
 
 6722 . 8 
 
 6708 . 4 
 
 6724.1 
 
 6739.3 
 
 6761.0 
 
 6774.3 
 
 2000 
 
 6680.6 
 
 6704 . 3 
 
 6731 . 6 
 
 6750.3 
 
 6795.4 
 
 6826 . 1 
 
 6830.0 
 
 6874 . 9 
 
 6900.7 
 
 6884.4 
 
 1000 
 
 6870.7 
 
 6874 . 9 
 
 6880.0 
 
 6896.4 
 
 6911.3 
 
 6928.7 
 
 6951.1 
 
 6953.9 
 
 6965.8 
 
 6955.6 
 
 000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Formula: (Wi - Wo) = R'IO (T* - To) - Plo .... (333). 
 
 Cp'io . . . (349) and (350). 
 
128 
 
 THERMODYNAMIC METEOROLOGY 
 
 o 
 
 <N CO 
 
 00 O O O 
 
 O CO CD 00 O 
 00 (N CO 00 t^ 
 
 00 <N 
 
 1 l> O5 'f O l> CO TH Ol 1>- ^ Oi T}1 OT> CD O OU 
 
 _iiOiO <* rfi CO CO CO <N <N <N TH TH OOOO5 
 
 t> !> t^* I s * t^ t^ t^* i"** * I s * JS t"* IS* t > * 1^ t > * 1 s * CD 
 
 I I I I I .Mill I I I I I III! 
 
 co op co p i> 
 
 %%%E% 
 
 t l> I 
 
 I I I I I 
 
 CO C^ C<l Ci 
 
 CO CO TH 
 
 ~fi r^ j^ LJ L^J 
 ^i CQ CQ CQ CQ 
 
 77777 
 
 CO O5 t^ (N ^ 
 O^ CO C^ C^ "^ 
 (N (N CQ <N TH 
 
 77777 
 
 CO T*< Tt< CO 
 
 (N CO (N Tjn 
 
 TH 10 o^ oo 
 
 i-H ^^ CO O^ 
 
 l> 1> I> CD 
 
 I I I I 
 
 O l> CD O TH 
 O^ 00 00 CO ^^ 
 
 77777 
 
 O O 00 (M 00 
 00 CO 00 ^ O 
 Tt< CO CO CO CO 
 
 77777 
 
 COOO(NC<I CDCOO^O 
 
 N CO O Cfl CO 
 O t^ CO O5 iO 
 
 ro <M c^ TH TH 
 
 77777 
 
 ^ TH iO 00 
 O 00 CO 00 
 rH O O O5 
 
 I I I I 
 
 OS N. lO 00 CD 
 
 ^ t>- Oi CD CD 
 
 O5 O !> ^ OO 
 
 CD 00 CO t^- 1O 
 
 !>. t^ l>- l> l> 
 
 I I I I I 
 
 IO p TH CC| rH 
 
 l> O t^- TH rt< 
 CO IO !> 1C CO 
 
 CD O CO 
 
 O5 TtH TjH CD 
 
 . t^ l> 
 
 I I I I I 
 
 IO CD t^- O5 T-H 
 rH CO t~ O5 CO 
 TfH CO <M i-l T-H 
 
 -# O <N 
 O O5 <M 
 
 T I O O 
 
 I 1 I I I I I I I 
 
 TH O 00 CD 
 
 O5 (M 00 O OS 
 
 1O O5 OS 00 
 
 TH CO CO 00 
 TH CO TH <N 
 ' 00 1> 
 
 O O 
 
 00 00 1> 
 
 I I I I I 
 
 00 (N O5 CO O 
 O iO CO <M 00 
 
 i> t>- 1> i> co 
 
 I I I I I 
 
 00 00 <M O 
 l> 00 OO CO 00 
 CO cq T-H 
 
 CO 
 
 Tt< 
 
 u 
 
 I 
 
 O.J, 
 
 ?S8 R^S2 ^g^2 
 
 rHO5O51>. I>TJHIOCOOO 
 
 !> t^* t*^* t^* V s * 1^* r** CO 
 
 X) 00 
 
 I I I I I 
 
 I I I I I 
 
 r^ CO CO (M 
 
 l> t^ 1> 1> t- 
 
 I I I I I 
 
 O O5 
 
 co >o co co co 
 
 <N <N 
 
 OOOCOOO OTH 
 
 C*5 CO CO I s * ^D t^* C^ 
 
 TH TJH TfH TH Oi CO TH 
 
 ^ ^^ Oi ^^ !> !> 1 s * CO C^ O^ 
 
 ooot^oot^ t^t^t^-t^co 
 
 I I I I I I I I I I 
 
 CO t> CO O5 
 
 00 co o TH 
 
 01 OS t^- CO 
 
 I I I I I I I I I 
 
 COCOOOO5O5 
 
 lOTHOOiOTH COiOOOOOiO 
 
 OCOIN-^iO OOCOO5O 
 
 COTHTHO5O5 OOt^-COCOCO 
 
 0000001>1>- 1>1>1>1>I> 
 
 I I I I I I I I I I 
 
 CQ I> d TH TH 
 
 i> co o "* oo 
 
 Tt< -^ rfi CO <N 
 
 77777 
 
 (N TH p 00 
 
 i> F^ t^ i> 
 
 I I I I 
 
 00 O5 TH p Tj< 
 
 l> ^ 00 C5 
 00 00 t>- J> 
 
 Mill 
 
 p Oi Tf< TH 
 
 CO 00 00 TH 
 l>* O^ Oi ^^ 
 l> CO CO CO 
 
 7 I I M 
 
 <N 
 
 IO O5 CO O5 00 
 
 8 
 
 O5 CO CO Tt< 
 00 O TH 
 
 CO H/l CO (M 
 
 > t^ 1> l> 1> 
 
 I I I I I 
 
 TH CO 1C TH 
 
 CO CO GO O 
 
 OJ <M CO O 
 
 TH TH O O 
 
 CD 05 (N CD b- TH 00 (N CO CO 
 
 lOOOCOCOCO i-HCOl-^C^CO 
 
 C^ C^ TH TH CO OS 00 t^ CO IO 
 
 oooooooooo i>i>t^-r>i> 
 
 I I I I I I I I I I 
 
 TH CO CO CD <N 
 
 CD 00 TH 10 O 
 
 t^ CO 00 CO 00 
 
 T}H TfH CO CO (N 
 
 CO O O5 TH 
 
 CO O O TH 
 
 TH IO 00 TH 
 
 C<l TH O O 
 
 I I I I I I I I I 
 
THERMODYNAMIC TABLES 
 
 129 
 
 88 
 
 TH C3 Tf 
 
 t>* CQ CO TH cO CO 
 
 T^H CO CO Tt^ CO Oi 
 
 00 TH Tj< 1>- TH IO 
 
 -i TH TH TH TH <N <N CQ CO CO 
 
 I I I I I I I I I I 
 
 00 iO TH CO 
 
 <N CN O TH 
 
 TH CO C^l 00 ^i^ 
 
 rfri rj< iO iO CO 
 
 I I I I I 
 
 t^OOOO 
 00 t^ OO T- i 
 
 oo <M i t co 
 
 O OO CO iO 
 l> 1> OO O5 
 
 I I I I 
 
 ^^ O^ 
 
 l> O^ 
 
 00 <M 
 
 TH CO 
 
 CO Ci 00 l> CO TH O 
 Oi CO Tt< t^ <N CO TJH 
 
 i> TH 
 
 00 lt TjH 1> TH CO 
 
 HrHTHi-HTH <N<N<NCOCO 
 
 I I I I I I I I I I 
 
 (M >O (M TH rfi 
 TH CO C^ 00 ^ 
 
 7 7 tTT 
 
 ^Soi 
 i i i 
 
 I I I I I 
 
 (N CO 00 iO 
 
 Oi i i >O O5 
 
 CO C<1 O5 O5 
 
 i i Oi CO >O 
 
 O5 CO CO I> CO CO O 
 
 Oi (N t^ 00 TH 
 
 _ . rj< I> (N <N O 
 iO t^ Oi 
 
 <N O 10 
 !> i I >O 
 
 I I I I I I I I I I 
 
 I I I I I 
 
 t^ 00 O IO 
 
 M O5 t^ CO 
 I> t>- 00 OS 
 
 I I I I 
 
 Mil! 
 
 OO CO C^ CO CO 
 (N <M (N (N CO 
 
 I I I I I 
 
 OOOO^CO CONOCO 
 
 I I I I I 
 
 I I I I 
 
 rJH 00 
 
 ?SS8 
 
 ^ 
 
 TH TH TH TH C^ 
 
 I I I I I 
 
 <N (N <M CO CO 
 I I I I I 
 
 Oi IO l> 00 TH CO 
 
 co^oioco t^t^ooo 
 
 I I I I I I I I I 
 
 TjH TH Oi 
 
 &~~ 
 
 CO 
 
 7 I I i I 
 
 rt< 1> 00 
 
 CO 
 
 CO 
 
 00 _ 
 
 Tt^OOSTHCO f~-<McOOO C<l 
 
 <M>OI^OTH OO-^Ot* 1 ^ THOi|>cO 
 
 C^IMC^COCO COr^iOiOCO 1>-I>OOO5 
 
 I I I I I I I I I I I I I I 
 
 00 
 CO 
 
 .._323 
 
 O5 iO O O> 00 
 CO iO l> OO O 
 rH TH TH TH C^l 
 
 I I I I I 
 
 OTH OOI>COcOiO lOCOTtn 
 t--O COI>C^COI> 00<N(N 
 (MO5 O5CO-^OOOO OOcO 
 
 OO500OTH OOI>COCO 
 
 t^co-^t^o coi>coco 
 
 O5 co O5 c^ Oi Oi co ^* oo 
 
 c^i *o t^" co ^f oo ^^ co co co TH Oi t*** 
 
 d(M<NCOCO COTtHiOiOCO l>t>-00 
 
 I I I I I I I I 1 I I I I I 
 
 iO 
 
 IO 
 
 1-1 O 
 
 I I I I I 
 
 (M O (N 
 CO iO CO !> C^ 
 CO 00 CO CO "* 
 Tt< IO l> O5 t-H 
 TH i-H ^H TH C<l 
 I I I I I 
 
 rH Tf t^ C^ CO Oi C^ ^^ C^l CO C^ CO C^l CO 
 
 OilS-OOOiO COOOi^fb- THOO^f'^f 
 
 ^COOOi-HO Oi-^Ot^CO THOit>-cO 
 
 NC^C<JCOCO COrtiiOtOcO l>I>OOOi 
 
 I I I I I I I I I I MM 
 
 s 
 I I 
 
130 
 
 THERMODYNAMIC METEOROLOGY 
 
 TABLE 41 
 
 THE PLANETARY CIRCULATION AND RADIATION 
 The coefficient log c in log K 10 = log c + a log T 10 . . . (344) 
 
 e 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 
 
 19000 
 
 -5.383 
 
 -5.416 
 
 -5.386 
 
 -5.380 
 
 -5.381 
 
 -5.378 
 
 -5.361 
 
 -5.387 
 
 -5.419 
 
 -5.433 
 
 18000 
 
 -5.404 
 
 -5.434 
 
 -5.408 
 
 -5.406 
 
 -5.404 
 
 -5.404 
 
 -5.387 
 
 -5.409 
 
 -5.426 
 
 -5.449 
 
 17000 
 
 -5.423 
 
 -5.445 
 
 -5.431 
 
 -5.426 
 
 -5.418 
 
 -5.424 
 
 -5.418 
 
 -5.439 
 
 -5.454 
 
 -5.465 
 
 16000 
 
 -5.436 
 
 -5.474 
 
 -5.454 
 
 -5.449 
 
 -5.444 
 
 -5.447 
 
 -5.435 
 
 -5.454 
 
 -5.469 
 
 -5.480 
 
 15000 
 
 -5.459 
 
 -5.486 
 
 -5.475 
 
 -5.468 
 
 -5.460 
 
 -5.466 
 
 -5.477 
 
 -5.463 
 
 -5.481 
 
 -5.491 
 
 14000 
 
 -5.481 
 
 -5.503 
 
 -5.496 
 
 -5.489 
 
 -5.484 
 
 -5.489 
 
 -5.490 
 
 -5.496 
 
 -5.504 
 
 -5.500 
 
 13000 
 
 -5.503 
 
 -5.522 
 
 -5.520 
 
 -5.513 
 
 -5.494 
 
 -5.504 
 
 -5.503 
 
 -5.518 
 
 -5.521 
 
 -5.510 
 
 12000 
 
 -5.526 
 
 -5.526 
 
 -5.542 
 
 -5.538 
 
 -5.510 
 
 -5.523 
 
 -5.507 
 
 -5.505 
 
 -5.507 
 
 -5.515 
 
 11000 
 
 -5.559 
 
 -5.571 
 
 -5.56'0 
 
 -5.544 
 
 -5.571 
 
 -5.540 
 
 -5.514 
 
 -5.510 
 
 -5.513 
 
 -5.518 
 
 10000 
 
 -5.578 
 
 -5.580 
 
 -5.589 
 
 -5.560 
 
 -5.547 
 
 -5.531 
 
 -5.512 
 
 -5.511 
 
 -5.513 
 
 -5.519 
 
 9000 
 
 -5.606 
 
 -5.601 
 
 -5.595 
 
 -5.589 
 
 -5.569 
 
 -5.552 
 
 -5.526 
 
 -5.511 
 
 -5.513 
 
 -5.519 
 
 8000 
 
 -5.614 
 
 -5.611 
 
 -5.596 
 
 -5.603 
 
 -5.563 
 
 -5.555 
 
 -5.524 
 
 -5.514 
 
 -5.517 
 
 -5.524 
 
 7000 
 
 -5.622 
 
 -5.618 
 
 -5.608 
 
 -5.608 
 
 -5.576 
 
 -5.556 
 
 -5.557 
 
 -5.524 
 
 -5.526 
 
 -5.531 
 
 6000 
 
 -5.628 
 
 -5.626 
 
 -5.610 
 
 -5.607 
 
 -5.577 
 
 -5.558 
 
 -5.536 
 
 -5.532 
 
 -5.537 
 
 -5.543 
 
 5000 
 
 -5.637 
 
 -5.627 
 
 -5.615 
 
 -5.615 
 
 -5.581 
 
 -5.564 
 
 -5.547 
 
 -5.531 
 
 -5.527 
 
 -5.536 
 
 4000 
 
 -5.638 
 
 -5.635 
 
 -5.622 
 
 -5.600 
 
 -5.589 
 
 -5.574 
 
 -5.534 
 
 -5.537 
 
 -5.542 
 
 -5.548 
 
 3000 
 
 -5.656 
 
 -5.646 
 
 -5.628 
 
 -5.616 
 
 -5.588 
 
 -5.561 
 
 -5.546 
 
 -5.554 
 
 -5.555 
 
 -5.561 
 
 2000 
 
 -5.664 
 
 -5.652 
 
 -5.634 
 
 -5.614 
 
 -5.597 
 
 -5.582 
 
 -5.561 
 
 -5.588 
 
 -5.546 
 
 -5.549 
 
 1000 
 
 
 
 
 
 
 
 
 
 
 
 000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 The minus sign affects only the characteristic log c = 5.664. 
 
 c = 4.61 X 10 ~ 
 
 TABLE 42 
 THE EXPONENT a IN K w = c T i0 a . . . (344) 
 
 z 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 
 
 19000 
 
 3.740 
 
 3.755 
 
 3.741 
 
 3.739 
 
 3.739 
 
 3.738 
 
 3.730 
 
 3.742 
 
 3.757 
 
 3.763 
 
 18000 
 
 3.750 
 
 3.763 
 
 3.752 
 
 3.750 
 
 3.750 
 
 3.749 
 
 3.742 
 
 3.752 
 
 3.764 
 
 3.770 
 
 17000 
 
 3.758 
 
 3.768 
 
 3.762 
 
 3.760 
 
 3.756 
 
 3.759 
 
 3.756 
 
 3.766 
 
 3.772 
 
 3.777 
 
 16000 
 
 3.764 
 
 3.781 
 
 3.772 
 
 3.770 
 
 3.768 
 
 3.769 
 
 3.764 
 
 3.772 
 
 3.779 
 
 3.783 
 
 15000 
 
 3.775 
 
 3.786 
 
 3.782 
 
 3.778 
 
 3.775 
 
 3.778 
 
 3.782 
 
 3.776 
 
 3.784 
 
 3.789 
 
 14000 
 
 3.784 
 
 3.794 
 
 3.791 
 
 3.788 
 
 3.785 
 
 3.788 
 
 3.788 
 
 3.791 
 
 3.796 
 
 3.793 
 
 13000 
 
 3.794 
 
 3.803 
 
 3.802 
 
 3.798 
 
 3.790 
 
 3.794 
 
 3.794 
 
 3.801 
 
 3.802 
 
 3.797 
 
 12000 
 
 3.804 
 
 3.804 
 
 3.812 
 
 3.810 
 
 3.797 
 
 3.803 
 
 3.796 
 
 3.795 
 
 3.796 
 
 3.799 
 
 11000 
 
 3.820 
 
 3.825 
 
 3.820 
 
 3.813 
 
 3.825 
 
 3.811 
 
 3.799 
 
 3.797 
 
 3.799 
 
 3.801 
 
 10000 
 
 3.828 
 
 3.829 
 
 3.833 
 
 3.820 
 
 3.814 
 
 3.807 
 
 3.798 
 
 3.798 
 
 3.799 
 
 3.801 
 
 9000 
 
 3.841 
 
 3.838 
 
 3.835 
 
 3.833 
 
 3.824 
 
 3.816 
 
 3.804 
 
 3.798 
 
 3.799 
 
 3.802 
 
 8000 
 
 3.844 
 
 3.843 
 
 3.836 
 
 3.833 
 
 3.822 
 
 3.818 
 
 3.804 
 
 3.799 
 
 3.800 
 
 3.804 
 
 7000 
 
 3.848 
 
 3.846 
 
 3.841 
 
 3.833 
 
 3.823 
 
 3.818 
 
 3.818 
 
 3.804 
 
 3.804 
 
 3.807 
 
 6000 
 
 3.850 
 
 3.850 
 
 3.842 
 
 3.831 
 
 3.828 
 
 3.819 
 
 3.809 
 
 3.807 
 
 3.810 
 
 3.812 
 
 5000 
 
 3.854 
 
 3.850 
 
 3.844 
 
 3.837 
 
 3.829 
 
 3.822 
 
 3.814 
 
 3.807 
 
 3.805 
 
 3.809 
 
 4000 
 
 3.855 
 
 3.854 
 
 3.848 
 
 3.838 
 
 3.833 
 
 3.826 
 
 3.808 
 
 3.810 
 
 3.812 
 
 3.814 
 
 3000 
 
 3.863 
 
 3.858 
 
 3.850 
 
 3.845 
 
 3.832 
 
 3.820 
 
 3.814 
 
 3 . 817 
 
 3.818 
 
 3.820 
 
 2000 
 
 3.867 
 
 3.862 
 
 3.853 
 
 3.844 
 
 3.836 
 
 3.830 
 
 3.820 
 
 3.832 
 
 3.814 
 
 3.815 
 
 1000 
 
 
 
 
 
 
 
 
 
 
 
 000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 The exponent a - 3.50 in dry air; a = 4.00 in a perfect radiator. 
 
THERMODYNAMIC TABLES 
 
 131 
 
 V) 
 
 d 
 1 
 S 
 
 00 00 TH t^. TH 
 
 IO tO TH TH O 
 
 CM CM CO CM O 
 CO CO CO CO CO 
 
 II II 1 
 
 O O TH o t> 
 
 TH T-H 00 CO CO 
 
 777 i i 
 
 
 t> CO CO CM CO 
 
 00 t^ <0 T) rH 
 
 
 
 O 
 
 Oi TH CO CM to 
 i 1 CO TH TH CO 
 CO CO CO CO CM 
 
 II II 1 
 
 77 ii i 
 
 
 TH CO t 00 b- 
 
 CT. CO 00 TH O> 
 
 
 
 T 1 tO TH CO TH 
 
 CO TH TH CO CO 
 CO CO CO CO CM 
 
 II II 1 
 
 gggSS; 
 
 777 i i 
 
 o 2 
 
 CM 
 
 
 
 00 TH CM O CO 
 O5 Ci 1> to CM 
 CO CO CO CM CM 
 II II 1 
 
 TJH CO CO t* lO 
 
 ggss^ 
 
 77 i i i 
 
 o 2 
 
 O 
 co w 
 
 i 
 
 -* 00 00 O5 to 
 
 TjH CM CO CO CO 
 
 CO CO CO CO CM 
 
 II II 1 
 
 to 0> CD 
 
 ^- CO 01 CO CO 
 10 OJ TH 0* 
 
 7 i ii i 
 
 g 
 
 CO t^ CO CM CM 
 
 CM T^ TH IO Tf 
 
 CO CO CO CO CO 
 
 II II 1 
 
 to 0> CD i-l 
 
 to O CD IO IO 
 CO CM iH W * 
 
 1 II + 1 
 
 en 
 
 W 
 
 CO CM CO ^ to 
 
 TH CO TH 00 CD 
 
 * 
 
 to til 
 
 < 
 
 
 
 i 
 
 CO CO CO CO CO 
 
 CO TH co O t- 
 
 777 i i 
 
 3 
 
 w 
 
 T* to to to CO 
 
 CM CO tO CO CD 
 
 ^ *- 
 8 * 
 
 5 
 
 O CM CO CO CO 
 CO CO CO CO CO 
 
 TH CO to CO CO 
 CO t>- !> O 00 
 
 7777 i 
 
 1. 
 
 10 CM 05 TH TH 
 
 t^. o TH to oo 
 
 fe w 
 
 t, 
 
 U 
 
 CO to O5 Oi ^ 
 
 co co co co co 
 
 II II 1 
 
 OO CO 0^ TH CO 
 tO to t TH CM 
 
 CO CM TH TH TH 
 
 Mill 
 
 
 CO l> CM CM 00 
 
 CO CM CO TH O5 
 
 b 
 
 00 !> CO to TH 
 CM CM CO CO CO 
 
 II II 1 
 
 77777 
 
 
 CO CM O CM tO 
 
 00 CO CO CO CO 
 
 o 
 
 8 
 
 CM CO O5 CM t^ 
 00 O^ ^^ CM CO 
 CM CM CO CO CO 
 
 Mill 
 
 to O5 CO O5 CO 
 
 TH CO O5 00 *> 
 
 CO CM TH TH TH 
 
 II II 1 
 
 N 
 
 O5 00 t>- CO to 
 
 TH CO CM TH O 
 
 
 
 
132 
 
 THERMODYNAMIC METEOROLOGY 
 
 
 CO r^ rH 
 
 CO <M iO O 00 
 
 rH CO CO O5 <N 
 
 T-H CD CO 00 
 
 rH T-H T-H 
 
 
 GO !> O5 O O5 
 
 rH t>- iO CO O5 
 
 l>- CD CO O5 CO 
 
 rH CO (M rH O5 
 
 O C^ to !> iO 
 
 O O (M l>- 
 
 CO CO t^- 
 
 T-H 00 to 
 
 
 M M 1 
 
 M M 1 
 
 Mill 
 
 i M i : 
 
 1 1 1 
 
 
 rH (M O 00 CO 
 
 GO T-H O5 CO t* 
 
 Oi CO TH CN! CO 
 
 s ^ g g 
 
 O CO CO tO (M 
 CO 0i 1 s ** 1 s * CO 
 
 GO 00 t^ ^' 
 
 co co i> e* ' 
 
 00 O 00 
 
 co oo ^ 
 
 
 M M 1 
 
 M M 1 
 
 M M 1 
 
 MM: 
 
 1 1 1 
 
 
 <NTi^<N CO 
 
 O O OS t- iH 
 
 00 O 00 CO 00 
 
 t-t-(NlO 
 
 <N CO (N 
 
 
 82^SS? 
 
 t2*oco8 
 
 t* CO r^ (M t^ 
 t T-H rJH CO CO 
 
 S^SS : 
 
 rH CO t^ 
 
 
 M 1 M 
 
 M M 1 
 
 M M 1 
 
 MM: 
 
 1 1 1 
 
 
 
 <N rh C5 GO O 
 
 t- * 
 
 1-1 tO rH Tfrl O5 
 
 CO rfl to tO 
 
 <*1U3 
 
 
 O 
 
 ^OGoScO 
 
 o^ooco^ 
 
 O rJH Oi (N CO 
 
 00 (M iO *tf 
 
 tO 00 (N CO 
 
 tO TjH 00 
 
 w 
 
 & 
 
 M M 1 
 
 M M 1 
 
 M M 1 
 
 MM: 
 
 1 1 1 
 
 d 
 
 tO O to ^ ^O 
 
 T-H XO t- 00 O* 
 
 O CO (M 1> tO 
 
 tOOOlO 
 
 CO to 00 
 
 M 
 M 
 
 O Tt< TH T-H O 
 CO >O ^t 1 Tf t>- 
 
 oo co *-i oa oo 
 
 O Tt< CO >0 (M 
 
 iF2^^ : 
 
 23 
 
 1 
 
 M M 1 
 
 M M + 
 
 M M 1 
 
 MM: 
 
 1 1 1 
 
 HH 
 
 w 
 
 1 
 
 O5 O5 O5 CO CO 
 
 to CO T-H T-H CO 
 GO O O (N T-H 
 
 <N CO l> CO CO 
 
 1- i CO O Tji Tjf 
 
 T-H iO CD O t" 
 
 r-i rH CO <N to 
 IO to OO tO rfi 
 
 O5 O5 CO iH 
 rH O5 O C* 
 
 CO CO TJ< <O 
 
 CO O5 to 
 CO 00 t^ 
 
 Tf r-H CD 
 
 g 
 
 i 1 1 1 1 
 
 1111 + 
 
 M M 1 
 
 MM: 
 
 1 1 1 
 
 H 
 
 M 
 
 O5 CO CO C^l *O 
 
 ^H 10X00X0 
 
 t- IO 00 rH 00 
 
 (M l> CO O> 
 
 00 rH |>. 
 
 5 
 
 H 
 
 <; 
 
 CO CO O^ ^^ CO 
 
 t^ 00 CO O 00 
 (N l> (M ^ O 
 
 |> IO O T-H (N 
 
 IO 00 (M to CO 
 
 O5 CO OO TH 
 to to CO O 
 
 >O 00 (M 
 CO CO 2 
 
 M 
 i 
 
 M M 1 
 
 11-11 + 
 
 M M 1 
 
 MM: 
 
 1 1 1 
 
 a 
 
 rH O5 IO (N 10 
 
 O5 ^^ CO C^ CO 
 
 o e* co co GO - 
 
 Tt< t^ l> 
 
 O l> CO 
 
 t 1 
 
 w 
 
 i 1 Tt< CO -tf CO 
 
 t^ !> GO GO CO 
 
 M S 0> W 
 
 CO 00 CO CO CO 
 
 O> t* ^f Th CO 
 
 T^ Oi ^^ O5 
 
 tO 00 CD 
 (M T-H 10 
 
 O 
 
 
 
 M M 1 
 
 1 1 1 1 + 
 
 M M 1 
 
 MM: 
 
 1 1 1 
 
 1. 
 
 b- co TJH co oo 
 
 CO CO 00 l> t^ 
 
 to <M i> co co 
 
 CO ^t 1 to CO 
 
 <Nt- to 
 
 1 
 
 w 
 
 to to CO CO 00 
 
 c^^ooc^w 
 
 rH CO 00 CO 00 
 
 -tf co TH co oo 
 
 CD CD rH IO 
 
 O5 rH CO C?fl 
 
 00 C<1 00 
 
 rH O5 t^ 
 
 en 
 
 3 
 
 M M 1 
 
 M M 1 
 
 M M 1 
 
 MM: 
 
 1 1 1 
 
 
 T-H T 1 !> GO Tf 
 
 00 l> O CO 00 
 
 05 05 rH ^HT^ 
 
 rH^ <N 
 
 CO T-H Tj< 
 
 
 co2o! 
 
 O W T-H 00 t^ 
 (M rf iO (M l> 
 
 (N TJH (N CO O 
 
 to to iO to OO 
 
 o coS 
 
 T^oJoo 
 
 
 M M 1 
 
 M M 1 
 
 M M 1 
 
 MM: 
 
 1 1 1 
 
 
 O5 O5 CO O5 t^- 
 
 rt< l> CO O5 (M 
 
 <N TH 05 
 
 00505-* 
 
 T-H CO 
 
 
 TH Tji CO rH CO 
 -# TjH t^ 00 Oi 
 
 M M 1 
 
 ^^ C^l CO *-O CO 
 CO CO CO CO (M 
 
 M M i 
 
 l> 00 rH (M tO 
 CO CO l> O^ ^H 
 
 1 1 T 1 T 1 1 i C^ 
 
 M M 1 
 
 ?ff!i 
 
 777 
 
 
 O5 00 l> CO >O 
 
 ^ CO C^ rH O 
 
 O5 00 l>- CD >O 
 
 rj< CO <M rH O 
 
 
 
 
 
 
 
 
SEVERAL IMPORTANT CONCLUSIONS 133 
 
 The Case Ii of low temperatures in the isothermal region 
 shows more rapid radiation than Case L of high temperatures, 
 but both indicate about twice as much radiation as in the con- 
 vectional region C. There are two special sources of heat, (1) 
 in the upper cirrus region, and (2) at the surface of the earth, 
 or in the lower cumulus region. The incoming radiation of the 
 sun is divided into two nearly equal parts, the effective radiation 
 penetrating to the surface, but diminishing in intensity in and 
 below the cirrus region, 30 to 10 kilometers, and the returning 
 radiation neutralizing an equal amount of the solar radiation. 
 Pyrheliometer observations at the surface give 2.00 calories per 
 square centimeter per minute, and the bolometer observations 
 probably require the solar constant of 4.00 calories. Hence 
 2.00 calories are penetrating to the surface and 2.00 calories are 
 returning to space. 
 
 Several Important Conclusions 
 From the formulas, 
 
 g * - * = - ~ - <?0 - - , 
 
 g Pio (2i -Z ) = - (Pi - Po) ~ J Pio (?i 2 - ?o 2 ) - Pio (Qi-Qo), 
 we must lay down several propositions. 
 
 (1) There is properly no such a thing as purely dynamic 
 meteorology, which is defined as a balance between the three 
 terms, 
 
 g 
 
 pio 
 
 since these conditions are fulfilled in nature only under tem- 
 porary circumstances. 
 
 (2) In Mar gules' paper, "Ueber die Energie der Sturme," 
 Jahrbuch der K. K. Central-Anstalt fur Meteorologie und 
 Erdmagnetismus, Wien, 1903, the formulas are nearly all 
 adiabatic, the gas coefficient R and the specific heat Cp are 
 constant, so that the radiation term (Qi Qo) cannot be com- 
 puted, and there is no balance possible among the other terms. 
 
 (3) In V. Bjerknes' paper, " Dynamic Meteorology and 
 
134 THERMODYNAMIC METEOROLOGY 
 
 Hydrography," Carnegie Institution of Washington, D. C., 1910, 
 the density is computed by formula (175) instead of by (176), 
 and it is, therefore, a mixed system, since the pressure depends 
 upon (172), and R, Cp, are taken constant, so that there is no 
 theoretical circulation and radiation to be computed. 
 
 (4) In Gold's paper, "The Isothermal Layer of the Atmo- 
 sphere and Atmospheric Radiation," Proceedings of the Royal 
 Society, A, Vol. 82, the assumption is made that the mass is 
 proportional to the pressure. This omits the important terms 
 I PIO (<?i 2 <?o 2 ), PIO (Qi Qo), involving the circulation and 
 radiation, so that the dependent formulas are not properly sup- 
 ported, because the adiabatic case is in reality assumed. 
 
 (5) There is a very large literature in meteorology based upon 
 attempts to make a balance of the equation, 
 
 P - - P 
 
 g (zi - * ) = - - - - i (?i 2 - ?o 2 ), 
 
 Pio 
 
 but it is in reality without substantial importance. In spite 
 of the serious difficulty that exists in determining the (QiQo) 
 term, it is necessary that this should be done. There are several 
 large observatories for balloon and kite ascensions which record 
 pressure and temperature, but not humidity and wind velocity, 
 and these entirely fail of their purpose in advancing the interests 
 of meteorology. 
 
 (6) In spite of the fact that only thermodynamic meteor- 
 ology can have any permanent value in science, there are yet 
 many subordinate problems in the atmosphere which are to be 
 studied without the heat term, such as the stream lines of the 
 circulation, however their forces may have been developed. Ac- 
 cordingly, we proceed, under dynamic meteorology, to deduce 
 the general equations of motion, in the rectangular, the cylin- 
 drical, and the polar co-ordinate systems, together with several 
 minor terms, in order to study their application in local storms 
 and circulations of various types. 
 
CHAPTER III 
 
 The Hydrodynamics of the Atmosphere 
 The Co-ordinate Axes 
 
 THE general equations of motion will be assumed from 
 hydrodynamics, because they are well known, and the proof is 
 
 Zenith 
 
 y / 
 
 South 
 
 x, y, z (Rectangular) 
 
 accessible in many treatises, but 
 the equations needed in meteor- 
 ology will be deduced from them 
 as briefly as possible. There are 
 three systems of co-ordinates, the 
 rectangular, the cylindrical, and 
 the polar, as represented in the 
 diagrams of Fig. 14. 
 
 Starting at the point as the 
 origin of co-ordinates, one can 
 reach the point P directly along 
 the line P, or indirectly, in 
 rectangular co-ordinates along the 
 
 distances x, y, z, in succession parallel to the axes; in cylindrical 
 co-ordinates along the radius or, through an angle <f> counted 
 
 135 
 
 r, \ B (Polar) 
 
 FIG. 14. Three systems of co- 
 ordinate axes. Compare Fig. 2. 
 
136 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 from an initial line of reference where = 0, and along z to 
 P; in polar co-ordinates along a line r whose position in space 
 is determined by the angle A in the plane x y counted from an 
 initial line at the axis of x on that plane, and in the plane z v 
 at the angular distance 6 from the axis of rotation z. These 
 systems are convenient in different problems and must each 
 be developed. 
 
 The Co-ordinate Velocities 
 
 If q is the velocity along the line O P with which a mass 
 is moving, the co-ordinate velocities are as follows: 
 
 Rectangular Cylindrical Polar 
 
 (352) u = -jfi (353) u=^' (354) u - r^y 
 
 dy d<t> d^ 
 
 v = dT "*'*Tf V = rsm9 d-i 
 
 
 
 dz dz dr 
 
 The Co-ordinate Accelerations 
 
 d t d t 2 d ' 
 
 dw d*z . d z z . ' d z r 
 
 The Constituents of the Force in Any Direction 
 
 Force is measured by the acceleration of a mass in any 
 direction, because it takes force to change the velocity of the 
 
THE CONSTITUENTS OF THE FORCE IN ANY DIRECTION 137 
 
 mass m at a given point. This will be taken as unity, m = 1, in 
 the preliminary equations. A mass of gas or liquid can undergo 
 changes in inertia, changes in volume by expansion or contrac- 
 tion, and changes in figure by rotation, and these three types of 
 forces must be placed in the general equations of motion. The 
 causes that produce these forces in the atmosphere of the earth 
 are external and internal. The external forces are due to 
 gravitation or the potential changes with position; the internal 
 forces are due to pressures which vary in different directions, 
 and cause motion during the restoration to normal equilibrium. 
 The primary source of these pressure forces is the distribution of 
 the thermal energy derived from the solar radiation, or trans- 
 ported in currents of circulation. 
 
 The Force of Inertia 
 
 This is a partial differential of the velocity in the direction 
 of the co-ordinate axes, and is 
 
 du dv dw 
 
 in each of the co-ordinate systems. 
 
 The Forces of Expansion or Contraction 
 
 It is convenient to express the differentials of the linear dis- 
 placements in each co-ordinate system, referred to the x, y, z 
 system. 
 
 Rectangular Cylindrical Polar 
 
 (359) 3 # = 3 # (360) 3 x = 3 w. (361) 3 x = r 3 fr 
 
 dy = d y> dy = ttr 30. 3 ? = r sin 03 A. 
 
 82 = 32. 32 = 32. 3^ = 3 r. 
 
 Using these values of 3 x, 3 y, 3 2, we obtain for the accelera- 
 tions due to expansion or contraction: 
 
138 
 
 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 Rectangular 
 
 , x 3# du 3w 
 
 (362) u + v^- + w. 
 
 dx dy 82 
 
 3fl dv dv 
 
 u + v -r 1- w ^ . 
 80; dy 32 
 
 u 
 
 Cylindrical 
 
 dv 
 
 3 z> 
 v - 
 
 dw 
 
 ^ 
 dy 
 
 . 
 
 dz 
 
 + V 
 
 to- 90 
 
 dz' 
 
 d~z 
 dw 
 d~z 
 
 (364)^+,-, 
 
 du 
 
 32; 
 
 dw dw dw 
 
 U h V ; ~ + W . 
 
 rd& r sin 3 A 3 r 
 
 In order to give some practical idea of the meaning of these 
 terms, the following example is taken from the Cottage City 
 water-spout, without explanation. 
 
 TABLE 44 
 THE RADII IX AND VELOCITIES , v, w, IN (1) OUTER AND (2) INNER TUBES 
 
 Height 
 
 Radius W 
 (1) (2) 
 
 Velocity u 
 
 (1) (2) 
 
 Velocity v 
 (1) (2) 
 
 Velocity w 
 (1) (2) 
 
 az 90 
 
 83.3 
 
 51.9 
 
 
 
 
 
 
 11.52 
 
 18.49 
 
 0.77 
 
 1.98 
 
 80 
 
 84.0 
 
 52.3 
 
 - 2.00 
 
 - 3. 
 
 21 
 
 11.34 
 
 18.20 
 
 0.76 
 
 1.95 
 
 70 
 
 86.0 
 
 53.6 
 
 - 3.94 
 
 - 6. 
 
 32 
 
 10.82 
 
 17.37 
 
 0.72 
 
 1.86 
 
 60 
 
 89.6 
 
 55.8 
 
 - 5.76 
 
 - 9. 
 
 24 
 
 9.97 
 
 16.01 
 
 0.66 
 
 1.71 
 
 50 
 
 95.2 
 
 59.3 
 
 - 7.40 
 
 -11. 
 
 88 
 
 8.82 
 
 14.16 
 
 0.58 
 
 1.52 
 
 40 
 
 103.9 
 
 64.8 
 
 - 8.82 
 
 -14. 
 
 16 
 
 7.40 
 
 11.88 
 
 0.50 
 
 1.27 
 
 30 
 
 117.9 
 
 73.4 
 
 - 9.97 
 
 -16. 
 
 01 
 
 5.76 
 
 9.24 
 
 0.38 
 
 0.99 
 
 20 
 
 142.5 
 
 88.8 
 
 -10.82 
 
 -17. 
 
 37 
 
 3.94 
 
 6.32 
 
 0.26 
 
 0.68 
 
 10 
 
 200.0 
 
 124.6 
 
 -11.34 
 
 -18. 
 
 20 
 
 2.00 
 
 3.21 
 
 0.13 
 
 0.34 
 
 
 
 00 
 
 00 
 
 -11.52 
 
 -18. 
 
 49 
 
 
 
 
 
 
 
 
 
 It is seen that the velocities in meters per second in the 
 lower half of the dumb-bell vortex of this water-spout undergo 
 changes from the vortex tube (1) to the vortex tube (2), and that 
 as the radius w and the height az change, these velocities change 
 by certain laws. By taking the differences 3w, 3, 3w, 3 -or, 
 
THE FORCES OF ROTATION 139 
 
 or 3 0, 3 z in cylindrical co-ordinates, and using the mean values 
 of u m , v m , w m , the forces that caused these changes from point 
 to point in the vortex can be computed. 
 
 The Forces of Rotation 
 
 If a mass moves relatively to fixed axes without rotation there 
 are no forces other than inertia and compression or expansion. 
 If the mass also rotates, a new set of forces of rotation is intro- 
 duced which may be analyzed as follows: Assume that there is 
 a set of rectangular axes fixed in space, and that another set of 
 
 4-zo> 2 
 
 VU 1 
 
 +y<i 
 
 FIG. 15. Angular velocities of motion about fixed axes. 
 
 axes attached to the rotating body moves relatively to the fixed 
 axes with the co-ordinate angular velocities wi about the axis 
 x, o) 2 about the axis y, and ^ 3 about the axis z. The entire 
 system of rotational velocities at the distances x, y, z, from the 
 origin of rotation is shown on Fig. 15. 
 
 The right-handed rotation is defined as that in which a 
 right-handed screw is turned to advance while the axis x moves 
 toward y, the axis y toward z, the axis z toward x, in transla- 
 tion along z, x, y, respectively, and all co-ordinates, of fixed and 
 moving axes, are so related. Thus, in rotation about z, with 
 the angular velocity, ^ 3 , at the distance x there is instantaneous 
 
140 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 velocity + x co 3 parallel to the axis y } and at the distance y 
 there is instantaneous velocity y co 3 parallel to the axis x, 
 that is, in the negative direction. The same considerations give 
 the component linear velocities parallel to the axes. 
 
 (365) Parallel to the axis x, y co 3 + z o> 2 , 
 Parallel to the axis y, z coi + x co 3 , 
 Parallel to the axis z, x 0)2 + y + y "i- 
 
 These symbols are all arranged in the cyclic order, and are easily 
 verified from Fig. 15. 
 
 Similarly the component accelerations are found by substi- 
 tuting u, v, Wj for x, y, z, in succession, and we have the accelera- 
 tions parallel to the axes: 
 
 (366) Parallel to the axis x, v co 3 + w co 2 , 
 Parallel to the axis y, w wi + u co 3 , 
 Parallel to the axis z, u o> 2 + v coi. 
 
 Since these forms are entirely general, it is only necessary 
 to substitute the special values of i, co 2 , co 3 , for given cases, to 
 apply the formulas to particular problems. If only fixed axes 
 are employed, we have i = 0, co 2 = 0, co 3 = 0. If cylindrical 
 axes are employed, there is rotation about the axis z only, so 
 
 u 
 that coi = 0, co 2 = 0, co 3 = -f- . If polar co-ordinates are 
 
 employed the angular velocities become i = -- , co 2 = + 
 
 o> 3 = + - -. Placing these results in tabular form, 
 Y tan u 
 
 have, as can be seen by the definition of angular velocity, 
 
 Rectangular 
 Fixed Axes 
 
 Cylindrical 
 Co-ordinates 
 
 Polar 
 Co-ordinates 
 
 (367) 
 
 coi == 
 
 (368) i = 0. 
 
 (369) coi = - - |. 
 
 
 C0 2 = 0. 
 C0 3 = 0. 
 
 co 2 = 0. 
 V 
 
 w 
 
 <* = + 7- 
 
 y l 
 
 0)8 + w' 
 
 
THE FORCES OF ROTATION 
 
 141 
 
 In cylindrical co-ordinates the angle increases with the 
 angular velocity ^3 = ^* In polar co-ordinates the angular 
 
 velocities may be illustrated by Fig. 16. 
 
 Let a mass move from P to P 3 relative to the rotating earth, 
 whose axis of rotation is Z. Lay down the fixed axes x, y, z, 
 
 Pole 
 
 FIG. 1 6. Angular velocities of moving axes relative to fixed 
 
 at the center O, draw the radius r to P Q at the polar distance 0, 
 in longitude A counted from an initial meridian. The angular 
 velocity of the rotating earth is ^ 3 ; r sin 6 is the perpendicular 
 distance of P Q from Z, and r tan 6 is the tangential distance from 
 P Q to O Z. If the mass moves from P Q to PI it rotates about the 
 
 u 
 axis y with the angular velocity co 2 = -, since it moves in the 
 
 positive direction of x; if it moves from P to P% it rotates about 
 
 v 
 z with the angular velocity o/ 3 = rzrr^J ^ ^ moves from 
 
 rtan0 
 
142 
 
 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 to 
 
 P 2 it rotates about x with angular velocity coi = -, 
 
 because in the motion from P to PZ the velocity is in the negative 
 direction of rotation. If the mass moves from P to P 3 along the 
 radius there is no rotation. These rotational relations can be 
 rigorously proved by analytical demonstration, but as the analysis 
 is rather complicated the reader is referred to the standard 
 treatises by Basset, Lamb, W. Wien, and others. The formulas 
 for the angular velocities in terms of the linear velocities are as 
 follows : 
 
 (370) Rectangular co-ordinates. 2 coi = TT . 
 
 (372) Polar co-ordinates. 
 
 8 u 'dw 
 
 o ^ 
 
 8 z 8#* 
 8 v du 
 
 2 COs 7T ~^ . 
 
 8* dy 
 
 (371) Cylindrical co-ordinates. 2 
 
 co 2 
 
 8 co - 8 v 
 iff 8</> 8 z 
 
 du 8_co 
 8-or* 
 
 8 v du v 
 2 cos = ~ ^ + 
 
 2coi = 
 
 8 co 8 v 
 
 r sin 6 8 A 8 r 
 
 du dw 
 2 co 2 = ^- - -^-2 
 
 2 0) 3 = 
 
 8 v 8 u 
 
 r'dB ~~ r sin 8 
 
 r tan 
 
 If the linear velocities u, v, co, at the point x, y, z, are known, 
 the angular velocities coi, co 2 , co 3 , can be computed in the three 
 systems of co-ordinates. 
 
THE PRESSURE GRADIENTS 143 
 
 It should be observed that since, 
 
 v . v z;cos0 
 
 " 3 = rlfo*' and " = TtoTo = ,- 
 
 (373) 0/3 = o) 3 cos 0, 
 
 so that at the pole for = 0, cos = 1, o/ 3 = co 3 , and at the 
 equator for = 90, cos = 0, o/ 3 = 0. At the pole a point 
 rotates about the axis of rotation with the angular velocity of 
 the rotating earth, but at the equator it does not have angular 
 velocity about z. 
 
 The Pressure Gradients 
 
 The earth in its rotation, acted upon by the gravitation of 
 its own mass, has assumed the form of an oblate spheroid, being 
 flattened at the poles, and if the atmosphere were not acted 
 upon by thermal forces it would be arranged in layers of density 
 parallel to those of the earth's solid mass. The heating of the 
 atmosphere in the tropics by solar radiation, and secondarily in 
 the other latitudes, disturbs these level surfaces of pressure, by 
 lifting some areas and depressing other areas during the processes 
 of heating, cooling, and circulation. The change in pressure 
 from the normal pressure in a given linear distance is the pressure 
 
 gradient -7-, and since the forces are directed from the higher 
 
 to the lower pressure the minus sign is used, -7-. Finally, the 
 
 forces are all to be reduced to the unit density so that the forms 
 become, for the three co-ordinate systems, respectively, 
 
 Rectangular Cylindrical Polar 
 
 Co-ordinates Co-ordinates Co-ordinates 
 
 (374) - (375) - (376) - 
 
 - 
 
 . 
 
 3 x p a-or pr 30 
 
 ap iap iap 
 
 py p t*r< 
 
 iap iap i ap 
 
 p dz' p dz' p a r * 
 
144 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 These expressions can be evaluated into many forms, which are 
 convenient for practical computations, as will be shown in a later 
 section. 
 
 The Potential Gradient 
 
 If the potential of the external forces of attraction of the 
 earth's mass is V, then the forces due to such a potential are, 
 
 8 V 8 V 
 
 In the case of the earth's mass the forces -r and -^ , 
 
 dx dy 
 
 on the meridians and on the parallels of latitude, respectively, 
 are chiefly concerned with the determination of its existing 
 figure. In the problems of meteorology these forces can be 
 neglected, so that there remains only the vertical potential 
 gradient, 
 
 (378) -!--* 
 
 There has been considerable confusion in the literature of 
 this subject because the positive direction is taken upward 
 by some authors, but downward by others. If the positive 
 direction is upward, as where the positive motion along the 
 radius is outward from the center of the earth, we have 
 
 (379) ^ = + g, and V = + g z. 
 
 If the positive direction is inward, as along the path of a 
 falling body, we have, 
 
 d V 
 
 (380) -j^ = - g, and V = - g z. 
 
 The positive direction upward is used by Ferrel, Sprung, 
 Oberbeck, Basset in some sections, Helmholtz, Bigelow; the 
 positive direction inward is used by Basset in some sections, 
 Lamb, V. Bjerknes. 
 
 The force of gravity is. however, modified on the rotating 
 
EQUATIONS OF CONTINUITY 145 
 
 earth by the centrifugal force, which acts only in planes per- 
 pendicular to the axis of rotation. These forces can be resolved 
 along and perpendicular to the axis, as follows, since the centrif- 
 ugal force is J co 2 w 2 . 
 
 The total potential is, 
 
 (381) V = g r + i coo 2 w 2 , 
 
 where co is the angular velocity, and co = n in the notation 
 for the rotating earth. By (66) we have 
 
 R 2 
 
 (382) g = go , and 
 
 R 2 
 
 (383) 'gr = g , so that 
 
 (384) V 
 
 Taking the differential along the axis z, and perpendicular 
 to it along -or, there results: 
 
 dV R 2 dr 
 
 -77 -+?-*;' 
 
 dV R 2 dr 
 
 (386) -^ = + so-^-" 2 o*r. 
 
 We have: 
 
 (387) r 2 = x 2 + y 2 + z 2 = -or 2 + z\ Differentiating, 
 
 (388) 2rdr = 2wd'&+2zdz, and 
 
 (389) Tz = 7* = 7' Hence > for * = >* 
 
 (390) - -^ = g ' j = g cos e. 
 
 (391) - -7 = g ' - co? 2 w = g sin - co 3 2 or. 
 
 a w Y 
 
 Equations of Continuity 
 
 When a mass of air streams through a given space, as a cubic 
 meter at a given place, as much air must pass out of it as enters 
 
146 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 it, or else there will be congestion, and a change in the con- 
 tinuity. The equations which finally control any solution of 
 current functions must satisfy these equations of continuity. 
 
 (392) Rectangular du 8_y dw = 
 Co-ordinates, ox ~ 
 
 (393) 
 
 (394) Cylindrical 
 Co-ordinates. 
 
 (395) Polar f 9 ( sin 6) + f dv + ^ $ 3 (>>) = Q 
 Co-ordinates. o 8 A 8 r 
 
 Operator V 2 
 
 The sum of the second differentials in three co-ordinates is 
 often used in analytical discussions, and the symbol V 2 has been 
 adopted for this process. 
 
 (396) Rectangular y2 = 8 2 _, 9 2 _, 8 2 
 Co-ordinates. ' 3*. 8/ ,8*" 
 
 (397) Cylindrical y2= _?L_,J__3_ + 1 9 2 , S 2 
 Co-ordinates. 3^ 2 tcr 8-ar -or 2 8 ^ 2 8 2 2 ' 
 
 (398) Polar y.'^J!. + ljL. + lI _*.-.,. 2* 4- + 
 
 Co-ordinates. dr 2 r dr r 2 8 0* r 2 d 
 
 1 _^_ 
 r 2 sin 2 8 A 2 ' 
 
 d 
 
 jTAe Toto/ Differential -r. 
 at 
 
 The symbol -7^ is often used to include the terms of the 
 inertia and expansion or contraction. 
 
 (399) A = 3- + u JL + v Ar w A 
 
 <// 8 8ic 83; 82* 
 
 There are a series of complicated terms used to express the 
 internal forces caused by the stretches, the shears, the dilatation, 
 the tractions due to elasticity and viscosity within the masses, 
 but these will be omitted in this place. They are summarized 
 on pages 499-501 of the Cloud Report. 
 
SUMMARY OF EQUATIONS OF MOTION 147 
 
 Summary of the Equations of Motion 
 
 By putting together the terms that have now been explained, 
 there result the general equations of motion which are at the 
 basis of all the dynamic meteorology, omitting the heat term d Q. 
 
 Rectangular Co-ordinates 
 
 /,\ 18P du du du t du 
 (400) -- = =-^T:+W +v + w a co 3 + w w 2 . 
 p dx dt dx d y dz 
 
 1 8P dv dv dv dv 
 
 - P dj=^i +u -d'x +v d-y + w d- z - w " l+ua3 - 
 1 8P dw dw dw dw 
 
 Cylindrical Co-ordinates 
 
 , . 18P8w 8w 8w du 
 
 (401) -- - = + ^ + v r- + w ^ 
 P 8zD" 3/ dv- ttdp 82 
 
 ^ 
 "JJ 1 8P 8^ 
 
 -, + o -- r- ^~ + w ^ 
 ^ P -nr8^ 8^ 9w WO^P 80 
 
 ^"3 1 8P dw dw dw dw 
 
 ^ .5 -- = - -U 11 - -\- 71 - -I- 7i -- 
 
 ^ 
 
 Co-ordinates 
 8P 8w 8w 
 
 s r r 
 
 g 1 8P dv dv dv dv 
 
 uv t wv 
 
 cot + . 
 
 _ ^ _ __ _ 
 
 " ''^ U ~* V dl^ W 3r ~ 
 
 u 2 v 2 
 
148 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 Equations of Motion for the Rotating Earth 
 
 The equations of motion for moving axes are further modified 
 when the axes are attached to the earth which is rotating with the 
 constant angular velocity oj s . 
 
 Cylindrical Co-ordinates on the Rotating Earth 
 
 The linear velocities remain the same except that the linear 
 velocity eastward is increased by the term w 3 .ttr cos 6. The 
 angular velocity about z is changed by the addition of the term 
 
 co 3 COS 6. 
 
 (403) Linear velocities, u' = u. 
 
 i) r = v+(*> 3 . IB cos e. 
 
 w' = w. 
 Angular velocities, a/i = 0. 
 
 /2 = 0. 
 
 6 H . 
 
 iff 
 
 The partial differentials are also modified: 
 
 8 z/ 8 (v + o) 3 . ttr cos 0) dv , 
 
 97-- ^T ;-9 F + ..cos. 
 
 8 w' 
 
 Substituting these terms in the general equations (400), 
 and using (399), we have: 
 
 (405) - ^ = -r! - (ojs.-ar cos0 + fl)(co 3 cos + ). 
 p ow at \ TX J 
 
 1 dP dv / v\ 
 
 ^T = -r* + u - ^ cos + u co 3 cos + ) . 
 
 p wo$ at \ TX J 
 
 _1_8^P dw 
 ' p dz = dt +g ' 
 
POLAR CO-ORDINATES ON ROTATING EARTH 149 
 
 Performing the algebraic work, and substituting for the 
 
 v 
 relative angular velocity eastward v = , we find: 
 
 1 9P du v 2 du 
 
 (406) -ZT- = -77 2 o> 3 cos B.v = -77 
 p Qfff a t tff a t 
 
 f^ (2o? 3 cos B -f- v) v. 
 
 i 
 V- 1 9P dv uv dv 
 
 z* 5^ = Ti + 2a> 3 cos O.u + = -T-, + 
 
 p tcrc/0 d f *& a i 
 
 (2 o> 3 cos -f v) w. 
 
 _1_^P_^ dw 
 
 'p92 ~ dt S ' dt S ' 
 
 Co-ordinates on the Rotating Earth 
 
 The linear velocity eastward is increased by the term o> 3 r sin 6, 
 and consequently the angular velocities o>i and co 3 are modified 
 to conform with it. 
 
 (407) Linear velocities, southward, u f = u. 
 
 eastward, v' = v + co 3 . r sin 6. 
 zenithward, w' = w. 
 
 v + co 3 r sin 
 
 (408) Angular velocities, about the axis re, a? i = - - . 
 
 axis y, 0/2 = + . 
 
 , v + co 3 r sin B 
 axis 0, w i = 
 
 r tan 
 
 (409j The partial differentials ^ = ^--. 
 
 u t o t 
 
 *dv' 9 (v + o> 3 r sin 0) _ 9 
 F7 = 9/ = 97 " 
 
 u . o) 3 cos + w . w 3 sin B. 
 
 d~t = 97* 
 
150 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 Substituting these values in the equations (400), we have, 
 1 8P du 
 
 1 8 P d v /v + co 3 r sin 
 
 rtane 
 
 w . 
 
 or _av /v + cc 3 r sin 0\ 
 
 7rsin08A = ~57~ *V r J" 
 
 /v + co 3 r sin 0\ 
 M ( rung ) 
 
 / -x 
 
 = -77 w . (v + co 3 r sm 0) 
 
 p 8 r dt r 
 
 i) -f- o> 3 r sin 
 
 /^ -T co 3 : 
 \ r 
 
 Performing the multiplications and reductions, 
 18P d u v 2 uw 
 
 dt r r 
 
 2 co 3 cos . . . r co 3 2 sin cos 0. 
 
 *S* a 1 8P dv UV WV 
 
 2 co 3 cos . w + 2 co 3 sin . w. . . . 
 
 ^SP _ dw u* + v 2 _ 
 
 p d r dt r 
 
 2 co 3 sin . v . . . r co 3 2 sin 2 0. 
 
 The terms in 2 co 3 represent the deflecting forces due to the 
 earth's rotation, which always act at right angles to the linear 
 velocity q with which a mass is moving in any direction. They 
 deflect a moving body to the right in the northern hemisphere, 
 but to the left in the southern hemisphere. The deflecting force 
 is a maximum and equal to 2 co 3 q on the horizontal plane at the 
 poles; it is equal to zero at the equator, for all velocities in the 
 horizontal plane. If the velocity is vertical the deflecting force 
 due to this term is zero at the pole and equal to 2 cos w at the 
 equator. The terms in r co 3 2 represent the forces which change 
 the figure of the earth from a sphere to an oblate spheroid. 
 
CONNECTION BETWEEN THE GENERAL EQUATIONS 151 
 
 They need not be considered in practical meteorology, but are 
 important in geodesy. 
 
 The derivation of the general equations of motion given 
 above is exceedingly simple and direct, showing immediately 
 where all the terms come from which are concerned with gen- 
 eral motions. They are equally true for the sun, the earth, and 
 the planets, and all observed motions must conform to them. 
 
 Connection Between the General Equations of Motion and the 
 Thermal Equations of Energy 
 
 We found that with changes of the temperature gradient 
 the variation of the pressure is expressed by equation (190), 
 
 when there is no change in heat Q not otherwise accounted for 
 in this process of motion. But in case all the heat energy is 
 not expended in motion, as where a part escapes in radiation or 
 in internal molecular or atomic motions, as in ionization, a new 
 term must be added to take this into the account, so that 
 o -p 
 
 (412) = -J + nCpdTt+CpTlogT.dn. 
 
 In order to avoid confusion of symbols in this set of equations, 
 take v = o>3 the angular velocity of the rotating earth, and 
 combine the equations (411) and (412). It should be noted 
 
 v 
 
 that since V = : -, by Fig. 16, we have 
 r sin 0' 
 
 , ^ v v 2 cot uv cot 6 
 
 (413) = v sin 0, = v cos . v, = u cos . v, 
 
 and we shall make use of them in the following transformations. 
 In order to connect together the hydrodynamic and the ther- 
 modynamic systems by the law of the conservation of energy, we 
 
 have the same in both systems. Combining these terms 
 P 
 
 and making the reductions, substituting d x t dy, d z for the 
 corresponding expressions of the linear displacements in polar 
 co-ordinates, and adding d Q for the change in the heat contents, 
 
152 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 and d J for the energy in the form of electric and magnetic forces, 
 there result for the forces of acceleration, 
 
 The General Hydrodynamic and Thermodynamic Equations of 
 
 Motion 
 
 , A1A . 1 dP du x uw 
 
 (414) - = cos 6 (2 co 3 + v) v + 
 
 o o x a t r 
 
 ^ 
 
 - g = -TT + cos 6 (2 w 3 + v) u + sin 9(2 w 3 + 
 
 
 1 QP dw 
 
 Corresponding with these co-ordinate equations is the differ- 
 ential equation, 
 
 dP dq dQ dJ , dT 
 
 (415) ~- = ^^ + + - = -^-- 
 
 This equation has been already discussed in its vertical 
 variations, but the more difficult task is to explain its meaning 
 
 in the horizontal directions (#, y). The terms -77, -T-, -7 
 
 U / U / U / 
 
 contain the inertia and the forces of expansion and contraction 
 as expressed in equations (358)-(364), (396)-(398), and these 
 must generally be employed in the study of tornadoes, water- 
 spouts, hurricanes, ocean and land cyclones. The terms in / 
 can hardly be discussed until the subjects of absorption in the 
 spectrum, scattering in the atmosphere, electric and magnetic 
 
THE EQUATIONS FOR THE WORK OF CIRCULATION 153 
 
 forces can be more thoroughly worked out. The difficulties in 
 determining the value of Q have already been stated. 
 
 The Equations for the Work of Circulation 
 
 If the equations for the force of acceleration (414) are multi- 
 plied by d x, d y, d z, respectively, that is, if the force is multi- 
 plied by the distance through which it acts, they give the work 
 expended in transporting the mass from one point to another. 
 Still retaining the unit mass, m = 1, the work-equations become, 
 
 (415) -- = du-j- cos d (2 co 3 -f- v) v d x -f 
 
 --- = dv -77 + cos (2 co 3 + v) udy + 
 p at 
 
 ! Q d y. 
 
 3P dz 
 
 = d w -j-. sin 6 (2 oj 3 -f v) v d z 
 p at 
 
 uudz 
 
 r dz 
 
 We have the auxiliary equations, 
 (416) v d x = u d y, w d x = u dz, w d y = v d z, 
 
 and it is seen that by substitution all the terms cancel except 
 the following, after using the total differential, 
 
 - 
 
 The integral equation is known as Bernoulli's, 
 
 /cL P f* 
 
 = 1 ( W 2 + Z ,2_|_ W 2)_|_ / gdz + (Q + J), one limit, 
 
 (419) - = | (qi 2 - o 2 ) + g (zi - z ) + (Qi - Qo) + 
 
 PIO 
 
 (/i / ), two limits. 
 
154 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 The other combination, in (414), gives, 
 
 (420) - f = - fnCpdT + f Cp T log T d n. as (202). 
 J p J J 
 
 In case / J = 0, we have, in practice, for the static 
 pressure P' and the dynamic pressure P", if P = P' + P". 
 
 (421) / = g (ZIZQ) = Cpni(Ti T<?), static pressure, 
 
 J p 
 
 (422) - f = J (gi 2 - g 2 ) + (Qi - Qo), circulation and 
 
 J p 
 
 radiation, 
 
 //*** .A. /"""* A \ /T* T 1 \ //"* A. /"^ t \ 
 
 = w^a ~~ Lpio) (J- a J- o) n\ \Lpa ~ ^pio) 
 
 (T, - r ). 
 
 The equation (421) has already been discussed under Barom- 
 etry (39) with others; also, under gradients (12) with others. 
 The equation (422) remains to be considered under circulation 
 and the variation of the gradients. 
 
 /dP 
 . 
 p 
 
 There are several methods of treating the computations 
 involved in the thermodynamic and static equation (419), 
 which considers the circulation and the vertical pressure along 
 with the thermodynamic sources of the pressure differences. 
 Since all the terms in (415), which represent the deflecting and 
 the centrifugal forces, disappeared on the substitution of the 
 values in (416), it follows that these forces are always acting at 
 right angles to the direction of the motion of the mass moving 
 with the velocity q in any direction relatively to the surface of 
 the rotating earth. The deflecting and the centrifugal forces 
 have, therefore, no effect upon the work of the circulation of the 
 atmosphere, any more than the central forces do upon the work 
 of the motion of a planet in its orbit. They change the direction 
 of the motion by the composition of the forces, but it requires 
 no additional pressure to overcome these forces at right angles 
 to the path of motion of a current of air. 
 
rdP 
 
 EVALUATION OF TERM - / - 155 
 
 J p 
 
 /d P . 
 in terms of 
 p 
 
 the temperature, eliminating the density. This can be done by 
 substituting from (62), 
 
 p ~ po P To po T Q P~ T P' 
 
 Substituting and using the values of the constants in Table 
 3, omitting henceforward (Qi Qo), 
 
 ^ ~f '--.*?<*-'> + ' $<>-* 
 
 Integrating between limits for natural logarithms, 
 
 (425) log Po - log P = ^^ (f-qf) + ( - so). 
 
 (426) lo g Po-logP= I ^ 5 .^(f- ?0 2 ) 
 
 7991.04 T 
 The same formulas in common logarithms become. 
 
 1 
 
 (427) log Po - log P = 
 
 1321837 
 
 (428) log Po - log P = 3^2 ' T 
 
 18400 T ' g Q 
 (429) The constants are derived in succession: 
 
 574.067 = . 287.033 = 
 
 ./ ^ 
 
 156720 = 2 g l Q . 7991.04 = 1 . 
 
 1321.837=^. 660.919 = ^ 
 TO M 1 o 
 
 360862 = -. 18400 = . 
 
156 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 The second term of (428) is the same as in formula (159), to 
 which static barometry is usually confined, but the term in the 
 velocity should be added for accurate computations. In the 
 integrations between two points the mean temperature, JIG = T m , 
 and the mean gravity, gi = g m , of the air column should be 
 employed, as already explained. 
 
 II. Since by the Boyle-Gay Lussac Law, we have: 
 
 I _ RT 
 P " P 1 
 
 this value can be introduced into the equation (419), so that 
 (430) -/ -j- = ~ (f - <?o 2 ) + ^ (* - *o). 
 
 This is correct because 2R = 574.067, by Table 3, and 
 
 79904' 
 
 III. Since, by equations (176) and (178), we have, for n = 1, 
 
 this value can be introduced into the equation (419). Then, 
 
 /fi P i 
 
 - = 
 
 /P\AzI (T\ n 
 Since (-5-) * = l^r) , we find again, for n = 1, 
 
 V -t ' ^ ' 
 
 -t 0' ^ 
 
 (433) -^-= 
 
 In case w is not equal to unity, and the non-adiabatic tem- 
 perature distribution of the air is considered, we have: 
 
 (434) - / ^ = j^ [i (? 2 -?o 2 ) + g (z - to) ]. 
 
 IV. In case it is desired to reduce the equation of motion to 
 a form where the standard density is unity, we shall have, p = 1, 
 and 
 
/dP 
 157 
 P 
 k-\ 
 
 (435) = P 17 * . P~ l/k = (R, r ) v *. j " . Hence, 
 
 1-fe 
 
 (436) - flj- = (1 g o r o )i/* [J (? 2 - go 2 ) + * (* - *o)l. 
 
 This form eliminates the temperature and throws all the 
 discussion upon P, being an impure form, since P can be found 
 
 only by trials. 
 
 CdP 
 V. It is often desirable to integrate the term J -- with- 
 
 out using the mean temperature of the column, and this can 
 be done by determining the mean density of the column p m . 
 Then, 
 
 (437) -=- 
 
 P Pm 
 
 (Qi - Co), and, 
 
 (438) - Pl ~ P = + Hl (Cp a - Cp 10 ) (7\ - To) - 
 
 Pm 
 
 i C#io (Ti - To). 
 
 Numerical Check on the Two Systems of Formulas 
 
 We have already found, if the humidity, gravity, and moun- 
 tain terms are now omitted, that 
 
 (439) log f = log f = 18400 2 7 67<6 ,, by (159), and that 
 
 (440) log J = log |? = - n ^ (log T - log T ), by (182), 
 
 so that the two methods of computation, through the mean 
 temperature of the air column 6 and the gradient ratio n can be 
 compared and checked. They prove by trial examples to be in 
 agreement. 
 
 Numerical Evaluations of the Pressure Gradient 
 
 It is necessary to reduce the difference on the barometric 
 pressure, at any distance apart on a horizontal level, to a 
 
158 
 
 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 standard distance which is taken as 1 degree = 111 111 meters 
 on the surface of the earth. 
 
 G = the barometric difference for the distance 111111 
 meters. 
 
 D Q = 1 degree or 111 111 meters on the surface of the earth. 
 
 A = any distance on the surface. 
 
 (441) = 
 
 D 111111 
 DC 
 
 dB 
 dx 
 
 A 
 
 ?! ~ B 
 
 A 
 
 . Hence, 
 
 (442) G = ^ (ft - 5). 
 
 Lower pressure 
 
 Higher pressure 
 
 B dx 
 
 FIG. 17. The derived gradient for the distance of I degree. 
 
 The term can be evaluated in many ways. 
 Since P = g Pm B = go p, we have: 
 
 fAA\ 
 
 (443) = goPm-^=S<>2- x - 
 
 (444) 
 
 = 1.200 G (meter) = 
 
 0.0012 G (mm.). 
 
 dp 
 
 dx 
 
 
 
 111 111 
 
 (445) 5f = Pm - = 0.12236 G (meter) 
 
 0.00012236 G (mm.). 
 Hence, from these formulas and others preceding, 
 
 (446) -7t^ = -7F 
 
 P U *v Po - 
 
 dP 
 
 = _ gQ 
 
 'o dx 
 
 go p m B n Td log P 
 
 Po TO dx 
 
NUMERICAL EVALUATIONS OF PRESSURE GRADIENT 159 
 
 7 1 = - 7 ^" H = ~ 7 Lao G (meter) - 
 
 - - 0.0012 G (mm.). 
 p 
 
 (448) -7H = ~7 p -lf = -7 - 12236 G (meters) = 
 
 - - 0.000 12236 G (mm.). 
 P 
 
 T/fe Evaluation of the Ratios -j-^ and -7 . 
 
 a r> ax 
 
 We have, from Fig. 17, 
 
 (450) -j = y^ = jr = jj (for the top of the homogeneous 
 
 atmosphere). 
 
 dh_ h _Do Pi I I 7991.04 
 
 d B ~ B Q - B ~ G Di ~ Bi - BD t ~ B t ~ 0.760 
 
 10514.5. 
 
 The pressure B = at the top of the homogeneous atmo- 
 sphere. BI is the pressure of the atmosphere 0.760 m., and / = 
 R T = the height of the homogeneous atmosphere 7991.04. 
 
 For the ratio -7 , we obtain, by (449) and (451), 
 a x 
 
 dh _d_h L Bo- B __ dh^ t G_ _ 10514.5 
 ' dx~dB' D ~dB'D<>~niin 
 
 0.09463 G. 
 
 This makes it possible to compute approximately the height 
 of the required isobar above the surface, at the horizontal 
 distance from the place of 1 degree = 111 111 meters. 
 
 To Find the Difference of Pressure (Bi B) at the Distance A 
 that will just balance the Eastward Velocity v 
 
 The eastward velocity of the general circulation v produces a 
 pressure directed southward along the meridian, and it is required 
 
160 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 to find the difference of pressure in the meridian, higher to the 
 south in the northern hemisphere, that will keep the velocity of 
 motion directed exactly eastward. By equation (414), for 
 
 steady motion, -77 = 0, and neglecting the term in w, we have 
 for v in meters per second, 
 
 (453) ; = cos 6 (2 co 3 +v ) v. 
 p d oo 
 
 Since d P = g p d k, this becomes: 
 
 (454) g -T- = cos d (2 co 3 + v) v. 
 
 d oc 
 
 By equations (452) and (442), 
 
 f . 10514.5 D (Bi-B) , N 
 
 (455) g . - -~ - = cose (2 co 3 + v) v. Hence, 
 
 (456) Bi-B 
 
 For any temperature and pressure other than the standard, 
 T = 273 and B = 760 mm., if we take A = D Q = 111 111 111, 
 so that BI = B Q , that is, DI = the pressure at a distance of 
 1 degree southward on the meridian, we have, for g = go, 
 
 B " " B = Wo cos e (2 
 
 The angular velocity of the rotating earth is 2 co 3 = 0.0001458 
 and if v is neglected, we find, 
 
 T> 
 
 (458) B Q - B = 0.05644 ^ v cos (in millimeters). 
 
 It should be learned from this example how to apply the 
 general equations of motion on the rotating earth to special 
 cases, by making the proper limitations in the use of the terms. 
 
 The Angular Velocity of the Earth's Rotation, a> 3 
 
 Allowing for the sidereal time the angular velocity is, 
 o 2 V ^ 141 'iQ 
 
 {(23X60) +56} 60 660 
 
LINEAR, ABSOLUTE AND RELATIVE VELOCITIES 161 
 
 (460) 2 co 3 = 0.000145846. 
 
 The Linear, Absolute and Relative Velocities 
 
 The absolute linear velocity at any latitude is 
 
 (461) v' = r o> 3 sin 
 
 at the distance r = R -f- h from the center of the earth, where 
 R = 6370191 meters of 208996600 feet, and h is the height in 
 the atmosphere above the surface. The relative linear velocity 
 of a body moving eastward over the surface of the earth is 
 
 (462) v = r v sin 0. Hence, 
 
 , Aao . v' 2v f cos 2v'cot0 
 
 (463) co 3 = r - , and 2 co 3 cos = 
 
 ' 
 
 ,._.* 
 
 (464) 
 
 (465) 
 
 , 
 r sin ' r sin 
 
 z> cos v cot 
 
 It follows that formulas (414) can be written in another 
 form, remembering that co = n for the angular velocity, 
 
 18P du cot , w 
 
 (466) ---- = --- r (2 9*4- 9 ) v + -> r . 
 
 3^ f/ ^ r r 
 
 rf z; cot , .w 
 
 ay= 
 
 9P 
 
 In some respects the fourth form of polar equations offers 
 distinct advantages in practical computations, especially by the 
 aid of a few auxiliary tables, such as Tables 104, 105, 106 of the 
 Cloud Report. 
 
162 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 Evaluation of the Barometric Gradients in the Fourth Form of the 
 Polar Equations of Motion 
 
 From equations (446), (5), and (449), we have: 
 
 1 8P 1 BoT G x 
 
 ( ^ 7) 7 9^ == PO ~B fi ' * Pm ' Do" 
 , , _!_ 8P _ 760 X 9.806 X 13595.8 T 
 1 ' P 9# " 1.29305 X 273 X 111 111 111 B x = 
 
 0.0025833 ^G x . 
 
 > 
 
 d u 
 
 For steady motion, -7-7=0, and neglecting the term in w, 
 a t 
 
 in (466) 
 
 (469) - 0.0025833 ^ G x = - ^^ (2 v' + ) Hence, 
 
 r> K 
 
 (470) G, = 387.1 1 ~ (2 ^ + ) . 
 
 G z = 387.1 (2 '+)r + - *. 
 
 The gradients are, in millimeters of mercury, 
 G x = (Bi B) X) along the meridian, 
 G y = (Bi - B) y , along the parallel of latitude, 
 G z = (Bi - B\ = along the vertical. 
 As an example of the computation take, 
 
 The north polar distance, = 90 - = 30, 
 The radius of the earth, R = 6370191, B = 700 mm., T = 
 260 C. 
 
 The angular velocity, 2 co 3 = 0.00014584, 
 - The eastward velocity of the earth at d, v r = Rn sin 0, 
 
 The relative eastward velocity over the earth, v = 40 meters 
 per sec. 
 
EVALUATION OF BAROMETRIC GRADIENTS 
 
 163 
 
 
 Logarithms 
 
 By Tables 
 
 104 and 106 
 
 387.1 
 
 2.58782 
 
 v' 
 ^ cot 0.2v 
 
 0.005052 
 
 700 
 
 2.84510 
 
 cot . v v 
 
 0.000435 
 
 cot 30 
 
 0.23856 
 
 
 0.005487 
 
 (2 i/ +v)v 4.30492 
 
 
 Logarithms 
 
 
 9.97640 
 
 cot0.v [-j 4- 
 
 ) 7.73933 
 
 
 
 378.1 
 
 2.58782 
 
 r 
 
 2.41497 
 
 B 
 
 2.84510 
 
 R 
 
 6.80415 
 
 
 3.17225 
 
 0.75728 
 
 9.21912 
 
 T 
 
 2.41497 
 
 
 
 5.717 
 
 0.75728 
 
 Logarithms 
 
 R = 6370191 6.80415 
 
 2 u>3 = 0.000145846.16388-10 
 sin = sin 30 9.69897 
 
 2v' = 464.5 2.66700 
 
 v = 40.0 
 
 2 v' + v = 504 . 5 
 (2v f + v)v = 2018.0 
 
 C x = (Bi - B) x = 5.717 
 
 Application of the General Equations of Motion to the Local Circula- 
 tions in the Earth's Atmosphere 
 
 The Local Circulations 
 
 The circulations of the earth's atmosphere can be con- 
 veniently analyzed under two classes: the first, or general 
 circulation, including the large movements that are primarily 
 related to the axis of the earth as the line of reference for the 
 angular motion, and the second, or local circulation, including 
 the minor movements that are referred to axes which are wander- 
 ing over the surface of the earth. The general circulation takes 
 account of the great polar whirls covering the entire hemisphere, 
 one north and one south of the equator, including the trade 
 winds in the tropics, the eastward drifts in the temperate zones, 
 and the minor circulations near the poles. These great zonal 
 currents break up into localized circulations, as determined by 
 the ocean and land areas, which constitute the first disintegration 
 of the general circulation into smaller circulations. The true 
 local circulations are commonly known as cyclones, anticyclones, 
 hurricanes, tornadoes, and waterspouts, and these are referred 
 to axes which move over the earth in paths that are determined 
 by their relation to the breaks in the normal general circulation. 
 Finally, there are very numerous minor whirls, as eddies, small 
 vortices of many types, which constitute the effectual internal 
 friction through the operation of the law of inertia in the moving 
 masses. The analysis of these motions, by the application of 
 the laws of hydrodynamics and thermodynamics given in this 
 chapter, determines the principal problems in theoretical meteor- 
 
164 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 ology. It is proposed to set forth the main features of this 
 subject with sufficient fulness to guide other students to the 
 problems of the research which are pressing for solution. 
 
 Discussion of the Cylindrical Equations of Motion 
 
 The cylindrical equations of motion are most convenient 
 for application to the discussion of the phenomena of the local 
 circulations in cyclones, hurricanes, and tornadoes. There 
 have been several attempts to adapt these equations to the 
 observed data, and the two best-known systems, that of Ferrel 
 and that of the German School of Meteorologists, will be briefly 
 mentioned before taking up the form of vortex that I have been 
 led to adopt in my researches. It will be desirable, at the outset, 
 to assume that the motion is symmetrical about the z-axis in 
 cylindrical co-ordinates, and that the isobars are centered as 
 circles upon this axis, though this is not the case in nature, 
 except for waterspouts, tornadoes, and hurricanes. The pure 
 vortex law does not apply directly to cyclones and anticyclones, 
 and the disturbing terms which make the transition between pure 
 and impure vortices can be studied only by comparing the pure 
 vortex underlying a cyclone with the data obtained by observa- 
 tion. It has been difficult for meteorologists to do this, because 
 the actual conditions in the free air above the surface are found 
 only indirectly by computation, or in an inadequate manner by 
 occasional ascensions with kites and balloons. In recent years, 
 however, enough data have been accumulated to make it possible 
 to advance these studies in the right direction. It will greatly 
 assist those who are engaged in the study of the atmosphere 
 above the ground, if, in planning and executing the observations, 
 the fundamental principles upon which the actual motions must 
 depend are clearly understood. 
 
 Ferret's Local Cyclone 
 
 If the vortex is assumed to be symmetrical about the z-axis, 
 and the friction terms k u, k v can be neglected, the second 
 
FERREL'S LOCAL CYCLONE 165 
 
 equation in the second form of cylindrical equations (406) be- 
 comes: 
 
 (471) - + (2 co 3 cos 6 + v) u = 0. 
 
 w 
 -7- 
 d t 
 
 v dw 
 
 Substituting, v = , u = -7-, and multiplying by tcr, this 
 " 
 
 becomes, 
 
 ozo' d v 
 
 (472) 2 co 3 cos 0. zrr -7- + tzr -y- + = 0. 
 
 Integrating, we have for each particle in gyration: 
 
 (473) ttr 2 co 3 cos 6 + ttr y = -or 2 (o> 3 cos + v) = c. 
 
 fed 
 initial value of v, 
 
 m 
 (474) Take C = - for the entire gyrating mass, if Vo is the 
 
 f or 2 (w3 cos + v) dm 
 
 (475) C = - - = I w 2 (co 3 cos + Vo). 
 
 7W 
 
 If the initial gyration is zero, v = 0, and we have: 
 
 (476) -or 2 ( co 3 cos + ~ = ^o 2 co 3 cos ^. 
 
 This is equal to the moment of inertia of the whole mass at 
 the distance \ tffo. We obtain: 
 
 (477) = ~^- 2 o) 3 cos 6 co 3 cos 0, and 
 
 / 7K 2 \ 
 
 v = \^~~ 2 l) ^co 3 cos0, the tangential velocity at to- . 
 
 If the tangential velocity vanishes, v = 0, for co = R, we have: 
 
 (478) R 2 = ^, and R = 0.707 
 
 If a cylinder with the radius W is drawn around the gyrating 
 mass, then at the distance 0.707 ttr the velocity is zero; inside 
 this R the fluid rotates in one direction and outside of it the 
 rotation is in the opposite direction. The general circulation of 
 
166 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 the atmosphere is arranged to operate in this way, the air north 
 of the latitude 35 16' rotating eastward, and the air south 
 of this parallel rotating westward. If water be enclosed in a 
 cylindrical vessel, and an upward current be formed in the center, 
 by heating the lower surface, or by a wheel operating in the 
 water to raise a central column, it will circulate in this way. 
 The water will gyrate about the central axis as it rises, till at 
 the distance 0.707 or from the center the gyration ceases, while 
 beyond this radius the gyration reverses and the water descends 
 in a ring bounded on the outside by the vessel. Ferrel supposed 
 that this type of vortex would represent the local cyclone as well 
 as the general circulation, but this is not the case. In the 
 general circulation there is a fixed mass in gyration, as there is 
 in the cylindrical vessel of the experiment, but observations show 
 that the cyclone, hurricane, and tornado are constructed by 
 means of another type of vortex in which the mass is continuously 
 changing. Other mechanical difficulties are mentioned in the 
 Cloud Report, 1898. 
 
 The German Local Cyclone 
 
 In the second form of cylindrical equations (406), the second 
 equation becomes: 
 
 (479) fi + ** + **.+-**--(*, 
 
 where /I = 2 co 3 cos 9 and k is the coefficient of friction. 
 This equation has two solutions, and the vortex has been divided 
 into an inner and an outer part to correspond with them. Thus 
 
 First Solution Second Solution 
 
 (Inner Part) (Outer Part) 
 
 (480) Radial velocity = - - or. u = . 
 
 "ZET 
 
 A c * c 
 
 (481) Tangential velocity v = 7 ; -U . v = -r . 
 
 K C A K> to 
 
 (482) Vertical velocity w = c z. w = 0. 
 
THE GERMAN LOCAL CYCLONE 167 
 
 The constant c depends upon the dimensions of the vortex, 
 and must be determined by observations. These values of u, 
 v, w, can be readily verified by substituting in equation (479). 
 
 If we take the following current functions, called the Stokes 
 functions, and the vortex law, 
 
 (484) w = + - 
 
 (485) v W = $ 
 
 there results for the two solutions, 
 
 First Solution Second Solution 
 
 (Inner Part) (Outer Part) 
 
 (486) Current c 
 
 i-i A '/'i = T* tCT^ 2 \l/\ = -f- C 2. 
 
 Function. ! 2 
 
 (487) Radial 1 8^1 c 1 9^i c 
 Velocity. u ~ " -or 3z ~ ~ ~2' w " ~ttr8lT = ~m" 
 
 (488) Vertical 
 
 iri v W = + 7^-=- = + C Z. 
 
 Velocity. -or 9w 
 
 (489) Vortex I c 
 Law. ~~ k c 2 
 
 (490) V 
 
 -c 2* A; 0* 
 
 It is seen that two forms of the current function are required 
 to satisfy the general vortex law which will be deduced later. 
 Even if the constants k and c could be determined the solution 
 is not consistent for the entire vortex in either the inner or the 
 outer part. There is in nature no such division of the vortex, 
 that is, there is no outward part without vertical velocity, as 
 compared with an inward part having vertical velocity, w = + c z. 
 It is for this reason that the application of these formulas to 
 the cyclone has not been successful. The cyclone is constructed 
 upon quite different principles. The solutions of the second 
 equation of motion can be satisfied by yet other values, which 
 
168 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 give a consistent current function for all values of the currents 
 and the vortex law. 
 
 The General Equation of Cylindrical Vortices 
 
 If the definition of a cylinder be extended to include any 
 figure of revolution formed by rotating any line as a generatrix 
 about an axis, the vortices of meteorology can be designated 
 as cylindrical or columnar vortices, the axis being approximately 
 vertical in direction. The cylindrical equations of motion will 
 therefore be adopted, and they are transformed in the following 
 manner. In discussing problems in vortex motion, it is con- 
 venient to use the current function ^, which is deduced from 
 the equation of continuity (394) : 
 
 This may be put into another form, 
 
 (493) J- -- (w) + ^ - 0. 
 ttr v 
 
 This is satisfied by substituting the velocities, 
 
 (494) =--p, 
 
 ZD- 8s 
 
 1 8^ 
 
 (495) *=+-^, 
 
 which are known as Stokes's functions. 
 
 In order that the equation of continuity may satisfy the 
 
 second equation of motion, assuming steady motion and = 0, 
 
 u t 
 
 this becomes, from (40l) 2 , with no deflecting force and no friction, 
 
 , . dv dv uv 
 
 (496) w+w +--=0, 
 
 dur 90 nr 
 
 and it is sufficient to make 
 
 (497) , = , 
 
GENERAL EQUATION OF CYLINDRICAL VORTICES 169 
 
 so that vor = ^ = constant is the usual vortex law, or in its 
 most general form, v ttr = a . \J/, where the \l/ of the Stokes func- 
 tions is made to cover the vortex law by the constant factor a. 
 By differentiation, 
 
 9w vs w w 
 
 (499) |?-^t*. 
 
 dz & 3 z 
 
 Substituting these values in (496), it becomes: 
 
 .j ,t _ t 
 
 w 2 aw 9 z """ w 82 y w aw a 3 w 3 a 3 ^ ' 
 
 If the equation (496) is multiplied by zcr, since ^ = 0, it 
 can be written, 
 
 (501) uv -f- ztr- + wttr - -f- ^ y = 0. and, therefore, 
 
 otir 02 oz . , 
 
 in the form, 
 
 p. c\ 
 
 (502) u (ttrz;) + w (tcrz;) = 0. 
 
 This shows that tffv = \f/ = constant is a solution of the 
 equation of continuity. Any function of \l/ which satisfies this 
 equation will be a solution of the second equation of motion. 
 Inasmuch as there are several such values of ^ known, it is only 
 necessary to choose the one which is in harmony with the observed 
 phenomena in the earth's atmosphere in order to obtain the 
 solution of the motions found in cyclones, hurricanes, and 
 tornadoes, or waterspouts. Hence an arbitrary function of ^, 
 
 (503) wz> =/(*), 
 
 is a solution of the second equation of motion. 
 
 The potential and the pressure terms in the first and third 
 equations can be eliminated by the following process. From 
 (401) in the case of symmetry about the 2-axis, 
 
 8V dP du . du du v* 
 
 -- 
 
 8V 8P dw dw dw 
 
170 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 Differentiate the first equation to 3 z, the second to 3 w, 
 and subtract with the result, 
 
 , v __ _ _ 
 
 = at \d~z ~ dv) + d^ z 9 
 
 3 w\ 3 / fl 2 \ 
 ary 
 
 dm dz 
 We derive the following auxiliary differentiations: 
 
 _- _ 
 
 - " " - 
 
 _ _ JL 
 
 82 "9w" " w 
 
 v z 
 Since v w = / (\l/), v 2 w 2 = [f (^)] 2 , = L>/ vy/J ? we have, 
 
 W "*" 
 
 dz zcr w 
 
 Making these substitutions in (505), we obtain, 
 
 + ' T 
 ^ _9 
 
 2 fU) 9 /^ 
 
 ia^.a 2 ^\~i az 
 
 vaw 2 "' 
 
 Any function of ^ satisfying this equation is capable of 
 giving a vortex motion. In the application to the atmosphere 
 some simple forms will be considered and illustrated by examples. 
 The first form, 
 
 (511) t = Aw 2 z, 
 
 gives a funnel-shaped vortex, and the second form, 
 
 (512) ^ = A w 2 sin a z, 
 
 gives a dumb-bell-shaped vortex. These are the common ones 
 
GENERAL EQUATION OF CYLINDRICAL VORTICES 171 
 
 in the atmosphere, as will be shown by the observations. Un- 
 fortunately the motions under the complex local forces that 
 generate storms do not often produce pure vortex motion, but it is 
 the province of meteorology to consider the perturbations as 
 observed and to give an account of their causes. 
 
 The Angular Velocity 
 By formula (371), the angular velocity is, 
 
 and if an arbitrary function of ^ is taken, 
 
 it follows that, 
 
 By differentiations it follows that 
 
 f ^ R \ 8 
 (516) 
 
 - 
 
 Pressure 
 Since by formula (418), omitting Q and 7, 
 
 (518) - - = i ( W 2 + ^2 + ^2) + g z> 
 
 it follows, by using Stokes's functions, that, 
 
 /KinN P X f/ 8 ^\ 2 , / r9 *\ 2 n , i a2 ^ 2 
 
 (519) 7 ".? Ks) + (a"J J + * -iT 
 
 for one limit. The difference of pressure between two points, 
 designated by n and n + 1, becomes, by using the mean density 
 
172 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 (520) P P . -f- \ -1 fY^V + /^ 
 
 (520) P n+1 -P n - 2 -( 
 
 _, /i\ /a\ 
 
 2 UwJ. ~ U 2 J.+ 
 
 
 
 The Application of the Vortex Formulas to the Funnel-shaped Tube 
 
 Employing formulas (494), (495), (497), (503), we readily 
 obtain the following group of relations, 
 
 r w Current 
 
 ^ ' ^ " "2 ' function. 
 
 (w>\ r JL JL ^L _^ Vortex 
 
 = Tff*z = vz ' iff 2z constant. 
 
 1 8i \L> ww v Radial 
 
 ( 523 ) u = 4- = C tcr = = = . , 
 
 -or 82; ttrz 222 velocity. 
 
 . . \1/ \1/ w Tangential 
 
 (524) v = =Ciff z = =- -& = uz. , .. 
 
 -or -or 2 velocity. 
 
 (525) 
 
 2 ttr 
 
 Vertical velocity. 
 
 The Application of the Vortex Formulas to the Dumb-Bell-Shaped 
 
 Tube 
 
 (526) ^ = A TB* sin az. 
 
 /rr _ N 18^ A aw 2 cos #3 
 
 (527) w= ^ = - - = A a TX cos az. 
 
 TX d z iff 
 
 / C oo\ a $ A aw 2 sin az 
 
 (528) v = = + - - = + A a iff sin az. 
 
 TX Tff 
 
 /__ N 1 8 ^ 2 A ttr sin 
 
 (529) * = 
 
 
APPLICATION OF VORTEX FORMULAS 173 
 
 The Total Pressure 
 For the funnel-shaped tube, omitting the expansion terms, 
 
 (530) - - = J (u 2 + v 2 + w 2 ) + gz = | C 2 (ttr 2 + w 2 z 2 + 
 
 4s 2 ) + g 2 -f- constant. 
 
 For the dumb-bell-shaped tube, 
 
 p 
 
 (531) -- = (A 2 a 2 m 2 cos 2 az + ,4 2 a 2 tcr2 s in 2 az + 
 
 4 yl 2 sin 2 az) + gz + constant. 
 
 p 
 
 (532) - = A 2 a 2 iff 2 +2A 2 sin 2 as + g z + constant. 
 
 p 
 
 (533) - = A 2 a 2 tcr 2 + 4 2 (1 '- cos 2az) + gz + constant. 
 
 It should be noted that the signs of the Stokes functions 
 have been taken opposite to one another in the funnel-shaped 
 and the dumb-bell-shaped vortices. This is because it is more 
 convenient to place the plane of reference for the funnel-shaped 
 vortex at the base of the cloud from which it is developed, with 
 the positive direction of the z-axis downward, while in the 
 dumb-bell-shaped vortex the first plane of reference is taken at 
 or below the surface of the sea or ground, and the positive 
 direction of the z-axis is upward to the second plane of reference. 
 These will be explained further by diagrams and examples. 
 
 The Relations Between Successive Vortex Tubes 
 
 A vortex is so constructed that a section through it perpendi- 
 cular to the 2-axis at any height, z or az } cuts off a series of rings 
 so regulated in size that the successive radii stand in a constant 
 ratio to each other. Take this ratio, 
 
 (534) P = ~, and 
 
 (535) 
 
174 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 If tff n is the radius of the outer ring Wi, and or 2j or 3 , zcr 4 ... of 
 the successive rings inward, then, 
 
 / KQA X 1 2 3 
 
 (536) p= _ = _ = _ 
 
 This constant ratio p plays a very important part in the 
 computation of these vortices, and it is found that we can pass 
 from one value of the radius and the velocities to the w others 
 in succession by employing the following formulas: 
 
 (535) Ratio of the radii. log p = log -^-. 
 
 Wn+l 
 
 (536) Vortex constant, log C n = log d + 2 n log p. 
 
 (537) Radii of rings, log w n = log^i n log p. 
 
 (538) Radial velocity, log u n = log HI + n log p. 
 
 (539) Tangential velocity, log v n = log Ui + n log p. 
 
 (540) Vertical velocity, log w n = logwi + 2n log p. 
 
 (541) Horizontal angle i, log tan i = constant. 
 
 (542) Vertical angle >?, log tan y = log tan ^ + w log p. 
 
 (543) Time of one rotation /, log / = log ti 2 n log p. 
 
 (544) Volume through rings, V = TT ( 2 ttr n ttr 2 n+ i) w m = 
 
 constant. 
 
 (545) Centrifugal force, log (-) = log (-) -f 3 n log p. 
 
 \w/.n Vw/i 
 
 (546) Barometric pressure, logEj = log - + log 
 
 D n -\ B n Wn+i 
 
 (547) Total velocity g, # = (w 2 -f v* + ^ 2 ) J = z; sec i sec w. 
 
 The relations shown by these formulas will be made clearer 
 by a diagram giving the connection between the angles and the 
 velocities. 
 
 (548) tani-. (549) 
 (550) tan y = 
 
 a sin 
 
 tfi 
 
 - .. 
 v sec^ 
 
RELATIONS BETWEEN SUCCESSIVE TUBES 
 
 175 
 
 ttr, = the cylindrical co-ordinates of a point on the x y plane 
 q (u, v, w) = the co-ordinates of the velocity at the point (zEr,0,z), 
 ff (u } v) = the component of q on the horizontal plane; i = the 
 
 FIG. 1 8. The relations of the angles and velocities in the formulas. 
 
 angle from v to <r, positive outward from the tangent; y = the 
 angle from <r to q, positive upwards. 
 
 The Second Form of the Cylindrical Equations of Motion (406) in 
 
 Terms of the Current Function \f/ 
 
 The equations of motion for cylindrical vortices can be readily 
 transformed into terms depending upon the current function ^. 
 Writing equations (406) in their full forms, they become, 
 
 , fn . 
 (552) 
 
 du , du , du v 2 
 + u - h w^- --- 2 o> 3 cos 6 . v + k u. 
 pottr ot out oz iff 
 
 1 dP dv 
 
 1 dP dw 
 
 du> 
 
 dv uv 
 
 dw 
 
 It is only necessary to develop the differential terms from 
 the velocities given in (523)-(525) and (527)-529). 
 
 (553) 
 
 For the Funnel-Shaped Vortex 
 u = C iff. v = C iff z. w 
 
 - 2Cz. 
 
 dv 
 
 = C z. 
 
 
 dz 
 
176 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 8w 
 
 8w 
 
 w 2 
 
 u = C 2 Tff . 
 
 O'Ztf 
 
 w = 0. 
 82 
 
 - = C 2 W 2 2 . 
 
 -or 
 
 u 1) 
 
 8y 
 
 z> 
 
 u fa =0ws 
 
 . w; ;r- = 2 C 2 ttr 
 82 
 
 z. = C 2 or 3. 
 -or 
 
 dw 
 u = 0. 
 
 8w 
 w = 4 C 2 2. 
 
 
 (554) 
 
 Hence the second form of the cylindrical equations of motion 
 becomes, for the funnel-shaped vortex, 
 
 (555) - - = + C 2 w- C 2 z 2 -os - 2 tcr 3 cos . v + k u. 
 
 8P 
 
 ^~ ~~ ' == ~~^ ~i~ 4 O 2 "T* ^ r~ k IV . 
 
 p u Z 9 / 
 
 Multiply the equations (555), respectively, by 8 -or, -or 8$, 
 and 82, and integrate for the total pressure, and we obtain, 
 omitting the friction terms, 
 
 (556) - f = = J (w 2 +z; 2 +w 2 )-f J C 2 ttr 2 - J C 2 2 2 ^ 2 + 
 
 / p p 
 
 Substituting the values of u\ v 2 i, r w i \ this becomes : 
 
 p 
 
 (557) = C 2 or, 2 + 4 C 2 2 2 + g 2, at the point (ttr, 2). 
 
 The difference of pressure between two points (&i z) n and 
 w+1 may be expressed, 
 
 rt 
 
 = (C 2 w 2 ) n +1 - (C 2 w 2 ) n + 4 [(C 2 2 2 ) n + i- (C 2 s 2 )J + 
 
 Pm 
 
 It has been customary in meteorology to use the formula 
 (530) as an expression for the total pressure integral, but it is 
 evident that (556) and (557) are the complete forms for the 
 funnel-shaped vortex. If the inertia terms are omitted the 
 formula becomes, without friction, 
 
 (558) -- = |C 2 *r 2 -K 2 * 2 ^ 2 
 
FOR THE FUNNEL-SHAPED VORTEX 
 
 177 
 
 Hence, we can summarize the result for : 
 
 p 
 
 (556) includes the inertia and the expansion terms; (519) 
 contains the inertia, but omits the expansion; (558) omits the 
 inertia but contains the expansion. 
 
 For the Dumb-Bell-Shaped Vortex 
 
 From the equations (527)-(529) we have by differentiation and 
 substitution, using \J/ = A ttr 2 sin az, 
 u = A a or cos az. 
 
 (559) 
 
 (560) 
 
 = A a cos az. 
 
 du 
 
 = A a 2 w sin az. 
 oz 
 
 v = A a -or sin az. 
 = A a sin az. 
 
 oz 
 
 cosaz. 
 
 w = z A sin az. 
 
 dw 
 
 = 2 A a cos az. 
 
 o z 
 
 u ~ = A z a z tff cos 2 az. 
 
 dv 
 u = A 
 
 dw 
 
 ;^- = 2 A 2 a? iff sin 2 oz. 
 
 O 
 
 ^- 
 oz 
 
 dw 
 
 -^- =4A 2 asm az cos az. 
 
 dz 
 
 = A 2 a 2 -or sin 2 
 
 or 
 
 wz ., 2 
 
 = A a W sm az cos <zz. 
 
 or 
 
 Substitute these values in the general equations (552) : 
 1 dP 
 
 (561) 
 
 otcr 
 
 = ^-. -f A 2 a 2 tff 2 -or 3 cos . v 
 
 w. 
 
 1 9P 3w 
 
 - = ^ + 2^4 2 . 2 sin a z cos a z. a + F + kw. 
 p o z o t 
 
178 THE HYDRODYNAMICS OF THE ATMOSPEEfcE 
 
 Multiply by 3 ztr, iff 3 <, 3 z, respectively, omit the &-terms, 
 and integrate for the total pressure, remembering that, 
 
 (562) 2J sin az cos a 8 az = sin 2 az, 
 
 (563) - - = \(tf + i; 2 + w 2 ) + ^ 2 a 2 to-2 + 2,4 2 sin 2 az + g z. 
 
 p 
 
 The term for the velocity square is, 
 
 (564) i (w 2 + ^ + w 2 ) = | ^ 2 a 2 w 2 + 2 ^ 2 sin 2 az, 
 
 so that the total pressure with the inertia and expansion becomes: 
 
 p 
 (565) = A 2 a 2 iff* + 4 A 2 sin 2 az + gz, for the point (of z). 
 
 p 
 
 The total pressure without the expansion is: 
 
 (566) - - = A 2 a? w 2 + 2 A 2 sin 2 az + gz. 
 
 p 
 
 The total pressure without the inertia is : 
 
 (567) -- = |^ 2 a 2 ^ 2 + 2A 2 sm 2 az + gz. 
 
 It is apparent that in the dumb-bell-shaped vortex the pressure 
 p p 
 
 difference , required to overcome the inertia resist- 
 
 Pm 
 
 ance, is the same as that which is needed to overcome the resist- 
 ance to expansion. 
 
 The Deflecting Force 
 
 The terms in 0, the polar distance, 2 w 3 cos . v and 
 + 2 co 3 cos 6 . u, disappear from the equation of total pressure 
 in the summation because we have, 
 
 (568) vdiff = urn d(f>, 
 
 just as in the rectangular co-ordinates, 
 
 (569) v d x = u d y, 
 
 which shows that the deflecting force is at right angles to the 
 
 ifi uv 
 direction of motion. The centrifugal force , , being at right 
 
 angles to the direction of motion and induced by the velocities 
 w, v y together with the deflecting force, has no influence upon 
 
THE DEFLECTING FORCE 179 
 
 the circulation except to change the direction without producing 
 acceleration. In the same way a planet falls toward a body ex- 
 erting central force, and thus moves in an orbit about it, but 
 the velocity in the orbit is not changed by these forces acting at 
 right angles to the direction of motion. 
 
 The Force of Friction 
 
 The viscous friction in the atmosphere is a very small quan- 
 tity, and k would be a small coefficient were it not that in all 
 large movements of the air there are numerous small vortices 
 produced within and carried along in the great current. These 
 minor whirls have a strong force of resistance and they are 
 largely concerned in frittering down the energy of motion 
 contained in the large current. It is customary to take the 
 term expressing friction proportional to the velocity, k u, k v, k w. 
 
 This is a subject that has not been satisfactorily cleared up, 
 and it will require much careful research. There can be no doubt 
 that k is a variable coefficient, and differs widely in tornadoes 
 passing over a city or rough country from that in a cyclone 
 over an ocean area. It is not certain that the velocity enters 
 the equations as the simple first power, but that remains to be 
 determined. The resistance due to the friction, whatever function 
 may be found to express it, acts along the line of motion to 
 retard the velocity, so that the pressure difference must increase 
 to overcome this type of resistance. 
 
 We may write the final equation for all the terms, when 
 the pressure-difference between two points is required, using 
 (v t z) n + i and (w, z ) n , 
 
 p _n+i n+i n 
 
 (570)-- =^ 2 a 2 ^ 2 + 4^ 2 sin 2 0z 
 
 Pm J n J n _J n 
 
 The mean density p m along the path between the two points 
 must be used, and in general the mean conditions of all the 
 terms along the path of the integration must be carefully con- 
 sidered. 
 
180 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 The Transformation of Energy in the Circulation of the Atmosphere 
 
 The circulation of the atmosphere is the process of the trans- 
 formation of energy, the transportation of warm and cold masses 
 of air from one place to another being the evidence that a dis- 
 turbed thermal condition is seeking its normal equilibrium. 
 These currents are so complex that at present there is no possi- 
 bility of working out a comprehensive system of equilibrium. 
 The direction and velocity of the currents in all levels and in 
 all latitudes and the temperatures of the masses, must be worked 
 out by numerous observations before that can be undertaken. 
 All the integrations heretofore proposed assume that a nearly 
 perfect vortex law can be laid at the base of the discussion of the 
 general and the local circulations, but as the vortices on the 
 hemisphere and in the cyclones are very imperfect a more com- 
 plicated treatment is necessary. At present it is possible to 
 lay down only some isolated, detached propositions which con- 
 tribute to the ultimate solution of the problems of atmospheric 
 circulation. The following discussions merely introduce a 
 subject of great value, which is capable of unlimited development. 
 
 CASE I. The Change of Position of the Layers in a Column of Air 
 
 When a layer of air in a column is not at the temperature 
 which belongs to its elevation it must move upward or down- 
 ward in order to gain a position of equilibrium, upward if too 
 warm, and downward if too cold for its place. This occurs when 
 a cold sheet overruns a warm layer, when there will be an inter- 
 change of position in certain streams, which may have a vortical 
 structure more or less fully developed. The following proposi- 
 tions take no account of the form of the current lines, but they 
 explain the amount of energy that can be transformed into a 
 velocity q. The chief imperfection in these propositions consists 
 in the omission of the powerful heat terms (Q\ QQ). 
 
 From the equations (196) to (199), we find, 
 
 (571) - - = g fe- 
 Pio 
 
THE CHANGE OF POSITION OF LAYERS 
 
 181 
 
 so that the velocity equation becomes, for the mass M, 
 
 (572) J (q? - ? 2 ) M = -g( Zl -Zo)M- (! - n ) Cpu (7\ - T ) M 
 
 M-(Q 1 -Q )M. 
 
 The evaluation of the term (HI n Q ) Cpi (Ti TO) m 
 is difficult, because the moment a mass of air moves up or down, 
 it at the first instant has an adiabatic gradient, w = 1, of ex- 
 pansion or contraction, which sets up a minor circulation within 
 the mass whose gradient is HI, so that this internal circulation 
 cannot be followed, and it is necessary to treat it as a resultant 
 mass whose general gradient is n\. We, therefore, omit this 
 
 clM 2 
 
 pfc 
 
 fnitial Final 
 
 FIG. 19. Change of position of the layers in a column of air. 
 
 term, also the initial velocity g 2 and the initial height z for 
 convenience, and have for the kinetic energy, for several masses, 
 
 (573) I m f 2 [ n\ Cpio (Ti T ) m g z m]. 
 
 These terms must be evaluated and substituted in the general 
 formula. (Compare Margules' "Energie der Stiirme.") 
 
 Change of Position of the Layers in a Column of Air 
 
 Suppose that the thin layer mi at the height Zi, pressure Pi, 
 temperature Ti, is too warm for its place, but that it must rise 
 to the height 2 to be in equilibrium, while a column M z of height 
 h falls through a short distance. The mass Mo is not affected 
 
182 
 
 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 while the mass M h above z 2 falls as in a piston without changing 
 the pressure or temperature. The changes in the mass M 2 
 
 must be integrated through the layers, M 2 = / d M 2 . In the 
 
 /o 
 
 exchanges the pressure of mi changes from PI to PI, and the 
 temperature from TI to 7Y; the pressure of the differential 
 layer d M 2 changes from P 2 to P 2 ', and the temperature from T 2 
 to Tz, while the height h changes slightly as the large mass con- 
 tracts in falling. 
 
 Layer 
 (574) 
 
 Initial 
 
 Pi' = 
 
 Final 
 g mi 2Y = 
 
 To evaluate T 2 we have 
 
 ,/ fc-l 
 
 /r~r\ rp , rp /* 2 \ IT rp I + . 6_lllf A n k T 
 
 (575J 1 2 /i(-"l*" = ^2 I 1 +^ WA! = ^ 2 
 
 _ 
 
 " l2 
 
 R 
 
 wyfe 
 Substitute these values in equation (573). 
 
 /Vi-r/^N 1 . ^2 * {~* J\ 
 
 \oi\)j ~2 mi c[ = ni ^P] 
 
 nCpP 2 ' 
 
 (577) | wi g 2 = % Cp w 
 since in the d M2-term, 
 
 (578) m CAO (r, - r 2 -/g 
 
 
 )"* J wi - g (2 - 
 
 f dM 2 r 
 
 l -- =- gmi) dz=-gmih. 
 / p 2 * 
 
 The two gravity terms in (576) nearly disappear by the 
 summation. The available kinetic energy ^ mi q 2 caused by 
 displacing a thin layer by a thick layer can be computed in this 
 way, but there is no account given of the form of the currents 
 produced by the transformation, nor of the energy lost in the 
 
EVALUATION IN TEMPERATURE CHANGES 183 
 
 small internal vortices with the accompanying inertia and 
 friction, nor the energy lost by radiation. 
 
 The Evaluation of JT dm in Linear Vertical Temperature 
 
 Changes 
 
 Since the integration of the term JT dm may frequently 
 occur for a simple linear vertical gradient, it is proper to secure 
 the general auxiliary theorem that will express this term when 
 the temperature is defined by 
 
 (579) T = To - a z. 
 
 If a is not constant, as is seldom the case except for short 
 vertical distances, then another solution will be required. 
 We have to evaluate, for T = T Q a z, 
 
 (580) $Tdm=$T P dz. 
 
 It is convenient to have before us the equivalents, 
 
 P /T\ nCp /T\ /r\ * /T\ 8/Ra 
 
 By substitutions, we find, 
 
 r z 1 f* 1 r z t T \ g/Ra 
 
 (582) 
 
 -g/Ra g/Ra 
 
 T dz. 
 
 o 
 
 Change the limits of integration from z to T. Since, 
 
 jrrt 
 
 (583) T = To - a z, dT = - adz, - ~- = dz, we have: 
 
 (584) T P dz = --Po T,- ga T 8a d T = - P T ~ g/Ra 
 
 r jK# 
 
 1 F T g/Ra +1 Tg/Ra+l^\ 
 
 a+ll 10 ]' 
 
 g/Ra+l 
 
 T 
 
 (P T -PT). Hence, 
 
 g + Ra 
 (586) fldm = CT pdz = r : (P T - PT), 
 
 /n Jo . K 1 
 
 t^TI 
 
184 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 The difference of the products of the pressure and the tem- 
 perature at two points, multiplied by the coefficient depending 
 on the w-coefficient of the gradient of the temperature between 
 them, is the integral of this term. It is, however, much simpler 
 to integrate by means of T and p for the stratum (zi 2 ), 
 
 (587) fT dm** fr p d z = Tio Pw Oi - * ), 
 
 /0 /0 
 
 which gives close approximate values. 
 
 CASE II. Effect of an Adiabatic Expansion or Contraction 
 in a N on- Adiabatic Temperature Gradient 
 
 Since a mass in moving from one level to another level in 
 the atmosphere begins to change adiabatically, while the pre- 
 vailing temperature gradient is non-adiabatic, it becomes 
 desirable to define the relation of these facts to the velocities 
 which are immediately set up in the mixing medium. The 
 equation (573) is to be evaluated under adiabatic conditions, 
 by which it becomes, 
 
 (588) i fin g 2 = Cp 
 
 From = ' we have ' f r r * = Tl " a h ' 
 
 From the binomial theorem, we have the formula, 
 
 1 
 
 (590) (l- X ) n =l- 
 
 nx* 
 
 --, so that 
 
 (591) i m, q* = Cp ^-T, (1 -m + n* . | - 
 
 2 
 
EFFECT OF EXPANSION OR CONTRACTION 185 
 
 (592) 1 
 (593) 
 
 The mass mi is driven from its position with a velocity energy 
 inversely proportional to the temperature, so that warm air has 
 less driving power than cold air. The drive depends upon the 
 departure-ratio n and vanishes when n = 1, that is, for adiabatic 
 expansion in an adiabatic gradient. When a > do the mass m : 
 is in unstable equilibrium, that is, too cold for its position and 
 tends to fall. Example, for n = 0.5, a = 19.747, a = 9.87. 
 When a < a Q the mass mi is in stable equilibrium. Example, for 
 n = 2, a = 4.94 <0 = 9.87. It is not possible to drive the 
 small mass mi through any great height in the atmosphere, 
 because the differential energy of the expanding mass sets up 
 minor whirls which tend to interchange the heat energy by 
 mechanical effects and internal friction and radiation. The 
 
 result is to change the gradient from a Q to. If the displace- 
 
 ment of the mass mi takes place in the medium of gradient a, 
 then the drive may be expressed by terms of the form 
 
 /r^,\ i i h* /a a Q \ i h* 
 
 (594) zmif = $g Yi mi (- - -) = ^g^mi 
 
 where HI is the effective temperature ratio of the moving mass, 
 and n is that of the prevailing general temperature gradient. 
 
 There are two primary type cases of the distribution of the 
 masses of different temperatures: (1) That in which they are 
 superposed, and (2) that in which the masses are located side 
 by side on the same levels. 
 
 CASE III. The Cher turn of Deep Strata in the Column 
 
 Let the pressures, temperatures, and heights be arranged 
 in the initial and final states as indicated in Fig. 20. 
 
185 
 
 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 The Overturn of the Deep Strata in the Column 
 
 The greatest entropy in the initial state in 1 is less than the 
 least in 2, so that the cold mass 1 will fall beneath the warm 
 
 T ftl 
 
 hi 
 
 ICold 
 
 
 2' Warm 
 
 Pi Tfi 
 
 
 p / T . 
 
 Pi Ti 2 
 
 P< T'^ 
 
 2 Warm 
 
 
 Cold 
 
 Po To 2 
 
 
 Po Toi 
 
 Initial Final 
 
 FIG. 20. The overturn of the deep strata in the column. 
 
 mass 2. The heights of the masses will change, as well as the 
 pressures and temperatures. Assume that P , r 02 , h 2 , Ti\, hi 
 are known in the initial states. 
 
 We shall have the following formulas for computing the other 
 required terms, in a non-adiabatic atmosphere. 
 
 
 T T 
 T ~ Tm ~ 
 
 (595) Pi = F 
 
 (596) P h = 1 
 
 Substitute mCp(jTdm jTi d nti) successively, using (715) . 
 
 (597) Initial. (7+ U)a = i m q 2 = Cp JT d m = 
 
 ^^ (Po T 02 - Pi T i2 + P, T a - P h T hl ) + const. 
 
 (598) Final. (V + U) e = = 
 
 ^p-j (PO r i'- P/ r a '+p/ T#-P H T W '. 
 
TRANSFORMATION OF MASSES 187 
 
 (599) Kinetic energy = (7 + U) a - (V + U) e = \ M q 2 = 
 
 ' 2 (approx.). 
 
 6 
 
 (600) Heights, h' = ^ (TV - TV). 
 
 o 
 
 (601) hi = ^f (TV - 2V). 
 
 O 
 
 The approximate solution of this case (Margules) is 
 (602) Velocity, i q 2 = & fr ? +~h T 
 
 CASE IV. The Transformation of Two Masses of Different 
 Temperatures on the Same Levels into a State of Equilibrium. 
 
 Tft2 
 
 F M 
 
 1 Cold (Sj) 
 
 POI T ol 
 
 2 Warm (S 2 ) 
 P o2 T 02 
 
 Initial 
 
 2' Warm 
 
 
 P/ 
 
 T/ 2 
 
 l' Cool 
 
 T 
 
 Po 
 
 To 
 
 Final 
 
 FIG. 21. Transformation of two masses on the same levels. 
 
 Transformation of Two Masses on the Same Levels 
 
 Given the initial data at the height h, Thi, TV, PH, the areas 
 Bi, B 2 , and the entropies Si < S 2 . Hence by the formulas we 
 shall have, 
 
 .T N fife 
 
 (603) Pm = Ph (^ = 
 
 (604) P ^ Ph (^ = 
 
 (605) P; = P A + \ (P M - 
 
 (606) Po' - P + *(**- 
 
 (607) r.- 
 
 (608) 
 
188 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 (609) Initial. (V+lT) a = ^r r f (Poi T 01 - P h T kL + 
 
 O -j I . 
 
 P 02 T Q2 - P h T h2 ) + const. 
 
 (610) Final. (V + U) e = \_.B (P ; T ' - P/ T,' + 
 
 i i _ _ 
 
 P/TV - P A r w ) + const. 
 
 (611) Kinetic energy = J M ? 2 = (F + Z7) a - (V + t/) e . 
 
 (612) Mass. Jkf = - (Po ; - P h ). 
 
 o 
 
 (613) Height. V = ^ (r c ' - zy). 
 
 /^ 
 
 (614) k,'^(T ff '-T u ). 
 
 An approximate solution is given (Margules). 
 
 (615) Take r = 2 ~ \ T* = TtTt, M = B P h -^, = 
 
 j. j\. 
 
 B p h (approx.) 
 
 (616) ^ M q z = ^ M-^^-ghr. 
 
 These solutions must be handled cautiously in practice, 
 because the internal motions of the atmosphere introduce ele- 
 ments of pressure, temperature, and velocity which it is very 
 difficult to follow, and take account of in forming the elements 
 of the integrals, and there is no term for the radiation. 
 
 CASE V. For local changes between two strata of different tem- 
 peratures, where on the boundary the pressure P = PI' = P 2 ' and 
 the temperature is discontinuous 
 
 Take the following conditions: 
 
 (617) w 2 , P 2 T 2 , P 2 ' = P 2 + g m 1} T 2 ' = T 2 {~JT^ 
 
 (618) m lt Pi T\, Px' = Px - g. m,, TV = ^ I ^ \ ^ 
 
 k-< 
 J 
 
LOCAL CHANGES 189 
 
 The condition of equilibrium becomes, for PI = P 2 ' = P, 
 (619) Kinetic energy = Cp [mi (7\ - TV) + m* (T 2 - T 2 ')] 
 
 (.20) , c 
 
 (621) = mi fl 
 
 r> 'T 1 -| T> 'T 1 
 
 (622) Since ^ = and ^ = , therefore, 
 
 fl Pl -T2 P2 
 
 /OO\ 1 1/f .-,2 L " 
 
 (\)Z6) 2^ M ^ = Wl W 2 ~ . 
 
 H pi P2 
 
 The kinetic energy inducing an interchange is proportional 
 to the difference of the densities, and inversely proportional to 
 the product of the densities. Hence, if strata of different den- 
 sities are flowing over one another in the general circulation, 
 which is temporarily stratified, these two strata tend to mix by 
 interpenetration according to this law. 
 
 There are numerous other cases which can be worked out in 
 accordance with the law which may be assumed for the distribu- 
 tion of the temperature in a vertical and in a horizontal direction. 
 
 Compare " Ueber die Energie der Stiirme," von Max Mar- 
 gules. Jahrbuch der k.k. Cent.-Anst. fur Meteo. u. Erdeng. 
 Wien, 1903. 
 
 '' The Thermodynamics of the Atmosphere," F. H. Bigelow. 
 Monthly Weather Review, 1906, and Bulletin W. B. No. 372. 
 
 The General Circulation on a Hemisphere of the Earth } s Atmosphere 
 
 While it is impracticable to take up the problems of the 
 general circulation with the purpose of forming integrals that 
 will take account of the entire circulation, there are yet a few 
 propositions which are of interest in the premises. 
 
 Resume equation (414) 2 , and limit it by assuming symmetry 
 
 about the axis of rotation, so that -~ = 0; also, by omitting 
 
190 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 the small term in w, so that we shall have as a special case, 
 wherein the vertical current and the friction are omitted from 
 the general motion, 
 
 (624) cos (2 co 3 + v) u + -T- = 2 cos ( 3 +v) w - v cos . w+ 
 
 Multiply this equation by r sin 6, and substitute from (413), 
 
 (625^ -7- V cos . u = r sin , so that, 
 at ot 
 
 (626) 2 r sin (w 3 + v) r cos + (r sin 0) 2 = 0. 
 
 Integrating for each gyrating particle, 
 
 (627) r 2 sin 2 (co 3 + v) = c. 
 
 Let C = constant for the entire rotating mass, if Vo is the 
 initial relative angular velocity, 
 
 m fr 2 sin 2 (o> 3 + V ) d m 
 
 ) 
 
 (628) c 
 
 This is equal to the moment of inertia of the entire mass at 
 the distance f r. If the initial state is that of rest on the rotat- 
 ing earth V = 0. Finally, we shall have: 
 
 (629) r 2 sin 2 ( <o 3 + ^-J = f r 2 , and 
 
 (630) rco 3 sin0 + z^ = f>or 
 
 (631) v= 
 
 This is the eastward velocity at the north polar distance 0. 
 (632) If v = 0, sin 2 = f , sin = 0.8165, = 54 44'. 
 
 At latitude = 90 - - 90 - 54 44'= 35 16' the east- 
 ward velocity vanishes at the surface of the earth. Observations 
 indicate that v = extends upward in a direction sloping to- 
 wards the equator from latitude 35 north and south, its position 
 being indefinite above 20,000 meters. 
 
TEMPERATURE GRADIENTS AND VELOCITY 191 
 
 The equation (624) can be obtained as follows: Assume the 
 vortex principle of the conservation of the momenta of inertia, 
 
 (633) or 2 (co 3 + v) = (r sin 0) 2 (o> 3 + v) = c a constant. 
 
 This is not strictly true in the atmosphere, because it is not 
 circulating in a perfect vortex, and this faulty assumption has 
 been generally made in discussing this subject. How far it 
 departs from a vortex law remains to be determined by the 
 observations. Differentiate, divide by r sin d /, and we obtain 
 
 (634) 2 cos 6 (co 3 + v) r-j-. + r sin B-r-. = 0. 
 
 d t (It 
 
 Since r-j-. = u, we find from (624), 
 
 dv dv 
 
 (635) v cos 6. Jj.+ f $m f jj - 0, 
 
 and this is the same as (625). 
 
 Ferrel discusses these equations, and gives some approxi- 
 mately correct views regarding the general circulation. Ober- 
 beck's treatment embraces the three equations of motion, and 
 the solution approaches more closely to the flow of currents 
 actually observed. The complete integration of the system is, 
 however, more complex than has been admitted, and the problem 
 awaits a better treatment. The actual velocities and direction 
 of motion, together with the temperatures, must be so handled 
 as to embrace the general and the local circulations in a single 
 comprehensive solution. 
 
 Three Cases of the Slope of the Temperature Gradients and the 
 Resulting Velocity of the East and West Circulations 
 
 In the earth's atmosphere there are three general cases of 
 the distribution of the temperature gradients and the resulting 
 circulation which can be distinguished, though the solution will 
 not be complete until the radiation term has been accounted 
 for in the equations of motion. These cases are: (l) for the 
 eastward drift in the temperate zone where the velocity increases 
 upwards, while the temperature decreases towards the pole in a 
 
192 
 
 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 line parallel to the axis of the earth's rotation; (2) for the lower 
 levels of the westward drift in the tropics, up to an altitude of 
 about 5,000 meters, wherein the westward motion increases with 
 the height, while the temperature increases towards the pole; (3) 
 for the upper layers of the westward drift, above 5,000 meters 
 into the isothermal region, wherein the velocity decreases with 
 the height, while the temperature increases towards the pole. 
 The case (3) seems to agree with the conditions observed in the 
 atmosphere of the sun, which has decreasing velocity from the 
 equator to the pole, and decreasing velocity from the surface 
 upwards in all latitudes, accompanied by increasing tempera- 
 tures towards the pole. The observations in the tropics on the 
 cloud velocities give an increasing velocity westward to 5,000 
 
 ii in 
 FIG. 22. The relative values of d r and dTJf in three cases. 
 
 meters, then decreasing to 11,000, then increasing to the limit of 
 
 balloon observations, so that cases (2) and (3) alternate to some 
 
 extent. 
 
 or = the distance from the axis of rotation of the earth. 
 
 r the radius from the center of the earth. 
 
 Draw a tangent to the circle at the initial point of the iso- 
 therm. Draw d TX and d r to second points on the same isotherm, 
 to show its slope relative to the horizon and axis of rotation. 
 We have to determine the relation of the temperatures and the 
 velocities of motion in space to each other at any point in the 
 earth's atmosphere. 
 
 Take the general integral of motion (417), omitting the 
 
TEMPERATURE GRADIENTS AND VELOCITY 193 
 
 (Q + J) term, and supply the centrifugal force in the gravitation 
 term, -- | v Q 2 , where v is the linear velocity of the rotating 
 surface at the given latitude, and we obtain, 
 
 (636) - f = | (u* + v z + w*) - ! i>o 2 + g r + constant. 
 
 */ p 
 
 1 7? T 
 
 Substitute - = p-, and pass to logarithms, also put 
 
 g r = from the general law of gravity. 
 
 (637) - logP . R T = | (u*+w*} + i (z; 2 -^ 2 ) + + C. 
 
 We give different values of the temperature (7\, T 2 ) to two 
 adjacent strata flowing over each other at different velocities 
 (fli, %), but since the pressures cannot be discontinuous at the 
 bounding surface, we take PI = P 2 . Hence, by substitution in 
 two strata, and transformation of the terms for differentiation, 
 
 l l 
 
 C, 
 
 This is the general equation to be fulfilled at every point. 
 Now differentiate (637) to r, the change along the radius, for 
 two adjacent strata at pressures PI, P 2 , and we have: 
 
 ^ (log Pi-log P 2 ) l l\ (r.-7\) 
 
 omitting the small terms in u and w. 
 
 Again differentiate (637) to -or, holding the angular momentum 
 (v&) constant in each stratum. At the surface of the earth 
 the velocity v<? = co 3 2 tcr 2 . Hence, 
 
 d V(? 2 co 2 3 zcr 2 2 v 2 
 
 (640) -T =2co 3 2 ttr=- - = -- . Using this form, 
 d iff tD" iff 
 
 and differentiating for two adjacent strata, 
 i -log A) 
 
 _ 1 /Ei 2 -Eo 2 _ ^2 2 -^o 2 \ 
 -or V Ti 2s/ 
 
 i rfa 2 -v)r 2 - (^- 
 
 orL ^r, 
 
194 
 
 THE HYDRODYNAMICS OF THE ATMOSPHERE 
 
 Divide (641) by (639) and the ratio -r becomes, 
 /~,\ dv 1 r~u>i 2 ZJo 2 ) T% (vz 2 Vo 2 ) '. 
 
 (642) ^=~^L Tv^rT 
 
 This equation connects the velocities and temperatures 
 with the slope of the isotherms, and it is capable of three solutions 
 which are expressed as follows: 
 
 Case I 
 
 Case II 
 
 Case III 
 
 dr _ (+) 
 drf (-) 
 
 T 2 <T 1 
 
 +dr _ ( ) 
 
 (+) 
 
 T 2 >T 1 
 
 East 
 
 West(lower) 
 
 "West (upper) and Sun 
 
 FIG. 23. The relative values of v , m, v 2 , Ti, T 2 in three Cases. 
 
 If dr = 0, and the isotherm is parallel to the surface, it 
 follows that fa - v 2 ) T 2 = (vf - vf) T! so that the crossed 
 products of the square of the relative velocities at any point 
 in the atmosphere by the alternating temperatures of the two 
 adjacent strata are equal. The warm stratum assumes greater 
 velocity than the cold stratum, in order to maintain a gradual 
 change in the value of the vertical hydrostatic pressure, such 
 as was developed in Chapter II. If d r changes from 0, in the 
 three cases described, and typically illustrated, the temperature 
 gradients take on slopes that respectively balance the velocities 
 of the air movements, generally above the series of tangents to 
 the horizon in the tropics, but below them in middle latitudes, 
 as have been found from the direct observations in balloon 
 ascensions. In Chapter II, it has been shown how powerfully 
 the (Qi Qo), the change of the heat contents per unit mass 
 
TEMPERATURE GRADIENTS AND VELOCITY 195 
 
 from one level to another reacts upon the velocity system, so 
 that this problem cannot be fully solved through velocity and 
 temperature functions. These theorems can be extended to very 
 useful inferences in the case of the sun where velocities can 
 be measured, but where it is very difficult to determine the ab- 
 solute temperatures prevailing in different strata. 
 
CHAPTER IV 
 
 Examples of the Construction of Vortices in the Earth's 
 Atmosphere 
 
 AN extensive computation on vortices has been published 
 in the Monthly Weather Review, October, 1907, and subse- 
 quent numbers, giving in sufficient detail the method of handling 
 the data. These comprise the funnel-shaped vortex of the 
 Cottage City waterspout, August 19, 1896, the dumb-bell-shaped 
 vortex of the same Cottage City waterspout, the truncated 
 dumb-bell-shaped vortex of the St. Louis tornado, May 27, 1896, 
 the De Witte typhoon, August 1-3, 1901, the impure dumb-bell 
 vortex in the ocean cyclone, October 11, 1905, and the very 
 imperfect vortices of the land cyclones of the United States. 
 In Figs. 24, 25, 26, are given typical (w . z) lines in the Cottage 
 City waterspout, the St. Louis tornado, and the De Witte 
 typhoon, respectively, to which further references will be made. 
 
 In the funnel-shaped vortex tube of the Cottage City water- 
 spout, the plane of reference is at the base of the cloud, 1,100 
 meters above the sea level, and the axis extends downwards, 
 this being the convenient form for tornadoes generally. In 
 the dumb-bell-shaped vortex there are two planes of reference, 
 and the lower one is placed below the sea level while the upper 
 one is placed at the cloud level. The axis between these planes 
 
 180 
 is divided into 180 degrees so that, a = 1 2QQ = 0.15, this being 
 
 the value in the current function \f/ for this case. It has been 
 found that these vortices are generated at the cloud base by 
 the thermodynamic action of strata of different temperatures, 
 and that they are propagated downwards to the sea level or to 
 the surface of the ground. These vortices seem to be cut off 
 or truncated at some distance above the lower plane of reference, 
 and on this supposition the vortex laws, when applied to the 
 observed phenomena, appear to meet satisfactorily all the 
 
 196 
 
EXAMPLES 
 
 197 
 
 requirements of the problem. A few details of the computations 
 will be introduced in this connection. 
 
 Funnel- shaped Water Spout 
 
 Cloud Level 
 
 looVN 
 
 200 * 
 
 300 
 
 400 
 
 500 
 
 600 
 
 700 
 
 800 
 
 900 
 1000 
 1100 
 
 Dumb-bell shaped Water Spout 
 0Cloud Level 
 
 160 
 140 
 120 
 100 
 
 80 
 
 60' 
 
 Se&.Level 
 
 1200 
 
 FIG. 24. The Cottage City, Mass., waterspout, August 19, 1896. 
 
 St. Louis Tornado, May 27, 1896. 
 FIG. 25. Illustrating the truncated dumb-bell-shaped vortex. 
 
 The vertical ordinate is magnified ten times. 
 
 100000 200000 300000 400000 
 
 FIG. 26. The De Witte typhoon, August 1-3, 1901, 
 
 In constructing a vortex of either type it is necessary to 
 know two facts from observations: (1) the tangential velocity 
 v at a point whose radius is S5 in meters from the axis, on a plane 
 defined by 2 in the funnel-shaped vortex, or by az in the dumb- 
 
198 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 bell-shaped vortex; (2) the ratio p, which is the ratio of the 
 successive radii -or in the tubes. By a series of studies on the 
 spacing of the isobars in the De Witte typhoon, the St. Louis 
 tornado, and the ocean cyclone of October 11, 1905, it was 
 found that log p = 0.20546 seems to comply with the positions 
 of the tubes in these vortices as developed in the earth's atmos- 
 phere. In the cyclones and anticyclones there is a wide de- 
 parture from this simple constant ratio, which indicates that 
 another source of energy is at work besides the one generating 
 these simple vortices, but this will require a fuller explanation. 
 In Tables 45-53, and on Figs. 24, 25, 26, are given the results 
 of the computations in sufficient detail to illustrate the scope of 
 the formulas, the dimensions of the vortices, and the velocities, 
 together with the angles of the helices which they make in the 
 tubes. The funnel-shaped tube of Fig. 24 is constructed from 
 or z in Table 45, using the tube (l) ; the truncated dumb-bell- 
 shaped vortex of the St. Louis tornado is constructed from Table 
 48. An examination of the tables of the velocities and the angles 
 suggests numerous remarks on their relations, but as they can 
 be very clearly perceived it is not necessary to write them down. 
 The vortices differ from one another in their dimensions, the 
 waterspout, tornado, and hurricane being illustrations of the 
 dumb-bell vortex. The cyclone shows a close relationship to 
 this type of vortex, but it is distinctly modified by a different 
 distribution of the thermal energy. The meteorological data 
 are so extensive as to make it impracticable to reproduce them 
 in this Treatise. 
 
 Table 45 contains the necessary initial data and the formulas 
 for developing a funnel-shaped vortex in all its tubes from the 
 outer to the inner in succession. Taking the assumed data 
 z = 100, iff = 60, v = 6.67 m/sec, \l/ = vw, we proceed through 
 the formula to construct for tube (1), which has the largest 
 radius, the constant of that tube C, the velocities u, v, w, and 
 the horizontal angle i and the vertical angle y as defined in Fig. 
 14 for cylindrical co-ordinates, and Fig. 18 for vortices. Then 
 follows the successive application of the formulas (535) to (550) 
 by which the data of tubes (2), (3), (4), (5), (6) are computed. 
 
EXAMPLES 199 
 
 Thus by subtracting 0.20546 from log or, in succession, the 
 radii of the other tubes ttr 2 , ttr 3j ztr 4 , or 5) ttr 6 , are obtained; similarly, 
 the log p as indicated is to be applied for C, u, v, w, i, y. 
 
 Table 48 contains the initial assumed data for the dumb-bell- 
 shaped vortex of the St. Louis tornado, truncated on the plane 
 az = 60. Thus, for az = 60, i = - 30, v = 960, v = 13.1 
 m/sec, we find in succession, a = 0.100, A, u, v, w, ttr, i, y, on 
 the tube (l), and on tubes (2), (3), (4), (5), (6), by applying 
 log p or its multiples as indicated by the working group of 
 formulas (535)-(550). In the same manner we proceed with 
 the De Witte typhoon, the ocean cyclone, and similar highly 
 developed vortices. The land cyclone and anticyclone are 
 imperfect vortices, and they involve a system of hydrodynamic 
 stream lines which are highly complex in their origin and develop- 
 ment. It must be constantly remembered that the important 
 radiation terms do not appear in these vortex formulas and 
 examples, so that a fuller treatment would be much more com- 
 plex than the one here briefly summarized. A further illustra- 
 tion will be added in discussing the origin of the cyclone. 
 
 TABLE 45 
 THE COTTAGE CITY WATERSPOUT, AUGUST 19, 1896 
 
 The Funnel-shaped Vortex Tubes, V = C tff z z 
 Collection of the Constants and Working Formulas 
 Assumed data, z = 100 meters, distance below cloud plane 
 Iff 60 meters, radius of cloud sheath. 
 v = 6.67 m/sec, tangential velocity at (z, tCC). 
 
 $ 
 
 Formulas. C = o~ the constant for each tube. 
 Wi 
 
 u C ttr the radial velocity. 
 
 ^ 
 
 v = the tangential velocity. 
 
 ttr 
 
 w = 2 C z the vertical velocity. 
 V = v Iff constant, log V = 2.60206. 
 
 The ratio of the successive radii, P = n ' log p = 0.20546. 
 
 n + 1 
 
 The successive radii, \og'0f n+l = logttr n log p. 
 The successive velocities, log w n + i = log u n + log P. 
 logfn+i = logfl n + log/ , 
 log w n +i = log w n + 2 log p. 
 
200 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 The successive angles, log tan i n = constant. 
 
 u 
 
 tan i = 
 v 
 
 log tan Jj n+l = log tan ?+ log p. tan y = 
 
 TABLE 46 
 THE VALUES OF tcr, C, u, v, w, ON THE PLANE z = 100 
 
 w. 
 
 V sec i 
 
 Formula 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 
 log P 
 
 log or 
 or 
 
 1.77815 
 60.0 
 
 1 . 57269 
 37.4 
 
 1.36723 
 23.3 
 
 1.16177 
 14.5 
 
 0.95631 
 9.0 
 
 0.75085 
 5.6 
 
 l 
 
 0.20546 
 
 log C 
 C 
 
 7.04576 
 0.001111 
 
 7.45668 
 0.002862 
 
 7.86760 
 0.007372 
 
 8.27852 
 0.01899 
 
 8.68944 
 0.04891 
 
 9.10036 
 0.12600 
 
 I + 
 
 0.41092 
 
 log M 
 
 U 
 
 8.82391 
 0.06667 
 
 9.02937 
 0.1070 
 
 9.23483 
 0.1717 
 
 9.44029 
 0.2756 
 
 9.6*575 
 0.4423 
 
 9.85121 
 0.7099 
 
 J + 
 
 0.20546 
 
 log i 1 
 
 V 
 
 0.82391 
 6.67 
 
 1 . 02937 
 10.70 
 
 1.23483 
 17.17 
 
 1.44029 
 27.56 
 
 1.64575 
 44.23 
 
 1.85121 
 70.99 
 
 1 + 
 I 
 
 0.20546 
 
 loga> 
 w 
 
 -9.34679 
 -0.222 
 
 -9.75771 
 -0.572 
 
 -0.16863 
 -1.474 
 
 -0.57955 
 -3.798 
 
 -0.99047 
 -9.783 
 
 - 1.40139 
 -25.199 
 
 B 
 
 0.41092 
 
 log tan i 
 i 
 
 8.00000 
 34' 
 
 8.00000 
 34' 
 
 8.00000 
 34' 
 
 8.00000 
 34 ' 
 
 8.00000 
 34' 
 
 8.00000 
 34' 
 
 s 
 
 o 
 U 
 
 
 log tan 77 
 
 8.52288 
 1 55 
 
 8.72834 
 3 4 
 
 8.93380 
 4 54 
 
 9 . 13926 
 7 51 
 
 9.34472 
 12 28 
 
 9.55018 
 19 33 
 
 + 
 
 0.20546 
 
 TABLE 47 
 THE VALUES OF ttr, u, v, w, i, f\ ON SEVERAL PLANES 
 
 The Radii Iff 
 
 z 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 
 
 00 
 
 CO 
 
 CO 
 
 oo 
 
 oo 
 
 00 
 
 1 
 
 600.0 
 
 373.8 
 
 232.9 
 
 145.1 
 
 90.4 
 
 56.3 
 
 10 
 
 189.7 
 
 118.2 
 
 73.7 
 
 45.9 
 
 28.6 
 
 17.8 
 
 25 
 
 120.0 
 
 74.8 
 
 46.6 
 
 29.0 
 
 18.1 
 
 11.3 
 
 50 
 
 84.9 
 
 52.9 
 
 32.9 
 
 20.5 
 
 12.8 
 
 8.0 
 
 100 
 
 60.0 
 
 37.4 
 
 23.3 
 
 14.5 
 
 9.0 
 
 5.6 
 
 200 
 
 42.4 
 
 26.4 
 
 16.5 
 
 10.3 
 
 6.4 
 
 4.0 
 
 300 
 
 34.6 
 
 21.6 
 
 13.5 
 
 8.4 
 
 5.2 
 
 3.3 
 
 400 
 
 30.0 
 
 18.7 
 
 11.7 
 
 7.3 
 
 4.5 
 
 2.8 
 
 500 
 
 26.8 
 
 16.7 
 
 10.4 
 
 6.5 
 
 4.0 
 
 2.5 
 
 700 
 
 22.7 
 
 14.1 
 
 8.8 
 
 5.5 
 
 3.4 
 
 2.1 
 
 900 
 
 20.0 
 
 12.5 
 
 7.8 
 
 4.8 
 
 3.0 
 
 1.9 
 
 1100 
 
 18.1 
 
 11.3 
 
 7.0 
 
 4.4 
 
 2.7 
 
 1.7 
 
EXAMPLES 
 TABLE 47. CONTINUED 
 
 201 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 The Radial Velocity u 
 
 
 
 1 
 
 00 
 
 0.667 
 
 oo 
 1.070 
 
 00 
 
 1.717 
 
 00 
 
 2.756 
 
 00 
 
 4.423 
 
 00 
 
 7.099 
 
 10 
 
 0.211 
 
 0.339 
 
 0.544 
 
 0.874 
 
 1.402 
 
 2.250 
 
 25 
 
 0.133 
 
 0.214 
 
 0.343 
 
 0.551 
 
 0.885 
 
 1.420 
 
 50 
 
 0.094 
 
 0.151 
 
 0.243 
 
 0.390 
 
 0.626 
 
 1.004 
 
 100 
 
 0.067 
 
 0.107 
 
 0.172 
 
 0.276 
 
 0.442 
 
 0.710 
 
 200 
 
 0.047 
 
 0.076 
 
 0.121 
 
 0.195 
 
 0.313 
 
 0.502 
 
 300 
 
 0.039 
 
 0.062 
 
 0.099 
 
 0.159 
 
 0.255 
 
 0.410 
 
 400 
 
 0.033 
 
 0.054 
 
 0.086 
 
 0.138 
 
 0.221 
 
 0.355 
 
 500 
 
 0.030 
 
 0.048 
 
 0.077 
 
 0.123 
 
 0.198 
 
 0.318 
 
 700 
 
 0.025 
 
 0.040 
 
 0.065 
 
 0.104 
 
 0.167 
 
 0.268 
 
 900 
 
 0.022 
 
 0.036 
 
 0.057 
 
 0.092 
 
 0.147 
 
 0.237 
 
 1100 
 
 0.020 
 
 0.032 
 
 0.052 
 
 0.083 
 
 0.133 
 
 0.214 
 
 The Tangential Velocity v 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 0.7 
 
 1.1 
 
 1.7 
 
 2.8 
 
 4.4 
 
 7.1 
 
 10 
 
 2.1 
 
 3.4 
 
 5.4 
 
 8.7 
 
 14.0 
 
 22.5 
 
 25 
 
 3.3 
 
 5.4 
 
 8.6 
 
 13.8 
 
 22.1 
 
 35.5 
 
 50 
 
 4.7 
 
 7.6 
 
 12.1 
 
 19.5 
 
 31.3 
 
 50.2 
 
 100 
 
 6.7 
 
 10.7 
 
 17.2 
 
 27.6 
 
 44.2 
 
 71.0 
 
 200 
 
 9.4 
 
 15.1 
 
 24.3 
 
 39.0 
 
 62.6 
 
 100.4 
 
 300 
 
 11.6 
 
 18.5 
 
 29.7 
 
 47.7 
 
 76.6 
 
 122.9 
 
 400 
 
 13.3 
 
 21.4 
 
 34.3 
 
 55.1 
 
 88.5 
 
 142.0 
 
 500 
 
 14.9 
 
 23.9 
 
 38.4 
 
 61.6 
 
 98.9 
 
 158.7 
 
 700 
 
 17.6 
 
 28.3 
 
 45.4 
 
 72.9 
 
 117.0 
 
 187.8 
 
 900 
 
 20.0 
 
 32.1 
 
 51.5 
 
 82.7 
 
 132.7 
 
 213.0 
 
 1100 
 
 22.1 
 
 35.5 
 
 57.0 
 
 91.4 
 
 146.7 
 
 235.7 
 
 The Vertical Velocity w 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 -0.0022 
 
 -0.0057 
 
 - 0.0147 
 
 - 0.0380 
 
 - 0.0978 
 
 - 0.2520 
 
 10 
 
 -0.0222 
 
 -0.0572 
 
 - 0.1474 
 
 - 0.3798 
 
 - 0.9783 
 
 - 2.520 
 
 25 
 
 -0.0556 
 
 -0.1431 
 
 - 0.3686 
 
 - 0.9495 
 
 - 2.446 
 
 - 6.300 
 
 50 
 
 -0.1110 
 
 -0.2862 
 
 - 0.7372 
 
 - 1.889 
 
 - 4.891 
 
 - 12.60 
 
 100 
 
 -0.222 
 
 -0.572 
 
 - 1.474 
 
 - 3.798 
 
 - 9.785 
 
 - 25.20 
 
 200 
 
 -0.444 
 
 -1.145 
 
 - 2.949 
 
 - 7.596 
 
 - 19.57 
 
 - 50.40 
 
 300 
 
 -0.667 
 
 -1.717 
 
 - 4.423 
 
 -11.39 
 
 - 29.35 
 
 - 75.60 
 
 400 
 
 -0.889 
 
 -2.290 
 
 - 5.898 
 
 -15.19 
 
 - 39.13 
 
 -100.80 
 
 500 
 
 -1.111 
 
 -2.862 
 
 - 7.372 
 
 -18.99 
 
 - 48.91 
 
 -126.00 
 
 700 
 
 -1.556 
 
 -4.007 
 
 -10.32 
 
 -26.59 
 
 - 68.48 
 
 -176.40 
 
 900 
 
 -2.000 
 
 -5.152 
 
 -13.27 
 
 -34.18 
 
 - 88.05 
 
 -226.80 
 
 1100 
 
 -2.444 
 
 -6.297 
 
 -16.22 
 
 -41.78 
 
 -107.61 
 
 -277.20 
 
202 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 The Horizontal Angle i 
 
 
 
 
 / 
 
 
 
 
 / 
 
 o 
 
 90 
 
 
 
 Constant . 
 
 90 
 
 o 
 
 10 
 
 5 
 
 43 
 
 
 
 5 
 
 43 
 
 50 
 
 1 
 
 9 
 
 11 
 
 1 
 
 9 
 
 100 
 
 
 
 34 
 
 11 
 
 
 
 34 
 
 300 
 
 
 
 11 
 
 
 
 
 
 11 
 
 500 
 
 
 
 7 
 
 tt 
 
 
 
 7 
 
 700 
 
 
 
 5 
 
 " 
 
 
 
 5 
 
 900 
 
 
 
 4 
 
 
 
 
 
 4 
 
 1100 
 
 
 
 3 
 
 
 
 
 3 
 
 The Vertical Angle rj 
 
 
 / 
 
 / 
 
 / 
 
 / 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 36 
 
 58 
 
 1 33 
 
 2 29 
 
 3 59 
 
 6 22 
 
 50 
 
 1 18 
 
 2 10 
 
 3 28 
 
 5 34 
 
 8 53 
 
 14 5 
 
 100 
 
 1 55 
 
 3 4 
 
 4 54 
 
 7 51 
 
 12 28 
 
 19 33 
 
 300 
 
 3 18 
 
 5 18 
 
 8 28 
 
 13 25 
 
 20 58 
 
 31 35 
 
 500 
 
 4 16 
 
 6 49 
 
 10 52 
 
 17 8 
 
 26 19 
 
 38 26 
 
 700 
 
 5 2 
 
 8 3 
 
 12 48 
 
 20 2 
 
 30 20 
 
 43 12 
 
 900 
 
 5 43 
 
 9 7 
 
 14 27 
 
 22 28 
 
 33 34 
 
 46 48 
 
 1100 
 
 6 19 
 
 10 4 
 
 15 54 
 
 24 34 
 
 36 16 
 
 49 39 
 
 All the data on the level z = 100 meters have been computed in Table 46. 
 
 The St. Louis Tornado, May 27, 1896 
 The Truncated Dumb-bell-shaped Vortex Tubes, ip = A z Tff* sin az 
 
 TABLE 48 
 COLLECTION OF THE CONSTANTS AND WORKING FORMULAS 
 
 Assumed data a z = 60 i = 30 on the sea-level plane. 
 ttT = 960 meters, radius of the outer tube. 
 v =13.1 m/sec. tangential velocity at (ttr . az). 
 
 a - I20ST600 * - 10 log P = 0.20546. 
 
 log a sin az = 8.93753 log a cos az = 8.69897. 
 
 A = : , constant for each tube. 
 
 atCTsm az 
 
 u = A a tcr cos az, the radial velocity. 
 
 aV 
 v = , the tangential velocity. 
 
 w = + A a Iff sin az, the vertical velocity. 
 
 Ztr = ( : = ) , the radius on different levels. 
 
 \A a sm az) 
 
EXAMPLES 
 
 203 
 
 tan * = tan (90 + az) = cot az = . 
 
 v 
 
 tan ^ = :, vertical angle. 
 
 vsect 
 
 q = v sec i sec v, total velocity. 
 
 TABLE 49 
 THE VALUES OF -or, A, u, v, w ON THE PLANE a 2 = 60 
 
 Formula 
 
 (i) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 logP 
 
 log Iff 
 
 2.98227 
 
 2.77681 
 
 2.57135 
 
 2.36589 
 
 2.16043 
 
 1.95497 
 
 - 0.20546 
 
 TV 
 
 960.0 
 
 598.2 
 
 372.7 
 
 232.2 
 
 144.7 
 
 90.2 
 
 
 log A 
 
 9.19672 
 
 9.60764 
 
 0.01856 
 
 0.42948 
 
 0.84040 
 
 1.25132 
 
 + 0.41092 
 
 A 
 
 0.1573 
 
 0.4052 
 
 1.0437 
 
 2 . 6883 
 
 6.9247 
 
 17.8371 
 
 f 
 
 log 
 
 -0.87796 
 
 -1.08342 
 
 -1.28888 
 
 -1.49434 
 
 -1.69980 
 
 -1.90526 
 
 Y - 0.20546 
 
 u 
 
 -7.6 
 
 -12.1 
 
 -19.5 
 
 -31.2 
 
 -50.1 
 
 -80.4 
 
 *G 
 
 
 
 
 
 
 
 
 8 
 
 log v 
 
 1.11652 
 
 1.32198 
 
 1.52744 
 
 1.73290 
 
 1.93836 
 
 2 . 14382 
 
 ~ + 0.20546 
 
 V 
 
 13.1 
 
 21.0 
 
 33.7 
 
 54.1 
 
 86.8 
 
 139.3 
 
 1 
 
 log w 
 
 9.43528 
 
 9.84620 
 
 0.25712 
 
 0.66804 
 
 1 . 07896 
 
 1.48988 
 
 -f 0.41092 
 
 w 
 
 0.27 
 
 0.70 
 
 1.81 
 
 4.66 
 
 12.00 
 
 30.89 
 
 M 
 
 
 
 
 
 
 
 
 8 
 
 log tan t 
 
 9.76144 
 
 
 Constant 
 
 
 
 9.76144 
 
 a 
 
 
 30 
 
 
 
 
 
 30 
 
 
 
 
 
 
 
 
 
 u ' 
 
 log tan ?/ 
 
 8.25629 
 
 8.46175 
 
 8.66721 
 
 8.87267 
 
 9.07813 
 
 9.28359 
 
 + 0.20546 
 
 ? 
 
 1 2' 
 
 1 39' 
 
 2 40' 
 
 4 16' 
 
 6 50' 
 
 10 53' 
 
 
 
 
 
 
 
 
 
 
 THE 
 
 TABLE 50 
 VALUES OF iff, u, v, w, i, ~n ON SEVERAL PLANES 
 
 a z 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 The Radii 
 
 180 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 170 
 
 2143.9 
 
 1335.8 
 
 832.3 
 
 518.6 
 
 323.1 
 
 201.3 
 
 160 
 
 1527.6 
 
 951.8 
 
 593.0 
 
 369.5 
 
 230.2 
 
 143.5 
 
 150 
 
 1263.4 
 
 787.2 
 
 490.5 
 
 305.6 
 
 190.4 
 
 118.6 
 
 140 
 
 1114.3 
 
 694.3 
 
 432.6 
 
 269.5 
 
 167.9 
 
 104.6 
 
 130 
 
 1020.7 
 
 636.0 
 
 396.3 
 
 246.9 
 
 153.8 
 
 95.9 
 
 120 
 
 960.0 
 
 598.1 
 
 372.7 
 
 232.2 
 
 144.7 
 
 90.2 
 
 110 
 
 921.6 
 
 574.2 
 
 357.8 
 
 222.9 
 
 138.9 
 
 86.5 
 
 100 
 
 900.2 
 
 560.9 
 
 349.5 
 
 217.8 
 
 135.7 
 
 84.5 
 
 90 
 
 893.4 
 
 556.6 
 
 346.8 
 
 216.1 
 
 134.6 
 
 83.9 
 
 80 
 
 900.2 
 
 560.9 
 
 349.5 
 
 217.8 
 
 135.7 
 
 84.5 
 
 70 
 
 921.6 
 
 574.2 
 
 357.8 
 
 222.9 
 
 138.9 
 
 86.5 
 
 60 
 
 960.0 
 
 598.1 
 
 372.7 
 
 232.2 
 
 144.7 
 
 90.2 
 
204 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 TABLE 50. CONTINUED 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 The Radial Velocity u 
 
 180 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 170 
 
 33.2 
 
 53.3 
 
 85.5 
 
 137.3 
 
 220.3 
 
 353.6 
 
 160 
 
 22.6 
 
 36.2 
 
 58.2 
 
 93.4 
 
 149.8 
 
 240.4 
 
 150 
 
 17.2 
 
 27.6 
 
 44.3 
 
 71.2 
 
 114.2 
 
 183.3 
 
 140 
 
 13.4 
 
 21.6 
 
 34.6 
 
 55.5 
 
 89.1 
 
 143.0 
 
 130 
 
 10.3 
 
 16.6 
 
 26.6 
 
 42.7 
 
 68.5 
 
 109.9 
 
 120 
 
 7.6 
 
 12.1 
 
 19.5 
 
 31.2 
 
 50.1 
 
 80.4 
 
 110 
 
 5.0 
 
 8.0 
 
 12.8 
 
 20.5 
 
 32.9 
 
 52.8 
 
 100 
 
 2.5 
 
 4.0 
 
 6.3 
 
 10.2 
 
 16.3 
 
 26.2 
 
 90 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 80 
 
 -2.5 
 
 - 4.0 
 
 - 6.3 
 
 -10.2 
 
 -16.3 
 
 -26.2 
 
 70 
 
 -5.0 
 
 - 8.0 
 
 -12.8 
 
 -20.5 
 
 -32.9 
 
 -52.8 
 
 60 
 
 -7.6 
 
 -12.1 
 
 -19.5 
 
 -31.2 
 
 -50.1 
 
 -80.4 
 
 The Tangential Velocity v 
 
 180 
 
 
 
 
 
 
 
 
 
 
 
 
 
 170 
 
 5.9 
 
 9.4 
 
 15.1 
 
 24.2 
 
 38.9 
 
 62.4 
 
 160 
 
 8.2 
 
 13.2 
 
 21.2 
 
 34.0 
 
 54.5 
 
 87.5 
 
 150 
 
 9.9 
 
 16.0 
 
 25.6 
 
 41.1 
 
 65.9 
 
 105.8 
 
 140 
 
 11.3 
 
 18.1 
 
 29.0 
 
 46.6 
 
 74.8 
 
 120.0 
 
 130 
 
 12.3 
 
 19.7 
 
 31.7 
 
 50.9 
 
 81.6 
 
 131.0 
 
 120 
 
 13.1 
 
 21.0 
 
 33.7 
 
 54.1 
 
 86.8 
 
 139.3 
 
 110 
 
 13.6 
 
 21.9 
 
 35.1 
 
 56.3 
 
 90.4 
 
 145.1 
 
 100 
 
 13.9 
 
 22.4 
 
 35.9 
 
 57.7 
 
 92.5 
 
 148.5 
 
 90 
 
 14.1 
 
 22.6 
 
 36.2 
 
 58.1 
 
 93.2 
 
 149.6 
 
 80 
 
 13.9 
 
 22.4 
 
 35.9 
 
 57.7 
 
 92.5 
 
 148.5 
 
 70 
 
 13.6 
 
 21.9 
 
 35.1 
 
 56.3 
 
 90.4 
 
 145.1 
 
 60 
 
 13.1 
 
 21.0 
 
 33.7 
 
 54.1 
 
 86.8 
 
 139.3 
 
 The Vertical Velocity -w 
 
 180 
 
 
 
 
 
 
 
 
 
 
 
 
 
 170 
 
 0.06 
 
 0.14 
 
 0.36 
 
 0.93 
 
 2.41 
 
 6.20 
 
 160 
 
 0.11 
 
 0.28 
 
 0.71 
 
 1.84 
 
 4.74 
 
 12.20 
 
 150 
 
 0.16 
 
 0.41 
 
 1.04 
 
 2.69 
 
 6.93 
 
 17.84 
 
 140 
 
 0.20 
 
 0.52 
 
 1.34 
 
 3.46 
 
 8.90 
 
 22.93 
 
 130 
 
 0.24 
 
 0.62 
 
 1.60 
 
 4.22 
 
 10.86 
 
 27.96 
 
 120 
 
 0.27 
 
 0.70 
 
 1.81 
 
 4.66 
 
 11.99 
 
 30.90 
 
 110 
 
 0.30 
 
 0.76 
 
 1.96 
 
 5.05 
 
 13.01 
 
 33.52 
 
 100 
 
 0.31 
 
 0.80 
 
 2.06 
 
 5.30 
 
 13.64 
 
 35.13 
 
 90 
 
 0.32 
 
 0.81 
 
 2.09 
 
 5.38 
 
 13.85 
 
 35.67 
 
 80 
 
 0.31 
 
 0.80 
 
 2.06 
 
 5.30 
 
 13.64 
 
 35.13 
 
 70 
 
 0.30 
 
 0.76 
 
 1.96 
 
 5.05 
 
 13.01 
 
 33.52 
 
 60 
 
 0.27 
 
 0.70 
 
 1.81 
 
 4.66 
 
 11.99 
 
 30.90 
 
EXAMPLES 
 The Horizontal Angle i 
 
 205 
 
 180 
 
 +90 
 
 Constant 
 
 +90 
 
 160 
 
 +70 
 
 
 +70 
 
 140 
 
 +50 
 
 
 
 +50 
 
 120 
 
 +30 . 
 
 
 
 +30 
 
 100 
 
 + 10 
 
 
 
 +10 
 
 90 
 
 
 
 a 
 
 
 
 80 
 
 -10 
 
 H 
 
 -10 
 
 70 
 
 -20 
 
 II 
 
 -20 
 
 60 
 
 -30 
 
 II 
 
 -30 
 
 The Vertical Angle rj 
 
 180 
 
 0' 
 
 0' 
 
 0' 
 
 0' 
 
 0' 
 
 0' 
 
 170 
 
 6 
 
 9 
 
 14 
 
 23 
 
 37 
 
 59 
 
 160 
 
 15 
 
 25 
 
 40 
 
 1 4 
 
 1 42 
 
 2 44 
 
 150 
 
 27 
 
 44 
 
 1 10 
 
 1 52 
 
 3 
 
 4 49 
 
 140 
 
 40 
 
 1 4 
 
 1 42 
 
 2 44 
 
 4 23 
 
 7 
 
 130 
 
 52 
 
 1 23 
 
 2 13 
 
 3 38 
 
 5 49 
 
 9 17 
 
 120 
 
 2 
 
 1 39 
 
 2 40 
 
 4 16 
 
 6 50 
 
 10 53 
 
 110 
 
 10 
 
 1 52 
 
 3 
 
 4 49 
 
 7 42 
 
 12 15 
 
 100 
 
 15 
 
 2 1 
 
 3 14 
 
 5 10 
 
 8 16 
 
 13 7 
 
 90 
 
 17 
 
 2 3 
 
 3 18 
 
 5 17 
 
 8 27 
 
 13 25 
 
 80 
 
 15 
 
 2 1 
 
 3 14 
 
 5 10 
 
 8 16 
 
 13 7 
 
 70 
 
 10 
 
 1 52 
 
 3 
 
 4 49 
 
 7 42 
 
 12 15 
 
 60 
 
 1 2 
 
 1 39 
 
 2 40 
 
 4 16 
 
 6 50 
 
 10 53 
 
 The data on the level a z have been computed in Table 49. 
 
 TABLE 51 
 
 THE DE WITTE TYPHOON, AUGUST 1-3, 1901, 
 IN THE CHINA SEA 
 
 Results from the plane a z = 60 
 
 Initial z = 12000 meters a = 
 
 180 C 
 
 12000 + 600 
 
 = 0.010 
 
 z 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 A 
 
 .002016 
 
 .005193 
 
 .013375 
 
 .034452 
 
 .088744 
 
 .248922 
 
206 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 The Radii W in Meters 
 
 a z = 180 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 170 
 
 887600 
 
 534338 
 
 332938 
 
 207443 
 
 129253 
 
 80534 
 
 160 
 
 611071 
 
 380742 
 
 237232 
 
 147813 
 
 92098 
 
 57384 
 
 150 
 
 505389 
 
 314900 
 
 196205 
 
 122250 
 
 76172 
 
 47460 
 
 140 
 
 445740 
 
 277727 
 
 173044 
 
 107820 
 
 67180 
 
 41858 
 
 130 
 
 408310 
 
 254406 
 
 158515 
 
 98766 
 
 61539 
 
 38344 
 
 120 
 
 384018 
 
 239272 
 
 149083 
 
 92890 
 
 57878 
 
 36062 
 
 110 
 
 368655 
 
 229700 
 
 143120 
 
 89174 
 
 55563 
 
 34619 
 
 100 
 
 360117 
 
 224379 
 
 139803 
 
 87108 
 
 54275 
 
 33818 
 
 90 
 
 357367 
 
 222663 
 
 138735 
 
 86444 
 
 53860 
 
 33559 
 
 80 
 
 360117 
 
 224379 
 
 139803 
 
 87108 
 
 54275 
 
 33818 
 
 70 
 
 368655 
 
 229700 
 
 143120 
 
 89174 
 
 55563 
 
 34619 
 
 60 
 
 384018 
 
 239272 
 
 149083 
 
 92890 
 
 57878 
 
 36062 
 
 The Radial Velocity u in Meters per Second 
 
 az = 180 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 170 
 
 17.03 
 
 27.32 
 
 43.86 
 
 70.38 
 
 112.96 
 
 181.30 
 
 160 
 
 11.58 
 
 18.58 
 
 29.82 
 
 47.85 
 
 76.80 
 
 123.27 
 
 150 
 
 8.82 
 
 14.16 
 
 22.72 
 
 36.47 
 
 58.53 
 
 93.94 
 
 140 
 
 6.88 
 
 11.05 
 
 17.73 
 
 28.46 
 
 45.67 
 
 73.30 
 
 130 
 
 5.29 
 
 8.49 
 
 13.63 
 
 21.87 
 
 35.10 
 
 56.34 
 
 120 
 
 3.87 
 
 6.21 
 
 9.97 
 
 16.00 
 
 25.68 
 
 41.22 
 
 110 
 
 2.54 
 
 4.08 
 
 6.55 
 
 10.51 
 
 16.86 
 
 27.07 
 
 100 
 
 1.28 
 
 2.02 
 
 3.25 
 
 5.21 
 
 8.36 
 
 13.42 
 
 90 
 
 0.00 
 
 0.00 
 
 0.00 
 
 0.00 
 
 0.00 
 
 0.00 
 
 80 
 
 -1.28 
 
 -2.02 
 
 -3.25 
 
 - 5.21 
 
 - 8.36 
 
 -13.42 
 
 70 
 
 -2.54 
 
 -4.08 
 
 -6.55 
 
 -10.51 
 
 -16.86 
 
 -27.07 
 
 60 
 
 -3.87 
 
 -6.21 
 
 -9.97 
 
 -16.00 
 
 -25.68 
 
 -41.22 
 
 The Tangential Velocity v in Meters per Second 
 
 a z = 180 
 
 0.00 
 
 0.00 
 
 0.00 
 
 0.00 
 
 0.00 
 
 0.00 
 
 170 
 
 3.00 
 
 4.82 
 
 7.73 
 
 12.41 
 
 19.92 
 
 31.97 
 
 160 
 
 4.21 
 
 6.76 
 
 10.85 
 
 17.42 
 
 27.95 
 
 44.86 
 
 150 
 
 5.09 
 
 8.17 
 
 13.12 
 
 21.06 
 
 33.79 
 
 54.24 
 
 140 
 
 5.78 
 
 9.27 
 
 14.88 
 
 23.88 
 
 38.32 
 
 61.50 
 
 130 
 
 6.31 
 
 10.12 
 
 16.24 
 
 26.07 
 
 41.84 
 
 67.14 
 
 120 
 
 6.70 
 
 10.76 
 
 17.27 
 
 27.72 
 
 44.48 
 
 71.39 
 
 110 
 
 6.98 
 
 11.21 
 
 17.99 
 
 28.87 
 
 46.34 
 
 74.36 
 
 100 
 
 7.15 
 
 11.47 
 
 18.42 
 
 29.55 
 
 47.43 
 
 76.13 
 
 90 
 
 7.20 
 
 11.56 
 
 18.56 
 
 29.78 
 
 47.80 
 
 76.71 
 
 80 
 
 7.15 
 
 11.47 
 
 18.42 
 
 29.55 
 
 47.43 
 
 76.13 
 
 70 
 
 6.98 
 
 11.21 
 
 17.99 
 
 28.87 
 
 46.34 
 
 74.36 
 
 60 
 
 6.70 
 
 10.76 
 
 17.27 
 
 27.72 
 
 44.48 
 
 71.39 
 
THE OCEAN AND THE LAND CYCLONES 
 
 The Vertical Velocity iv in Meters per Second 
 
 207 
 
 a z = 180 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 0.0000 
 
 170 
 
 0.0007 
 
 0.0018 
 
 0.0046 
 
 0.0120 
 
 0.0308 
 
 0.0794 
 
 160 
 
 0.0014 
 
 0.0036 
 
 0.0091 
 
 0.0236 
 
 0.0607 
 
 0.1564 
 
 150 
 
 0.0020 
 
 0.0052 
 
 0.0134 
 
 0.0345 
 
 0.0887 
 
 0.2286 
 
 140 
 
 0.0026 
 
 0.0067 
 
 0.0172 
 
 0.0443 
 
 0.1141 
 
 0.2939 
 
 130 
 
 0.0031 
 
 0.0080 
 
 0.0205 
 
 0.0528 
 
 0.1340 
 
 0.3502 
 
 120 
 
 0.0035 
 
 0.0090 
 
 0.0232 
 
 0.0597 
 
 0.1537 
 
 0.3959 
 
 110 
 
 0.0038 
 
 0.0098 
 
 0.0251 
 
 0.0648 
 
 0.1668 
 
 0.4296 
 
 100 
 
 0.0040 
 
 0.0102 
 
 0.0263 
 
 0.0679 
 
 0.1748 
 
 0.4502 
 
 90 
 
 0.0040 
 
 0.0104 
 
 0.0268 
 
 0.0689 
 
 0.1775 
 
 0.4572 
 
 80 
 
 0.0040 
 
 0.0102 
 
 0.0263 
 
 0.0679 
 
 0.1748 
 
 0.4502 
 
 70 
 
 0.0038 
 
 0.0098 
 
 0.0251 
 
 0.0648 
 
 0.1668 
 
 0.4296 
 
 60 
 
 0.0035 
 
 0.0090 
 
 0.0232 
 
 0.0597 
 
 0.1537 
 
 0.3959 
 
 The Ocean and the Land Cyclones 
 
 The tornadoes and hurricanes always occur in strata of air 
 which are practically quiescent in the vertical direction, the 
 tornadoes in the lower levels of stagnant air during hot summer 
 afternoons, and the hurricanes in the neighborhood of the 
 latitudes 30 to 35, where the east and west movements in the 
 general circulation practically disappear. Should hurricanes 
 move into higher latitudes, where the eastward drift prevails 
 with an increase of its velocity in proportion to the height above 
 the ocean, the nearly perfect dumb-bell vortices which represent 
 them are transformed into imperfect vortices of the same general 
 type. The penetration of the head of the vortex into the midst 
 of the eastward drift introduces components of resistance which 
 deplete and even destroy the type in the upper levels, so that 
 it is degraded to a cyclone, or imperfect dumb-bell vortex by the 
 mere mechanical action. Furthermore, the temperature dis- 
 tribution is distinctly different in hurricanes and in cyclones. 
 In the former the temperature differences are separated by 
 horizontal planes in the upper levels, while in the latter the 
 temperatures are separated chiefly in a vertical direction. The 
 hurricanes have a symmetrical horizontal distribution of tem- 
 peratures, but in cyclones the temperature distribution is de- 
 cidedly asymmetrical, as is well known from the weather maps 
 on the surface. The same asymmetry of temperature continues 
 
208 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 to the levels as high as 10,000 meters, warm on the east and cold 
 on the west side of cyclones in the United States. These broad, 
 thin sheets of warm and cold air, under the action of gravity, 
 tend to return to a horizontal symmetry by the production of 
 stream lines, whereby the cold air underruns the warm stream 
 to the east and to the west by dividing into two branches, while 
 the warm air overruns the cold air to the east and to the west 
 similarly in two branches. Complicated stream lines are thus 
 produced, which are those observed in the free air, after entering 
 into composition with the velocities of the general circulation 
 of the locality. This complex subject will require much more 
 study than has been possible up to the present time in order to 
 secure a complete analysis of the data, but it is clear that the 
 research must proceed along certain lines which can be briefly 
 indicated. 
 
 The first problem is to separate the imperfect from the 
 perfect vortices, and to assign the components of resistance, 
 that is, to construct a reverse vortex which is practically equiva- 
 lent to the system of reactions that prevents the dumb-bell vortex 
 from developing into a pure form. The second problem is to 
 determine the stream lines by which the masses of air at different 
 temperatures are drawn by the force of gravity into these im- 
 perfect cyclonic vortices. The ocean cyclone, October 11, 1905, 
 has been taken to illustrate the composition of vortices, and 
 the land cyclones must be studied more at length from the data 
 provided by balloon and kite ascensions in Europe and the 
 United States. The ocean cyclone is more highly developed 
 than the land cyclone, and affords a convenient transition from 
 the hurricane to the ordinary cyclonic storm. The cyclone of 
 October 11, 1905, has been reduced to an equivalent cylindrical 
 vortex by taking the mean radii as measured in four directions 
 at right angles to each other. This mechanical process need 
 not be repeated here, but the result is that the radii are not 
 spaced in the vortical geometrical ratio. They diverge from that 
 model which belongs to the perfect vortex. The corresponding 
 velocities tangential to the equivalent circular isobars were 
 constructed from the observed values in different parts of the 
 
THE OCEAN AND THE LAND CYCLONES 
 
 209 
 
 cyclone as reported by the 110 vessels that made observations 
 on that date. 
 
 TABLE 52 
 
 THE OCEAN CYCLONE, OCTOBER 11, 1905 
 
 The Imperfect Dumb-bell-shaped Vortex V'l 
 
 Results for the Plane a z = 60 
 
 180 
 Initial z = 8000 meters, a = QQQ + 4QQQ = 0.015, log p =0.10600. 
 
 z 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 (7) 
 
 (8) 
 
 A, 
 
 .00089 
 
 .00127 
 
 .00184 
 
 .00265 
 
 .00382 
 
 .00551 
 
 .00795 
 
 .01146 
 
 The Radii Iff in Kilometers 
 
 
 
 
 
 
 
 
 
 
 
 a z = 180 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 170 
 
 3079.9 
 
 2412.9 
 
 1890.3 
 
 1480.6 
 
 1160.2 
 
 909.0 
 
 712.1 
 
 557.9 
 
 160 
 
 2194.6 
 
 1719.3 
 
 1346.9 
 
 1055.2 
 
 826.7 
 
 647.7 
 
 507.4 
 
 397.5 
 
 150 
 
 1815.0 
 
 1422.0 
 
 1114.0 
 
 872.8 
 
 683.7 
 
 535.7 
 
 419.6 
 
 328.8 
 
 . 140 
 
 1600.8 
 
 1254.1 
 
 982.5 
 
 769.7 
 
 603.0 
 
 472.4 
 
 370.1 
 
 290.0 
 
 130 
 
 1466.4 
 
 1148.8 
 
 900.0 
 
 705.1 
 
 552.4 
 
 432.8 
 
 339.0 
 
 265.6 
 
 120 
 
 1379.2 
 
 1080.5 
 
 846.5 
 
 663.2 
 
 519.5 
 
 407.0 
 
 318.9 
 
 249.8 
 
 110 
 
 1324.0 
 
 1037.2 
 
 812.6 
 
 636.6 
 
 498.8 
 
 390.7 
 
 306.1 
 
 239.8 
 
 100 
 
 1293.3 
 
 1013.2 
 
 793.8 
 
 621.9 
 
 487.2 
 
 381.7 
 
 299.0 
 
 234.3 
 
 90 
 
 1283.4 
 
 1005 . 5 
 
 787.7 
 
 617.1 
 
 483.5 
 
 378.8 
 
 296.7 
 
 232.5 
 
 80 
 
 1293.3 
 
 1013.2 
 
 793.8 
 
 621.9 
 
 487.2 
 
 381.7 
 
 299.0 
 
 234.3 
 
 70 
 
 1324.0 
 
 1037.2 
 
 812.6 
 
 636.6 
 
 498.8 
 
 390.7 
 
 306.1 
 
 239.8 
 
 60 
 
 1379.2 
 
 1080.5 
 
 846.5 
 
 663.2 
 
 519.5 
 
 407.0 
 
 318.9 
 
 249.8 
 
 The Radial Velocity u\ in Meters per Second 
 
 a z = 180 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 170 
 
 40.2 
 
 45.4 
 
 51.3 
 
 58.0 
 
 65.5 
 
 74.0 
 
 83.6 
 
 94.4 
 
 160 
 
 27.3 
 
 30.9 
 
 34.9 
 
 39.4 
 
 44.5 
 
 50.3 
 
 56.8 
 
 64.2 
 
 150 
 
 20.8 
 
 23.5 
 
 26.6 
 
 30.0 
 
 33.9 
 
 38.3 
 
 43.0 
 
 48.9 
 
 140 
 
 16.2 
 
 18.4 
 
 20.7 
 
 23.4 
 
 26.5 
 
 29.9 
 
 33.8 
 
 38.1 
 
 130 
 
 12.5 
 
 14.1 
 
 16.0 
 
 18.0 
 
 20.4 
 
 23.0 
 
 26.0 
 
 29.4 
 
 120 
 
 9.1 
 
 10.3 
 
 11.7 
 
 13.2 
 
 14.9 
 
 16.8 
 
 19.0 
 
 21.5 
 
 110 
 
 6.0 
 
 6.8 
 
 7.7 
 
 8.7 
 
 9.8 
 
 11.0 
 
 12.5 
 
 14.1 
 
 100 
 
 3.0 
 
 3.4 
 
 3.8 
 
 4.3 
 
 4.8 
 
 5.5 
 
 6.2 
 
 7.0 
 
 90 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 80 
 
 - 3.0 
 
 - 3.4 
 
 - 3.8 
 
 - 4.3 
 
 - 4.8 
 
 - 5.5 
 
 - 6.2 
 
 - 7.0 
 
 70 
 
 - 6.0 
 
 - 6.8 
 
 - 7.7 
 
 - 8.7 
 
 - 9.8 
 
 -11.0 
 
 -12.5 
 
 -14.1 
 
 60 
 
 - 9.1 
 
 -10.3 
 
 -11.7 
 
 -13.2 
 
 -14.9 
 
 -16.8 
 
 -19.0 
 
 -21.5 
 
210 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 The Tangential Velocity vi in Meters per Second 
 
 az = 180 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 170 
 
 7.1 
 
 8.0 
 
 9.0 
 
 10.2 
 
 11.6 
 
 13.0 
 
 14.7 
 
 16.7 
 
 160 
 
 9.9 
 
 11.2 
 
 12.7 
 
 14.3 
 
 16.2 
 
 18.3 
 
 20.7 
 
 23.4 
 
 150 
 
 12.0 
 
 13.6 
 
 15.4 
 
 17.3 
 
 19.6 
 
 22.1 
 
 25.0 
 
 28.3 
 
 140 
 
 13.6 
 
 15.4 
 
 17.4 
 
 19.7 
 
 22.2 
 
 25.1 
 
 28.4 
 
 32.0 
 
 130 
 
 14.9 
 
 16.8 
 
 19.0 
 
 21.5 
 
 24.3 
 
 27.4 
 
 31.0 
 
 35.0 
 
 120 
 
 15.8 
 
 17.9 
 
 20.2 
 
 22.8 
 
 25.8 
 
 29.1 
 
 32.9 
 
 37.2 
 
 110 
 
 16.5 
 
 18.6 
 
 21.0 
 
 23.8 
 
 26.9 
 
 30.3 
 
 34.3 
 
 38.7 
 
 100 
 
 16.9 
 
 19.1 
 
 21.5 
 
 24.3 
 
 27.5 
 
 31.1 
 
 35.1 
 
 39.7 
 
 90 
 
 17.0 
 
 19.2 
 
 21.7 
 
 24.5 
 
 27.7 
 
 31.3 
 
 35.4 
 
 40.0 
 
 80 
 
 16.9 
 
 19.1 
 
 21.5 
 
 24.3 
 
 27.5 
 
 31.1 
 
 35.1 
 
 39.7 
 
 70 
 
 16.5 
 
 18.6 
 
 21.0 
 
 23.8 
 
 26.9 
 
 30.3 
 
 34.3 
 
 38.7 
 
 60 
 
 15.8 
 
 17.9 
 
 20.2 
 
 22.8 
 
 25.8 
 
 29.1 
 
 32.9 
 
 37.2 
 
 The Vertical Velocity w\ in Meters per Second 
 
 o 
 
 az = 180 
 
 .0000 
 
 .0000 
 
 .0000 
 
 .0000 
 
 .0000 
 
 .0000 
 
 .0000 
 
 .0000 
 
 170 
 
 .0003 
 
 .0004 
 
 .0006 
 
 .0009 
 
 .0013 
 
 .0019 
 
 .0028 
 
 .0040 
 
 160 
 
 .0006 
 
 .0009 
 
 .0013 
 
 .0018 
 
 .0026 
 
 .0038 
 
 .0054 
 
 .0078 
 
 150 
 
 .0009 
 
 .0013 
 
 .0018 
 
 .0026 
 
 .0038 
 
 .0055 
 
 .0079 
 
 .0115 
 
 140 
 
 .0011 
 
 .0016 
 
 .0024 
 
 .0034 
 
 .0049 
 
 .0071 
 
 .0102 
 
 .0147 
 
 130 
 
 .0014 
 
 .0020 
 
 .0028 
 
 .0041 
 
 .0059 
 
 .0084 
 
 .0122 
 
 .0176 
 
 120 
 
 .0015 
 
 .0022 
 
 .0032 
 
 .0046 
 
 .0066 
 
 .0095 
 
 .0138 
 
 .0198 
 
 110 
 
 .0017 
 
 .0024 
 
 .0035 
 
 .0050 
 
 .0072 
 
 .0104 
 
 .0149 
 
 .0215 
 
 100 
 
 .0017 
 
 .0025 
 
 .0036 
 
 .0052 
 
 .0075 
 
 .0109 
 
 .0157 
 
 .0226 
 
 90 
 
 .0018 
 
 .0025 
 
 .0037 
 
 .0053 
 
 .0076 
 
 .0110 
 
 .0159 
 
 .0229 
 
 80 
 
 .0017 
 
 .0025 
 
 .0036 
 
 .0052 
 
 .0075 
 
 .0109 
 
 .0157 
 
 .0226 
 
 70 
 
 .0017 
 
 .0024 
 
 .0035 
 
 .0050 
 
 .0072 
 
 .0104 
 
 .0149 
 
 .0215 
 
 60 
 
 .0015 
 
 .0022 
 
 .0032 
 
 .0046 
 
 .0066 
 
 .0095 
 
 .0138 
 
 .0198 
 
 TABLE 53 
 
 THE OCEAN CYCLONE, OCTOBER 11, 1905 
 The Perfect Dumb-bell-Shaped Vortex Vo 
 The Radii ZCT remain the Same 
 
 z 
 
 A, 
 
 (1) 
 .00089 
 
 (2) 
 .00144 
 
 (3) 
 .00235 
 
 (4) 
 .00382 
 
 (5) 
 .00623 
 
 (6) 
 .01014 
 
 (7) 
 .01653 
 
 (8) 
 .02699 
 
 The Radial Velocity 
 
 
 
 
 
 
 
 
 
 
 
 a z = 180 
 
 00 
 
 00 
 
 00 
 
 00 
 
 oo 
 
 00 
 
 00 
 
 00 
 
 170 
 
 40.2 
 
 51.3 
 
 65.5 
 
 83.6 
 
 106.7 
 
 136.2 
 
 173.8 
 
 221.9 
 
 160 
 
 27.3 
 
 34.9 
 
 44.5 
 
 56.8 
 
 72.5 
 
 92.6 
 
 118.2 
 
 150.9 
 
 150 
 
 20.8 
 
 26.6 
 
 33.9 
 
 43.3 
 
 55.3 
 
 70.6 
 
 90.1 
 
 115.0 
 
 140 
 
 16.2 
 
 20.7 
 
 26.5 
 
 33.8 
 
 43.1 
 
 55.1 
 
 70.3 
 
 89.7 
 
 130 
 
 12.5 
 
 15.9 
 
 20.4 
 
 26.0 
 
 33.2 
 
 42.3 
 
 54.0 
 
 69.0 
 
 120 
 
 9.1 
 
 11.7 
 
 14.9 
 
 19.0 
 
 24.3 
 
 31.0 
 
 39.5 
 
 50.4 
 
 110 
 
 6.0 
 
 7.7 
 
 9.7 
 
 12.5 
 
 15.9 
 
 20.3 
 
 26.0 
 
 33.1 
 
 100 
 
 3.0 
 
 3.8 
 
 4.8 
 
 6.2 
 
 7.9 
 
 10.1 
 
 12.9 
 
 16.4 
 
 90 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
THE OCEAN AND THE LAND CYCLONES 
 
 211 
 
 The Tangential Velocity v 
 
 az = 180 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 170 
 
 7.1 
 
 9.0 
 
 11.6 
 
 14.7 
 
 18.8 
 
 24.0 
 
 30.7 
 
 39.1 
 
 160 
 
 9.9 
 
 12.7 
 
 16.2 
 
 20.7 
 
 26.4 
 
 33.7 
 
 43.0 
 
 54.9 
 
 150 
 
 12.0 
 
 15.4 
 
 19.6 
 
 25.0 
 
 31.9 
 
 40.8 
 
 52.0 
 
 66.4 
 
 140 
 
 13.6 
 
 17.4 
 
 22.2 
 
 28.4 
 
 36.2 
 
 46.2 
 
 59.0 
 
 75.3 
 
 130 
 
 14.9 
 
 19.0 
 
 24.2 
 
 31.0 
 
 39.5 
 
 50.4 
 
 64.4 
 
 82.2 
 
 120 
 
 15.8 
 
 20.2 
 
 25.8 
 
 32.8 
 
 42.0 
 
 53.6 
 
 68.5 
 
 87.4 
 
 110 
 
 16.5 
 
 21.0 
 
 26.9 
 
 34.3 
 
 43.8 
 
 55.9 
 
 71.3 
 
 91.0 
 
 100 
 
 16.9 
 
 21.5 
 
 27.5 
 
 35.1 
 
 44.8 
 
 57.2 
 
 73.0 
 
 93.2 
 
 90 
 
 17.0 
 
 21.7 
 
 27.7 
 
 35.4 
 
 45.1 
 
 57.6 
 
 73.6 
 
 93.9 
 
 The Vertical Velocity 
 
 a z = 180 
 
 .0000 
 
 .0000 
 
 .0000 
 
 .0000 
 
 .0000 
 
 .0000 
 
 .0000 
 
 .0000 
 
 170 
 
 .0003 
 
 .0005 
 
 .0008 
 
 .0013 
 
 .0022 
 
 .0035 
 
 .0057 
 
 .0094 
 
 160 
 
 .0006 
 
 .0010 
 
 .0016 
 
 .0026 
 
 .0043 
 
 .0069 
 
 .0113 
 
 .0184 
 
 150 
 
 .0009 
 
 .0014 
 
 .0024 
 
 .0038 
 
 .0062 
 
 .0101 
 
 .0165 
 
 .0269 
 
 140 
 
 .0011 
 
 .0018 
 
 .0030 
 
 .0049 
 
 .0080 
 
 .0130 
 
 .0212 
 
 .0346 
 
 130 
 
 .0014 
 
 .0022 
 
 .0036 
 
 .0058 
 
 .0095 
 
 .0155 
 
 .0253 
 
 .0412 
 
 120 
 
 .0015 
 
 .0025 
 
 .0041 
 
 .0066 
 
 .0108 
 
 .0176 
 
 .0286 
 
 .0466 
 
 110 
 
 .0017 
 
 .0027 
 
 .0044 
 
 .0072 
 
 .0117 
 
 .0191 
 
 .0311 
 
 .0506 
 
 100 
 
 .0017 
 
 .0028 
 
 .0046 
 
 .0075 
 
 .0123 
 
 .0202 
 
 .0326 
 
 .0530 
 
 90 
 
 .0018 
 
 .0029 
 
 .0047 
 
 .0076 
 
 .0124 
 
 .0203 
 
 .0331 
 
 .0538 
 
 TABLE 54 
 
 THE REVERSING OR COMPONENT VORTEX, Vi 
 
 At. = Ai Ao 
 
 .00000 -.00017 
 
 -.00051 
 
 -.00117 
 
 -.00241 
 
 -.00463 
 
 -.00858 
 
 -.01553 
 
 The radial velocity of the reverse vortex, uz 
 
 az = 180 
 
 0.0 
 
 
 
 
 
 
 
 
 170 
 
 0.0 
 
 -5.9 
 
 -14.2 
 
 -25.6 
 
 -41.2 
 
 -62.2 
 
 -90.2 
 
 -127.5 
 
 160 
 
 0.0 
 
 -4.0 
 
 - 9.6 
 
 -17.4 
 
 -28.0 
 
 -42.3 
 
 -61.4 
 
 - 86.7 
 
 150 
 
 0.0 
 
 -3.1 
 
 - 7.3 
 
 -13.3 
 
 -21.4 
 
 -32.3 
 
 -47.1 
 
 - 66.1 
 
 140 
 
 0.0 
 
 -2.3 
 
 - 5.8 
 
 -10.4 
 
 -16.6 
 
 -25.2 
 
 -36.5 
 
 - 51.6 
 
 130 
 
 0.0 
 
 -1.8 
 
 - 4.4 
 
 - 8.0 
 
 -12.8 
 
 -19.3 
 
 -28.0 
 
 - 39.6 
 
 120 
 
 0.0 
 
 -1.4 
 
 - 3.2 
 
 - 5.8 
 
 - 9.4 
 
 -14.2 
 
 -20.5 
 
 - 28.9 
 
 110 
 
 0.0 
 
 -0.9 
 
 - 2.0 
 
 - 3.8 
 
 - 6.1 
 
 - 9.3 
 
 -13.5 
 
 - 19.0 
 
 100 
 
 0.0 
 
 -0.4 
 
 - 1.0 
 
 - 1.9 
 
 - 3.1 
 
 - 4.6 
 
 - 6.7 
 
 - 9.4 
 
 90 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 The tangential velocity of the reverse vortex, 
 
 a z = 180 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 170 
 
 0.0 
 
 -1.0 
 
 -2.6 
 
 - 4.5 
 
 - 7.2 
 
 -11.0 
 
 -16.0 
 
 -22.4 
 
 160 
 
 0.0 
 
 -1.5 
 
 -3.5 
 
 - 6.4 
 
 -10.2 
 
 -15.4 
 
 -22.3 
 
 -31.5 
 
 150 
 
 0.0 
 
 -1.8 
 
 -4.2 
 
 - 7.7 
 
 -12.3 
 
 -18.7 
 
 -27.0 
 
 -38.1 
 
 140 
 
 0.0 
 
 -2.0 
 
 -4.8 
 
 - 8.7 
 
 -14.0 
 
 -21.1 
 
 -30.6 
 
 -43.3 
 
 130 
 
 0.0 
 
 -2.2 
 
 -5.2 
 
 - 9.5 
 
 -15.2 
 
 -23.0 
 
 -33.4 
 
 -47.2 
 
 120 
 
 0.0 
 
 -2.3 
 
 -5.6 
 
 -10.0 
 
 -16.2 
 
 -24.5 
 
 -35.6 
 
 -50.2 
 
 110 
 
 0.0 
 
 -2.4 
 
 -5.9 
 
 -10.5 
 
 -16.9 
 
 -25.6 
 
 -37.0 
 
 -52.3 
 
 100 
 
 0.0 
 
 -2.4 
 
 -6.0 
 
 -10.8 
 
 -17.3 
 
 -26.1 
 
 -37.9 
 
 -53.5 
 
 90 
 
 0.0 
 
 -2.5 
 
 -6.0 
 
 -10.9 
 
 -17.4 
 
 -26.3 
 
 -38.2 
 
 -53.9 
 
212 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 The vertical velocity of the reverse vortex, 
 
 az = 180 
 
 .0000 
 
 .0000 
 
 .0000 
 
 .0000 
 
 .0000 
 
 .0000 
 
 ,0000 
 
 .0000 
 
 170 
 
 .0000 
 
 -.0001 
 
 - . 0002 
 
 - . 0004 
 
 - . 0009 
 
 -.0016 
 
 - . 0029 
 
 - . 0054 
 
 160 
 
 .0000 
 
 -.0001 
 
 - . 0003 
 
 - . 0008 
 
 -.0017 
 
 -.0031 
 
 -.0059 
 
 -.0106 
 
 150 
 
 .0000 
 
 -.0001 
 
 - . 0005 
 
 -.0012 
 
 - . 0024 
 
 -.0046 
 
 - . 0086 
 
 -.0154 
 
 140 
 
 .0000 
 
 -.0002 
 
 - . 0006 
 
 -.0015 
 
 - . 0031 
 
 -.0059 
 
 -.0110 
 
 -.0199 
 
 130 
 
 .0000 
 
 - . 0002 
 
 -.0008 
 
 -.0017 
 
 - . 0036 
 
 - . O0'< 1 
 
 -.0131 
 
 - . 0236 
 
 120 
 
 .0000 
 
 - . 0003 
 
 - . 0009 
 
 - . 0020 
 
 - . 0042 
 
 -.0081 
 
 -.0148 
 
 - . 0268 
 
 110 
 
 .0000 
 
 - . 0003 
 
 - . 0009 
 
 - . 0022 
 
 -.0045 
 
 - . 0087 
 
 -.0162 
 
 -.0291 
 
 100 
 
 .0000 
 
 -.0003 
 
 -.0010 
 
 - . 0023 
 
 - . 0047 
 
 - . 0091 
 
 -.0169 
 
 - . 0301 
 
 90 
 
 .0000 
 
 - . 0004 
 
 -.0010 
 
 -.0023 
 
 - . 0048 
 
 -.0093 
 
 - . 0172 
 
 -.0309 
 
 The Composition of Vortices 
 
 By subtracting the computed velocities of the perfect vortex 
 from those of the imperfect vortex, u\ U Q = u 2 , Vi V Q = v 2 , 
 Wi WQ = w 2 , we have a component vortex which, added to 
 the perfect vortex, will produce the observed imperfect vortex, 
 u\ UQ -f ^2, v\ = VQ + v 2 , Wi = w -\- w 2 , the signs being added 
 algebraically. The corresponding values of the constants A, 
 AI AQ = A 2 , can be found by computing the values of A for 
 the derived velocities by the formulas, 
 
 (643) A 2 = 
 
 U 2 
 
 -W 2 
 
 a w cos a z 
 
 sin a z 2 sin a z' 
 
 More simply, the algebraic values of A 2 are derived im- 
 mediately from AI (imperfect vortex) A (perfect vortex), 
 whence the corresponding velocities u 2 , v 2 , w 2 can be computed 
 in the usual manner. 
 
 Table 52 gives the values of AI, MI, Vi, w\ in the imperfect 
 vortex; Table 53 those of A , u , V Q , w d in the perfect vortex, 
 and Table 54 those of A 2j 2, v 2 , w 2 in the component reversing 
 vortex. A comparison of the velocities in these tables shows 
 that, by starting with the same radius and velocity on the outer 
 isobar (1), the observed imperfect vortex departs more and more 
 from the corresponding perfect vortex in proportion as the veloci- 
 ties approach the axis. The component vortex which is equiva- 
 lent to these differences is a vortex reversed in all respects to the 
 original vortex, revolving in the opposite direction and directed 
 downward from the clouds to the surface of the sea. This 
 principle of the composition of vortices through the constants A 
 
THE REVERSED DUMB-BELL VORTEX 
 
 213 
 
 of the successive tubes is very important, and leads to many 
 practical researches in the theory of cyclones, because it enables 
 us to take account of the numerous departures from the pure 
 vortex law, without giving up the advantages of the method of 
 vortex computations. 
 
 The Reversed Dumb-bell Vortex 
 
 A very erroneous impression would be left if it were sup- 
 posed that the imperfect dumb-bell vortex could be applied 
 directly to the study of the common cyclones in the atmosphere. 
 The dumb-bell vortex seems to be essentially reversed, turned 
 
 Height 
 
 in 
 meters 
 
 (6) (5) (4) (3) (2) (I) 
 
 10000 
 0000 
 8000 
 7000 
 6000 
 5000 
 1000 
 3000 
 2000 
 1000 
 000 
 
 11500 
 
 FIG. 27. The constant a A lines as derived from observations in the land cyclones. 
 
 inside out, as can be seen by Table 55, and Fig. 27. The dis- 
 cussion of the velocity components resulted as given in the 
 Cloud Report, in Table 126. Taking the velocities in that 
 table and plotting them on diagrams, a consistent system was 
 deduced which conforms on the average to that there given. It 
 
214 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 is produced on Table 55, I, II, for every 1,000-meter level, and 
 for the radial distances given in the normal land cyclone. Since 
 
 tan a z = , by (527) and (528), we can compute a A for each 
 
 level and tube. The result appears in Table 55, III. On Fig. 
 27, at the radial distances, 80, 300, 508, 733, 975, 1,250 kilometers, 
 these values of a A were plotted down, and lines of equal a A 
 were drawn and they are given on Fig. 27. They contain the 
 surprising result that the radial distances W are arranged nearly 
 on a geometrical ratio system, as can be readily seen* by making 
 the tests, and that the old ratio value log p = 0.20546 is quite 
 competent to satisfy the average conditions. The lines a A 
 are, however, concave towards the axis, instead of convex as in 
 the hurricane, and the lines are closed up on the outer circles 
 rather than on the inner, this being a complete reversal of the 
 configuration. The theoretical and the thermodynamical con- 
 ditions which produce this circulating structure have been 
 indicated in the series of papers on the " Thermodynamics of the 
 Atmosphere," W. B. 372, 1907. The subject will require further 
 study and investigation. 
 
 TABLE 55 
 
 THE OBSERVED RADIAL AND TANGENTIAL VELOCITIES IN CYCLONES. 
 INTERNATIONAL CLOUD REPORT 
 
 I. u = the observed radial velocities 
 
 Height 
 in Meters 
 
 (i) 
 1250000 
 
 (2) 
 
 975000 
 
 (3) 
 
 733000 
 
 (4) 
 
 508000 
 
 (5) 
 300000 
 
 (6) 
 
 80000 
 
 10000 
 
 -0.5 
 
 -0.7 
 
 -1.0 
 
 -1.3 
 
 -1.6 
 
 -2.0 
 
 9000 
 
 -1.5 
 
 -2.1 
 
 -2.7 
 
 -3.0 
 
 -3.7 
 
 -4.5 
 
 8000 
 
 -2.0 
 
 -3.0 
 
 -4.0 
 
 -4.5 
 
 -5.5 
 
 -6.0 
 
 7000 
 
 -3.0 
 
 -4.0 
 
 -5.0 
 
 -6.0 
 
 -7.0 
 
 -7.5 
 
 6000 
 
 -3.0 
 
 -4.0 
 
 -5.0 
 
 -6.0 
 
 -7.0 
 
 -8.0 
 
 5000 
 
 -2.5 
 
 -3.0 
 
 -4.0 
 
 -4.5 
 
 -6.0 
 
 -7.0 
 
 4000 
 
 -2.0 
 
 -2.5 
 
 -3.0 
 
 -4.0 
 
 -5.0 
 
 -6.0 
 
 3000 
 
 -2.0 
 
 -2.0 
 
 -2.0 
 
 -2.5 
 
 -3.5 
 
 -4.5 
 
 2000 
 
 -2.0 
 
 -2.0 
 
 -2.0 
 
 -2.0 
 
 -2.0 
 
 -2.0 
 
 1000 
 
 -3.5 
 
 -3.5 
 
 -4.0 
 
 -4.0 
 
 -4.0 
 
 -4.0 
 
 000 
 
 -4.5 
 
 -5.0 
 
 -6.0 
 
 -6.5 
 
 -6.5 
 
 -6.0 
 
II. 
 
 THE REVERSED DUMB-BELL VORTEX 
 
 v = the observed tangential velocities 
 
 215 
 
 10000 
 
 +2.0 
 
 + 2.5 
 
 + 3.0 
 
 + 4.0 
 
 + 4.5 
 
 + 5.0 
 
 9000 
 
 +4.0 
 
 + 6.0 
 
 + 8.0 
 
 . + 8.5 
 
 + 8.5 
 
 + 9.0 
 
 8000 
 
 +4.0 
 
 + 7.0 
 
 + 10.0 
 
 + 12.0 
 
 + 12.0 
 
 +11.0 
 
 7000 
 
 +5.0 
 
 +10.0 
 
 + 13.0 
 
 +14.0 
 
 +14.0 
 
 +14.0 
 
 6000 
 
 +6.0 
 
 +11.0 
 
 + 14.0 
 
 +15.0 
 
 +15.0 
 
 +16.0 
 
 5000 
 
 +7.0 
 
 +12.0 
 
 + 15.0 
 
 +16.0 
 
 + 18.0 
 
 +19.0 
 
 4000 
 
 +7.0 
 
 +11.0 
 
 +14.0 
 
 +17.0 
 
 + 19.0 
 
 +21.0 
 
 3QOO 
 
 +6.0 
 
 + 10.0 
 
 +13.0 
 
 +16.0 
 
 +19.0 
 
 +23.0 
 
 2000 
 
 +5.0 
 
 + 9.0 
 
 +12.0 
 
 +15.0 
 
 +18.0 
 
 +22.0 
 
 1000 
 
 +5.0 
 
 + 6.0 
 
 + 8.0 
 
 +10.0 
 
 + 12.0 
 
 +14.0 
 
 000 
 
 +5.0 
 
 + 5.5 
 
 + 6.0 
 
 + 6.0 
 
 + 6.5 
 
 + 7.0 
 
 III. 
 
 The computed a A = 
 
 u 
 
 Iff sin az Iff cos a z 
 
 10000 
 
 165 
 
 266 
 
 431 
 
 828 
 
 1593 
 
 6732 
 
 9000 
 
 342 
 
 652 
 
 1152 
 
 1775 
 
 3091 
 
 12577 
 
 8000 
 
 358 
 
 781 
 
 1469 
 
 2523 
 
 4401 
 
 15659 
 
 7000 
 
 467 
 
 1105 
 
 1900 
 
 2998 
 
 5217 
 
 19852 
 
 6000 
 
 537 
 
 1201 
 
 2028 
 
 3180 
 
 5518 
 
 24801 
 
 5000 
 
 595 
 
 1269 
 
 2118 
 
 3270 
 
 6325 
 
 25306 
 
 4000 
 
 585 
 
 1157 
 
 1953 
 
 3439 
 
 6548 
 
 27298 
 
 3000 
 
 506 
 
 1046 
 
 1794 
 
 3187 
 
 6441 
 
 29300 
 
 2000 
 
 431 
 
 946 
 
 1659 
 
 2978 
 
 6040 
 
 27598 
 
 1000 
 
 488 
 
 713 
 
 1220 
 
 2120 
 
 4216 
 
 18198 
 
 000 
 
 538 
 
 762 
 
 1158 
 
 1741 
 
 3064 
 
 11524 
 
 The unit = .00000001 = 1 X 10 - 8. 
 
 It is obvious that the velocities u, v can be computed from 
 the formulas, knowing the values of the constants a A, or, on 
 the other hand, the constants can be computed from the observed 
 u, v velocities. The results of this computation, Table 55, Fig. 
 27, show that the dumb-bell vortex system has been entirely 
 reversed. The a A lines are concave toward the axis, they are 
 geometrically spaced but closed up on the outer rather than on 
 the inner areas of the cyclone. The temperature distribution 
 conforms to this arrangement, and the cause is probably due to 
 the penetration of the vortex of the lower strata into the rapidly 
 moving drift of the upper strata. 
 
216 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 Historical Review of the Three Leading Theories Regarding the 
 Physical Causes of Cyclones and Anticyclones in 
 the Earth's Atmosphere 
 
 In the Astronomical and Astro physical Journal, January, 
 1894, the writer made a summary of the three most important 
 general theories regarding the physical causes which generate 
 the local storms, called cyclones, in the earth's atmosphere, and 
 the following extracts from that review are sufficiently explicit 
 for ordinary purposes. The three theories are: (l) Ferrel's 
 warm-center and cold-center cyclones; (2) Hands dynamic 
 production of temperatures as found by observation; (3) Bige- 
 low's asymmetric cyclone with warm and cold currents arranged 
 in ridges, or streams of different densities, and driven into local 
 cyclonic and anticyclonic circulations by the force of gravitation 
 acting upon them. Ferrel had in mind for his cyclone the type 
 of the general circulation of the atmosphere, and conceived that 
 the same principles dominate both of them. The general cir- 
 culation is described as a cold-center cyclone, with eastward 
 movement from the pole to the high-pressure belt in latitude 33, 
 and a warm ring of westward movement in the tropics; the local 
 cyclone is warm at the center and has right-handed rotation from 
 the axis to a ridge of high pressure, outside of which a cold ring 
 circulates in the anti-right-handed direction in the northern 
 hemisphere, these directions of motion being reversed in the 
 southern hemisphere. Ferrel's practical difficulty was to 
 account for the originating heat energy in the central column 
 of the cyclone, and this must precede any criticism of the circula- 
 tion that depends upon it. He writes, "In the ordinary cyclonic 
 disturbances of the atmosphere, the causes are similar to those 
 in the general circulation but more local, and consist of a differ- 
 ence of density arising mostly from a difference of temperature 
 between some central area and the external surrounding parts 
 of the atmosphere." This dominant idea proved fatal to Ferrel's 
 successful development of sound fundamental principles, and 
 has greatly influenced many students to travel a road whose 
 
HISTORICAL REVIEW OF LEADING THEORIES 217 
 
 end has never been found. He was evidently unable to account 
 satisfactorily for the energy implied in the temperature 
 difference required to do the work observed in the motions of 
 the cyclones and anticyclones. In his Coast and Geodetic 
 Survey Report, p. 183, he remarks: "If for any reason there 
 is kept up a continued interchange of air between the central 
 and exterior part"; p. 201, "The condensation of aqueous 
 vapor plays an important part in cyclonic disturbances, but is 
 by no means either a primary or a principal cause of cyclones"; 
 p. 239, "Rainfall is not essential to the formation of areas of 
 low barometer, and is not the principal cause of their formation 
 or of their progressive motion"; in Waldo's edition of FerrePs 
 "Hydrodynamics," p. 39, "The theory which attributes the 
 whole of the barometrical oscillations to the rarefaction of the 
 atmosphere produced by the condensation of vapor in the forma- 
 tion of clouds and rain cannot be maintained." However, being 
 hard pushed to find a cause for his central area of high tempera- 
 ture in cyclones, he gradually weakened from this position, en- 
 dorsed Espy's condensation theory of the development of latent 
 heat by the formation of clouds and rain, and in the last year 
 of his life could write in Science, December 19, 1890, "All this 
 has been done in the condensation theory of cyclones, with 
 results so satisfactory as scarcely to leave a doubt as to the 
 truth of the whole theory." This was written in reply to Dr. 
 Hann's revolt against 'the sufficiency of this cause to produce 
 cyclones as observed, who took the ground that these local 
 gyrations are only subordinate whirls in the general circulation, 
 which depend upon the effects of the equatorial radiation only, 
 and are independent of any local cause. Hann even went so 
 far as to conclude, that "the actual motion of the atmosphere is 
 not a product of the temperature (FerrePs idea), but is in spite 
 of it; the temperature is a product of the motion," Science 
 May 30, 1890. Ferrel was loyal to the theory that temperature 
 differences cause the motion always and everywhere, and Hann, 
 in adopting the inverse proposition, has surely erred against first 
 principles. 
 
 FerrePs mechanical theory urged him to adopt a ridge of 
 
218 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 high pressure surrounding every cyclone, as indicated in formulas 
 (471) to (478), but this is opposed by several fundamental con- 
 ditions which show that it does not conform with modern observa- 
 tions. Another solution for the warm-center cyclone is given in 
 equations (479) to (490), but this is equally opposed by the 
 observed conditions. Both of these mathematical solutions 
 have been practically abandoned, chiefly because there are, in 
 fact, no warm-centered cyclones and no cold-centered anti- 
 cyclones in existence. The hurricane probably has a warm- 
 centered system of motion, but it is entirely different in structure. 
 When a cold sheet of air overflows a warm sheet, the warm sheet 
 flows outward radially from a central point in all directions, like 
 the spokes on a wheel, and this outward movement in the high 
 cloud levels drags behind it the vortex tube described in Fig. 26, 
 and Tables 51. This, however, is entirely different from the 
 temperature conditions of motion in cyclones and anticyclones. 
 Bigelow writes in the same paper of the Astronomical and 
 Astrophysical Journal, "I must admit freely that I am unable 
 to see in the daily weather maps that formation as fundamental 
 which Ferrel and meteorologists generally assume to be the pri- 
 mary state. I propose to see in temperature differences, ar- 
 ranged in waves or ridges, the true cause of the observed pressures 
 and the antecedent of the precipitation. It is therefore necessary 
 to account for cold and warm temperature waves passing over 
 the United States." "The passage of winds past each other in 
 opposite directions tends toward local gyrations, which all drift 
 eastward with the prevailing component in middle latitudes. 
 All this simply depends upon the difference between the polar 
 and the equatorial temperature, and is fully in accordance with 
 the views of Ferrel and the latest expressions by Dr. Hann." 
 "The formation of these low- and high-pressure areas is the 
 result of the existence of the warm or cold sections of waves 
 lying athwart the maximum crest. From first principles the 
 warm and cold masses will be impelled toward each other, 
 because of the action of gravitation on media of differing density. 
 They will tend to encounter along or near the ridge of greatest 
 temperature variation. Along the line of greatest temperature 
 
HISTORICAL REVIEW OF LEADING THEORIES 219 
 
 change, with cold air to the west and warm air to the east of 
 it, the gyrating cyclone is formed, the couple existing from 
 the system of causes thus described. Likewise, along the next 
 ridge, with cold air to the east and warm air to the west, and 
 often to the south of the maximum crest, the anticyclone is 
 produced. A corollary remark is that the storm track along 
 the north United- States seems to be the effort of the general 
 circulation to restore the permanent polar low-pressure belt 
 which is interrupted by the continent. Another is that tornadoes 
 and hurricanes are due to precisely the same cause, namely, 
 the juxtaposition of masses of air having great temperature 
 differences." The ideas were illustrated by the cyclone of 
 November 16, 17, 1893. 
 
 The origin of these cold and warm waves, or ridges of different 
 densities, has been discussed at great length in the International 
 Cloud Report, and the streams from the cold north and warm 
 south were called "leakage" currents, because these are in fact 
 sporadic offshoots from the general circulation into middle lati- 
 tudes. The warm currents in the United States are thrown off 
 by the Atlantic center of action, from the Gulf of Mexico to the 
 north; those upon southeastern Asia from the Pacific center 
 of action ; those upon northwestern United States from the same 
 Pacific center of action, or else from the Arctic zone over British 
 America; those upon northwest Europe, from the Atlantic center 
 of action, or else from the Arctic circulation. The continents 
 and the oceans react upon the general circulation in such a 
 manner as greatly to disturb and distort its free operation, so 
 that finally southerly currents prevail in certain regions and 
 northerly currents are dominant in other regions. In the 
 United States the southerly warm currents and the northerly 
 cold currents encounter in long streams, flowing past each other 
 in waves or ridges of density, and under the force of gravitation 
 they are compelled to flow in cyclonic and anticyclonic circula- 
 tions toward a thermal equilibrium. The exact mathematical 
 conditions prevailing at every point have been indicated in 
 Chapter II of this Treatise, and in that place, and in Bulletin 
 No. 3, Argentine Meteorological Office, 1912, the practical details 
 
220 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 have been illustrated. Bigelow, in 1894, laid down the principle 
 of the force of gravitation acting upon density masses in alternate 
 juxtaposition, whether side by side as in cyclones, anticyclones, 
 and tornadoes, or in vertical superposition as in hurricanes, 
 with intermediate cases between these principal positions; and 
 in 1912 he worked out a method of computing the very important 
 terms of the radiation energy in the general equations of motion, 
 which had heretofore been entirely omitted from the discussions. 
 A very brief summary of some of the most important features 
 of the general and the local circulations are added in this place, 
 though the student must consult the weather maps of various 
 countries for any complete knowledge of such a complex subject 
 as the actual circulation. 
 
 The General and Local Components of the Velocities, Pressures, 
 and Temperatures in the Circulation of the Atmosphere 
 
 The General and the Local Components 
 
 There are certain distributions of the temperature, pressure, 
 wind direction, and velocity, which are characteristic of the 
 general motions of the atmosphere, and others which belong to 
 the local circulations peculiar to the cyclones and anticyclones. 
 It is necessary to separate them from the observed values that 
 are the resultants of those two components. It is practicable 
 to observe certain values of the velocities, pressures, and tem- 
 peratures at a given station, from which the general or normal 
 values are computed, so that by subtracting in the form of 
 vectors the normal values from the observed values, the local or 
 component terms may be found. This great labor has been 
 performed for the United States and the West Indies, the results 
 for the velocities being recorded in the Cloud Report, 1898, for 
 the pressures in the Barometry Report, 1901, and for the tem- 
 peratures in the Report on Homogeneous Normals, 1909, 
 together with numerous papers in the Monthly Weather 
 Review. This Treatise is concerned with the methods of com- 
 putation and discussion appropriate to meteorology, rather 
 than with statistical results, so that only a brief summary of 
 
NORMAL AND LOCAL VELOCITIES IN STORMS 
 
 221 
 
 these data can be presented in this connection. Unfortunately, 
 the data of meteorology are so bulky that it becomes very difficult 
 to do justice to the subject within the limits of a reasonable 
 volume. 
 
 The Normal and Local Velocities in Storms 
 
 In the Cloud Report are to be found the resultant velocities 
 and directions of the wind at the surface, in the cumulus levels 
 (1,000-2,000 meters), and in the cirrus levels (8,000-10,000 
 meters), for all parts of cyclones and anticyclones, when the 
 centers of these areas are located in different parts of the United 
 States, as the Dakotas, the Lake region, New England, Colorado, 
 Texas, and East Gulf States, respectively. They were obtained 
 in the several areas by making a composite chart from about 
 
 FIG. 28. Observed stream lines of air in the cumulus cloud level (2,000) 
 over a cyclone whose center is in the Lake Region. 
 
 fifty charts for each type of storms. For this purpose the United 
 States was divided into small areas by the parallels of latitude and 
 the meridians, the centers of the fifty storms were made to coin- 
 cide, and the vectors or arrows were transferred to a common 
 chart, from which the resultant vector was carefully computed. 
 These charts are of great theoretical value for the student, as 
 well as of practical value for the forecaster, and they should be 
 
222 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 thoroughly examined. Specimens of these charts are given on 
 Fig. 28 for a cyclone centered in the lake region, transcribing the 
 
 FIG. 29. Observed stream lines in the cumulus cloud level (2,000) over a 
 cyclone whose center is in the West Gulf States. 
 
 FIG. 30. Observed stream lines in the cirrus cloud level (10,000) over 
 a hurricane whose center is in the South Atlantic States. These cur- 
 rents all show that there is a U-shaped formation in the circulation gen- 
 erally opening to the northeastward, though it is also found pointing 
 westward and southward. 
 
 lower cloud level; on Fig. 29 for a cyclone in the West Gulf 
 States for the cumulus cloud level; and on Fig. 30 for a hurri- 
 
NORMAL AND LOCAL VELOCITIES IN STORMS 223 
 
 cane in the East Gulf States, upon the upper or cirrus cloud 
 level. It is seen in all these cases that the currents of air form 
 U-shaped figures like the isobars and isotherms, in entering a 
 cyclonic vortex, and that the eastward drift is locally diverted 
 into this configuration. There is a bridge across the top of 
 the U-shaped vortex, and in the isobars a well-defined saddle 
 is always constructed, where the high-pressure areas are tem- 
 porarily broken through in the construction of a vortex circu- 
 lation. 
 
 Fig. 31 gives a representation of a typical circulation of air 
 in connection with the isobars in three levels: sea level, 3,500-foot 
 level, and 10,000-foot level, showing the relation of the currents 
 to the isobars. The high-pressure cusps tend to approach over 
 a bridge or saddle at C S C, the pressure being lower to the 
 north and to the south of it. The number of- the closed isobars 
 decreases with the height, and it is usual for them to disappear 
 at the level of 3,000-4,000 meters, and sometimes even lower. 
 This is a proof that the dumb-bell vortex which dominates in 
 hurricanes has almost entirely vanished in cyclones except in 
 the lowest levels, the top being entirely depleted in the higher 
 levels. This throws back the theory of cyclones into quite a 
 different category of imperfect vortices and, considering the 
 asymmetrical distribution of the temperature, it is evident that 
 the currents are due to pressure gradients in the thin sheets of 
 air of different temperatures in the process of mixing in the 
 middle latitudes. The tendency for the currents to divide into 
 two streams mutually underrunning and overrunning each other 
 should be carefully noted. 
 
 In the Cloud Report are contained the data from which the 
 average heights were computed where the several clouds are 
 formed, and the average velocities were deduced at which they 
 move nearly eastward, after the cyclonic and the anticyclonic 
 components have been eliminated. Table 56 contains a summary 
 of the velocities in high and low areas, the northward and south- 
 ward components over high and low areas, the northward and 
 southward components between the centers in the warm and 
 cold streams, and the seasonal velocities. See Fig. 32. 
 
224 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 The cloud forms, stratus, cumulus, strato-cumulus, alto- 
 cumulus, alto-stratus, cirro-cumulus, cirro-stratus, cirrus, occur 
 at certain well-defined heights on the average, and it is found by 
 observation that they drift over the earth's surface at certain 
 average velocities as shown on the diagram. This increase of 
 velocity upward, from 7 meters per second at the surface to 40 
 
 Typical abnormal Isobars, (Sea Level) 
 
 FIG. 3 1 a. 
 
 FIG. 3 1 a, b, c. Showing the relation of the local circulation to the typical 
 isobars in high or low areas of pressure. The closed isobars form a rough 
 vortex, which is supplied by the two-branched stream-lines and gradually 
 dies out in the higher levels. 
 
 meters per second, is called the eastward drift in middle latitudes. 
 This is the normal velocity component which must be eliminated 
 from the observed component to produce the local disturbing 
 component of velocity due to the cyclones and anticyclones 
 proper. 
 
 The eastward and the westward drifts, in the middle latitudes 
 and the tropics, respectively, are shown on Figs. 33, 34, 35. 
 
Typical abnormatlsobars, (8,500 foot) 
 
 FIG. 316. 
 
 Typical-abnormal Isobars, (10,000 foot) 
 
 FIG. 3 ic. 
 
226 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 TABLE 56 
 
 SUMMARY OF THE VELOCITIES OF THE MOTIONS OF CLOUDS IN THE DIFFERENT 
 LEVELS FOR THE MIDDLE ATLANTIC STATES. THE UPPER CLOUDS INCLUDE 
 Ci., Ci.S., Ci.Cu., A.S., A.Cu., WITH THE MEAN HEIGHT 8.4 KILOMETERS, 
 AND THE LOWER CLOUDS INCLUDE S.Cu., Cu., S., AT THE MEAN HEIGHT 2.4 
 KILOMETERS. 
 
 Symbols: Ci. = Cirrus. S. = Stratus. Cu. = Cumulus. A. = Alto. 
 L TOTAL VELOCITY IN HIGHS AND Lows WITHOUT REGARD TO DIRECTIONS 
 
 Clouds 
 
 Ci. 
 
 Ci.S. 
 
 Ci.Cu. 
 
 A.S. 
 
 A.Cu. 
 
 S.Cu. 
 
 Cu. 
 
 S. 
 
 Wind 
 
 
 Height in kilo- 
 meters 
 
 9.8 
 
 9.8 
 
 8.1 
 
 5.9 
 
 4.5 
 
 2.5 
 
 1.5 
 
 0.9 
 
 
 
 
 High Areas 
 
 34 9 
 
 39 1 
 
 33 5 
 
 30 2 
 
 23 5 
 
 23 3 
 
 11 2 
 
 11 4 
 
 4 8 
 
 Per cent 
 
 Northern 
 Southern 
 Low Areas 
 Total motion 
 
 38.3 
 30.4 
 
 40.8 
 44 6 
 
 42.6 
 34.8 
 
 39.8 
 42 5 
 
 33.9 
 30.5 
 
 39.3 
 43 8 
 
 31.1 
 24.1 
 
 36.0 
 39 4 
 
 26.6 
 19.7 
 
 29.2 
 32 6 
 
 22.7 
 18.5 
 
 28.6 
 32 9 
 
 10.9 
 10.4 
 
 14.6 
 17 4 
 
 12.2 
 9.5 
 
 11.1 
 13 2 
 
 4.9 
 
 4.8 
 
 5.4 
 5.3 
 
 "19" 
 15 
 
 Southern 
 
 28.3 
 
 36.3 
 
 34.8 
 
 30.5 
 
 24.4 
 
 21.1 
 
 11.8 
 
 8.6 
 
 5.9 
 
 28 
 
 II. SOUTHWARD AND EASTWARD COMPONENTS OF VELOCITIES IN HIGHS AND Lows 
 
 Clouds 
 
 Ci. 
 
 Ci.S. 
 
 Ci.Cu. 
 
 A.S. 
 
 A.Cu. 
 
 S.Cu. 
 
 Cu. 
 
 S. 
 
 Wind 
 
 
 High Areas 
 
 + 1 Q7 
 
 + 1 ft** 
 
 60 
 
 37 
 
 07 
 
 32 
 
 13 
 
 1 22 
 
 69 
 
 
 +0 IN 
 1 T? W 
 
 4-33 7 
 
 +32 
 
 +32 6 
 
 +27 2 
 
 +22 1 
 
 +16 
 
 + 51 
 
 + 58 
 
 + 11 
 
 
 Low Areas 
 
 10 VT 
 
 c OR 
 
 9 24 
 
 3 00 
 
 4 60 
 
 2 38 
 
 4 00 
 
 11 
 
 1 32 
 
 40 
 
 
 1 c> VXT 
 
 _i_qq A 
 
 iqc q 
 
 +37 2 
 
 +31 3 
 
 +24 3 
 
 4-24 3 
 
 +11 4 
 
 + 78 
 
 + 15 
 
 
 
 
 
 
 
 
 
 
 
 
 
 III. MEAN NORMAL COMPONENTS OF VELOCITY FOR THE UNITED STATES 
 
 
 1 6 
 
 3 8 
 
 1 8 
 
 2 5 
 
 1 2 
 
 2 2 
 
 1 
 
 1 3 
 
 05 
 
 
 or war 
 
 +36 6 
 
 +34 
 
 4-34 9 
 
 +29 2 
 
 +23 2 
 
 +20 2 
 
 + 83 
 
 + 68 
 
 + 13 
 
 
 
 
 
 
 
 
 
 
 
 
 
 IV. COMPONENT VELOCITIES IN SELECTED AREAS BETWEEN HIGH AND Low 
 
 CENTERS 
 
 Selected Areas 
 
 + 66 
 
 2 11 
 
 + 4 95 
 
 4-2 79 
 
 + 6 24 
 
 +10 22 
 
 + 6 52 
 
 + 5 25 
 
 + 2 23 
 
 
 
 +40 1 
 
 +36 9 
 
 +38 7 
 
 +26 5 
 
 +23 7 
 
 +22 1 
 
 + 96 
 
 + 75 
 
 + 32 
 
 
 Selected Areas 
 
 3 75 
 
 3 89 
 
 7 34 
 
 7 47 
 
 7 78 
 
 11 13 
 
 8 13 
 
 7 97 
 
 3 25 
 
 
 
 +32 7 
 
 +36 9 
 
 +32 1 
 
 +31 
 
 +21 9 
 
 +17 1 
 
 + 65 
 
 + 51 
 
 + 02 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V. SEASONABLE VELOCITIES OF THE UPPER AND LOWER CLOUDS 
 
 Clouds 
 
 Upper clouds 8.4 
 
 Lower clouds 2.4 
 
 Seasons 
 
 June 
 
 Sept. 
 
 Dec. 
 
 March 
 
 Ann. 
 
 June 
 
 Sept. 
 
 Dec. 
 
 March 
 
 Ann. 
 
 20.0 
 17.2 
 
 27.7 
 19.6 
 
 High Areas 
 Northern 
 Southern 
 Low Areas 
 Northern 
 Southern 
 
 30.8 
 24.2 
 
 39.7 
 25.9 
 
 33.7 
 35.6 
 
 45.2 
 30.9 
 
 37.5 
 36.0 
 
 47.4 
 33.1 
 
 41.7 
 27.7 
 
 37.1 
 34.8 
 
 35.5 
 29.1 
 
 42.6 
 32.4 
 
 12.8 
 16.3 
 
 19.2 
 12.9 
 
 23.2 
 17.8 
 
 32.8 
 21.9 
 
 24.9 
 20.0 
 
 31.5 
 
 18.7 
 
 21.8 
 18.2 
 
 27.5 
 16.1 
 
 All velocities in meters per second (1 m.p.s. = 2.2 miles per hour) 
 
NORMAL AND LOCAL VELOCITIES IN STORMS 
 
 227 
 
 The scale on Fig. 35 is twice as great as on Fig. 33 or Fig. 34. It 
 is seen that at San Juan, and generally in the West Indies, the 
 westward drift in the lower levels reverses into an eastward 
 drift in the upper levels, the transition occurring in the A.Cu. 
 and A. St. levels. Hence, the westward trade winds are shallow, 
 
 Height 
 
 Velocity Scale 
 
 
 Metere 
 
 Feet 
 
 Miles 
 
 m.p.h. 22.2 44.7 67.1 90 
 
 
 10000 
 9000 
 8000 
 7000 
 6000 
 5000 
 4000 
 3000 
 
 32808 
 
 m.p.s. 10 20 30 40 
 
 Ci.CiS 
 Ci.Cu 
 
 "A.St. 
 A.Cu 
 
 S.Cu 
 
 Cu 
 
 St. 
 Wind 
 
 6.21_ 
 29527- 
 
 
 
 
 
 - 
 
 
 
 
 \\ 
 
 6.59- 
 26247 - 
 
 4.97 _ 
 22966- 
 
 
 
 $ 
 
 1 
 
 4.35 _ 
 
 19685- 
 
 16404- 
 
 
 
 1 
 
 I 
 
 - 
 
 3.11- 
 
 13123- 
 
 
 
 } J 
 
 
 2.49 _ 
 98431 
 
 1.86 _< 
 6562- 
 
 
 x^ 
 
 / / 
 
 - 
 
 1000 
 
 1.24- 
 3281- 
 
 
 V 
 
 0.62 _ 
 
 / 
 
 FIG. 32. The total eastward velocity in high and low areas. Cloud heights. 
 
 the greatest velocity westward being in the S.Cu. level (2,000- 
 3,000 meters), and that they give way to the eastward drift 
 which prevails over the sub tropics in the higher levels. At 
 Key West, in the midst of the North American high-pressure 
 belt, the eastward and the westward drift is small in velocity, 
 a similar reversal taking place in the middle levels. In the middle 
 latitudes of the United States the eastward drift prevails in all 
 
228 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 levels, the velocities increasing from the surface upward. On 
 these diagrams / stands for January and D for December, and 
 all the intermediate months of the year are given in the line of 
 
 FIG. 33. The eastward drift above Washington, D. C., for each month 
 
 in the year. 
 
 SCALE OF VELOCITY ^ 20 40 60 
 
 METERS PER SECOND 
 ^ ^D 
 
 Ci,Ci.S,Ci.Cu,. J ^^^, 
 
 
 D 
 
 A.S, A.Cu,. ^S 1 *. 
 
 
 D 
 S.Cu, Cu,S,. ** <-*x 
 
 
 Wind ^ 
 
 \ 
 
 FIG. 34. The eastward and the westward drift above Key West, Florida. 
 
 vectors. In summer the velocities in the tropics for the upper 
 levels are small and disturbed in direction, showing that the 
 circulation is diminished when the sun is north of the equator. 
 In all cases circulation depends upon contrasts in temperature, 
 so that a vigorous circulation occurs in winter rather than in 
 
ANALYTIC CONSTRUCTION OF RESULTANTS 
 
 229 
 
 summer, when the temperatures of the air in the northern hemis- 
 phere are much more nearly equal than they are in winter. 
 
 SCALE Of VELOCITY V *" METERS ER SECOND 
 
 FIG. 35. The westward drift in the lower levels at San Juan, W. I., revers- 
 ing into the eastward drift in the A.Cu., A.St. levels and the eastward drift 
 in the upper levels. 
 
 The Normal and the Local Isobars in Cyclones and Anticyclones 
 
 The Analytic Construction of the Resultants 
 
 The general theory of the separation of components can be 
 illustrated by the following figures. 
 
 Draw circles about the pole (see Fig. 36) representing baro- 
 metric pressures at some level above the surface, ranging from 
 25.4 inches near the pole to 27.2 in latitude 20. At two points 
 
230 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 superpose a series of local circles, 1, 2-8, representing a defect of 
 pressure at (1), and an excess of pressure at (4). Add the respec- 
 tive values together at every point of intersection, and connect 
 up the pressures having the same isobaric value. The resultant 
 lines for (1) are seen at (2), and for (4) they are seen at (5). 
 
 Latitude 20 c 
 
 FIG. 36. The formation of cyclones in the general circulation about the 
 poles of the earth. 
 
 Putting (2) and (5) in continuous figures the resultant disturbed 
 values are found at (3) and (6), the ordinary form of the isobars 
 observed in the atmosphere somewhat above the surface, as on 
 the 3,500-foot and the 10,000-foot planes. The same facts can 
 be determined analytically as illustrated by Fig. 37. 
 
 Take the co-ordinate systems as shown on the diagram. 
 
 Let R = the radius of the circle, (a. b) the co-ordinates 
 
ANALYTIC CONSTRUCTION OF RESULTANTS 
 
 231 
 
 of the center, (x. y) the co-ordinates of any point on the 
 circle. 
 
 The general equation of the circle is 
 
 (644) (x - a) 2 + (y - b) z = R*. 
 Take 5 = and transpose the terms, so that, 
 
 (645) y 2 = - x z + 2ax + R 2 - a*. 
 
 The equation of condition for the isobar which is the resultant 
 of successive circular isobars added to successive straight-line 
 
 -1 
 
 FIG. 37. The composition of right lines and circles where the gradients 
 are twice as great on the lines as on the circles. 
 
 normal isobars, is that the sum of certain pair numbers shall be 
 constant on the same line. Thus, A + B = constant, where 
 A = n x, some multiple of the ordinate x, and B = the gradient 
 number on the circles. For example, take the gradient on the 
 normal straight lines one-half that on the normal circles, so that 
 n HJ which is about the average in highly developed storms. 
 Take successive circles, R = 6, 5, 4, 3, 2, whose gradient numbers 
 are respectively = 0, -1, -2, -3, -4. Take a = 6, A = 
 Y^ x, A + B = for the 0-line, and n = %. 
 
 Similarly, by taking the proper groups of R, B, x for the 
 1, +1 . . . 2, +2, . . . lines in low and high areas, we 
 
232 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 obtain the resultants shown on Fig. 37. Curves can be com- 
 pounded analytically through trieir equations, when the equations 
 of the lines are known, but as this is not usually the fact for the 
 lines representing meteorological gradients, and lines of equal 
 meteorological values, resort must be had to some graphic proc- 
 ess of construction. 
 
 Graphic Construction of Resultants 
 
 The first step is to determine the normal isobars, isotherms, 
 and wind vectors on certain selecte'd planes. For the purpose 
 we have chosen, as appropriate to forecasting requirements in 
 the United States, the 3,500-foot and the 10,000-foot planes, 
 besides the usual sea-level plane; and the Barometry Report 
 contains the normals of pressure, temperature, and vapor pressure 
 on these three planes, while the Cloud Report contains the 
 normal vectors for the velocities of the air motions. We have, 
 therefore, to reduce the observed data to these three planes, and 
 subtract graphically the normals from them to obtain the local 
 disturbing terms of the given element. Thus, in Fig. 38, if we 
 obtained the oval lines from observation and subtracted the 
 right lines from them, we should recover the circles or local 
 components. Practically, lay down the normal values of the 
 isobars or isotherms on a given plane, using transparent paper, 
 superpose this upon the chart of observed values on that plane, 
 and draw the diagonals of the quadrilateral figures that cover 
 the diagram. In this way the charts of barometric pressure 
 for the year 1903 on three planes (of which an example is 
 given in Fig. 38) have been decomposed into the elements of 
 the general and the local circulations from which we can study 
 the general circulations on the one hand, and the local cir- 
 culations on the other hand, without confusion. 
 
 The observed pressures at the stations of the United States 
 can be reduced to these planes by means of suitable tables, so 
 that they are easily embodied in a telegraphic report without 
 delay to the forecast message. This was done for a few weeks 
 in a preliminary study. It is to be noted that the closed isobars 
 of the lower planes soon expand into loops in the upper level. 
 
115~ 110 105 s 100 3 95" 90 5 85 80 
 
 125 120 115 110 105 100 95" 9Q- 1 85 80 7 
 
 115 J 110 J 105 J 100 J 
 
 90 J 85 80 75 
 
 FIG. 38. The systems of isobars on three planes for the 
 storm of February 27, 1903. 
 
115 110 -.1 105 100 90 90 85 
 
 ,".-- J6^ 70 65 
 
 FIG. 39. The normal isobars (continuous lines) and the disturbing local isobars (dotted 
 lines) in the storm of February 27, 1903. The normal lines were laid upon the observed lines 
 of Fig. 38, and the dotted diagonals of Fig. 39 drawn. There are five closed isobars on the sea- 
 level plane and only two closed isobars on the io,ooo-foot plane. The system of high and low 
 areas on the sea-level charts soon opens up into sinuous lines in the upper levels. A general 
 view of this fact is given on Fig. 40. 
 
ISOTHERMS IN CYCLONES 
 
 235 
 
 The Normal and the Local Isotherms in Cyclones and Anticyclones 
 The isotherms as observed at a given time in the United 
 States are separated by the same process into the two compo- 
 nents, the normal and the local disturbance isotherms. Lay a 
 chart containing the normal isotherms over the observed map 
 and draw diagonal lines connecting up points having the same 
 
 Observed Isobars Local Component Isobars 
 
 Pressure 
 
 Pressure 
 
 Low 
 
 High 
 
 FlG. 40. Scheme of the distribution of the pressures in high and in low areas, 
 in the observed and in the component isobars, on the levels up to 10,000 meters. 
 
 These isobars are somewhat ideal, but they conform to conditions existing 
 up to the top of the local disturbances in the atmosphere, that is to the cirrus 
 region. The winter storms are cut off at 6,000 meters and even lower, while the 
 summer storms can be traced much higher, on account of the relative retreat of 
 the low temperatures to the higher levels. The U-shaped loops of the high 
 areas open southward, and those of the low area open northward, so that in the 
 upper levels there are sinuous, not closed isobars. Progress in forecasting con- 
 sists in studying these upper plane auxiliary charts, in connection with the 
 corresponding sea-level charts. 
 
236 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 30.30 
 
 30.20 
 
 ,29.80 7() '29. 90 30.00 30 ' 10 
 Isobars and Isotherms on the Weather Map for February 27, 1903 
 
 FIG. 41. The weather map of February 27, 1903, showing the observed 
 isobars and isotherms. 
 
 -10 
 
 -10' 
 Temperature Components for February 27, 1903 
 
 FIG. 42. The normal isotherms (full) and the local disturbance isotherms 
 (dotted) which added together produce the observed isotherms of Fig. 41, 
 Fahrenheit degrees of temperature. 
 
Height in 
 
 Meters 
 
 6000 
 
 High 
 
 Low 
 
 High 
 
 5000 
 
 4000 
 
 3000 
 
 2000 
 
 1000 
 
 000 
 
 FIG. 43. Distribution of the high and low temperatures in cyclones and 
 anticyclones up to the height of 6,000 meters, showing the tendency to divide 
 into two branches with the maximum departure near the border of the high- 
 and low-pressure areas. Centigrade degrees of temperatures. 
 
238 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 difference of temperature, which are the local disturbing iso- 
 therms. Such normal charts of temperature on three planes 
 are found in the Barometry Report. On Fig. 48 the composite 
 of such a disturbance temperature system is given for nine 
 cyclones, similar to that of February 27, 1903 (Fig. 4l). Fig. 42 
 shows how the disturbance isotherms cover adjacent high- and 
 low-pressure areas. These cold and warm areas, as distinguished 
 from the normal temperatures of the season and place, are what 
 accompany all anticyclonic and cyclonic disturbances. The 
 wind currents are simply the effect of the force of gravity trans- 
 porting these masses of air of different temperatures, so that the 
 cold mass underruns the warm masses, and the warm mass over- 
 runs the cold masses to the right and the left hand, on all the 
 levels simultaneously from the surface to the top of the disturb- 
 ance. The campaign of extending the temperature observations 
 into the higher levels is going on in different parts of the world, 
 but definitive results have not been reached. There is no very 
 general and fixed system of temperature values to be expected, 
 because the incessant circulation, due to the annual change in 
 declination of the sun, prevents the atmosphere from settling 
 down into a simple thermal equilibrium. Fig. 43 gives an 
 example of the distribution of the warm and cold areas in cyclones 
 and anticyclones up to 6,000 meters. There is a tendency for 
 the warm area to divide into two branches to the northward, 
 and for the cold area to divide into two branches to the 
 southward. The maximum departure of the temperature is 
 somewhere between the centers of the low- and high-pressure 
 areas, and it is not distributed symmetrically about the center 
 as was assumed by Ferrel, and by the early German meteorol- 
 ogists, in the construction of their theories of vortex motion. 
 This asymmetric theory of vortices was first discussed in 1894 
 and in the Cloud Report of 1898, and the defects of the other 
 theories were pointed out. The pressure and temperature data 
 of observation were then entirely lacking in the upper strata, so 
 that several years were allowed to elapse before the subject 
 could be properly resumed, as was done in 1906 in the series of 
 papers on the "Thermodynamics of the Atmosphere," Monthly 
 
ISOTHERMS IN CYCLONES 
 
 239 
 
 Weather Review, and continued in 1907, 1908, in the series of 
 papers on the vortices in the atmosphere. The labor of securing 
 
 Observed isotherms 
 
 Local component isotherms 
 
 Temperature 
 
 Temperature 
 
 High 
 
 Low 
 
 High 
 
 High 
 
 Low 
 
 High. 
 
 000 
 
 FIG. 44. The observed isotherms and the local disturbing isotherms in high 
 and low areas of pressure from the surface to 10,000 meters. There is a ten- 
 dency for the warm mass to ascend and rotate through about one quadrant, 
 changing the direction of the horizontal axis from N.E. to N.W., and for the 
 cold mass to descend and rotate through one quadrant changing the direction 
 of the axis from the S.E. to the S.W. The sinuous lines in the upper levels 
 deepen in the lower levels, chiefly because the rapid eastward drift in the upper 
 levels, which smooths out all kinds of disturbances, relaxes in the lower levels, 
 and permits the disturbance components to dominate more fully. Compare 
 these diagrams with the Figs. 41, 42, 4.8, and note the position of the line of 0- 
 departure. Further observations will improve the accuracy of these diagrams. 
 
 a sufficiently large amount of data in the upper levels, by balloon 
 and kite ascensions, in order to eliminate temporary local con- 
 ditions and secure average values, is so great that many years 
 
240 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 must elapse before meteorology will possess data for computa- 
 tions of precision in this field of research. Meanwhile it is very 
 important for students to have in mind a picture of the general 
 phenomena so that useless discussions may be avoided. It is 
 particularly necessary that the temperature values due to the 
 location of the observatory, as a hill or mountain, should be 
 thoroughly eliminated, because the temperatures on an elevation 
 of land are not the same as that in the free air at the same height. 
 Otherwise, errors at the base station would go into the entire 
 series of gradients through a mistaken process of computation. 
 
 The Normal and the Local Velocity Vectors in Cyclones and Anti- 
 cyclones 
 
 The results of the observations made on the velocities of the 
 cloud motions at Washington, D. C., are embodied in Fig. 45 
 for the anticyclones, in Fig. 46 for the cyclones, and in Fig. 47 
 the entire system of stream lines is laid down as a whole. The 
 complete vector as observed in different areas surrounding the 
 center is first plotted, and then the component vector after the 
 mean eastward drift has been eliminated. The resultant vectors 
 are long in the upper levels and short in the lower levels; the 
 currents are slightly sinuous in the upper levels, and as they 
 escape from the eastward drift they become more nearly cyclonic. 
 In the component vectors there is a maximum velocity in the 
 3,000-meter level, and a decrease in velocity upward and down- 
 ward. The cyclonic disturbances often penetrate to the 10,000- 
 meter level, but in many cases they cease before the cirrus level 
 is reached, and as a practical matter they do not retain an im- 
 portant value above 4,000-5,000 meters, because the eastward 
 drift there predominates. It should be noted that in the cyclonic 
 components the U-shaped opening at A is in the northwest 
 quadrant of the upper levels, but in the northeast quadrant of the 
 lower levels; in the anticyclonic components it is in the south- 
 west quadrant in all levels. As can be readily inferred, it becomes 
 a very difficult task for the meteorologist to construct analytical 
 formulas to cover these complex curves, and to include the press- 
 ures, temperatures, and vectors under the general equations 
 
High Area Vectors Anticyclonic Components 
 
 76 M P 
 
 4.66 miles 
 
 0.62 mUea 
 
 FIG. 45. Mean vectors of velocity and direction in high areas. 
 
 I cm. = 500 kilometers for the distances, 
 i mm. = 2 meters per second = 4.48 miles per hour for the velocity vectors. 
 
Low Area Vectors 
 
 Cyclonic Components 
 
 4.66 milet 
 
 0.62 miles 
 
 Surface 
 
 FIG. 46. Mean vectors of velocity and direction in low areas, 
 i cm. = 500 kilometers i mm. = 2 meters per second = 4-48 miles 
 
 for the distances. per hour for the velocity vectors. 
 
THE LAND CYCLONE 
 
 243 
 
 of motion requires unusual skill. Some years may, therefore, 
 elapse before a satisfactory general theory can be perfected. In 
 what follows there is only possible a series of fragmentary prop- 
 ositions regarding circulations in the atmosphere. 
 
 Observed total wind vectors 
 
 Local component vector 
 
 Velocity 
 
 Velocity 
 
 High 
 
 Low 
 
 High 
 
 Low 
 
 High 
 
 FIG. 47. Observed total and observed local component wind vectors connect- 
 ing high and low areas. 
 
 The Land Cyclone 
 
 It has been shown that the ocean cyclones are imperfect 
 vortices of the dumb-bell-shaped type, which depart from the 
 nearly perfect forms found in hurricanes and tornadoes, through 
 the effect of certain resistances that are represented by com- 
 ponent reversing vortices. These departures may become very 
 
244 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 irregular, leaving only the remnants of pure vortex motion 
 from point to point in the cyclone, in proportion as the new 
 system of thermodynamic forces, due to gravity acting on masses 
 of air of different temperatures, are not symmetrically distributed 
 about an axis. In the pure vortex motion of the tornado and 
 the hurricane the* e was no need to consider specifically the action 
 of gravity on the vortex motion, because of the symmetrical 
 disposition of the air masses in superposed horizontal layers. 
 In the cyclones, on the other hand, the differential action of 
 gravity on adjacent air masses of different densities becomes the 
 primary consideration, as demonstrated in Chapter II, so that 
 the vortex action, though still of influence, becomes of secondary 
 dynamic value. The study of temperature distributions in 
 cyclones and anticyclones, together with the corresponding 
 velocity vectors and pressures, must be first determined by 
 observations before the dynamic theories can be suitably 
 applied. The ocean cyclone has been used as a transition be- 
 tween the hurricane and the land cyclone, in order to bring out 
 the method of the composition of vortices. In the land cyclone 
 the departures from the perfect dumb-bell vortex are very great, 
 especially in the upper strata, where the head of the vortex is 
 depleted by its intrusion into the rapid eastward drift whose 
 average velocities increase with the height above the surface of 
 the ground. This subject is so very voluminous that only the 
 leading features can be brought out in this place. 
 
 The cloud observations, made by the United States Weather 
 Bureau, 1896-1897, showed that when the true cyclonic com- 
 ponents of velocity are eliminated from that of the eastward 
 drift, there remain a cold current on the western side of a cyclone 
 and a warm current on the eastern side, and that this arrange- 
 ment persists in a general way from the ground up to the cirrus 
 levels, 10,000 meters. The tangential velocities v are at a maxi- 
 mum in the strato-cumulus level, 3,000 meters, and they decrease 
 downward and upward, the lower part being the truncated 
 portion of the vortex, while the longer upper part is gradually 
 destroyed by degradation in the eastward drift. The radial 
 velocities seem to be inward from top to bottom, taking the 
 
THE LAND CYCLONE 245 
 
 cyclone as a whole; or rather the inward flow on the west and 
 the outward flow on the east side do not appear to balance in 
 the different levels, so that the mean velocity shall become 
 inward below and outward above, as required by the perfect 
 dumb-bell vortex. In this respect the funnel-shaped vortex, 
 with the tube pointing upward, was suggested as the proper 
 mode of analysis, but the analogy does not hold in its details. 
 The determination of these radial velocities, upon which so much 
 depends, in the upper strata is really very difficult, and some 
 suitably located observatory might properly devote several years 
 of observations to the elucidation of this point with precision. 
 It has been proper to make a resume of the observations, so far 
 as was required to bring out the theory of the subject. We 
 began with the pressures and then took up the temperatures 
 and the velocities in the levels up to 10,000 meters. 
 
 The land cyclone differs from the ocean cyclone especially 
 in the fact that it is not so highly developed as a dumb-bell- 
 shaped vortex. The barometric pressure in the ocean cyclone 
 sometimes falls to 28.00 inches (711 mm.), while in the land 
 cyclone it seldom falls below 29.00 inches (737 mm.). This 
 deficiency of the central areas in the vortex tubes is due to a 
 variety of causes, but the principal fact is that the air masses 
 of different temperatures are placed side by side on the same 
 horizontal plane instead of being superposed; and the second 
 point is that the penetration of the head of the vortex into the 
 eastward drift of the general circulation is followed by its de- 
 pletion, which is caused by stripping away from the vortex of 
 fragments of the masses of ascending air. The meteorological 
 data that serve to illustrate these facts can be briefly presented. 
 
 A study has been made of the location of the isobars, the 
 variations in the temperature, and the wind velocity and direction 
 in a large typical land cyclone, by constructing the mean values 
 for a composite of nine selected cyclones, March 16, 1876, 
 March 27, 1880, April 18, 1880, January 12, 1890, December 3, 
 1891, November 17, 1892, April 20, 1893, January 25, 1895, 
 November 22, 1898. They were chosen such that the cyclonic 
 center occupied nearly the same place in the United States, 
 
246 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 namely, the lower Ohio Valley, and they were about equally 
 developed at the axis. On the weather map the scale is 1 mm. = 
 10,000 meters. The linear distances of the radii to each isobar 
 
 TABLE 57 
 The Land Cyclone 
 
 I. THE RADIAL DIMENSIONS 
 
 
 (i) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 B 
 
 760 
 
 755 
 
 750 
 
 745 
 
 740 
 
 735 
 
 25 
 
 1250000 
 
 975000 
 
 733000 
 
 508000 
 
 300000 
 
 110000 
 
 log ztr 
 
 6.09691 
 
 5.98900 
 
 5.86510 
 
 5.70586 
 
 5.47712 
 
 5.04139 
 
 logp 
 
 
 0.10791 
 
 0.12390 
 
 0.15924 
 
 0.22874 
 
 0.43573 
 
 II. A/ VARIATION OF THE TEMPERATURE FROM THE MEAN DISTRIBUTIONS 
 
 s 
 
 -1.3 
 
 -1.5 
 
 -1.1 
 
 -0.5 
 
 +0.2 
 
 +0.9 
 
 S30E 
 
 + 1.3 
 
 +1.7 
 
 +1.9 
 
 +1.4 
 
 +2.3 
 
 +2.7 
 
 S60E 
 
 +2.9 
 
 +2.8 
 
 +2.9 
 
 +2.9 
 
 +3.0 
 
 +3.2 
 
 E 
 
 +2.1 
 
 +2.1 
 
 +2.8 
 
 +2.8 
 
 +2.4 
 
 +2.1 
 
 E30N 
 
 +1.7 
 
 + 1.8 
 
 +1.7 
 
 + 1.7 
 
 + 1-2 
 
 +1.0 
 
 E60N 
 
 +2.2 
 
 +2.3 
 
 +2.1 
 
 + 1.8 
 
 +1.6 
 
 + 1.1 
 
 N 
 
 0.0 
 
 +0.1 
 
 +0.1 
 
 +0.7 
 
 +0.8 
 
 +1.3 
 
 N30W 
 
 -1.1 
 
 -1.2 
 
 -1.0 
 
 -0.4 
 
 +0.4 
 
 +1.3 
 
 N60W 
 
 -3.2 
 
 -3.0 
 
 -2.8 
 
 -2.4 
 
 -1.0 
 
 +0.8 
 
 W 
 
 -6.4 
 
 -5.5 
 
 -4.3 
 
 -2.7 
 
 -0.7 
 
 +1.7 
 
 W30S 
 
 -7.1 
 
 -6.1 
 
 -4.7 
 
 -3.3 
 
 -1.1 
 
 +1.0 
 
 W60S 
 
 -5.6 
 
 -4.6 
 
 -3.3 
 
 -2.1 
 
 -0.4 
 
 +1.1 
 
 III. WIND VELOCITY AND DIRECTION WITH THE ISOBAR 
 
 S 
 S30E 
 S60E 
 
 6.4 47 
 6.0 45 
 6.7 37 
 
 7.3 44 
 7.6 46 
 7.8 41 
 
 10.0 43 
 9.1 46 
 
 8.7 44 
 
 11.9 43 
 10.3 44 
 9.4 45 
 
 12.6 44 
 10.8 41 
 9.9 39 
 
 12.6 46 
 11.0 38 
 10.1 35 
 
 E 
 E30N 
 E60N 
 
 6.9 40- 
 7.9 42 
 7.8 29 
 
 7.6 42 
 8.1 45 
 7.6 35 
 
 7.2 43 
 8.5 46 
 7.8 41 
 
 6.8 41 
 8.2 46 
 8.0 44 
 
 7.3 39 
 7.9 45 
 7.8 43 
 
 8.5 37 
 8.2 45 
 7.6 40 
 
 'N 
 
 N30W 
 N60W 
 
 6.5 54 
 6.6 55 
 8.1 49 
 
 7.3 54 
 8.7 51 
 10.2 46 
 
 9.0 54 
 10.8 48 
 11.0 43 
 
 11.1 53 
 11.6 45 
 11.0 41 
 
 10.9 50 
 10.4 42 
 10.4 39 
 
 9.3 47 
 9.6 39 
 10.2 38 
 
 W 
 W30S 
 W60S 
 
 8.9 44 
 6.9 39 
 6.7 51 
 
 10.8 40 
 8.4 36 
 9.4 51 
 
 11.4 38 
 9.8 35 
 10.7 52 
 
 11.5 35 
 10.8 34 
 11.2 50 
 
 11.3 33 
 11.1 34 
 11.1 45 
 
 11.5 31 
 11.2 34 
 10.7 39 
 
THE LAND CYCLONE 
 
 247 
 
 were scaled in the N.W. - S.E. and S.W. - N.E. directions, 
 and the means were taken for the equivalent circular isobars. 
 All the data of Table 57 are given in the metric measures. The 
 section I contains the barometric pressure B, the circular radii 
 
 N30 W 
 
 E60N 
 
 N60VV 
 
 E30N 
 
 -7 
 
 W30S 
 
 S60E 
 
 -3 
 
 FIG. 48. Land cyclone with circular isobars equivalent to the elongated 
 cyclones of the United States, with the temperature disturbances and the 
 wind vectors, the center being located in the central valleys. 
 
 ZD-, and the log p = log ^ n . In a pure vortex log p = constant, 
 
 but in the land cyclone log p is not a constant. Hence it follows 
 that the pure vortex laws no longer prevail, though their influence 
 continues to be felt. 
 
 On Fig. 48 the isobars are laid down, and radii are drawn for 
 every 30 degrees, making points of intersection where the com- 
 putations can be concentrated. The isobars are spaced too 
 widely near the center of the land cyclone, that is, the barometric 
 pressure does not fall near the axis sufficiently to conform to 
 
248 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 the outer isobars from which the vortex is to be constructed. 
 The temperatures were scaled from the weather maps at the 72 
 points of intersection just indicated, and the means for each 
 point were taken and plotted on diagrams, one for each radius. 
 As a matter of fact, the compilation was made in the English 
 system, and this involved 132 readings for each of the nine 
 cyclones. Similarly, the wind directions relative to the isobars, 
 and the velocities were measured at the same points. The re- 
 sults transformed to metric measures appear in sections II, III, 
 of Table 57. After the temperatures at the several points had 
 been found, it was necessary to subtract from them the average 
 undisturbed temperature of the region, that is, the normal tem- 
 perature for the average of the dates of the years in question. It 
 is desired to know what disturbance of temperature accompanies 
 the cyclonic movements of the air, as distinguished from the normal 
 temperatures which are due to the general circulation taken by it- 
 self. The section II contains these differences, which are also 
 plotted on Fig. 48, together with the vectorsof the wind circulation. 
 It shows that the maximum departure for the cold area is 
 on the S.W. edge of the cyclone, and that the maximum depart- 
 ure for the warm area is in the S.E. quadrant generally, the line 
 of 0-departure running nearly due north and south through the 
 center of the cyclone. The mean angle of the vector is i = 43, 
 though it ranges from 27 to 54 in an irregular fashion from 
 one point to another. It has been shown that a similar asym- 
 metric distribution of the temperature prevails within the lower 
 levels, being at a maximum of departure in the strato-cumulus 
 level, 3,000 meters, and disappearing above in the cirrus level, 
 10,000 meters. When there are masses of air of different tem- 
 peratures on the same level, the densities are different, and the 
 action of gravity is to set up currents which cause the cold cur- 
 rents to underflow the warm currents, and the warm currents to 
 overflow the cold currents. The effort of gravity is to restore 
 the isobars to a normal value when they have been disturbed 
 by abnormal temperature densities. The air masses are trans- 
 ported from the north or from the south into some middle 
 latitude, where this underflowing and overflowing process sets 
 
THE LAND CYCLONE 
 
 249 
 
 up the cyclonic and the anticyclonic circulations. This prin- 
 ciple can be illustrated by a vertical section running from west 
 to east through a series of cold and warm masses of air. In a 
 cold mass the isobars are concentrated near the surface and 
 opened in the upper levels; in a warm mass the upper isobars 
 are concentrated and the lower are opened. When these cold 
 and warm masses alternate with one another the cold underflows 
 in two opposite directions, and the warm overflows in two 
 opposite direcions. In effect in nature, the cold mass from the 
 
 "Warm 
 
 Vertical Section West to East 
 Cold Warm Cold 
 
 Warm 
 
 W 
 
 E. 
 
 Horizontal Section South to North 
 
 FIG. 49. Model of the action of gravity G in forming streams of air which 
 underflow and overflow the warm and cold masses on either side. The isobars 
 in warm masses are relatively open below and closed above ; in cold masses they 
 are relatively closed below and open above. Gravity tends to restore them to 
 the same barometric levels, and the cyclones and anticyclones are the effect of 
 this process of circulation in the impure vortices of the dumb-bell-shaped type. 
 A thermodynamic discussion of the cyclone and the anticyclone has been 
 given in Chapter II, showing the interplay of the general forces: gravity, 
 pressure, circulation, and radiation. We shall next give a summary of the data 
 for the land cyclone, which will include the entire series of terms in the equa- 
 tions of motion, transformed to the vortex type, as in Formulas (561), and 
 their various modified forms. 
 
 north flows southward and divides, underflowing two warm masses 
 on either side, while the warm mass flows northward and over- 
 flows two cold masses on either side. The result of this complex 
 system of currents is to produce the cyclones and anticyclones, 
 and the tendency is to approach a dumb-bell vortex, though the 
 resistance is too great in general to permit this to be done. 
 
250 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 Recapitulation of the Formulas for the Dumb-B ell-Shaped Vortex, 
 (526)-(550), Fig. 18 
 
 We resume the formulas for the dumb-bell-shaped vortex 
 in connection with the cylindrical equations of motion, (526) 
 to (550) and (406), illustrated by Figs. 14, 18, including all the 
 terms: inertia, expansion, deflection, friction, radiation, circula- 
 tion, pressure, and gravitation. If + i is the angle between the 
 tangent and the horizontal velocity of motion, positive (+ ) on 
 the outside, negative ( ) on the inside of the circle, we have 
 az = 90 + i, the angle from the radius, which in the complete 
 vortex passes from a z = and i = 90 on the lower reference 
 plane for the air inflowing along the radius, to a z = 90 at the 
 middle height and i tangential, to az 180 and i = + 90 
 on the upper reference plane for air outflowing along the radius. 
 The intermediate inflowing and outflowing angles are all de- 
 termined by the relation of the line integral of the air flowing 
 radially to or from the axis, and the surface integral of the air 
 that rises in the vortex from the surface upwards. For a z = 60, 
 i = - 30 and for az = 120, i = + 30. Hence, we have for 
 
 (646) Angles, cos az = -f- sin i, 
 
 + sin a z = + cos i, 
 
 u 
 cot a z = + tan ^ = . 
 
 (647) Velocities, u = A aw cos a z = -f A a w sin i, 
 
 v = A a iff sin az = -j- -A a iff cos i, 
 w = 2 A sinaz = -f 2 A cos i, 
 
 u 
 
 = cot a z = + tan ^, 
 
 u aw aw 
 
 = -- cot a z = tan ^, 
 w Z Z 
 
 2u 2u 
 
 w = --- tan a z = cot i. 
 aw aw 
 
 (648) Line Integral, 2x w u = 2 ir A a w 2 cos a z = 
 
 2 IT A aw 2 sin i. 
 
 (649) Surface Integral, xwi w = + 2 n A w* sin a z = 
 
 2 x Aw 2 cos i. 
 
RECAPITULATION OF FORMULAS 
 
 251 
 
 (650) Ratio, 
 
 = - - = a cot <z z 
 
 W TffW 
 
 tan 
 
 u 
 a. 
 
 v 
 
 1 2 
 
 (651) Tangential Angle, tan i = . 
 
 2u 
 
 u 
 cot az = . 
 
 v 
 
 Meaning of the Tangential Angle i 
 The line integral of velocity is the product of the closed line, 
 circle, ellipse, or any other boundary line, multiplied by the mean 
 velocity at right angles to it, or more properly the integral of 
 the mass velocity per unit length around the boundary, 
 
 ff uds 
 
 where u is the velocity perpendicular to d s at every point. 
 For a circular vortex symmetrically disposed to an axis at the 
 radius uf the velocity u is radial and the same at every point of 
 the boundary circle 2 TT tcr, so that the line integral is, 
 
 S = 2 TT iff u 
 
 It is assumed that the inflowing air at the bottom of a vortex, 
 for example in a tornado, is not congested and compressed, and 
 therefore the inflowing mass must escape by rising upward 
 from one level plane to another in the surface integral I, = J J w 
 dS. 
 
 The inflowing mass 2 iriff u escapes vertically with the velocity 
 w through the plane whose area is irttr 2 , so that the surface integral 
 is simply Trttr 2 w, not now counting 
 the impermeable bottom, or the ^ 
 
 cylindrical surface. The same mass 
 of air entering the vortex ra- 
 dially and horizontally on one 
 plane escapes vertically on the 
 next adjacent horizontal plane, 
 and it is the vortex-constant a 
 and the tangential angle i that 
 controls this flow, through the 
 equation, 
 
 w 
 
 r 2 . w 
 
 FIG. 50. The line integral and 
 the surface integral in vortices. 
 
252 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 u aix v 
 
 = tan i = - tan ^ 
 
 w 2 w 
 
 1 Zirlffu 2u u 
 Hence, tan * = _ . -- r = - - = -, and a tan i = the 
 
 " 2 ' 
 
 ratio of the line integral to the surface integral. 
 
 Similarly from (523) to (525) for the funnel-shaped vortex, 
 and from (527) to (529) for the dumb-bell-shaped vortex, we 
 may summarize, 
 
 Funnel Vortex Dumb-bell Vortex 
 
 , . u 1 u 
 
 (652) - = - . - = cot a z = tan ^. 
 
 V Z V 
 
 u iff 1 u -or -of 
 
 = .- = a cot a z a tan ^. 
 
 w 2 z w 2 2 
 
 v w v tcr 
 
 w ~ ~ 2 ' w ~ 2 a ' 
 
 2 21 
 
 w = - . z .u. w = -. - . . u. 
 zcr -ay a tan i 
 
 The connecting link between these vortices becomes, 
 
 (653) - -- = tan*', 
 
 a z 
 
 the difference of sign depending upon the fact that the vertical 
 axis was assumed in opposite directions in these vortices. The 
 tangential angle i varies from one plane to another, 90 on 
 the lower reference plane, gradually changing to on the middle 
 plane where there is no inflowing or outflowing air, then con- 
 tinuing to + 90 on the upper reference plane. These are due 
 entirely to the supply of air needed to balance the inflowing or 
 outflowing line integrals with the increasing or decreasing vertical 
 surface integrals over the same planes in succession. 
 
 It has been customary for meteorologists to explain these 
 tangential angles in cyclones as the effect of the deflecting or 
 the friction forces. Thus, by equations (480) (481), 
 
 AM n _ s* 
 
 (654) tan 2*1 = - = j , for the inner part, 
 
 tan i 2 = - = -T-, for the outer part, 
 
 but in fact the theory is erroneous. The deflecting forces de- 
 pendent upon A are small, and those depending on the friction 
 
MEANING OF THE TANGENTIAL ANGLE * 253 
 
 k are nearly negligible. The inflowing tangential angle in 
 cyclones at the surface is due to the supply of air necessary to 
 compensate for the rising air over the entire surface of the 
 closed isobar, and therefore depends upon the integral of the 
 entire thermodynamic system. In the hurricane the inflowing 
 angle on the ocean shows the amount of air that is required to 
 supply the mass of air that is flowing away in horizontal radial 
 directions in the high levels, underneath the cold stratum that 
 has flowed as a sheet over the tropic region. Since tan i = 
 cot a z, the vertical distance of the azimuth plane from the ref- 
 erence planes can at once be found and thence the structure of the 
 entire cyclonic vortex can be deduced by the preceding methods. 
 Since we are not dealing with pure vortices in the case of cyclones, 
 these simple laws must be modified from point to point according 
 to conditions, and we proceed to evaluate the land cyclone in 
 the several complete terms of the equations of motion, applied 
 to the dumb-bell system. Making the substitutions indicated, 
 we find the following system of equations for a symmetrical 
 circular vortex with (406) in cylindrical co-ordinates. 
 
 3P du du du v 2 
 (655) Rad 1 al ) -^ = r ,+ ^+ W g- --- 
 
 2 w 3 cos 6 . v + ku+dQ & . 
 = + A 2 a 2 tff 2 o> 3 cos .A atxrcosi + 
 
 dv 8 v 80 uv 
 Tangential, =- + 
 
 2o> 3 cos . u + k v + 
 
 = r-T + . . . + 2 us cos B.A a -& sin i + 
 ot 
 
 dw dw dw , 
 
 Vertical, - = -^ + u - + w + g+ kw+dQ z . 
 
 -- ^- = - - 4 A 2 a sin i cos i +g+k w+d Q z 
 poz o t 
 
254 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 These equations contain the inertia, convergence, centrifugal, 
 deflecting, friction, and radiation terms in succession, reduced 
 to a form for computation at any point where the data are known. 
 
 Example of the Evaluation of the Terms in the Equations of Motion 
 
 for a Cyclone 
 
 As an example in the evaluation of the several terms in the 
 radial, tangential, and vertical equations of motion for a cyclone, 
 I have taken the data of Table 57 for B. P. T. p. R. -or. q. - i } as 
 in Table 58A, and have computed a, a ttr, A , u, v, w, A iff, A P, 
 p m , in succession. In Table 58s is a summary of the several 
 terms, as indicated in the equations of motion, for the pressure, 
 convergence, deflection, inertia, and friction combined with 
 radiation. In the vertical component the data and results are 
 from Table 25, (Qi - Qo) in the first column for the levels 000 
 to 500 meters, these data being divided by 500 to give the heat 
 losses per meter in a vertical direction. The other data of 
 Tables 58B are reduced to the unit length in all cases for compari- 
 sons. The pressure term radially is very much larger than the sum 
 of the convergence, deflection, and inertia terms, so that the 
 remainder friction plus radiation amounts on the average to 
 .0014582 per meter in mechanical units (M. K. S.) Since press- 
 ure acts inward these two terms act outward along the positive 
 radius. The result shows how impossible it is to balance the 
 terms of the equation of motion without friction and radiation. 
 It is not now known how to separate friction energy from radia- 
 tion energy directly, and thus evaluate them separately. These 
 terms in the tangential component are much smaller, and act 
 in the antirotational direction. The radial component averages 
 0.00146, and the tangential component is 0.00038, that is about 
 one-fourth the amount. It is probably true that most of the 
 tangential term is due to friction alone, and since i = about 45 
 we may suppose that the same amount of friction energy applies 
 to the radial component, leaving 0.00108 for the transported heat 
 energy towards the axis of the cyclone. The vertical component 
 is without friction and amounts to 0.11648, so that more than 
 100 times as much heat is transported upward vertically as 
 inward radially. 
 
EVALUATION OF TERMS IN EQUATIONS 
 
 255 
 
 TABLE 58A 
 
 SUMMARY OF THE DATA OF OBSERVATIONS FOR THE VORTEX TERMS IN A 
 
 TYPICAL CYCLONE 
 
 Quantities 
 
 (1) 1 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 Formulas 
 
 Barom. pressure, B 
 
 0.7600 
 
 0.7550 
 
 0.7500 
 
 0.7450 
 
 0.7400 
 
 0.7350 
 
 m mercury 
 
 Force pressure, P 
 
 101323 
 
 100656 
 
 99988 
 
 99323 
 
 98655 
 
 97990 
 
 (M.K.S.) system 
 
 Temperature, T 
 
 290.7 
 
 287.1 
 
 284.9 
 
 283.2 
 
 281.9 
 
 281.1 
 
 South 
 
 
 281.8 
 
 281.8 
 
 283.1 
 
 283.0 
 
 282.6 
 
 281.7 
 
 East 
 
 
 262.9 
 
 266.3 
 
 269.2 
 
 272.7 
 
 275.8 
 
 279.0 
 
 North 
 
 
 266.5 
 
 268.1 
 
 270.1 
 
 273.1 
 
 276.6 
 
 281.2 
 
 West 
 
 Density, P 
 
 1.2143 
 
 1.2215 
 
 1 . 2227 
 
 1.2219 
 
 1.2193 
 
 1.2145 
 
 South 
 
 
 1.2526 
 
 1 . 2444 
 
 1 . 2305 
 
 1.2227 
 
 1.2163 
 
 1.2119 
 
 East 
 
 
 1.3427 
 
 1.3475 
 
 1.2941 
 
 1.2689 
 
 1.2462 
 
 1.2237 
 
 North 
 
 
 1.3246 
 
 1 . 3080 
 
 1.2897 
 
 1.2613 
 
 1.2427 
 
 1.2140 
 
 West 
 
 Gas coefficient, R 
 
 287.033 
 
 287.033 
 
 287.033 
 
 287.033 
 
 287.033 
 
 287 . 033 
 
 Constant 
 
 Radii, u> 
 
 1250000 
 
 975000 
 
 733000 
 
 508000 
 
 300000 
 
 110000 
 
 Meters 
 
 
 OA1 K 
 
 
 180 
 
 
 180 
 
 
 
 Vortex constant, a 
 
 . UlO 
 
 = 
 
 12000 
 
 " 9000 + 3000 
 
 
 45 truncated 
 
 Vortex product, a w 
 
 18750 
 
 14625 
 
 10995 
 
 7620 4500 
 
 1650 
 
 
 Tube constants, A 
 
 . 000341 
 
 .000499 .000910 .001562 .002802 
 
 . 007633 
 
 g _ u cosec i 
 
 
 
 
 
 a co 
 
 
 357 
 
 519 655 
 
 986 
 
 1621 
 
 5156 
 
 E 
 
 
 347 
 
 500 
 
 829 
 
 1457 
 
 2422 
 
 5635 
 
 ^ ^ v see i 
 
 
 
 
 
 
 
 
 a to 
 
 
 475 
 
 738 
 
 1037 
 
 1510 
 
 2513 
 
 6966 
 
 W 
 
 Velocities, Q 
 
 6.4 
 
 7.3 
 
 10.0 
 
 11.9 
 
 12.6 
 
 12.6 
 
 S Meters per sec. 
 
 
 6.9 
 
 7.6 
 
 7.2 
 
 6.8 
 
 7.3 
 
 8.5 
 
 E 
 
 
 6.5 
 
 7.3 
 
 9.0 
 
 11.1 
 
 10.9 
 
 9.3 
 
 N 
 
 
 8.9 
 
 10.8 
 
 11.4 
 
 11.6 
 
 11.3 
 
 11.5 
 
 W 
 
 Tangential angle, i 
 
 47 
 
 44 
 
 43 
 
 43 
 
 44 
 
 46 
 
 S Inflowing angle 
 
 
 40 
 
 42 
 
 43 
 
 41 
 
 39 
 
 37 
 
 E 
 
 
 54 
 
 54 
 
 54 
 
 53 
 
 50 
 
 47 
 
 N 
 
 
 44 
 
 40 
 
 38 
 
 35 
 
 33 
 
 31 
 
 W 
 
 Radial velocity, u 
 
 -4.68 
 
 -5.07 
 
 -6.82 
 
 -8.12 
 
 -8.76 
 
 -9.06 
 
 S u = q sin i 
 
 
 -4.30 
 
 -5.08 
 
 -4.91 
 
 -4.46 
 
 -4.59 
 
 -5.12 
 
 E 
 
 
 -5.26 
 
 -5.91 
 
 -7.37 
 
 -8.87 
 
 -8.35 
 
 -6.80 
 
 N = A a 75" sin i 
 
 
 -6.19 
 
 -6.94 
 
 -7.02 
 
 -6.60 
 
 -6.16 
 
 -5.92 
 
 W 
 
 Tangential velocity, v 
 
 4.36 
 
 5.25 
 
 7.31 
 
 8.70 
 
 9.06 
 
 8.76 
 
 S v = qcoai 
 
 
 5.29 
 
 5.65 
 
 5.26 
 
 5.13 
 
 5.67 
 
 6.79 
 
 E 
 
 
 3.82 
 
 4.29 
 
 5.35 
 
 6.68 
 
 7.09 
 
 6.34 
 
 N = A a oT cos i 
 
 
 6.40 
 
 8.27 
 
 8.98 
 
 9.41 
 
 9.47 
 
 9.86 
 
 W 
 
 Vertical velocity, w 
 
 .000465 
 
 . 000718 
 
 .001458 
 
 .002283 
 
 .004027 
 
 .010618 
 
 s.-lt 
 
 a <o 
 
 
 564 
 
 773 
 
 957 
 
 1346 
 
 2520 
 
 8230 
 
 E 
 
 
 408 
 
 587 
 
 973 
 
 1753 
 
 3116 
 
 7685 
 
 N = 2 A cos * 
 
 
 683 
 
 1131 
 
 1633 
 
 2470 
 
 4209 
 
 11952 
 
 W = cot * 
 
 
 
 
 
 
 
 
 o5 
 
 Differences A w 
 
 
 275000 
 
 242000 
 
 225000 
 
 208000 
 
 190000 
 
 The mean values 
 
 
 
 
 
 
 
 
 of two successive 
 
 Differences A P 
 
 
 667 
 
 668 
 
 665 
 
 668 
 
 665 
 
 tubes placed under 
 
 
 
 
 
 
 
 
 the second of the 
 
 Means Pm 
 
 
 
 1.2179 
 
 1.2221 
 
 1.2223 
 
 1 . 2206 
 
 1.2169 
 
 pair from which they 
 
 
 
 1.2485 
 
 1.2375 
 
 1.2266 
 
 1.2195 
 
 1 2141 
 
 are computed. 
 
 
 
 1.3451 
 
 1.3208 
 
 1.2815 
 
 1 . 2576 1 2350 
 
 
 
 
 1.3163 
 
 1.2989 
 
 1.2755 
 
 1.2520 
 
 1.2284 
 
 
 
 The ratio of the line integral to the surface integral checks. 
 
 2u a u 
 
 a tan * = = = . 
 
 taw v 
 
256 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 
 
 
 ^S 
 
 j| 
 
 
 
 
 en 
 
 O 
 
 
 
 
 8 
 
 1 
 
 
 
 
 d TTI 
 
 + 
 
 
 
 en 
 
 ^?^ |^' 
 
 jj 
 
 
 
 3 
 
 8" 
 
 
 
 
 i 
 
 1 " W 
 
 
 
 3 
 
 
 S 
 to 
 
 ^ s e <S 
 
 S 
 
 s 
 
 
 
 I S i ^ 
 
 8 
 
 ^ 
 
 
 
 (M rH CO rH 
 
 1 1 
 
 ' + 
 
 
 
 l> 00 CO rH 
 
 00 O5 00 
 
 00 O . 
 
 ^ CO CO 
 
 
 e 
 
 i 
 
 8 8 
 
 ! i 
 
 
 
 i 1 rt< CO CO 
 
 CO CO 
 (M CO 
 
 o u 
 
 
 ^"^ 
 
 c^ ^^ 
 
 rH rH 
 
 
 
 ^ 
 
 8 8 
 
 8 8 
 
 a i 
 
 
 
 
 
 10 a rr. 
 
 
 
 
 
 H 
 
 W S so' 
 
 
 
 g ?22go. 
 
 (M CO 
 
 w 8 
 
 ^ 
 
 ^^ 
 
 i ^- ^ 
 
 rH rH 
 
 
 3 
 
 51- 
 
 8 8 
 
 8 8 
 
 H 1 1 
 
 
 
 ' 
 
 
 & 
 
 
 
 00 ^O 00 CO 
 
 S 2 
 S S 
 
 ^ 
 
 If 
 
 
 g 
 
 1 8 
 
 
 
 D 
 
 
 
 
 
 3 
 
 
 
 
 
 ^ ^s 
 
 
 
 lO I>- t^* ^^ 
 
 **"t^ i-H 
 
 > to 
 W (C f 
 
 1 1 
 
 
 
 rH CO O b- 
 
 O5 Tj< IO 
 
 I 
 
 iS 
 
 
 e 
 
 8 
 
 I 8 
 
 
 
 
 
 
 
 CX 
 
 
 
 
 
 1 
 
 
 
 
 
 ]a 
 
 
 rH 
 
 
 
 
 "O 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 d 
 
 
 
 
 
 o 
 
 
 
 
 
 | 
 
 
 
 H 
 
 2 &l rt 
 
 m > cu'S 
 
 d 
 O 
 
 
 
 
 
 d "u 
 
 .- 
 
 3 >- 
 
 
 . 
 
 
 .0017365 
 
 00 CO rH O 
 rH |>. TjH CO 
 
 CO *O 
 IO CO 
 
 rH CO 
 
 O 00 
 
 rH rH 
 
 I 
 
 
 IO CO TjH 00 
 
 00 b- 
 
 Oi co 
 
 rH ? 
 
 l> 10 
 
 CO Ci 
 
 8 3 
 
 8. 8 
 
 CO CO CO *O 
 O T-H CO O 
 
 O^ *H 
 
 IO 00 
 
 1 I 
 
 8 8 
 
 00 Oi CO ^O 
 
 1 1 
 
 i 
 
 8 
 
 i IP 
 
 IO rH 
 
 CO 00 
 IO CO 
 
 8 I 
 
 8 8 
 
 
 
 
 
 
 
 d 
 
 
 .2 
 
 
 i 
 
 8 
 
 i 
 
 1.2 
 3 b .S 
 
 tg > <U -M 
 
 d 
 
 .2 
 
 8 * | 
 
 UQrS 
 
 
 
 3 'i- 
 (/) b 
 
EVALUATION OF TERMS IN EQUATIONS 
 
 257 
 
 
 
 
 
 
 5l 
 
 
 
 
 
 
 8 
 
 
 
 8 
 
 
 
 rH 
 
 8 
 
 
 Tt< rH IO rH 
 
 t^- CO 
 
 
 (M 00 T^ CO 
 OS CO CO t> 
 
 c^ S 
 
 
 CO C^ C^l CO 
 00 tO <O I 
 
 rH CO 
 
 
 ^^ ^H ^^ 1 
 
 t^ rH 
 
 
 1 i 
 
 8 8 
 
 
 I i 
 
 8 8 
 
 
 00 10 l^ to 
 
 CO Oi rH rH 
 tO rH T^ <M 
 
 tO CO CO I 
 
 O5 CO 
 
 
 Cl to -^ to 
 
 ^f 00 
 CO 00 
 
 
 s i 
 
 8 8 
 
 
 = o 
 
 8 8 
 
 
 CO O CO rH 
 
 CO 
 
 
 t>- O5 CO (M 
 
 CO O5 
 Oi t*^ 
 
 " 
 
 o 
 
 O t^- CO to 
 
 CO rH IO 
 
 rH 
 8 
 
 
 
 0) 
 
 g 
 
 rH rH CO rH 
 
 8 
 
 
 1 1S1 
 
 t > * ^H 
 
 
 rH >O 00 CO 
 
 IO 00 00 CQ 
 
 CO to 
 
 
 
 O5 rH 
 
 
 8 8 
 
 8 8 
 
 
 i 
 
 O *H 
 
 8 8 
 
 
 <M CO rJH C^l 
 
 CO "^ ^D CO 
 
 O5 CO 
 N. to 
 
 
 CO CO rH O 
 
 i ^ 
 
 
 O 5 '^H OO rH 
 
 CO CO 
 
 
 00 O CO ^ 
 
 ^ o 
 
 
 8 8 
 
 O rH 
 
 8 8 
 
 
 rH O 
 
 8 1 
 
 1 8 
 
 
 | \ 
 
 \ 
 
 
 j 
 
 , j 
 
 
 
 a 
 
 
 
 d 
 
 
 
 o 
 
 
 
 
 
 
 
 
 
 .2 
 
 
 
 *o 
 
 
 
 "O 
 
 
 8 
 
 2 
 
 
 
 
 c 
 
 
 s g 
 
 K -S rt 
 1 ||l 
 
 1 all 
 
 1 1 
 
 
 S c 
 
 <u bfl.2 
 
 3 V t5 .2 
 
 c ~ 
 
 C^ flj QJ 
 
 0^ f ^) O II 
 
 d 
 
 o 
 
 en 
 
 s .a 
 
 3 u. 
 C/3 (x< 
 
258 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 8 
 
 colco 
 
 ff 
 
 
 
 ^ O 
 CO Tt^ 
 
 Oi 
 (N 
 
 CO 
 
 ^.oS 
 
 1 1 
 
 tj ctf o 
 a; '-M '^ 
 
 w 
 
 co o oo 
 
 O5 (N CO 
 
 IQ 
 
 <N 
 
 10 ^H 
 
 1 1 
 
 O5 
 
 + \ 
 
 + l 
 
 (M CO 
 CO t^* 
 
 1 1 
 
 
 rt o 
 
 II 
 
 
 + l 
 
 1 1 
 
 CO 1-1 
 
 1 1 
 
 + 
 
EVALUATION OF TERMS IN EQUATIONS 
 
 259 
 
 
 8 
 
 
 
 a|| 
 
 II 
 
 58 
 
 H.S 
 
 bfl 
 
 "^ G 
 
 s. 
 
 3 0) 'o ^ 
 _ -G 0) O 
 
 Jt & | 
 
 l^lllll 
 
 3 TO <-> ^ V LH 
 
 f3 . C Cti ^ ""^ ^^ 
 
 
 2 E 
 
 3 ,, 
 
 'S)>3 ^ 
 
 88-^ 
 
 Ctf c/3 CO 
 
 ^'^ S 
 
260 CONSTRUCTION OF VORTICES IN EARTH'S ATMOSPHERE 
 
 Similar studies of the relation of the heat and radiation terms 
 to the mechanical dynamic terms may be extended to the higher 
 levels of the cyclone, and to the anticyclone, but the final 
 conclusion to be remembered is that there is no possibility of 
 balancing in a dynamic system of equations the several terms of 
 motion, without including the radiation of heat energy, and its 
 convection from point to point. This branch of meteorology 
 will require much further study along the lines that have been 
 developed in this Treatise. 
 
CHAPTER V 
 
 Radiation, lonization, and Magnetic Vectors in the Earth's 
 
 Atmosphere 
 
 THE incoming solar radiation separates into two parts, the 
 first the irreversible heat that cannot be transformed back into 
 the original energy, the second the reversible energy which 
 appears as electrical and magnetic forces. The heat energy is 
 observed as the air temperature at different points, and its 
 effects are found in the general and the local circulations of the 
 atmosphere. The short waves of the solar radiation, at very 
 high temperatures, as 6700 to 7700, are capable of producing 
 ions of positive and negative electricity by the disintegration of 
 the atoms and molecules of the gases that compose the air, 
 whereby a part of the radiation energy reappears by transforma- 
 tion as free ions, or free electric charges, more abundantly in 
 some strata than in others. These ions tend to move in electric 
 streams, in certain general lines as controlled by a series of physical 
 conditions, and in their movement they induce magnetic deflect- 
 ing vectors, which disturb the earth's normal magnetic field, 
 through whose lines of magnetic force the electric ions move. 
 We have, therefore, to study the distribution of the solar radia- 
 tion in the atmosphere, the production of free electric charges 
 through ionization, and the dependent induced magnetic de- 
 flecting vectors. In spite of prolonged researches in these sub- 
 jects by many students, there is a wide discrepancy in the results, 
 as whether the solar intensity of radiation is 2.00 calories or 4.00 
 calories, whether the absolute coefficient of electric conduction 
 is 2X10~~ 5 or 6 X10~ 5 , whether the vectors that produce the 
 diurnal variations of the magnetic field originate in the higher 
 or the lower strata of the atmosphere, whether the sun is a star 
 with a variable periodic output of radiation or practically con- 
 stant, and whether the annual changes of the weather conditions 
 are dependent upon solar variability or are merely accidental. 
 
 261 
 
262 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 The Determination of the Intensity of the Solar Radiation by 
 Observations with the Pyrheliometer and the Bolometer 
 
 Measurements of the intensity of the solar radiation are made 
 by the pyrheliometer, which integrates in bulk the rays received 
 by the instrument, and by the bolometer which measures the 
 individual lines in the energy spectrum. The two instruments 
 supplement each other, because while the pyrheliometer gives 
 no account of the selective absorption of lines and bands in the 
 spectrum, the bolometer defines these depletions, and permits 
 the comparison of the observed spectrum energy with that of a 
 full radiator at the given temperature. Neither instrument 
 gives any account of that portion of the incoming solar radiation 
 which is reflected back to space, as the albedo of the earth, but 
 this can be found indirectly by thermodynamic computations 
 on the temperatures of the atmosphere, as observed in balloon 
 ascensions to great elevations. 
 
 The Pyrheliometer. This instrument consists of a chamber 
 for receiving a bundle of the solar rays, whose temperature can 
 be accurately measured at any time. The temperatures are 
 measured when the solar rays are shaded by a screen, and again 
 when exposed to the radiation, the sum of the changes of tem- 
 peratures in a given interval of time, as one minute, being the 
 effect of the radiation in temperature degrees. A factor can be 
 found by experiment which will convert these temperature 
 changes per minute into calories per square centimeter per 
 minute. There are many types of actinometers or pyrheliom- 
 eters, in which the different materials used for receiving the 
 radiation are involved with certain conversion coefficients. 
 The earliest form of pyrheliometer, by Pouillet, 1838, consisted 
 of a silver vessel rilled with a known volume of water, the surface 
 being blackened to absorb all the radiation, whose increase in 
 temperature in a given time could be measured by a thermometer 
 whose bulb was embedded in it. Silver box with mercury, copper 
 box with mercury, silver disk with no liquid, and many other 
 combinations have been employed. The electrical resistance 
 thermometer, the Angstrom double strip compensated pyrheliom- 
 
DETERMINATION OF INTENSITY OF SOLAR RADIATION 263 
 
 eter, in which the heat absorbed is measured by a compensating 
 electrical resistance, are used with success. We shall confine 
 our attention to Abbot's silver disk pyrheliometer, which has 
 been standardized against an elaborate Primary Standard 
 No. Ill, 1911, of the Astrophysical Observatory of the Smith- 
 sonian Institution, and furnishes the comparison factors for 
 copies of the secondary pyrheliometers. This instrument leaves 
 little to be desired for durability and accuracy of its operation. 
 More time is required for the observations with a thermometer 
 system than with an electrical resistance apparatus, but the latter 
 needs much more elaborate auxiliaries, battery, galvanometer, 
 current, and resistance apparatus, so that it is less readily 
 portable, and more liable to accidental inaccuracies of adjust- 
 ment. 
 
 Theory of the pyrheliometer. 
 
 Let S = the entire surface of the body receiving radiation. 
 
 5 = the cross-section of the rays falling upon it. 
 
 c = the coefficient of heat received referred to water, 
 
 h = the coefficient of heat lost by radiation, dQ 2 = 
 q = the intensity of the radiation received in dt, dQi = qsdt. 
 Hence, the general equation of equilibrium is, 
 
 (656) dQ = dQ 1 -dQ 2 = qsdt-hSTdt = cdT. 
 
 The shaded or cooling term. If the body is in the shade, 
 g = 0, and we have, 
 
 7 rr\ 1 Ql 
 
 (657) -=-= ---dt. Integrate for T=T , when /=0. 
 
 J. c 
 
 h S* h S* 
 
 (658) log T = - - - 1+ const. = - - - 1+ log T . 
 
 c c 
 
 T -* * -* t 
 
 (659) TT = e '< or T = T Q e ' 
 
 ^ 
 
 This is the cooling correction for T= the excess of the tem- 
 perature above the surrounding medium, when / is the interval 
 of time elapsed, and T is the initial excess at which the cooling 
 begins. 
 
264 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 The exposed or heating term. Divide the general equation by 
 c, and we have, 
 
 (670) dT= q dt---Tdt. 
 
 c c 
 
 From (656) the maximum temperature T m is obtained 
 when d T = 0, 
 
 (671) = qs-hST m , ~ = ~ T m . Substituting in (670), 
 
 (672) d 
 
 m . 
 
 C C L m L C 
 
 Integrate for T = T when dt = 0, since dT=-d (T m -T), 
 
 (673) log(r, n -r)= - ^l+const.-- --t+log(T m -T,). 
 
 C C 
 
 T m T hS T m T - t 
 
 (674) log ~^ ^r = - - t, and -~^ ^- = e c . Hence, 
 
 1 m 1 o c l m l o 
 
 _h^ t _hS t 
 
 (675) T=T e c +T m (l-e c ) for the total heating effect. 
 
 This is the equation for the total effect of radiation when 
 exposed to the solar rays. It consists of two parts: (1) the 
 cooling term from the initial temperature T 0} and the heating 
 term reckoned from the possible maximum temperature T m 
 relative to that of the surrounding medium, where T is the 
 effective difference of temperature at the time / above the sur- 
 rounding medium. It depends upon the total surface S of the 
 receiving body, the cooling coefficient h, and the heating co- 
 efficient of the body c, referred to water. 
 
 An Example of the Practical Observations with the Silver Disk 
 Pyrheliometer, S. I. No. 7, 1911 
 
 Several auxiliary tables must be prepared. 
 
 1. The table for the equation of time for every day of the 
 year. 
 
 2. The table of the declination of the sun for every day of 
 the year. 
 
 3. The table of 2 log r, for the radius vector of the earth. 
 
 4. The secant of the zenith distance, secz, should be com- 
 puted through the formulas leading to z. 
 
DETERMINATION OF INTENSITY OF SOLAR RADIATION 265 
 
 
 *o <u *o <u *o 
 
 III 1 1 
 
 13 
 
 
 B 
 
 g S 
 
 cog || 
 
 > m 
 
 4 rH 
 
 
 rH rH 
 
 N rH C 
 
 5 rH 
 
 c 
 
 i 
 
 s ss s ss 3 
 r if f r + r r 
 
 ^s K 
 < 
 
 ri 
 
 N 
 
 N 00 CO O N IO lOrHOJ t-t-CO OO^ 
 NrHrH rHOO COCON OJOO ****&> 
 
 
 
 i <i 
 
 1 1 1 If"-* 1 + III 
 
 
 
 
 gco oo w 
 COrHO OOOOOIO COOOt-00 00 UO <M lO 0000CTj< 
 
 O 00 
 
 . 
 
 
 W5NN NtMINN INNNIM CJNNN 
 
 
 
 to 
 
 OOOO OOOO OOOO OOOO OOOO 
 
 coco coco coco coco coco 
 
 c oo ^ 
 
 *, 
 
 I 
 
 "rji 10CO t-00 00 rHM 
 
 ".CO COCO COCO CO 1 ^ 1 '*^< 
 
 
 
 *s 
 
 OrH 
 
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 o. 
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 D IO 
 
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 5 
 
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 '+ f r + r r 
 
 I 
 
 < 
 
 F> 
 
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 ggrH OOOrH rH C, 3 ^ ^ ^ ^^ 
 
 
 
 < 
 
 1 + + 1 |' ' 
 
 
 
 
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 ^,^^4 t.3 fH NrHOOO TTrHOOiO USCOrHO 
 
 
 
 "^ 
 
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 c^ O 00 
 
 00 S 
 
 
 
 
 
 
 
 
 us 
 
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 COM CO COCO COCO CO 
 
 
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 *+ 
 
 = 
 
 
 
 
 
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 3 
 
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 j 
 
 
 * o v. 
 
 1 
 
 IN 
 
266 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 t x tan 5 tan'/ cos M tan(< M) 
 
 (676) tan M = , tan A = ^ ( 1/r , , tan z 
 
 cost 
 
 cos A 
 
 A carefully constructed diagram of curves is required for 
 the interpolations to two decimals of sec z. 
 
 Take four readings of the thermometer every 30 seconds 
 for the beam shaded and exposed alternately during ten minutes. 
 Compute the mean A7 for each group; take the means by pairs 
 of the shaded terms, and subtract from mean exposed term 
 lying between them; the sum of the two corrected A7 gives 
 the rise in the thermometer degrees per minute; multiply by 
 the pyrheliometer factor 0.610 to obtain I in calories; take 
 log 7 for the ordinate on the diagram of the Bouguer formula. 
 Adopt as epoch the middle minute of the watch, running on 
 mean time, and add the equation of time +7 W 8 s to reduce to 
 apparent time; subtract this from 12.00.00 for the hour angle 
 t; take the sun's decimation of date 6=+0 27', and with 
 /. 6 as arguments read off sec z on the scale diagram for La 
 Quiaca. Plot the points (log 7, sec z) on the diagram, and 
 draw the best mean line through the points of the day. Scale 
 off the point (log 7 , sec 2 = 0), and reduce to the mean solar 
 distance by the radius vector factor 2 log r to obtain log 7 . r 
 and 7 . r . 
 
 TABLE 60 
 
 LA QUIACA, ARGENTINA, SEPTEMBER 22, 1912 
 
 Similar pairs for September 22, 1912, at La Quiaca are as follows: 
 
 Time 
 
 Log/ 
 
 Sec z 
 
 Time 
 
 Log/ 
 
 Sec z 
 
 
 A.M. 
 
 6.48.00 
 
 0.018 
 
 4.65 
 
 P.M. 
 
 2.0.00 
 
 0.205 
 
 1.24 
 
 Plotted as o for the A.M. 
 
 6.58 
 
 0.060 
 
 3.89 
 
 2.10.0 
 
 0.182 
 
 1.33 
 
 Plotted as X for the P.M. 
 
 7.8 
 
 076 
 
 3.37 
 
 2.20.0 
 
 0.190 
 
 1.38 
 
 
 8.21 
 
 0.157 
 
 .82 
 
 4.0 .0 
 
 0.141 
 
 2.30 
 
 
 8.31 
 
 0.180 
 
 .73 
 
 4.10.0 
 
 0.124 
 
 2.52 
 
 
 8.41 
 
 0.180 
 
 .64 
 
 4.20.0 
 
 0.114 
 
 2.73 
 
 
 10.28 
 
 0.202 
 
 .17 
 
 
 
 
 
 10.38 
 
 0.205 
 
 .15 
 
 
 
 
 
 10.48 
 
 0.202 
 
 .13 
 
 
 
 
 
THE BOUGUER FORMULA OF DEPLETION 
 
 267 
 
 The most successful hours of observation are, weather per- 
 mitting, 
 
 1. Group of five, ten minutes each, 7.00 to 7.50 A.M. 
 
 2. Group of three, ten minutes each, 8.30 to 9.00 A.M. 
 
 logU 
 0.250 
 
 0.200 
 0.150 
 0.100 
 
 0.050 
 0.000 
 
 \ 
 
 
 loglo 
 
 lo.r 
 0.263 
 
 
 
 \ 
 
 V 
 
 lofiflo.r 
 
 0.267 1849 
 
 
 
 
 * \0 
 
 c IN 
 
 logic 
 Ii 
 
 I 2 
 
 Is 
 -Alogl 
 \ log p 
 
 0.263 for 
 .210 
 .157 
 .104 
 -.053 
 9.947 0.885 
 
 secz=0 
 =1 
 = 2 
 = 3 
 
 P 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 o^v 
 
 
 
 
 
 
 
 \ 
 
 
 sec z 
 
 0.00 1.00 2.00 3.00 4.00 5.00 
 
 FIG. 51. Plotting the pyrheliometer observations. 
 
 3. Group of three, ten minutes each, 11.00 to 11.30 A.M. 
 The corresponding computations can be completed in one 
 hour, so that three hours suffice to obtain I .r on any good day. 
 
 The Bouguer Formula of Depletion 
 
 The incoming solar radiation is subjected to two types of 
 depletion, (1) the reflection and scattering of the rays, on the 
 molecules, ice crystals, dust, and other constituents of the at- 
 mosphere, whereby a certain amount of energy is reflected back 
 
268 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 to space as albedo, and is not subject to measurement by the 
 pyrheliometer. This type of depletion affects chiefly the very 
 short waves in the energy spectrum, and its region of operation 
 is especially in the high cirrus region; (2) the other type of 
 depletion is due to selective absorption of certain lines and 
 bands of the spectrum, which can be determined specifically 
 by bolometer observations, with the sun at different zenith 
 distances for the same station, or by observations on the radia- 
 tion at the same zenith distance, from stations having different 
 heights above the sea level. The law of depletion is generally 
 expressed by Bouguer's formula, 
 
 (677) / = / sec2 , 
 
 where /o = the source of radiation before depletion, 
 
 ^ = the fraction which is transmitted for the unit dis- 
 
 tance, 
 sec z = the secant of the angular distance from the zenith 
 
 to the incoming ray. 
 
 / = the radiation measured at the instrument. 
 The formula is only correct for homogeneous rays. 
 
 I. Passing to logarithms. 
 
 (678) log / = log 7 + sec z. log p. 
 
 Taking two successive observations at Zi, z 2 , 
 
 (679) log /i - log 7 2 = (sec Zi - sec z 2 ) log p, hence, 
 
 (680) log p = 1 ~ 2 f or p constant in the interval. 
 
 sec zi - sec z 2 
 
 II. If it is assumed that sec z is constant and p variable, 
 First line, log 7 = log 7 + log p. sec z. 
 
 Second line, log /' = log /</ + log p f . sec z. 
 
 (681) (log / - log /') = (log 7 - log 7o') + (log p - log p') sec z. 
 
 (682) log j, = log -p + log j, sec z. 
 
 By the first type of formula, it is seen that log I decreases as 
 sec z increases. Taking (log /i log 1 2) for unit differences in 
 (sec Zi sec z 2 ), as in the example, we easily find its mean value 
 
THE BOUGUER FORMULA OF DEPLETION 269 
 
 0.053 for the given line as drawn, by reading the successive 
 values of log/ as the line crosses the sec z lines, 0.1.2.3 . . . 
 whence p is found from log p. 
 
 As the aqueous vapor rises from the surface upwards in the 
 diurnal convection the absorption (1.00 p) increases, and p 
 decreases towards the midday, so that the line joining the 
 observed values of log / is likely to be a curve which is slightly 
 convex toward the origin. In the dry climate at La Quiaca 
 such a curvature was not observed, though it is common at such 
 stations as Washington and Mt. Wilson. A refined treatment 
 of the variable p can be made for the same zenith distance by 
 the second form of the formula. 
 
 The Bouguer formula indicates that the depletion of the 
 incoming radiation is proportional geometrically to the length 
 of the path m which it traverses in the atmosphere, and it is 
 common to the large class of physical formulas which correspond 
 with similar conditions. It has been generally assumed, in 
 applying the formula to the earth's atmosphere, that the unit 
 distance is the depth of the atmosphere in the zenith mo, and that 
 what is measured by the pyrheliometer is the intensity of the 
 solar radiation at the actual distance of the earth, before any 
 depletion takes place. It is proposed to give several arguments 
 which show that this is a clear assumption, which does not 
 correspond with the facts of observation. 
 
 1. The Bouguer formula contains two unknown terms, 
 since sec 2 is a ratio of two distances, 
 
 m m' m" 
 
 (684) sec s = = = 77 =...- 
 mo mo mo 
 
 in which the denominator m 0) m f , mo" is undetermined; and 
 since only log / and log p can be deduced from the observations, 
 log IQ is also to be considered an unknown term. For example, 
 m may refer to one plane above the observer as the cumulus 
 stratum, m ' may refer to another plane as the cirrus stratum, 
 and m " may refer to any other plane as the outermost layer of 
 the earth's atmosphere. The two unknown terms (log 7 , m ) 
 refer to the source of the radiation 7 n on that plane whose unit 
 
270 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 distance above the surface m may be any plane in the atmosphere 
 satisfying these conditions. If the incoming radiation is partly 
 reflected back to space as albedo on the cirrus levels m Q ', then 
 
 Outer I ( 
 layer 
 
 Cirrus 
 layer 
 
 Cumulus 
 layer 
 
 FIG. 52. Illustrating the use of the Bouguer formula. 
 
 IQ is already diminished to that extent, and is not subject to 
 observations by the instruments. 
 
 /^o^\ W l W l W l 
 
 (685) sec zi = = 7 = 77 = . . . . 
 
 mo mo m 
 
 m<t m z ' m 2 " 
 sec z 2 = = 7 = 77 = . . . . 
 
 sec Zz = = 7 
 mo mo 
 
 m 
 
 The Bouguer formula contains, therefore, a double ratio, 
 
 (686) log -r = log p, that is -7- and 
 & /o m IQ mo 
 
 and it is quite indeterminate in itself, unless some means can 
 be found to fix the unit distance m , whether 7 emanates from 
 the cumulus, the cirrus, or the outermost layer of the atmos- 
 phere. It has been customary to make m = 1, refer it to the 
 outermost layer, and thus assume that the pyrheliometer meas- 
 ures the so-called "solar-constant" /o", or intensity of the 
 sun's radiation as it falls upon the earth's atmosphere, at W. 
 
THE BOUGUER FORMULA OF DEPLETION 
 
 271 
 
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272 
 
 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
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THE BOUGUER FORMULA OF DEPLETION 
 
 273 
 
 
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274 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 We shall now give strong evidence for thinking that the effective 
 radiation Id is on the lower levels, its value being less than 2.00 
 calories, while about 2.10 calories is reflected back to space and 
 escapes direct observation, so that the true "solar-constant" is 
 really more than 4.00 calories per square centimeter per 
 minute. 
 
 2. The depletion of the incoming radiation from a maximum value 
 on the cirrus levels, as determined by observations at different 
 heights. 
 
 Several series of pyrheliometer observations have been made 
 at stations having different heights above the sea-level, and after 
 they have been reduced to homogeneous data, and referred to 
 a given primary standard, they are comparable, and show an 
 increase in the value of 7 . r with the height. The primary 
 standard adopted is Abbot's Standard No. Ill, 1911, Smith- 
 sonian Astrophysical Observatory. The data of Table 13, 
 Vol. II, Annals, have been reduced by the factor 0.951 and the 
 data of the United States Weather Bureau (Mt.Weather Bulletin) 
 by the factor 1.039, those observations having been there re- 
 duced previously to the Angstrom pyrheliometer No. 104. In 
 order to exhibit the average conditions that prevail at different 
 elevations, several 10-day series have been impartially selected 
 from the large amount of material in hand. The individual 
 values (/o- r > P) are given in Table 61, and in the second section 
 their variations on the 10-day means. It is noted that the 
 variations of the coefficient of transmission p, do not materially 
 differ with the height, but there is a change in the mean values 
 of p, approximately 0.825 for all stations below the 1,500-meter 
 level, and 0.900 for all stations above that level. The variations 
 of A /o. r decrease steadily with the elevation, and 7 .r increases 
 with the height to a maximum value 1.918 which is to be identi- 
 fied as belonging approximately to the cirrus level. This is 
 explained more fully on Table 62, where the available annual 
 values are collected for eight stations. Special attention is 
 directed to the remarkable results at La Quiaca. 
 
THE BOUGUER FORMULA OF DEPLETION 
 
 275 
 
 It contains in section I, the station, observer, latitude <, 
 altitude in meters z, the individual annual values for the 10 
 years 1903-1912, and the mean for each station increasing from 
 1.709 at Washington to 1.848 at La Quiaca and 1.861 at Mt. 
 Whitney. These means are plotted on Fig. 53, as /o. r , alongside 
 of p the vapor pressure of the aqueous vapor in grams per cubic 
 meter /*, and the relative absorption of A/ . r by A/* = 1.00 
 gram per 500 meters of vertical path length as will be explained. 
 
 In order to determine the maximum value of 7 . r for the pyr- 
 heliometer observations, we need to know the bolometer factor 
 
 Cirrus 
 
 Mt. Whitney 
 La Quiaca 
 
 Maimara 
 Mt.Wilson 
 Jujuy 
 Mt. Weather 
 Cordoba 
 Washington 
 Vapor press 
 Radiation in 
 Relative ab 
 
 10000 
 
 
 
 
 
 
 9000 
 
 /* 
 
 [lo.^a 
 \ I 
 
 
 
 
 
 8000 
 
 
 u 
 
 
 
 
 
 7000 
 
 
 st 
 
 I^Sfc, 
 
 
 
 
 6000 
 
 
 X5- 
 
 ^ 
 
 X. 
 
 
 
 5000 
 
 
 \ 
 
 \ 
 
 
 \ 
 
 
 
 ( " 20) 4000- 
 
 - 
 
 \ 
 
 X \ 
 
 
 5 
 
 
 
 ^ 3000 
 
 - 
 
 
 \ 
 
 X^ 
 
 c 
 
 
 
 (2384) 2000 
 
 - 
 
 
 x \, 
 
 ^^--*. 
 x ^ 
 
 C^ 
 
 -^ 
 
 
 (1780) 
 (1302) 1000- 
 
 - 
 
 
 
 ^1, 
 
 "-v^ 
 
 N\ 
 
 
 (526) 
 
 ( ?I1! ooo: 
 
 : 
 
 
 
 
 "" 
 
 "\\ 
 
 
 are /J. 0123456789 10 
 Calories Io.r 1.900 1.850 1.800 1.750 1.700 
 sorption 0.1000 -0.0750 -0.0500 -0.0250 -0.0000 
 
 FIG. 53. The relation between the loss of radiation per i.o gram per cu. 
 meter for 500 meters, and the ratio between them, a. 
 
 F which will supply the amount depleted by line and band absorp- 
 tion. Mr. C. G. Abbot has determined these for Washington, 
 (1.123), and Mt. Wilson (1.094) from a long series of carefully 
 executed bolometer observations. These factors refer the 
 depletion to the adopted curve of a perfect radiator at 6,000 
 temperature. At present, similar factors are not available for 
 the other stations in this list, so that the data of the table in 
 reduced form are instructive rather than definitive, and it is 
 very important to extend accurate bolometer observations to 
 the other stations employed in making pyrheliometer observa- 
 
276 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 tions. In section II, the annual values are all reduced to the 
 sea level by the bolometer factors FI, and in section III to the 
 cirrus level or approximately to the 10,000-meter plane by the 
 factors F 2 . The mean value of I . r on the sea-level is 1.711 
 calories; and on the 10,000-meter plane it is 1.918 calories. 
 Section IV contains the mean annual values of p, approximately 
 0.900 above 1,500 meters, the top. of the diurnal convection, 
 and approximately 0.825 below that level. This difference in 
 the transmission factor p is due to the increase of density, p, 
 in the lower atmosphere. An example is taken from the data 
 derived from balloon ascensions for summer in the temperate 
 zone, which gives a fair distribution for the year in the low 
 latitude zones. 
 
 TABLE 63 
 
 RELATIVE EFFICIENCY OF 1 GRAM OF AQUEOUS VAPOR PER CUBIC METER IN 
 ABSORBING THE INCOMING RADIATION PER 500 METERS, MEASURED IN CALORIES 
 
 Elevation in Meters 
 
 Vapor 
 
 Pressure 
 
 Radiation 
 
 
 Absorption 
 
 z 
 
 P 
 
 A/z 
 
 I .r l 
 
 ^I .r 
 
 a 
 
 10000 
 
 0.03 
 
 0.02 
 
 1.918 -0 
 
 .002 
 
 -0.1000 
 
 
 0.05 
 
 0.02 
 
 1.916 - 
 
 .002 
 
 - .1000 
 
 9000 
 
 0.07 
 
 0.02 
 
 1.914 - 
 
 .002 
 
 - .1000 
 
 
 0.09 
 
 0.03 
 
 1.912 - 
 
 .003 
 
 - .1000 
 
 8000 
 
 0.12 
 
 0.03 
 
 1.909 - 
 
 .003 
 
 - .1000 
 
 
 0.15 
 
 0.05 
 
 1.906 - 
 
 .004 
 
 - .0800 
 
 7000 
 
 0.20 
 
 0.10 
 
 1.902 - 
 
 .006 
 
 - .0600 
 
 
 0.30 
 
 0.13 
 
 1.896 - 
 
 .007 
 
 - .0539 
 
 6000 
 
 0.43 
 
 0.17 
 
 1.889 - 
 
 .007 
 
 - .0412 
 
 
 0.60 
 
 0.20 
 
 1.882 - 
 
 .007 
 
 - .0350 
 
 5000 
 
 0.80 
 
 0.28 
 
 1.875 - 
 
 .009 
 
 - .0321 
 
 
 1.08 
 
 0.36 
 
 1.866 - 
 
 .011 
 
 - .0310 
 
 4000 
 
 1.44 
 
 0.46 
 
 1.855 - 
 
 .015 
 
 - .0326 
 
 
 1.90 
 
 0.60 
 
 1.840 - 
 
 .025 
 
 - .0417 
 
 3000 
 
 2.50 
 
 0.79 
 
 .815 - 
 
 .030 
 
 - .0380 
 
 
 3.29 
 
 0.93 
 
 .785 - 
 
 .025 
 
 - .0269 
 
 2000 
 
 4.22 
 
 1.54 
 
 .760 - 
 
 .016 
 
 - .0104 
 
 
 5.76 
 
 1.69 
 
 .744 - 
 
 .014 
 
 - .0083 
 
 1000 
 
 7.45 
 
 1.91 
 
 .730 - 
 
 014 
 
 - .0073 
 
 
 9.36 
 
 0.87 
 
 .716 - 
 
 .006 
 
 - .0069 
 
 000 
 
 10.23* 
 
 
 1.710 
 
 
 
 /* = the vapor pressure in grams per cubic meter. 
 A /* = the variation of n for 500 meters. 
 
THE BOLOMETER AND ITS ENERGY SPECTRUM 277 
 
 7 . r = the observed radiation in calories per square centi- 
 meter per minute, with maximum 1.918 on the 10,000- 
 meter plane. 
 
 A 7 . r = the variation in the radiation or absorption per 
 500 meters. 
 
 A/o . r 
 
 (687) a = the absorption by 1 gram per 500 meters. 
 
 The coefficient multiplied by the observed change in /* for 
 500 meters gives the amount of radiation which has been absorbed 
 by it. The curve of a shows that there is a gradual diminution 
 of A 7 . r per A /* = 1 gram from 10,000 meters to the sea level, 
 with a maximum of principal absorption, chiefly of short wave 
 lengths, in the stratum 10,000 to 8,000 meters, and a secondary 
 maximum in the stratum 4,000 to 2,000 meters. The laws of 
 selective absorption in the atmosphere for different wave lengths 
 are very complex, and they will require much more research for 
 their complete explanation. 
 
 It is quite evident that the curve of 7 . r , as derived from the 
 pyrheliometer and bolometer observations at different elevations 
 up to 4,500 meters, at Mt. Whitney, cannot be extended much 
 above 10,000 meters without exaggeration. Furthermore, since 
 the aqueous vapor, upon which the absorption chiefly depends, 
 does not extend above the 10,000-meter, or cirrus levels generally, 
 we are compelled to place the effective source of the radiation 
 observed by the instruments at the cirrus levels 10,000 to 
 20,000 m, rather than on the outermost layer of the atmosphere. 
 
 m f 
 Hence, in sec z = - f the unit length md is something like 13,000 
 
 m$ 
 
 meters, rather than the full depth of the atmosphere, and the 
 depletion, or albedo by reflection, is not observed. 
 
 3. The Bolometer and its Energy Spectrum of Radiation 
 
 The bolometer is a complex apparatus, of which the reader 
 can find excellent accounts in the Annals of the Astrophysical 
 Observatory of the Smithsonian Institution. It consists of a 
 siderostat or ccelostat for directing a beam of solar light in a 
 
278 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 fixed horizontal position, upon a slit which can be adjusted to 
 alter the quantity passing through it; a converging and a 
 collimating mirror to focus the slit upon a prism, which directs 
 the resulting spectrum upon a reflecting flat; a bolometer which 
 consists of a very fine filament of platinum, forming one branch 
 of a delicate Wheats tone balance, including a minute galvanom- 
 eter in one branch; the lines of the heat spectrum falling upon 
 the bolometer thread modify the current of electricity in the 
 circuits, and deflect the galvanometer which is registered photo- 
 'graphically in a manner to act in synchronism with the position 
 of deviation of the spectrum. The movement of the entire 
 spectrum across the bolometer produces a spectrum energy dia- 
 gram according to the prism, which must be transformed into 
 a normal spectrum of uniform dispersion, and upon this there 
 are very numerous line and band deficiencies of the ordinates 
 due to the selective absorptions that may have occurred. There 
 are numerous coefficients of absorption and reflection in the 
 mirrors and other parts of the apparatus, so that an unusual 
 degree of skill and experience is required for its successful 
 manipulation and the correct interpretation of the resulting 
 ordinates. This more or less defective energy spectrum is to 
 be compared with the full energy spectrum of a perfect radi- 
 ator, and that of 6,000 C. has been used by the Smithsonian 
 Observatory. 
 
 The energy spectrum of a full radiator at a given temperature 
 T may be computed by the Wien-Planck formula, in which 
 / = the energy of radiation at the wave-length X (x in microns, 
 V = 0.001 mm.), ci = 575,000, and c 2 = 14455, the adopted 
 constants. 
 
 (688) Wien-Planck formula,*/ = a /~ 5 (eTr- l)' 1 . 
 
 * Transformations of the (M. K. S.) and (C. G. S.) systems for the 
 mechanical and heat units. 
 
 (C. G. S.) System. 1 large calorie = 1 Kilogram degree 1 C. water = 426.8 
 Kilogram meters = 426.8 X 1000 X 100 X 980.60 = 4.1851 X 10 10 ergs 
 (Log. 10.62171). 1 small calorie = 1 gram degree 1 C. water = 426.8 
 gram meter = 426.8 X 100 X 980.60 = 4.1851 X 10 7 ergs (Log. 7.62171). 
 
THE BOLOMETER AND ITS ENERGY SPECTRUM 279 
 
 The total energy of radiation is proportional to the fourth 
 power of the temperature and would be expressed by 
 
 (689) Stefan formula, 
 
 / = a T* = . 6 4 T* Watts /cm. 2 deg. 4 (C. G. S.). 
 
 iv c% 
 
 Multiply by 10 7 for ergs/ cm. 2 (C. G. S.) Mech. units, and 
 divide by 4.1851 X 10 7 for gr. cal./cm. 2 sec. 
 c\ in absolute (C. G. S.) units is about sixty times too large, 
 and depends on the block (A A) included in the spectrum measures. 
 The constant is therefore more appropriate to a system with the 
 minute for the unit of time. 
 
 The displacement of the maximum ordinate is inversely 
 proportional to the temperature. 
 
 (M. K. S.) System. 1 large calorie = I Kilogram degree 1 C. = 426.8 kilo- 
 gram meter = 426.8 X 1 X 9.806 = 4.1851 X 10 3 (Log. 3.62171). 
 1 small calorie = 1 gram degree 1 C. = 426.8 gram meter = 0.4268 X 1 X 
 9.806 = 4.1851 (Log. 0.62171). 
 
 Kilogram Gr. X 10 3 Gr. N 
 Transformalvn Factor,. ^^, = ^^ = , X 10- 
 
 M i. KS ; ) X ^X6- . !iCaL_ Factor =0.000014336 (5.15644-10). 
 
 mech. units/ 41851000 cm. 2 mm. 
 
 Dimensions and transformation of the gravity equation from (M. K. S.) to 
 
 (C. G. S.}. Gravity equation, g fa - ) = - Pl ~ P - Y^ (tfi 2 - tfo 2 ) - 
 
 Pio 
 
 (Qi - Qo), (M. K. S.). All these terms as computed in the (M. K. S.) 
 system are transformed into the (C. G. S.) system of mechanical units by the 
 Factor 10 4 , as can be tested by substituting the terms dimensionally. 
 Similarly the equations (330) to (337) have the same factor 10000. This 
 equation is transformed from (M. K. S.) mechanical units by the factor 
 
 1.4336 X 10- 5 into _ 
 
 cm 2 , mm, 
 
 Dimensions and transformation of the radiation equation. 
 
 Radiation equation. Qi ~ Qo - P 10 = Ul ~ Uo = K lo = c T*. 
 v\ VQ Vi VQ 
 
 All these terms as computed in the (M. K. S.) mechanical units, are 
 transformed into (C. G. S.) mechanical units by the factor 10, and into 
 
 -M*. by the factor ",*!* = 0.000014336. 
 cm 2 , mm. 4.1851 X 10 7 
 
 The transformation of the coefficient in the Stefan Law, J = a T*. The co- 
 
 <c 3 X 10 10 6 ci , ci = 8 TT c h , ., , , 
 
 efficient a = 7 X p, where _ , ,, for the Planck constants 
 
 h = 6.545 X 10- 27 , k = 1.3606 X lO" 16 (C. G. S.). 
 
280 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 2891 
 (690) Wien displacement formula, ^ max = ~^~- 
 
 The constants used in these formulas have not been found 
 quite the same at all temperatures, especially c z seems to have 
 increasing values for higher temperatures, but they serve suffi- 
 ciently for illustrating the principles now discussed. 
 
 If the sun is a body which emits radiation at a given tem- 
 perature, it will have an efficient energy of radiation per square 
 centimeter per minute at the surface of the emission. 
 Take R = 694,800 the radius of the sun in kilometers, 
 
 D = 149,340,900 the distance to the earth in kilometers 
 and the effective radiation falling upon the earth's outermost 
 layer of atmosphere, before any reflection or absorption takes 
 place, is found by, 
 
 (691) 
 
 7.90 X 10" 11 T 4 . 
 
 (C. G. S.) ci = 8 TT c h = 8 X 3.14159 X 3 X 10 10 X 6.545 X 10~ 27 = 
 
 4.93456 X 10- 15 (-15.69326) 
 
 C2 = ch/k = 3 X 10 10 X 6.545 X 10- 27 /1.3606 x 10- 16 = 
 1.4433 cm. (0.15928) 
 (Mech. units) a = 3.10 10 X 6 X 4.93456 X 10- 15 /4 X 10 7 X (1.4433) 4 = 
 
 5.1210 X 10- 12 (-12.70935) 
 (Heat units) a = 5.121 X 10- 12 X 10 7 /4.1851 X 10 7 
 
 1.2236X 10- 12 (-12.08764) 
 
 (M.K.S.) ci = 8 * c h = 8 X 3.14159 X 3.10 8 X 6.545 X IO- 34 = 
 
 4.93456 X IO- 24 (-24.69326) 
 
 C2 = c h/k = 3 X 10 8 X 6.545 X 10- 34 /1.3606 X 1Q- 23 = 
 
 0.014433 (-2.15928) 
 (Mech. units) <r = 3 X 10 8 X 6 X 4.93456 X 10- 24 /4 X 10 7 X (0.014433) 4 = 
 
 5.1210 X 10- 13 (-13.70935) 
 (Heat units) a = 5.121 X 10~ 13 X 10 7 /4.1851 X 10 3 
 
 1.2236 X 10- 11 (-11.08764) 
 
 In the Wien-Planck Formula % in cm. becomes 10 4 ft. Since Ci is the 
 coefficient of A 5 = 10 20 /", and c 2 that of 2, = 10 4 /*, we shall have, 
 (C. G. S.) ci = 493456 (5.69326), c 2 = 14433 (4.15928). 
 (M.K.S.) cj. = 4934560 (3.69326), c 2 = 14433 (4.15928). 
 
 These values are computed for dry air. Kurlbaum's G for ether, from 
 recent experiments may be taken provisionally: 
 
 
 
 
 C. G. 
 
 S. 
 
 
 Log. 
 
 M. K. 
 
 S. 
 
 
 Log. 
 
 a 
 
 in 
 
 cal./sec. 
 
 1.3167 
 
 X 
 
 10 -12 
 
 12.11948 
 
 1 
 
 3167 
 
 X 
 
 io- 11 
 
 11.11948 
 
 a 
 
 in 
 
 cal./min. 
 
 7.900 
 
 X 
 
 io- n 
 
 11.89763 
 
 7 
 
 .900 
 
 X 
 
 10 -io 
 
 10.89763 
 
 a 
 
 in 
 
 ergs/sec. 
 
 5.510 
 
 X 
 
 io- 5 
 
 - 5.74119 
 
 5 
 
 510 
 
 X 
 
 1(H 
 
 - 8.74119 
 
 a 
 
 in 
 
 ergs/min. 
 
 3.3063 
 
 X 
 
 10- 3 
 
 3.51934 
 
 3 
 
 3063 
 
 X 
 
 10-* 
 
 6.51934 
 
THE BOLOMETER AND ITS ENERGY SPECTRUM 
 
 281 
 
 The following Table 64 gives the solar energy of radiation 
 for certain selected temperatures T. It is computed with the 
 
 value of ff = 1.316 X 10 C. G. S., which is not far from the 
 value given by Kurlbaum. In order to make the numbers 
 comparable with the usual convention by which the solar con- 
 stant is stated in equivalent gram calories for the time-unit of 
 1 minute, the C. G. S. value of ff is multiplied by 60 in (691). 
 
 TABLE 64 
 
 I 
 
 EVALUATION OF THE SOLAR RADIATION AT THE MEAN DISTANCE OF THE EARTH 
 FOR SELECTED TEMPERATURES 
 
 T 
 
 /o 
 
 T 
 
 /o 
 
 T 
 
 /o 
 
 8000 
 
 7.004 
 
 7000 
 
 4.106 
 
 6000 
 
 2.216 
 
 7900 
 
 6.661 
 
 6900 
 
 3.876 
 
 5900 
 
 2.072 
 
 7800 
 
 6.329 
 
 6800 
 
 3.656 
 
 5800 
 
 1.935 
 
 7700 
 
 6.011 
 
 6700 
 
 3.446 
 
 5700 
 
 1.805 
 
 7600 
 
 5.705 
 
 6600 
 
 3.245 
 
 5600 
 
 1.682 
 
 7500 
 
 5.410 
 
 6500 
 
 3.052 
 
 5500 
 
 1.565 
 
 7400 
 
 5.128 
 
 6400 
 
 2.869 
 
 5400 
 
 1.454 
 
 7300 
 
 4.856 
 
 6300 
 
 2.694 
 
 5300 
 
 1.349 
 
 7200 
 
 4.595 
 
 6200 
 
 2.527 
 
 5200 
 
 1.250 
 
 7100 
 
 4.345 
 
 6100 
 
 2.368 
 
 5100 
 
 1.157 
 
 According to this formula the equivalent "solar-constant" 
 1.918 would correspond with a temperature 5,787. At the time 
 the bolometer factors were determined it was supposed that the 
 value IQ.J. = 2.200 so that 6,000 was assumed to be the proper 
 temperature for the full radiator, with which to compare the 
 actual observed ordinates of the bolometer energy spectrum. 
 Successive improvements in the standardization of the pyrheliom- 
 eter has, however, reduced that value to 1.918, so that the 
 5,800 curve would be the proper one to use in determining the 
 bolometer factor at the several stations. 
 
 Table 65 contains the evaluation of the Wien-Planck formula 
 of radiation for temperatures from 7,700 to 5,800, together 
 with the sum of the ordinates from /I = 0.020 p to A = 2.500 p, 
 the maximum ordinate I max , and the maximum wave-length A m , 
 in the displacement formula. Abbot has determined the or- 
 dinates in certain arbitrary units for a number of wave-lengths, 
 
282 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 S 
 
 a 
 
 S^ 
 
 H* 
 
 ^ 
 
 <TO005-*0>^OOCOU300(NOOCO<NlOO>t-r-IOO>lOt-TJ<OONlO 
 
 0(NOOCO<NlOO>t-r-IOO>lOt-TJ<OO 
 
 <ot-'^<oaooooiot-T-i?OT-(ooiOD3i-io 
 
 NCU5NC<IC<lTj<O>Tj<lOOeO < '*t-lOOC>eOr-lTl<*O 
 OOiCiOS05CO(Nt>(NO>T--ICOi35NOOCOCOO>OOCSC<I 
 iHO>COOOOO>OC<IOlOOO5>OrHOt-lOTl*eOCO 
 
THE BOLOMETER AND ITS ENERGY SPECTRUM 
 
 283 
 
 and we have finally decided to reduce them to calories corre- 
 sponding approximately with 6,900, and for this purpose have 
 divided them by 1,000. This will be fully explained. 
 
 T 
 
 6700 
 
 6600 
 
 6500 
 
 6400 
 
 6300 
 
 6200 
 
 6100 
 
 6000 
 
 5900 
 
 5800 
 
 A. =0.20|* 
 
 0.779 
 
 0.662 
 
 0.559 
 
 0.470 
 
 0.393 
 
 0.326 
 
 0.270 
 
 0.222 
 
 0.181 
 
 0.146 
 
 .25 
 
 2.209 
 
 1.938 
 
 1.694 
 
 1.474 
 
 1.277 
 
 1.101 
 
 0.945 
 
 0.807 
 
 0.685 
 
 0.579 
 
 .30 
 
 3.743 
 
 3.357 
 
 2.999 
 
 2.671 
 
 2.370 
 
 2.095 
 
 1.844 
 
 1.616 
 
 1.411 
 
 1.226 
 
 .35 
 
 4.844 
 
 4.411 
 
 4.006 
 
 3.626 
 
 3.273 
 
 2.944 
 
 2.639 
 
 2.358 
 
 2.098 
 
 1.859 
 
 .40 
 
 5.382 
 
 4.958 
 
 4.555 
 
 4.177 
 
 3.817 
 
 3.478 
 
 3.160 
 
 2.862 
 
 2.584 
 
 2.322 
 
 .45 
 
 5.458 
 
 5.073 
 
 4.705 
 
 4.353 
 
 4.017 
 
 3.698 
 
 3.395 
 
 3.108 
 
 2.838 
 
 2.583 
 
 .50 
 
 5.232 
 
 4.896 
 
 4.574 
 
 4.264 
 
 3.965 
 
 3.681 
 
 3.408 
 
 3.147 
 
 2.898 
 
 2.661 
 
 .55 
 
 4.841 
 
 4.557 
 
 4.280 
 
 4.014 
 
 3.757 
 
 3.509 
 
 3.270 
 
 3.041 
 
 2.821 
 
 2.610 
 
 .60 
 
 4.378 
 
 4.140 
 
 3.909 
 
 3.683 
 
 3.465 
 
 3.253 
 
 3.049 
 
 2.851 
 
 2.660 
 
 2.477 
 
 .70 
 
 3.451 
 
 3.287 
 
 3.125 
 
 2.968 
 
 2.815 
 
 2.664 
 
 2.517 
 
 2.375 
 
 2.236 
 
 2.101 
 
 .80 
 
 2.662 
 
 2.548 
 
 2.436 
 
 2.326 
 
 2.218 
 
 2.112 
 
 2.009 
 
 1.907 
 
 1.807 
 
 1.710 
 
 .90 
 
 2.046 
 
 1.965 
 
 1.886 
 
 1.808 
 
 1.732 
 
 1.657 
 
 1.583 
 
 1.509 
 
 1.438 
 
 1.367 
 
 .00 
 
 1.577 
 
 1.521 
 
 1.464 
 
 1.409 
 
 1.353 
 
 1.298 
 
 1.245 
 
 1.191 
 
 1.139 
 
 1.089 
 
 .10 
 
 1.226 
 
 1.185 
 
 1.144 
 
 1.103 
 
 1.063 
 
 1.C23 
 
 0.983 
 
 0.944 
 
 0.906 
 
 0.869 
 
 .20 
 
 0.963 
 
 0.932 
 
 0.901 
 
 0.871 
 
 0.840 
 
 0.811 
 
 .782 
 
 .752 
 
 .724 
 
 .695 
 
 .30 
 
 .763 
 
 .740 
 
 .717 
 
 .694 
 
 .672 
 
 .649 
 
 .626 
 
 .604 
 
 .582 
 
 .560 
 
 .40 
 
 .611 
 
 .593 
 
 .576 
 
 .558 
 
 .541 
 
 .523 
 
 .506 
 
 .489 
 
 .472 
 
 .455 
 
 .50 
 
 .495 
 
 .481 
 
 .467 
 
 .453 
 
 .439 
 
 .426 
 
 .413 
 
 .399 
 
 .386 
 
 .372 
 
 1.60 
 
 .403 
 
 .392 
 
 .381 
 
 .371 
 
 .360 
 
 .349 
 
 .338 
 
 .328 
 
 .318 
 
 .307 
 
 1.70 
 
 .332 
 
 .323 
 
 .315 
 
 .306 
 
 .297 
 
 .289 
 
 .280 
 
 .272 
 
 .263 
 
 .255 
 
 1.80 
 
 .276 
 
 .268 
 
 .262 
 
 .255 
 
 .248 
 
 .241 
 
 .234 
 
 .227 
 
 .220 
 
 .214 
 
 1.90 
 
 .231 
 
 .225 
 
 .219 
 
 .214 
 
 .207 
 
 .202 
 
 .197 
 
 .191 
 
 .185 
 
 .180 
 
 2.00 
 
 .196 
 
 .189 
 
 .185 
 
 .180 
 
 .176 
 
 .171 
 
 .166 
 
 .162 
 
 .157 
 
 .152 
 
 2.10 
 
 .165 
 
 .161 
 
 .157 
 
 .153 
 
 .149 
 
 .145 
 
 .141 
 
 .138 
 
 .134 
 
 .130 
 
 2.20 
 
 .140 
 
 .137 
 
 .134 
 
 .131 
 
 .127 
 
 .124 
 
 .121 
 
 .118 
 
 .114 
 
 .111 
 
 2.30 
 
 .121 
 
 .118 
 
 .115 
 
 .113 
 
 .109 
 
 .107 
 
 .104 
 
 .101 
 
 .099 
 
 .096 
 
 2.40 
 
 .104 
 
 .102 
 
 .099 
 
 .097 
 
 .095 
 
 .092 
 
 .090 
 
 .088 
 
 .085 
 
 .083 
 
 2.50 
 
 .090 
 
 .088 
 
 .086 
 
 .084 
 
 .082 
 
 .080 
 
 .078 
 
 .076 
 
 .074 
 
 .072 
 
 Sums. 
 
 52.718 
 
 49.247 
 
 45.950 
 
 42.826 
 
 39.857 
 
 37.038 
 
 34.392 
 
 31 . 873 
 
 29.515 
 
 27.281 
 
 Max. 
 
 5.474 
 
 5.077 
 
 4.704 
 
 4.354 
 
 4.024 
 
 3.714 
 
 3.424 
 
 3.153 
 
 2.899 
 
 2.661 
 
 * max. 
 
 0.432 
 
 0.438 
 
 0.445 
 
 0.452 
 
 0.459 
 
 0.466 
 
 0.474 
 
 0.482 
 
 0.490 
 
 0.498 
 
 Fig. 54 contains the diagrams of the energy curves for 7,700, 
 6,900, 5,800, and Abbot's observed ordinates, which closely 
 agree with the 6,900 curve from wave-length A = 0.50 /* to 
 A = 1.00 p. It is seen that there is a heavy depletion from 
 X = 0.00 to 0.30 {J. or 0.35 ft, a slight excess from A = 0.50 n to 
 0.70 fi, a slight deficiency from A = 0.70 j to 1.00 j, and a rapid 
 increase in the ordinates to agree with those of the 7,700 curve 
 for A = 1.50 j to 2.50 /*. Whether the matching of the observed 
 with the computed ordinates is made for the curves 6,500, 
 6,600, 6,700, 6,800, 6,900, 7,000, the preceding description of 
 results holds true, but the best coincidence seems to be for 6,900. 
 
284 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 0.00/X 0.50 M 1.00/A 1.50 jU, 2.00 P 2.50 /* 3.00 /* 
 
 FIG. 54. Observed bolometer ordinates in their relation to the perfect 
 radiation spectrum at different temperatures. 
 
THE BOLOMETER AND ITS ENERGY SPECTRUM 285 
 
 Now this corresponds with 3.876 for the solar constant at the 
 earth, instead of 1.918 as given for the maximum from pyrheli- 
 
 Q Q7t 
 
 ometer observations, and this requires - - = 2.02 times as 
 
 many calories to fall upon the surface of the outer layer of the 
 atmosphere as upon the pyrheliometers, so that at least 1.958 
 calories must have been reflected back into space as albedo, 
 which would therefore be about 51 per cent of the radiation of 
 the sun received at the earth. If the bolometer ordinates can be 
 more accurately determined this ratio may be definitely found; 
 if the ratio varies from year to year at the same, or at several 
 stations, the proper distribution between the variability of the 
 solar radiation and the variability of the terrestrial reflection 
 can be further discussed. At present, one must be very con- 
 servative in attributing to solar variation the entire apparent 
 variations of the radiation measured by the pyrheliometer and 
 the bolometer. 
 
 The remarkable fact appears to be established by Abbot's 
 bolometer ordinates, Table 65, that these ordinates do not cor- 
 respond with any single temperature of emission. By comparing 
 the Abbot ordinate at any given wave-length with the ordinates 
 computed at different temperatures for the same wave-length, it 
 is not difficult to interpolate for a temperature of emission that 
 would produce the ordinate. Thus, they are 6,850 at A = 0.40 /*, 
 6,960 at A = 0.55 /*, 7,260 at A = 1.30 /*, 7,800 aU = 1.60 / to 
 2.00 p. On the face of it, the Abbot ordinates range through 1 ,000 
 temperature, 6,700 to 7,700, and this may have several inter- 
 pretations. (1) The solar envelope may consist of layers of 
 different temperatures, 7,700 at the photosphere, which would 
 be the general source of emission for all wave-lengths, gradually 
 diminishing to 6,700 at the top of the chromosphere, or possibly 
 the inner corona, in which envelope there is gradual selective 
 absorption of certain wave - lengths, so that the effective 
 emission of the sun to space is very complex, and corresponds 
 with the observed bolometer spectrum; (2) a similar selective 
 depletion of a uniform spectrum of 7,700-energy may occur in 
 the middle and lower region of the earth's atmosphere, so that 
 
286 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 i g 
 
 W 
 
 8 
 
 3 
 
 S 
 
 u 
 
 I 
 
 8 
 
 J j 
 g 
 
 1> CO CO CN I 00 
 OS T-I CO <N IO 
 
 rH CO TjH i-H CO 
 CO CO CO CO C^ 
 
 I I I I I 
 
 00 t- CO T*< rH 
 
 1 1 1 1 1 
 
 i-t CO 1> 00 t- 
 
 O> CO 00 rH O) 
 
 i 1 IO i 1 CO 
 CO "* Tj< O 
 CO CO CO CO 
 
 1 1 I 1 
 
 t- 
 
 CO 10 O C~ rjl 
 
 777 i i 
 
 oo TF CQ o co 
 
 OS OS t^ JO O3 
 
 CO CO CO <M C* 
 
 I I I I I 
 
 CO CD t- IO 
 
 77 
 
 i i i 
 
 ^ 00 00 OS O 
 CO CO CO CO C^ 
 
 I I I I I 
 
 U3 OS O OS O 
 
 i i i i i 
 
 CO t^. CO <M <N 
 
 ^ oo i> co ^ 
 
 CO CO CO CO CO 
 
 1 1 1 1 1 
 
 O O> vH 
 
 O) IO IO 
 
 1 1 1 + 1 
 
 CO CN CO "tf O 
 
 O (M (M CO 10 
 CO CO CO CO CO 
 
 I I I I I 
 
 1-1 CO T* CO O 
 
 CO ' I CO O t* 
 
 05 O ^ t- 10 
 
 I I I I I 
 
 CO 
 
 co os co 
 
 (M CO CO 
 CO CO CO 
 
 <M co o eo os 
 
 i I CO IO 
 
 I I I I I I I I I I 
 
 O CO 
 
 1 1 
 
 CO CO CO 
 
 1 1 1 
 
 l> O iH IO 00 
 
 1 1 
 
 CO C<1 iH 
 
 1 1 1 
 
 COJ>(M(NOO COCNCOiHO 
 
 OOI>COU7)Tf< i iiOrfr-iiH 
 
 <M<Mcococo CO<MTHTHTH 
 
 Mill I I I I I 
 
 00 O5 
 (M (N 
 
 1 1 
 
 (M CO 
 CO CO 
 
 1 1 
 
 00 CO CO O CO 
 
 10 os o o eo 
 
 1 1 1 1 1 
 
THE BOLOMETER AND ITS ENERGY SPECTRUM 
 
 287 
 
 rH C<l O 00 CO 
 00 rH OS CO t-i 
 
 (M <N <M <N O 
 
 lllll 
 
 OS CO <* C* CO 
 
 1 1 1 1 1 
 
 O CO CO *O (N 
 
 i> t~ co 
 
 00 00 t* ^1 
 
 CO CO l> N 
 
 77 i i 
 
 IO 00 Tfi 
 
 3^3 
 
 I I I 
 
 M CO <N CO <N 
 lllll 
 
 O O O> t iH 00 O 00 CO 00 
 
 77 i i i i 7777 
 
 iO 
 
 1 1 1 1 
 
 C^ CO <N 
 
 rH CO t>- tO tO O 
 
 CO 00 rt^ O3 OS rH 
 
 777 ~ 
 
 iH i I i I T^ O5 
 
 co co co co c5 
 lllll 
 
 \a *& a c* 
 
 00 1-1 -* 5 
 
 lllll lllll 
 
 to oo <M eo 
 
 777 i 
 
 1 1 1 
 
 00 Tfl rH 
 
 O O to rt< to 
 
 I-H tO t- 00 
 
 ro co co co (M 
 lllll 
 
 I I I I + lllll 
 
 to OS OS IO 
 
 |g! 
 
 777 i 
 
 CO to 00 
 
 !> c<i oo os os n< 
 
 C^ rH Tfrl rH O rH 
 
 777 w N 
 
 Os OS OS CO OS 
 
 IO CO rH i-H CO 
 
 c<i co i> eo eo 
 
 O5 Oi CO J-t 
 
 1 1 1 1 1 
 
 CO CO CO 10 * 
 
 I I I I + 
 
 IO tc 
 
 i 7 
 
 CO (N IO 
 
 oo >o TJ< 
 
 III I I I I 
 
 CO OS OS 
 CO 00 l> 
 CO COrH 
 
 I> (M O 
 
 rH O rH 
 
 (N IM' <N' 
 
 OSCOCOlMtO rHlOiOOSCO 
 
 |> IO 00 i-H 00 
 
 %%%%% 
 
 lllll 
 
 <N t> (N T}< IO 
 
 CO CO ^t^ CO CO 
 
 I I I I + 
 
 t- tO 
 tO 00 
 
 O I-H (N 
 (N tO CO 
 
 O5 CO 
 to to 
 
 lllll 
 
 d 
 
 r77 1 
 
 00 rH 1> 
 
 to 06 <M' 
 
 CO COrH 
 
 co to n< 
 
 rH rH rH 
 
 ' <N <N' 
 
 rH OS tO (N IO 
 
 Cjl (N<N <N CO 
 
 CO OO 
 
 O t- 
 
 CO CO 
 -^ CO 
 
 O IT- 
 CO C> 
 
 I I I I + lllll I I I I 
 
 O l> CO 
 
 to oo co 
 
 CO COrH 
 
 to to co co oo 
 
 tO t~- !> OS O 
 C^ (N (N (M CO 
 
 lllll 
 
 co co oo t> t> 
 
 rH N. CO 00 tO 
 
 1 1 1 1 1 
 
 00 CO CO rH IO 
 
 r^^^% Sc^^$H 
 
 lllll I I I I 
 
 (M l> to 
 
 00 (N 00 
 CO (N rH 
 
 I I I 
 
 rH rH 1> 00 
 
 O CO rH 00 CO 
 
 <N C<J (N <M (M 
 lllll 
 
 O^ O^ i~4 TH ^^ 
 CO CO *O CO OO 
 
 11111 77777 
 
 (N rH 00 t^ 
 
 T^l IO C^ t^- 
 
 CO CO CO (N 
 
 rH Tfl ^^ O^ 
 
 59 i i o 10 
 lo5^S 
 
 I I I I 
 
 CO rH 
 
 rH 
 
 00 
 
 OS 00 
 
 CO <N rH 
 
 I I I 
 
 ^ ^ JO rH CO 
 
 (N (N (N (M (N 
 
 lllll 
 
 t^ CO OS 
 
 S S J2 SS 
 
 Q CO CO CO CN 
 
 1 1 1 1 1 
 
 (N rF OS O O 
 
 ^0^ 
 rH rH rH rH Ol 
 
 lllll 
 
 O OS OS ^i 
 
 !i 
 i i i i 
 
 T-H CO 
 
 ^ = 
 
 (2 s 
 
288 
 
 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 the observed spectrum is only the remainder that has escaped 
 from the terrestrial depletion in the lower levels; (3) a mix- 
 ture of these two cases may be more closely related to the facts, 
 but if so, there will be great difficulty in disentangling the ele- 
 ments. Corresponding with these temperatures we may compute 
 the value of the radiation at the earth, on the first supposition, 
 that a depleted radiation escapes from the solar envelope, and 
 we find 3.765 calories for X = 0.45 /<, about 4.000 for A = 0.50 fJ. 
 to 0.60 /*, a lower variable amount from A = 0.70 /* to 1.00 /*, 
 4.751 at * = 1.30 /*, 5.705 at /l = 1.50 /, 6.329 at A = 1.60 p, 
 6.169 at A = 2.00 p. The evidences indicate that the solar 
 intensity of radiation at the earth is from 4.00 to 6.00 calories 
 per square centimeter per minute. 
 
 The isothermal region radiates 2.11 times as much heat as 
 does the convectional region. The isothermal region is separated 
 from the convectional region by a wedge-shaped layer, which 
 
 FIG. 55. The courses of the relative radiation. 
 
 partly reflects the incoming ray and partly absorbs it. The 
 remainder proceeds through the convectional region, suffering 
 depletion by minor absorptions to the cumulus layer, where it 
 is more rapidly absorbed, and the balance reaches the sea level 
 
THE BOLOMETER AND ITS ENERGY SPECTRUM 
 
 289 
 
 or land areas. The outgoing ray experiences a similar complex 
 series of absorptions and reflections. The absorption is accom- 
 panied by transformation of energy into ionization currents in 
 the cirrus layer and in the cumulus layer, the electric streams of 
 the former flowing to the poles cause the auroras, and by induc- 
 tion the aperiodic variations in the magnetic field; the ionization 
 currents of the cumulus region are controlled by the diurnal 
 convection circulation, and induce the diurnal variations of 
 the magnetic field. 
 
 TABLE 67 
 
 dQ 
 
 APPROXIMATE MEAN VALUES OF -~f IN THE RADIATION EQUATION jy- = 
 
 dz at 
 
 k 2 - PER 1000 METERS 
 
 a z 
 
 3 
 
 90 
 
 80 
 
 70 
 
 60 
 
 50 
 
 40 
 
 30 
 
 20 
 
 10 
 
 
 
 19000 
 
 -245 
 
 -250 
 
 -256 
 
 -271 
 
 -270 
 
 -286 
 
 -310 
 
 -325 
 
 -288 
 
 -248 
 
 18 
 
 -245 
 
 -266 
 
 -275 
 
 -275 
 
 -282 
 
 -304 
 
 -354 
 
 -367 
 
 -310 
 
 -241 
 
 17 
 
 -277 
 
 -272 
 
 -286 
 
 -283 
 
 -296 
 
 -302 
 
 -345 
 
 -386 
 
 -308 
 
 -259 
 
 16 
 
 -282 
 
 -289 
 
 -294 
 
 -284 
 
 -309 
 
 -321 
 
 -341 
 
 -357 
 
 -303 
 
 -244 
 
 15000 
 
 -294 
 
 -303 
 
 -309 
 
 -317 
 
 -336 
 
 -314 
 
 -320 
 
 -288 
 
 -254 
 
 -207 
 
 14 
 
 -306 
 
 -321 
 
 -322 
 
 -333 
 
 -327 
 
 -311 
 
 -284 
 
 -196 
 
 -176 
 
 -170 
 
 13 
 
 -322 
 
 -343 
 
 -348 
 
 -361 
 
 -379 
 
 -353 
 
 -233 
 
 -145 
 
 -148 
 
 -142 
 
 12 
 
 -335 
 
 -352 
 
 -384 
 
 -396 
 
 -427 
 
 -361 
 
 -113 
 
 - 87 
 
 -100 
 
 -103 
 
 11 
 
 -352 
 
 -329 
 
 -309 
 
 -200 
 
 -180 
 
 -110 
 
 - 49 
 
 - 40 
 
 - 37 
 
 - 52 
 
 10000 
 
 -262 
 
 -278 
 
 -236 
 
 -182 
 
 -108 
 
 - 25 
 
 - 23 
 
 - 79 
 
 - 59 
 
 - 52 
 
 9 
 
 -167 
 
 -163 
 
 -144 
 
 - 96 
 
 - 58 
 
 - 51 
 
 - 92 
 
 - 85 
 
 - 78 
 
 - 76 
 
 8 
 
 -168 
 
 -165 
 
 -136 
 
 - 78 
 
 - 86 
 
 -151 
 
 -145 
 
 -114 
 
 -113 
 
 -110 
 
 7 
 
 -172 
 
 -152 
 
 -149 
 
 -143 
 
 -121 
 
 -154 
 
 -160 
 
 -149 
 
 -145 
 
 -148 
 
 6 
 
 -192 
 
 -163 
 
 -166 
 
 -146 
 
 -151 
 
 -152 
 
 -156 
 
 -152 
 
 -162 
 
 -178 
 
 5000 
 
 -215 
 
 -180 
 
 -188 
 
 -164 
 
 -163 
 
 -146 
 
 -130 
 
 -157 
 
 -168 
 
 -173 
 
 4 
 
 -226 
 
 -202 
 
 -196 
 
 -134 
 
 -159 
 
 -162 
 
 -131 
 
 -159 
 
 -169 
 
 -164 
 
 3 
 
 -234 
 
 -211 
 
 -216 
 
 -140 
 
 -154 
 
 -170 
 
 -174 
 
 -182 
 
 -171 
 
 -164 
 
 2 
 
 -260 
 
 -239 
 
 -232 
 
 -210 
 
 -189 
 
 -140 
 
 -147 
 
 -126 
 
 -103 
 
 - 78 
 
 1 
 
 -140 
 
 -125 
 
 -126 
 
 -110 
 
 - 92 
 
 - 62 
 
 - 46 
 
 - 35 
 
 - 19 
 
 - 24 
 
 0000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 The mean values of -r* are collected in Table 67 and plotted 
 
 (l Z 
 
 in Fig. 55. The radiation increases downwards to a maximum 
 at the 12,000-meter level, then there is a large increase in the 
 
290 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 heat supply by absorption in the cirrus region, then an increase 
 in radiation to the cumulus level, and a second supply of heat is 
 at hand extending to the surface. 
 
 d 2 
 
 The Values of -y in the Radiation Equation 
 J dz 2 
 
 4. There is another important argument leading to the same 
 conclusion, that the so-called " solar-constant" exceeds 4.00 
 calories, derived from the thermodynamic computations described 
 in Chapter III, the results of the same being more fully published 
 in Bulletin No. 3, Oficina Meteorologica Argentina, 1912. 
 The value of (Qi Q ), the loss of heat by radiation for every 
 1,000 meters difference of level, and for every 10 degrees of 
 latitude from the equator to the pole, was derived from the 
 temperature data of the balloon ascensions. Table 66 contains 
 
 /d 2 O 
 -7-7, which are found in the radiation 
 
 formula, 
 
 r692) *Q__ k2 <?Q 
 
 (692) dt -- k dz2 , 
 
 and from which, having once obtained the coefficient k 2 , the 
 true radiation -~, loss of heat in the unit time, may be computed. 
 
 For our present purpose we need only admit that the rate of 
 radiation is proportional to the second differences A 2 (Qi Q Q ) 
 per thousand meters, to be able to make important inferences. 
 The two cases of low I\ and high 7 2 temperatures in the 
 isothermal region are considered in connection with average 
 conditions in the convectional region C; between the isothermal 
 and the convectional region there is a region of absorption and 
 reflection of the radiation R, wedge-shaped, about 6,000 meters 
 deep over the equator, and thin or vanishing in the arctic zone; 
 below the convectional region there is a second stratum of 
 absorption and reflection, less than 2,000 meters deep, and 
 occupying the region of the specific diurnal convection R 2 . We 
 are not now concerned with the absorption regions RI, R%, but 
 
VALUES IN THE RADIATION EQUATION 291 
 
 only with the radiation regions /i, 7 2 , C. Taking the mean 
 values of A 2 (Qi Q 2 ) in these regions, respectively, and the 
 ratios I\/C and /2/C, together with the means, it results that 
 about 2.11 times as much radiation is passing through the iso- 
 thermal region as passes through the convectional region. This 
 can only signify that of the incoming radiation 52 per cent is 
 reflected before reaching the surface of the earth, and that a 
 nearly equal amount returns to space as albedo. If we 
 again admit that the 1.918 calories determined as the max- 
 imum passes below the cirrus levels to the lower stations, it 
 follows that, 
 
 2.11 X 1.918 = 4.047 calories, 
 
 is the approximate average value of the "solar-constant." This 
 confirmation of the results from the bolometer ordinates also 
 strengthens the conclusion that the solar temperature of emission 
 is about 6,900. Since I . r = 1.918 calories, and this corresponds 
 nearly with the temperature 5,800 in Table 65, the sum of the 
 ordinates for a full radiator at 5,800, summing them from 
 /I = 0.20 // to A = 2.50 //, is about 27.28. Now, this sum is 
 proportional to the area of the energy spectrum between these 
 wave lengths, beyond which limits the energy not included is 
 very small in amount, and since we require to employ 2.11 times 
 this energy, 
 
 2.11 X 27.28 = 57.56, 
 
 it is seen that the sum of the ordinates equal to this amount 
 lies between 6,800 and 6,900 solar temperature. If an allow- 
 ance be made for selective absorptions in the several zones of 
 the earth's atmosphere, as in the cirrus region, it is evident that 
 these general conclusions may be extended to many important 
 special researches. 
 
 We have, therefore, besides the negative argument that the 
 Bouguer formula is indeterminate, so far as fixing the level from 
 which the observed radiation proceeds to the surface, the three 
 positive arguments that the solar constant is 4.00 calories up- 
 ward, and the temperature of the solar photosphere, 6,900 
 
292 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 upward, namely, (l) The maximum 1.918 calories on the 
 10,000-meter plane, from stations at different elevations above 
 the sea level. (2) The form of the bolometer ordinate curves 
 of the energy spectrum for 6,900 solar temperature. (3) The 
 thermodynamic ratio 2.11 of the radiation in the isothermal and 
 convectional regions. Compare in addition Chapter VII. 
 
 The Measures of the lonization of the Atmosphere 
 
 There are two types of instruments for measuring the ioniza- 
 tion of the atmosphere, (1) the Elster and Geitel apparatus for 
 the coefficient of dissipation of electric charges from a body 
 connected with an electroscope, (2) the Ebert ion-counter with 
 fixed capacity and variable charge, and the Gerdien ion-counter 
 with fixed charge and variable capacity. The former measures 
 the coefficient of dissipation A, and the two latter measure the 
 component parts of /, namely, the number of ions per cubic 
 centimeter of air n, and the velocity of motion of the ions which 
 dissipate the charge on the body attached to the electroscope. 
 For each kind of electricity, we have, 
 
 A = (e n u + 
 
 where e = 3.4 X 10~ 10 , the constant charge on the single atom. 
 These subjects have been discussed in many papers, and the 
 results are collected in: Die Atmospharische Electrizitat, H. 
 Mache und E. V. Schweidler, Braunschweig, 1909; Die Luftelek- 
 trizitat, Albert Gockel, Leipzig, 1908; Electricite Atmospherique, 
 Observatoire de 1'Ebre, 1910, par le P. /. Garcia Molld, S.J. 
 
 There is a fundamental discrepancy, amounting to about 
 300 per cent, between the resulting values of A observed and 
 computed by the two methods indicated, and as it is our purpose 
 to come to this problem, we shall as briefly as possible summarize 
 the formulas and the conditions of the instruments leading to 
 these conclusions. The elementary electrostatic formulas can 
 be verified by reference to numerous treatises, of which one of 
 the most convenient is, Elements of the Mathematical Theory of 
 Electricity and Magnetism, by /. /. Thomson, Cambridge, 1895. 
 
IONIZATION OF THE ATMOSPHERE 293 
 
 It is not necessary further to explain the individual formulas 
 which follow by simple transformations from the elementary 
 conditions that have been stated. 
 
 Notation and Elementary Relations 
 
 The sign -f signifies repulsion along the line joining two 
 charges of the same kind Q Q f . 
 
 Q = charge on a point, Qi = charge on unit line, Q 2 = charge 
 on unit area. 
 
 00' 
 
 (693) Force = F = ^^-. F n = the normal, and F t = the 
 
 tangential components. 
 
 (694) Total intensity = I = S F n S = / F n d S = 4 IT Q on 
 
 the closed surface S. 
 
 The induction is zero when the charge is outside S. 
 Special applications to given surfaces for unit lengths. 
 
 Sphere Cylinder 
 
 TotalforceF=F g .47rr 2 = IwQ F = F c 2irr = lirQi 
 
 (695) F.= - F c - 
 
 Infinite Plane 
 
 F p = 47r<2 2 
 
 The difference of potential A V = the work done in moving 
 the unit charge against the electric field of intensity F n through 
 the distant A r. 
 
 (696) Work. W = f*F n dr = F n (r 2 - r,) = (Vi - F 2 ) 
 
 for the unit charge. 
 
 for the charge Q. 
 
294 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 (697) Force. F n = - (F2 ~ Fl) = - ^. Mechanical 
 
 f 2 f\ a^ 
 
 force F = F n Q. 
 
 (698) Work. W = /F n dr = -- f^dr = y = 
 
 (Fi - F 2 ) for the sphere. 
 
 (699) Inner energy. U = J JF = J Q (7i - F 2 ) = 
 
 The following quantities, Q, Z>, 77, <r, w, are synonyms for 
 the same physical process in a dielectric, and differ only by 
 coefficients. 
 
 E = the electric force, c = the specific conductive capacity, 
 
 H = the magnetic force, /* = the magnetic permeability, 
 
 D = cE = the electric displacement; B = /J-H = the magnetic 
 
 induction. 77 = - F n = the polarization = <s the surface den- 
 sity = Q the charge. 
 
 (670) U = W = D E = \cE^=\^=\E(V, - F 2 ). 
 
 L> 
 
 In a medium other than air the specific inductive capacity 
 is K. The number of Faraday lines = n. 
 
 (671) 
 
 
 
 K K 
 
 (672) Ui = | . F n 2 = . n F n = the longitudinal tension. 
 
 K K 
 
 (673) U t % . j . F^ 2 = . w ^ = the transversal tension. 
 
 A very complete collection of electrical and magnetic for- 
 mulas may be found in Bulletin I, U. S. W. B., 1902, "Fclipse 
 Meteorology and Allied Problems." 
 
ELECTROSTATIC RELATIONS PER UNIT LENGTH 
 
 Electrostatic Relations per Unit Length 
 Q = c V = <*S 
 
 295 
 
 Formula 
 
 Sphere 
 
 Cylinder 
 
 Plane 
 
 (674) Charge Q = C V 
 
 =rF 
 
 Ol= 2lol~r 
 
 -if, 
 
 (675) Potential V = Q/C 
 
 (676) Capacity C = Q/V 
 
 (677) Potential 
 coefficient = V/Q 
 
 dV 
 (678) Force F = - 
 a r 
 
 (679) Work U = i F 
 
 C =r 
 
 1 
 
 * = 7 
 *-I| 
 
 F = 2 Oi log e r 
 
 F =4* r. Qj 
 
 ~21og e r 
 = 2 log e r 
 
 I/ = Oi 2 log e r 
 
 I/ = 2 * r Q,* 
 
 I/=i>Q'=*CF' 
 
 
 
 
 Formula 
 
 Two Concentric 
 Spheres 
 
 Two Coaxial 
 Cylinders 
 
 Two Parallel 
 Planes 
 
 (680) Potential Fi - Fj 
 
 Vl _ Vt = Q !ii n 
 
 Vi - Fz = 
 
 F! - Fz = 
 
 
 
 2 Q, i g 11 
 
 4 TT Qz (ri rz) 
 
 Q 
 
 nr* 
 
 1 
 
 1 
 
 (681) Capacity C - Fl _ Fj 
 
 C ' 
 rt-n 
 
 21ojr 
 
 4 TT (ri - rz) 
 
 (682) Charge Q = C (Fi - Fz) 
 
 Q= ^ (VI _ VI} 
 
 1 
 
 1 
 
 Ol " ri 
 
 4 TT (n - rz) 
 
 
 
 2 log e 
 
 (Fi - Fz) 
 
 
 
 (Fi - Fz) 
 
 
 (683) Potential Fi - Fz 
 
 ri - r\ 
 
 
 P = 4 TT (ri - rz) 
 
 coefficient Q 
 
 nrj 
 
 p - 2 log e rj 
 
 (684) Work U = $ Q (Vi -Fz) 
 
 n rz 
 
 U = log,, Q. 2 
 
 U - 2 rr ( n _ rj ) Qj* 
 
 U = i P Q 2 = i C (Fi -^Fz) 2 
 
 Q 
 
 Oi 
 
 02 
 
 (685) Surface density <r = 
 
 
 2 IT (ri - rz) / 
 
 ~ 5 
 
 (686) Force F = 4 TT o- 
 
 r Q 
 
 F = (n^i 
 
 4^Qz 
 
 (rt-r-P) 
 
 "S 
 
 Quantity neutralized in the 
 
 
 
 
 unit time 
 
 
 
 
 (687) j idS =4?rAQ =oQ 
 
 fidS-i*(e*u)Q 
 
 fidS = Mnu) Ql 
 
 JW5 -4- (.)& 
 
 = 4 * [() -t 
 
 
 
 
 + (nu)_jQ 
 
 
 
 
 = 4irA . C F - 
 
 
 
 
 Derivation of Coulomb's Law of Dissipation 
 (688) fidS = 4 TT A . Q = 4 w A . ff S = A. F. 5 = a (X 
 
296 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 (689) fidS = - -jj . The loss of charge with the time dt. 
 
 (690) 4 * * . Q = - ^2 . Combining (689) and (690) in 
 
 Coulomb's Law, 
 
 (691) nx.dt= 
 
 (692) Vi = V Q e~ a(tl ~^\ 
 
 (693) a (/i - t ) = C. log. nat -~ = ^ . C log ^- 
 
 Conduction lonization Currents 
 Number of ions per cu. cm. n + n__ 
 * Charge IP E. S. U. e + e, e = 3.4 X 10 ~ 10 . 
 
 Velocity of ions in cm/sec at - , u + u _ 
 
 (694) Conductivity = the positive current through 1 cm 2 in 
 unit time, A = (e + n + u + ) + (e^n_ w_). 
 
 Example of an Insulated Charged Sphere of Radius r 
 
 (695) Charge. Q = CV = rV. = <?S. 
 
 (696) Surface density, a = - - = -^. S = the surface of 
 
 the sphere. 
 
 (697) Potential. V = = 4 TT r. <r. 
 
 (698) Capacity. C = r = . 
 
 (699) Force. F n = -^ = 4 TT a, acting on the unit 
 
 quantity at distance r. 
 
 * Result of many experiments. Compare Table 95. 
 
THE ELSTER AND GEITEL APPARATUS 297 
 
 (700) Ion current i = A F = [(e n u) + + (e n w)_] 
 
 from the unit area. 
 
 (701) On whole sphere = 4 w r 2 . A F n = 4 TT [(e n u) + + 
 
 This ionization current neutralizes the charge of the body at 
 a certain rate by 4 w (e n u) + Q acting outward, and 4 x (e n u) _ 
 Q acting inward, as to the surface. 
 
 The dissipation coefficient A is the measure of the rate of the 
 neutralizing the charge. 
 
 (702) A + = 4 TT (neu)_ neutralized negative charge. 
 /_ = 4 TT (n eu) + neutralized positive charge. 
 
 (703) Quantity neutralized = XQ = Xr V = 47r Xr* <r. 
 
 Large bodies neutralize faster than small bodies. 
 The neutralization amount is proportional to a. 
 
 The Coefficient of Electrical Dissipation of the Atmosphere X 
 
 The Elster and Geitel apparatus consists of an electroscope, 
 whose capacity by itself is C, and a small dissipating body, 
 generally a copper cylinder 10 cms. long and 5 cms. in diameter, 
 whose capacity by itself is K. The dissipating body charged 
 with the quantity of electricity Q, in the time d t, loses the quan- 
 tity d Q, by the formula for voltage, 
 
 (704) --(C + *). 
 
 Since it is required to know the rate of dissipation from the 
 cylinder alone, where, 
 
 (705) Q = KV, 
 
 we construct the Coulomb law, by division, 
 
 , 7r ^ 1 dQ C + K dV 
 
 (706) 4,* = =--- -= -- -. 
 
298 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 Multiply by d i and integrate for common logarithms, 
 
 (707) ,-<)= ^J^ "^ ' log f. 
 
 This is the formula for dissipation from the body to the 
 free air, provided there is no leakage into the interior of the 
 electroscope. This is expressed, from parallel reasoning, for 
 C and K separately, 
 
 (708) <*(/!-/<>) =-^ lo g^7- 
 Combining these two parts into one formula, 
 
 (709) 4^-.-^.*..(iog;- 
 
 C F ' 
 
 If the instrument is charged with the electroscope alone for 
 Fo' at the time / , then the V\ is to be read at the time fc, gener- 
 ally an interval of 15 minutes for (/i / )- When the insulation 
 is good this correction is small, and the compensation can be 
 made by (Fi) = Vi + A Fi. It is customary to express the 
 constant a in terms of percentage of loss of charge per minute, so 
 that, 
 
 .. , 100 K + C 1 Fo 
 
 (710) a lg - T'M' log (FO- 
 
 Finally, the general formula for A becomes in E. S. U., 
 
 -2- 
 
 4 TT 4 TT X 60 X 100 4 TT X 60 X 100 K 
 
 - log = ( e n+ u++ e n_ u,) 300. 
 
 The Ebert ion counter consists of an electroscope to which a 
 condenser is attached as an integral part, and through this a 
 known volume of air A in cubic centimeters per second is drawn 
 by a revolving turbine. The capacity C of this combination is 
 known. In the axis of the long cylinder of radius r, a small 
 
THE EBERT ION COUNTER 299 
 
 cylinder of radius r z and length / is supported by an insulating 
 rod connecting with the electroscope. If the apparatus runs 
 for a given time (t\ t Q ) while the small cylinder is without 
 charge, there will be the loss in voltage 
 
 (712) C (Fo' - F/), (without charge). 
 
 When the rod is charged the loss in the same interval is, 
 
 (713) C (Fo - FO, (with charge). . 
 
 This change in the potential is caused by the passage of N 
 ions, each with charge e, in the volume of air A per second, so 
 that in volts with the factor 300, we have the equation, con- 
 taining correction for the leakage by (695) , for positive ions, 
 
 (714) + e N + A 300 = C [(F - FO - (F ' - F/)] in cubic 
 
 meters. 
 
 (715) n+=NX l<r 6 = C [(F. - F:) - (V t f - V,')} jj~^ in 
 
 cu. cms. 
 
 The passage of n + positive ions per cu. cm. has neutralized 
 a certain amount of the charge of the opposite sign, so that by 
 charging F , F ' for one sign, the number of ions of the opposite 
 sign is determined. 
 
 Example of the Computation 
 
 A 1 run = 0.185283 cubic meters. C = 9.69. 
 A b (560 seconds) = 0.926417 cubic C X 2941 
 
 meters. 300,4 
 
 (+) (Fo - FO = 9.0. (Fo' - F/) = 0.6. 
 
 For positive charge, w_ = 84 X 102.54 = 861. 
 
 (-) (F - FO = 10.2 (Fo' - F/) = 0.5. 
 
 For negative charge, n+ = 9.7 X 102.54 = 995. 
 
 The Formulas for the Velocity of the Ions 
 
 The velocity of motion of the positive and the negative ions 
 respectively depends upon a special proposition, as follows: 
 
300 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 J E* 
 
 (716) =-- U ' Gdr=-Fudx. 
 
 (717) F=-==- ^ by (678) and (682). 
 
 2 log e 
 
 H *~^fc * 
 
 FIG. 56. The parabolic motion of the ions in a condenser of co-axial cylinders 
 charged to V. u = the velocity of motion in cm/sec. F = the radial 
 component along r. G = the axial component along x. 
 
 (718) T~ = "TT ' ~ 1 > f r the unit charge. Integrate 
 
 ax Lr r\ 
 
 r log e after transposing, 
 
 ^2 
 
 2 ^ 7 _ Vi 
 
 (719) r 2 = -^r . ; x + C. C = r 2 on ion entering 
 
 log e the condenser field at r - 
 
 * 
 
 (720) 
 
 The maximum value of x is at a distance less than / from the 
 origin, where the ion entering at r Q falls finally upon the charged 
 surface of the inner electrode. 
 
 2 log e [Malls inside 11 Condition 
 
 , v r - x - , x ^2 , , that all the 
 
 (721) fr - (fi'-r^) rrr- 1 =falls on end / ionsfallupon 
 
 electrode. 
 
 L < falls outside / 
 
 ri 
 
 (722) ~ '= (r 2 - r 2 2 ) i ', equation of condition. 
 
 velocity apparatus consists of the primary condenser 
 and electroscope, used for counting the number of ions, and in 
 
EXAMPLE OF VELOCITY COMPUTATION 301 
 
 addition an auxiliary condenser in the same axial line, the inner 
 electrode charged to a few volts A V, and the outer connected to 
 earth. 
 
 Let VQ Vi = the loss in voltage with no charge on the 
 
 auxiliary. 
 
 Fo' Vi = the loss when the auxiliary is charged to 
 
 about A V = 30 volts. As some ions fall on the 
 
 auxiliary electrode, it follows that (F - FI) > (F '- F/) 
 
 The amount of electricity drawn down upon the auxiliary 
 
 electrode by its charge of opposite sign is, 
 
 (723) 
 
 We have, also, e n = C (F - 7i) = F ~ V \ Hence, 
 
 (Vo - Fi) - (Fo' - FiQ 
 
 Example of the Velocity Computation 
 A 0.85065 X 10 6 _ cubic centimeters 
 
 /i 1 515 per second 
 
 1 1 1 
 
 47T/.AF" 47TX12X27 4071* 
 ~ 2 log ^ = 2.3026 X 2 X (log 1.50 - log 0.25) = 3.584. 
 
 1652 X 3.584 
 Constant = - - = 1.45. 
 
 Q-CFo'-F/), Fo - V l auxiliary 
 
 T7 -, 7 
 
 Vo V\ not charged. 
 
 Fo' - Vi' auxiliary 
 charged. 
 
302 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 (190.7 - 177.1) - (184.9 - 177.6) 
 
 ' 
 
 (-) Charge. , + - 1.45 
 
 1.45 f = 0.67 cm/sec. 
 
 lo.O 
 
 (192.8 - 179.0) - (183.9 - 176.1) 
 (+) Charge, u. = 1.45 * - 192 .s _ \ 79 . - '- = 
 
 1.45 ' = 0.64 cm/sec. 
 
 lo.o 
 
 The Gerdien apparatus for the number and velocity of ions. 
 This consists of a double condenser, fitted with a turbine for a 
 measured quantity of air to be drawn through the tubes in the 
 unit time. There are two electrometers, / with a variable 
 capacity C", // with a constant capacity C. The outer cylinder 
 has the radius r\ = 4.9 cm., and the length / = 65 cm.; the 
 auxiliary with variable capacity C' has the radius r% = 0.5 cm., 
 and / = 20.1 cm.; the principal with constant capacity has 
 the radius r 2 = 1.0 cm., and / = 35 cm. C = 16.7, C = 20.2, 
 in some instruments. 
 
 The number of ions. The capacities are both at a minimum, 
 and the number of ions falling on both electrodes neutralizes a 
 quantity of electricity of the opposite sign, expressed by the 
 equation (725), where A = the number of cubic centimeters of 
 air that passes in the time (/i - J ) = 10800000 cm. 3 for 80 
 revolutions. 
 
 (725) e n A 300 = C (F ' - F/) + C (V Q - Fi). 
 
 (726) n = [(C (Fo' - F') X C (F - Fi)] 0.908. 
 
 The velocity of the ions per second. The capacity C' is 
 increased to C/, from 16.7 to 124, by raising the capacity 
 cylinder 25 divisions. From (722), 
 
 v ' v ' 2 lo S* 7" 
 
 (727) K^ZL = ( , 2 _ , 2) __- . 
 
 The quantity of electricity on w-ions entering the field 
 through the area TT (r 2 rf) falls on the electrode, so that, 
 
GERDIEN APPARATUS 303 
 
 F' 
 
 (728) TT (>o 2 - rf) = 4 TT /. u. C V = 4 w I. u. 
 
 2 log e 
 
 7-2 
 
 Hence, by substitutions, and for the interval d t, 
 
 V 
 
 (729) en .TT (r 2 - rf) G d t = e n . 4 TT A w 
 
 2 **. 
 
 C/ 7 T// 
 i a V . 
 
 (731) u + = 
 
 en + 
 
 u_ = 
 
 (e n u) - 
 
 Velocity in a 
 
 (732) A + = (en + w_j_) 300. 
 
 en- 
 field 1 volt/cm. E. S. U. 
 
 ?w_ _) 300. A = A , +A_. 
 
 EXAMPLE OF THE GERDIEN COMPUTATION 
 
 COMPARE MOLLA, TORTOSA OBSERVATORY 
 ( ) Negative charge for the positive ions neutralized. 
 
 I. Ci = 16.7 
 
 II. C = 20.2 
 
 H m s 
 
 Index 
 
 L 
 
 B 
 
 S I 
 
 ri log yi 
 
 L R 
 
 S 
 
 V 
 
 to 9 40 18 
 ti 10 6 33 
 
 26 15 
 li-h 1575 
 
 log (ti-io) 3.19728 
 
 logcn_j_3.7254 
 log_j_ 0.0912 
 log 300 2.4771 
 
 60 
 140 
 
 80 
 A 
 
 I 
 3 
 2 
 
 11.2 10.7 21.9 19 
 9.3 9.8 19.1 174 
 
 yi^yi 2 
 
 log. log VVVi 1 
 [Constant] 
 
 >.8 2.30060 
 LO 2.24055 
 
 5.8 0.06005 
 
 8.77851 
 8.23544 
 
 12.0 15.6 27.6 
 9.0 10.5 19.5 
 
 Vo-Vi 
 
 C i (V( C Vl \ 
 
 209.4 
 145.5 
 
 63.9 
 1290.8 
 430.9 
 
 1721.7 
 Factor 0.908 
 
 n_j. 156.3 
 Iogn + 3.19396 
 loge 0.53148 
 log (ti-to) 3.19728 
 
 log. Const, log VoVW 7 . 01395 
 
 log en i (<i-fo) 6 . 92272 
 log + 0.09123 
 tt + 1.234 
 
 = 1.967 X KM 
 
 log\ + 6.29379 X + 
 
 log. en _j_ (h-to) 6 
 
 . 92272 
 
 Read the scales for V ' (auxiliary), and F (principal), while 
 C is at minimum value 16.7. Raise the C to C/ = 124 by 25 
 divisions. Start the turbine and read the hour and index; after 
 an interval for 80 A read the hour and index. Lower the capa- 
 
304 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 city to minimum C' 16.7 and read V\ and V. The formulas 
 become by evaluation : 
 1 
 
 u = 
 
 e n ' ti 
 A = e n u 300. 
 4 TT A = a. 
 
 1 V r 
 
 0.0172 log T /7 (auxiliary). 
 
 Example Continued. 
 
 ( +) Positive charge for the negative ions neutralized. 
 
 16.7 
 
 II. C = 20.2 
 
 H m s 
 
 Index 1 
 
 R 
 
 S 
 
 V 
 
 logy 
 
 L R i 
 
 ? V 
 
 to 10 11 30 
 h 37 5 
 
 65 10 
 145 9 
 
 .0 10.7 20.7 
 .5 9.5 19.0 
 
 188.4 
 173.4 
 
 2.27508 
 2.23905 
 
 12.5 15.8 28 
 9.9 11.7 21 
 
 Vr-Vi 
 
 C(VoVi) 
 C l (Vo l Vi 
 
 Factor 
 w_ 
 log u 
 loge 
 log (fa-fo) 
 
 .3 213.9 
 .6 162.0 
 
 25 35 
 ti-to 1535 
 
 log '(h-to) 318611 
 
 log en 3.60299 
 logtt 0.00300 
 log 300 2.47712 
 
 logv_ 6.08311 
 
 80 Vo^-Vi 
 A 
 log. log. V^/Vi 
 [Constant] 
 
 log Const, log. 
 
 log en_ (ti-to) 
 log u_ 
 u_ 
 
 A_=A 1.211X10-4 A=A + 
 
 i 15.0 
 
 0.03603 
 
 8.55666 
 8.23544 
 
 51.9 
 
 1048.4 
 250.5 
 
 WVi 1 6.79210 
 
 6.78910 
 0.00300 
 1.007 
 
 + A_ =3.178X10- 4 
 
 1298.9 
 0.908 
 1179 
 3.07151 
 0.53148 
 3.18611 
 
 log e n_ (h-to) 6.78910 
 
 The Cause of the Discrepancy in the Values of the Conductivity 
 of the Atmosphere, as Determined by Two Methods 
 
 It will be seen from the collection of results in Table 68 that 
 we have arrived at two very different results for the coefficient 
 of atmospheric conductivity X, as determined by the two methods 
 that have been described. Section I is compiled in part from 
 Mache and Schweidler's volume, and in part from Bigelow's obser- 
 vations of 1905, made during the U. S. Eclipse Expedition to 
 Spain and Algeria, and of 1912 made at Cordoba, Argentina, 
 with the resulting value of /I = 5.575 X 10 ~ 5 E. S. U. Section 
 II is compiled from the same sources, with the result A = 20.49 X 
 10 ~" 5 , so that the ion counters give a value for X which is 3.68 times 
 greater than by the dissipation apparatus. The cause of this 
 
DISCREPANCY AS DETERMINED BY TWO METHODS 
 
 305 
 
 great discrepancy is not far to seek. The dissipation apparatus 
 is generally worked under a cylindrical hood, which converts it 
 into a condenser, through which the air current is not freely 
 passing, with the effect that the free dissipation is transformed 
 into a saturated or stagnant circulation of the ions. It is easy 
 to show that this apparatus with the hood does not respond to 
 the capacity formula upon which the computations are based. 
 We have used cylinders of different dimensions, with the hood 
 
 TABLE 68 
 
 COMPARISON OF THE COMPUTED VALUES OF A FROM THE DATA 
 OBSERVED BY THE Two METHODS 
 
 Section I. Elster and Geitel dissipation apparatus 
 
 Stations 
 
 a l + 
 
 a l _ 
 
 2 
 
 a + 
 
 0_ 
 
 A + 
 
 A_ 
 
 A 
 
 Lugano ... 
 
 1.87 
 4.22 
 3.02 
 3.76 
 2.88 
 1.17 
 1.26 
 0.84 
 0.72 
 0.97 
 1.26 
 0.55 
 3.33 
 
 1.36 
 1.45 
 3.63 
 1.87 
 0.73 
 1.50 
 
 2.59 
 
 1.87 
 4.44 
 3.38 
 5.86 
 4.62 
 1.54 
 1.34 
 1.33 
 1.00 
 1.29 
 1.43 
 0.60 
 3.82 
 
 1.39 
 1.55 
 3.77 
 1.90 
 0.79 
 1.56 
 
 2.53 
 
 1.00 
 1.05 
 1.11 
 1.55 
 1.60 
 1.32 
 1.06 
 1.58 
 1.43 
 1.36 
 1.13 
 1.09 
 1.15 
 
 1.02 
 1.07 
 1.04 
 1.02 
 1.08 
 1.04 
 
 0.98 
 
 3.12 
 7.04 
 5.03 
 6.27 
 4.80 
 1.95 
 2.10 
 1.40 
 1.20 
 1.62 
 2.10 
 0.92 
 5.55 
 
 2.27 
 2.42 
 6.05 
 3.12 
 1.22 
 2.50 
 
 4.32 
 
 3.12X10" 4 
 7.40 
 5.63 
 9.77 
 7.70 
 2.57 
 2.23 
 2.22 
 1.67 
 2.15 
 2.38 
 1.00 
 6.37 
 
 2.32 
 2.58 
 6.28 
 3.17 
 1.32 
 2.60 
 
 4.22 
 
 2.48 
 5.59 
 3.99 
 4.98 
 3.81 
 1.55 
 1.67 
 1.11 
 0.95 
 1.29 
 1.67 
 0.73 
 4.40 
 
 1.80 
 1.92 
 4.80 
 2.48 
 0.97 
 1.99 
 
 3.43 
 
 2.48 
 5.87 
 4.47 
 7.75 
 6.11 
 2.04 
 1.77 
 1.76 
 1.33 
 1.71 
 1.89 
 0.79 
 5.06 
 
 1.84 
 2.05 
 4.98 
 2.52 
 1.05 
 2.06 
 
 3.35 
 
 4.96 XlO- fi 
 11.46 
 8.46 
 12.73 
 9.92 
 3.59 
 3.44 
 2.87 
 1.28 
 3.00 
 3.56 
 1.52 
 9.46 
 
 3.64 
 3.97 
 9.78 
 5.00 
 2.02 
 4.05 
 
 6.78 
 
 Capri 
 
 Tromso 
 
 Spitzbergen, land. . . 
 Spitzbergen, sea .... 
 Juist 
 
 Wolfenbuttel 
 Misdroy 
 
 Ostsee 
 
 Potsdam 
 Kremsmiinster 
 Triest .... 
 
 Karasjok 
 
 Daroca, 1905 
 Porta Cceli 
 Guelma. . 
 
 Bona 
 
 Casar, bulkhead. . . . 
 Ccesar, hatch 
 
 Cordoba 1912. 
 
 
 5.575XKT 8 
 
 and without the hood, leading to the following results: 
 cylinders are as indicated, 
 
 The 
 
306 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 TABLE 68 (Continued) 
 
 Section II. Ebert's and Gerdien's Ion counters. A = (en+u+ + en_u_) 
 
 300 E. S. U. 
 
 Stations 
 
 n + 
 
 n _ 
 
 u + 
 
 u_ 
 
 * + 
 
 A_ 
 
 A 
 
 . dv 
 * = *dh 
 
 Helgoland, dune. . . . 
 Helgoland, oberland. 
 Swinemiinde .... 
 
 382 
 735 
 823 
 1088 
 1147 
 1323 
 1029 
 1264 
 1558 
 1793 
 559 
 1176 
 828 
 1117 
 1000 
 588 
 
 206 
 382 
 647 
 882 
 794 
 1117 
 853 
 382 
 1238 
 823 
 529 
 1205 
 676 
 970 
 735 
 588 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Potsdam 
 
 
 
 
 
 
 
 Atlantic Ocean 
 
 
 
 
 
 
 
 Seewalchen 
 Mattsee 
 
 1.02 
 
 1.25 
 
 13.76 
 
 14.24 
 
 28.00X1Q- 8 
 
 9.33X10- 7 
 
 Santis 
 
 
 
 
 
 
 
 Miinchen 
 
 
 
 
 
 
 
 Bayern Alpental 
 Golf von Lyon 
 
 
 
 
 
 
 
 
 
 
 
 
 
 IMallorca 
 
 0.83 
 
 0.90 
 
 9.96 
 
 11.06 
 
 20.02 
 
 6.67 
 
 Barcelona. . . . 
 
 Karasjok 
 
 
 
 
 
 
 
 Freiburg 
 
 1.00 
 
 1.11 
 
 10.20 
 
 8.32 
 
 18.52 
 
 6.17 
 
 Stiller Ocean 
 
 Gottingen 
 
 1 32 
 
 1 40 
 
 
 
 
 
 Daroca 1905 
 
 999 
 2940 
 1155 
 621 
 1013 
 1275 
 
 1558 
 1733 
 
 819 
 2960 
 972 
 626 
 1166 
 1206 
 
 1542 
 1522 
 
 0.684 
 
 0.771 
 
 6.97 
 
 6.43 
 
 13.40 
 
 4.47 
 
 Guelma. 
 
 Bona 
 
 
 
 
 
 
 
 CcBsar, bulkhead. . . . 
 CcKsar hatch 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Casar, outward 
 Cordoba, Sep. 12 to 
 Oct. 10, 1912 
 Oct. 11-26 
 
 0.701 
 0.715 
 
 0.613 
 0.766 
 
 
 
 
 
 10.77 
 11.83 
 
 10.30 
 10.08 
 
 21.07 
 21.91 
 
 7.02 
 7.30 
 
 
 20.49X1Q- 8 
 
 6.83X10" 7 
 
 The value of A by the Ebert or Gerdien methods is more than three times greater 
 than by the Elster and Geitel method. 
 
 (1). r = 0.25cm. (2). r = 2.50cm. (3). r = 2.50cm. 
 / = 5.00 cm. / = 10.00 cm. / = 20.00 cm. 
 
 K = 0.65. K = 5.46. K Q = 10.91. 
 
 The capacity of the electroscope is 4.36 cm. 
 The capacity coefficients without hood become, 
 K+C K + C K + 
 
 K 
 
 7.71. 
 
 K 
 
 1.80. 
 
 K 
 
 1.40. 
 
DISCREPANCY AS DETERMINED BY TWO METHODS 307 
 
 C C 
 
 K + C 
 
 0-87. 
 
 
 -0.44. 
 
 = 0.29. 
 
 The capacity coefficients with the hood become: 
 
 K = 1.34. K = 7.50. K =15.00. 
 
 K + C 
 K 
 
 C 
 
 = 4.25. 
 = 0.76. 
 
 K + C 
 K 
 
 C 
 K + C 
 
 = 1.58. 
 
 0.37. 
 
 K + C 
 K 
 
 C 
 K + C 
 
 = 1.29. 
 
 = 0.23. 
 
 Two sets of experiments have been carried out, one in 1905 
 and the other in 1912. In the Eclipse expedition the hood was 
 used throughout the observations, but the dissipation bodies 
 (1) and (2) were, used in frequent interchanges. The following 
 table shows the average fall in scale divisions during fifteen- 
 minute intervals, with (2) the cylinder, and (l) the small rod, 
 respectively: 
 
 TABLE 69 
 MEAN Loss IN SCALE DIVISIONS IN FIFTEEN-MINUTE INTERVALS 
 
 Station 
 
 Daroca 
 
 Porta Coeli 
 
 U. S. S. Ctesar 
 Bulkhead 
 
 U. S. S. Casw 
 Hatch 
 
 Cylinder 10x5 cm. 
 Rod 3x0.5 cm. 
 
 No. (+) (-) 
 657 -5.4 -5.4 
 139 -5.3 -5.5 
 
 No. + (-) 
 354 -5.8 -6.3 
 80 -5.1 -5.8 
 
 No. (+) (-) 
 147 -3.4 -3.6 
 32 -3.3 -3.4 
 
 No. (+) (-) 
 126 -6.5 -6.7 
 25 -6.6 -6.6 
 
 The observations at Daroca and Porta Cceli were continued 
 throughout the twenty-four hours. It is seen that the size of the 
 dissipating body under the hood does not effectively control the 
 rate of dissipation, and that the computed capacity coefficient 
 does not properly enter into the computation. 
 
 A series of experiments was made at Cordoba, 1912, with 
 the three dissipation bodies, in part with, and in part without, 
 the hood. The mean values of the loss in scale divisions are 
 proportional to the following data: 
 
308 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 TABLE 70 
 MEAN LOSSES IN SCALE DIVISIONS WITHOUT AND WITH HOOD 
 
 Dissipation Body 
 
 Without Hood 
 
 With Hood 
 
 
 (+) 
 
 (-) 
 
 Mean 
 
 (+) 
 
 (-) 
 
 Mean 
 
 (1) Rod 
 
 1 58 
 
 1.32 
 
 1.45 
 
 2.07 
 
 1.73 
 
 1.90 
 
 (2) Small cylinder . . . 
 
 4.82 
 
 3.98 
 
 4.40 
 
 1.63 
 
 2.07 
 
 1.85 
 
 (3) Large cylinder . . . 
 
 5.60 
 
 5.12 
 
 5.36 
 
 2.20 
 
 1.97 
 
 2.09 
 
 
 
 
 
 1.98 
 
 1.92 
 
 1.95 
 
 It is seen that with the hood the size of the dissipation body 
 is an indifferent matter, but that without the hood the size of 
 the body and the rate of dissipation progress together. 
 
 A special observing shelter at Cordoba was constructed of 
 canvas and netting, so that the dissipation observations could 
 be made without using the hood, and the results for the series 
 
 are recorded in Table 68. Referring to the capacity coefficients 
 
 /(" j C* 
 
 for the three bodies, it is seen that increases as K dimin- 
 ishes, the upper limit for K = being <*> , and for K = <*D being 
 1. We may, therefore, increase the coefficient by decreasing 
 the size of the dissipation body. Since most of the experiments 
 
 of Table 68 were made with the hood and for 
 
 K 
 
 = 1.58 
 
 (on the average), while, without the hood in Cordoba, 
 1.80, it is apparent that the comparatively large value A 
 
 K + C 
 K 
 
 6.78 X 
 
 1CT is due to this fact, at least in part. We can make a cylinder 
 for a proper dissipation body which will give 5.575 X 3.68 = 
 20.49. Its approximate size can be obtained by computation, 
 but its actual and correct size is a matter for experiment. Such 
 experiments were executed at La Quiaca, 1913, with the result 
 that a dissipating cylinder 52 centimeters long by 1 centimeter in 
 diameter will give nearly the same value of A as the ion-counters. 
 Similar results have been obtained in Potsdam. Such a long 
 cylinder is very difficult to charge to a sufficiently high potential 
 V and it is impractical. The Elster and Geitel apparatus with- 
 
DISCREPANCY AS DETERMINED BY TWO METHODS 309 
 
 out hood is valuable for giving relative values of (a+ a_) (A + A_), 
 and not their absolute values. These must be obtained by 
 means of the Ebert or Gerdien ion-counters. 
 
 In Terrestrial Magnetism, Vol. XVIII, No. 4, Vol. XIX, No. 1, 
 No. 2, No. 3, 1914, Mr. W. F. G. Swann discusses the theories of 
 the several ionization apparatus, and indicates that several cor- 
 rections must be applied in order to remove certain errors. 
 These depend upon the following conditions: 
 
 1. The variation 'of the atmospheric potential gradient with 
 the height above the surface to the axis of the horizontal cylinder. 
 The change may amount to 20%. 
 
 2. The change in the shape of the electric stream lines on en- 
 tering the cylinder, due the electrostatic attractions and repul- 
 sions, whereby the number of ions entering may vary as much 
 as 20% to 30%. 
 
 3. The modification of the capacity of the inner condenser, 
 which should contain the stub of the supporting rod outside the 
 electroscope as well as the small inner charged cylinder. This 
 may increase the "computed capacity" into the "measured 
 capacity" by as much as 30% to 40%. 
 
 4. The value of the unit electric charge, as determined by 
 several different lines of experiments, has been taken 3.4 X 10~ 10 , 
 but the electrical constant of Table 95, e = 4.774 X 10~ 10 , 
 seems to require this increase in the computation of the coefficient 
 of conductivity ^. It is evident that the entire subject is in 
 need of further discussion. 
 
 The Atmospheric Electric Potential' 
 
 The distribution of the electricity in the atmosphere consists 
 normally of a heavy positive charge of several thousand volts at 
 about 5,000 meters altitude, which diminishes toward the 
 earth, with an increasing potential fall near the surface, the 
 latter being charged with a negative potential. There are 
 continual fluctuations among these potentials in diurnal, annual, 
 and other periods, or in aperiodic variations as in thunder- 
 storms and minor changes. 
 
310 RADIATION, IONIZATION, AND MAGNETIC VECTORS 
 
 Let V = the potential at any height. 
 
 d V 
 
 (733) Force F = -rr = 4 TT <r, where <r is the surface den- 
 
 sity. (699) 
 
 (734) -T^T = + -jjp = 4 TT p, where p is the volume 
 
 density. 
 
 _&V__ volts volts E. S. U. 
 
 d h ~ ~ meter ~ " 100 cm. " 300 X 100 cm.' 
 
 Surface density for the average potential fall of 100 volts /m. 
 
 100 
 4 TT X 300 X 100 
 
 1 00 
 (735) * = - 7 = - 2.65 X 1<T 4 E. S. U. 
 
 d 2 V 
 The potential gradient changes at about 1/1000 = -T-VT. 
 
 d fl 
 
 (736) ^=-f|-4 1 ;=-1^56(l^)35o = +27X 
 
 10~ 9 E. S. U. 
 
 The total surface charge of the earth is computed from <r 
 with the radius of the earth r = 6.37 X 10 8 cms., by (696), 
 
 (737) Q E = 4 TT r 2 . * = 12.56 X (6.37 X 10 8 ) 2 . (- 2.65 X 10~ 4 ) 
 
 = - 1.35 X 10 15 . 
 
 Since 1 coulomb = 3 X 10 9 E. S. U., 
 Q E = - 4.5 x 10 5 E. S. U. 
 
 7^2 1 c* 2 2 
 
 The electric pressure U t = U t = - - = ^ - 2 TT <r 2 , by 
 
 O 7T O 7T 
 
 (672). 
 
 (738) 2 TT ** = 4.43 X 10~ 7 dynes. 
 
 The vertical electric current for A = 20.5 X 10~ 5 E. S. U. 
 
 - 
 
 E. S. U. by (700). 
 i = 2.05 X 10~ 15 amperes/cm. 2 , since 1 ampere = 
 
 3 X 10~ 9 E. S. U. 
 
 (730) ^.^xHT.-e. 83X10 
 
ATMOSPHERIC ELECTRIC POTENTIAL 
 
 311 
 
 The theory adopted in this work of the electric potential and 
 its gradient observed in the lower atmosphere is that the in- 
 coming radiation ionizes the aqueous vapor in the strata within 
 
 Height 
 Z 
 
 5000 
 
 4000 
 
 3000 
 
 2000 
 
 1000 
 
 000 
 
 Gradient per meter = 
 
 Voltage at different heights 
 
 1602 x 100 1236 x 100 
 
 60 
 
 Volts per meter = -r-^- 
 
 Volts 
 
 FIG. 57. The electric potential fall and the voltage at different heights. 
 
 a few thousand meters of the surface, so that the normal charge 
 is 150,000 volts at 5,000 meters, 100,000 volts at 1,400 meters, 
 and volts at the surface, with negative induction in the earth 
 itself. A study of the radiation data of Table 67, those of Table 
 63 for the distribution of the aqueous vapor, together with those 
 of Fig. 57 for the electric potential, may lead to the function 
 connecting radiation, ionization, aqueous vapor contents, electric 
 currents, and diurnal magnetic deflecting vectors. This subject 
 will require prolonged research in observation and analysis. 
 Table 57 gives the distribution in heights for two cases of the 
 voltage (160200, 123600), and the voltage gradient per 100 
 meters (-8 -5) at 5000 meters, with (-130 -90) at the 
 surface where V = o in both cases. 
 
CHAPTER VI 
 Terrestrial and Solar Relations 
 
 The Five Types of the Diurnal Convection in the Earth's Atmosphere 
 
 THE further analysis of the problems of electric and magnetic 
 variations depends upon the determination of the types of the 
 diurnal convections in the earth's atmosphere. There are five 
 of these types, distinct from one another: (1) In the Arctic 
 zone; (2) in the North Temperate zone, Lat. + 66 to + 30; 
 (3) in the Tropic zone, Lat. + 30 to - 30; (4) in the South 
 Temperate zone, Lat. - 30 to - 66, and (5) in the Antarctic 
 zone. What is needed is a complete determination of the 
 diurnal deflecting wind vectors, for each hour of the day and 
 night, and on several planes from the surface to 3,000 meters, as 
 000, 200, 400, 600, 800, 1,000, 1,500, 2,000, 2,500, 3,000 meters. 
 These can be obtained by kites or captive balloons, but the labor 
 will be not inconsiderable. Unfortunately the available material 
 is very meager, and it is almost wholly lacking during the night. 
 Without the night observations those made during the day, 6 
 A.M. to 6 P.M., are of quite subordinate value. At present we 
 have data made during the day and night only at the Blue Hill 
 Observatory, 1897-1902, analyzed in my papers, " Studies on 
 the Diurnal Periods in the Lower Strata of the Atmosphere," 
 Monthly Weather Review, February to August, 1905; at several 
 Mountain Observatories, J. Hann, K. Ak. Wiss., Wien, Bd. 
 CXI, Abth. Ha, December, 1902, and April, 1903; there are 
 several stations on the surface which can be utilized for pro- 
 visional discussions, Wien, Mauritius, Batavia, Cordoba, Cha- 
 carita, Laurie Island in the South Orkneys. The circulation in 
 the North Temperate zone can be quite accurately constructed, 
 while that in each of the other zones can only be provisionally 
 inferred. The available data have been thoroughly recomputed 
 with the results collected in Table 71, where s = the velocity 
 in meters per second, and & = the azimuth angle from 5 = 
 
 312 
 
FIVE TYPES OF DIURNAL CONVECTION 
 
 313 
 
 | 
 
 M 
 
 E 
 
 H 
 H 
 B 
 
 I 
 
 ex 6 
 3 o 
 O 2 
 
 <N 
 
 I 
 
 O CO 
 
 c/5 
 
 E 
 
 L 
 
 <NIOCOOOC5<MCOOCOOO<NIO 
 OOOOOOOOOOOO 
 
 w- vq w. ,.^ - co ^ ^ CO C^| CO C^ 
 
 OcoT^coeocot^.Oicot-HO5(N 
 
 iOCOCOiOOOGO^COCOCOCX)TtH 
 
 o'o'o'oooooo'ooo 
 
 o 
 
 I^i-i^COtNC^fNCN 
 
 OOOOOO'-ioOOOO 
 
 oooooooooooo 
 oooooooooooo 
 
 OOOOOOr-i^OOOO 
 
 rHi IT IOO*~<'-<OOOO 
 
314 
 
 TERRESTRIAL AND SOLAR RELATIONS 
 
 I H 
 
 u 
 
 PQ 
 
 < 
 I 6 
 
 a s 
 
 H w 
 
 ri. 8 
 
 I>OOOO 
 
 i 1 0} CO 00 CO CO r-(i-( 
 
 rHTtlOOOOlNC'O-tf 
 rH CO ^^ ^O ^O CO C^ ^^ 
 
 CO ^H ^H *"^ C^ CO CO 
 
 OOOOOOOi-HrHOOO 
 
 OOOOO^O 
 iOcOCDcOrJH 
 
 COlMOO^Oi 
 
 OOOOOO'-H'-HOOOO 
 
 ^ 
 <M 
 
 (N <M <N <M r-i 
 
 oooooooooooo 
 
 88 S 8 S 848 g 2 3 fc 8 
 
 CO (M (N C^ (N -r-i 
 
 oooooooooooo 
 
FIVE TYPES OF DIURNAL CONVECTION 
 
 315 
 
 I a 
 
 i\ 
 
 S i 
 
 PQ o 
 
 S 
 
 C/5 
 
 i 
 
 a 
 
 8 s 
 
 M C 
 
 S s 
 
 (/) 
 
 
 2* 
 
 H 
 
 ^CO 
 CO <N 
 
 000000000000 
 
 T~^ O^ CO CO ^H OO 
 
 i (JO^fCOOOs 
 
 COCOCMfNr-t 
 
 0000 
 
 ^D 
 (M 
 
 *O "^ CO CO ^"* 
 
 
 co co 1-1 
 
 Ot-HT-iC<ICO<N'*iOiOC5 
 
 THi IdCOCOlOi IT (OOS 
 THi-Hi li IT ICMCOCOCOC^ 
 
 t > * CO O^ i^ *O C^l ^O CO C^ CO O5 i^ 
 (N<N<MCOCOCOCOCO(N'-<COCO 
 ^Hr-li-lT-lrH(NCOCOCOCO i-l 
 
 
 CO CO CO CO CO 
 
316 
 
 TERRESTRIAL AND SOLAR RELATIONS 
 
 through E = 90, N = 180, W = 270. The vector direction 
 is that toward which the stream is moving. Having the mean 
 observed hourly vectors, D, and the mean 24-hourly resultant 
 R, these deflecting vectors D are such that R + D = 0, con- 
 structed as true vectors of velocity and direction. 
 
 MN. 
 
 10 Noon 2 
 
 10 MN. 2 
 
 From Stations in the Northern Hemisphere 
 
 \ 
 
 \\ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 From Stations in the Southern Hemisphere 
 
 FIGS. 58 and 59. The probable types of the mean diurnal circulation. 
 
 The data of Table 71 were plotted in diagrams and the 
 apparent circulation for the North Temperate zone was con- 
 structed, as in Fig. 58. By analogy, from the surface data of 
 Cordoba and Chacarita, the circulation is constructed for the 
 
DIURNAL VARIATIONS OF ELEMENTS 317 
 
 South Temperate zone, Fig. 59. From Mauritius and Batavia, 
 we have that from the Tropics; and those for the Arctic and 
 the Antarctic zones are made from other data to be mentioned 
 later. In the two Temperate zones the circulation is oppositely 
 directed, in general, with turning-points at 10 A.M. and 8 P.M., 
 with divergence at 2 to 4 P.M., and convergence at 6 to 8 A.M., 
 and 8 to 12 P.M. The air is rising in the afternoon and falling 
 in two streams during the night, as 10 to 12 P.M. and 2 to 8 A.M. 
 This is the circulation which is caused by the diurnal heating of 
 the earth's surface and the lower strata of the atmosphere with 
 rising air during the daytime, and with cooling and descending 
 air during the night. This circulation is limited to 3,000 meters 
 from the surface, and it is not vigorous above 2,000 meters. 
 The local conditions of mountain stations introduce many 
 minor modifications, and such stations are never fully equivalent 
 to ideal free-air conditions. The diagrams of Figs. 58, 59, 
 contain the horizontal component chiefly, but the vertical 
 component can be approximately inferred from the general 
 stream lines. Similarly, we have the horizontal and the vertical 
 circulations in the several zones, as may be seen in Fig. 64 in 
 connection with the magnetic vector systems which depend 
 upon them. The vectors of Fig. 64 are the ends of the stream 
 lines at the surface, as determined by the data of Table 71. 
 These data were actually applied to globe models, and from 
 them the adopted circulation was derived. 
 
 The Diurnal Variations of the Meteorological, Electrical, and 
 Magnetic Elements 
 
 The effect of the diurnal circulation on the several meteor- 
 ological elements is very complex, and especially so in view of 
 the incessant interchanges between the diurnal and the semi- 
 diurnal periods in the lower strata. Table 72 summarizes some 
 examples of this interconversion, which can be profitably studied 
 by transferring the data to suitable diagrams. 
 
 1. The temperature data for the lower strata, 000 to 2,500 
 meters, are taken from Table 24; that for B is from the same 
 
318 
 
 TERRESTRIAL AND SOLAR RELATIONS 
 
 table, by the conversion from P to B. The temperature has a 
 simple diurnal wave at the surface, as heretofore explained, but 
 a semidiurnal wave above 500 meters, diminishing to extinction 
 at about 3,000 meters. These results conform to the Blue Hill 
 direct observations of temperature in the free air. 
 
 TABLE 72 
 
 EXAMPLES OF THE TRANSITION FROM THE SEMIDIURNAL PERIODS TO THE 
 DIURNAL PERIOD OF DIFFERENT ELEMENTS. 
 
 1. Temperature T 
 
 2. Barometric 
 
 Pressure B 
 
 \ jf _4._ 
 
 Cordoba 
 
 TVyT^4- 
 
 Cordoba 
 
 Meters 
 
 000 
 
 500 
 
 1000 
 
 1500 
 
 2500 
 
 Meters 
 
 000 
 
 500' 
 
 1000 
 
 1500 
 
 2500 
 
 A.M. 
 
 292.8 
 
 295.0 
 
 291.0 
 
 287.7 
 
 280.9 
 
 A.M. 
 
 760.35 
 
 717.64 
 
 676.95 
 
 638.18 
 
 565.99 
 
 2 
 
 291.7,295.4291.9 
 
 288.6 
 
 281.0 
 
 2 
 
 760.13717.27676.72 
 
 638.10 
 
 565.98 
 
 4 
 
 291.1295.3,291.6288.0 
 
 281.0 
 
 4 
 
 760.40 717.42 676.85 
 
 638.18 
 
 565.98 
 
 6 
 
 291.0294.9290.6287.5 
 
 280.8 
 
 6 
 
 760.80 
 
 717.84 
 
 677.16 
 
 638.29 
 
 565.99 
 
 8 
 
 293. 2|294. 0290. 0287.1 
 
 280.8 
 
 8 
 
 760.93 
 
 718.22 677.30 
 
 638.33 
 
 566.00 
 
 10 
 
 297.71293.5289.5287.0280.7 
 
 10 
 
 760.95 
 
 718.28677.34 
 
 638.34 
 
 566.01 
 
 P.M. 
 
 300.2295.0290.7287.7,280.8 
 
 OP.M. 
 
 760.30 
 
 717.80677.06 
 
 638.28 
 
 565.99 
 
 2 
 
 301.3298.0292.61288.8 
 
 280.9 
 
 2 
 
 759.08 
 
 717.05676.71 
 
 638.13 
 
 565.97 
 
 4 
 
 300.7 
 
 298.6293.8289.7 
 
 281.2 
 
 4 
 
 758.56 
 
 716.72 676.43 
 
 637.93 
 
 565.95 
 
 6 
 
 299.0297.7 
 
 293.5 
 
 290.0 
 
 281.5 
 
 6 
 
 758.62 
 
 716.49 676.25 
 
 637.79 
 
 565.94 
 
 8 
 
 296.4296.1 
 
 292.2 
 
 288.9 
 
 281.2 
 
 8 
 
 759.70717.15676.60 
 
 638.02 
 
 565.97 
 
 10 
 
 294.3 
 
 294.7 
 
 290.7 
 
 287.1 
 
 280.8 
 
 10 
 
 760.58 
 
 717.91 
 
 677.21 
 
 638.27 
 
 566.00 
 
 3. Vapor Pressure e d 
 
 4. Vapor Pressure ej 
 
 Tower 
 
 Sal ton Sea, Tower No. 1 
 
 Tower 
 
 Salton Sea, Tower No. 4 
 
 Pans 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 Pans 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 Feet 
 
 00 
 
 10 
 
 20 
 
 30 
 
 40 
 
 Feet 
 
 2 
 
 10 
 
 20 
 
 30 
 
 40 
 
 A.M. 
 
 11.7 
 
 11.7 
 
 11.6 
 
 11.8 
 
 12.9 
 
 A.M. 
 
 16.0 
 
 14.3 
 
 14.8 
 
 14.2 
 
 14.3 
 
 2 
 
 10.7 
 
 11.0 
 
 11.0 
 
 11.0 
 
 12.7 
 
 2 
 
 14.8 
 
 13.8 
 
 14.1 
 
 13.3 
 
 13.5 
 
 4 
 
 10.0 
 
 10.2 
 
 10.3 
 
 10.6 
 
 12.5 
 
 4 
 
 13.7 
 
 12.8 
 
 13.2 
 
 12.0 
 
 12.7 
 
 6 
 
 9.8 
 
 10.2 
 
 10.1 
 
 10.4 
 
 12.3 
 
 6 
 
 12.6 
 
 12.7 
 
 12.9 
 
 11.6 
 
 12.5 
 
 8 
 
 12.4 
 
 13.2 
 
 12.2 
 
 12.3 
 
 13.3 
 
 8 
 
 14.6 
 
 14.6 
 
 14.0 
 
 13.7 
 
 14.1 
 
 10 
 
 14.3 
 
 14.7 
 
 13.6 
 
 13.7 
 
 13.9 
 
 10 
 
 16.9 
 
 16.5 
 
 15.8 
 
 15.0 
 
 15.4 
 
 P.M. 
 
 12.1 
 
 12.1 
 
 10.6 
 
 11.2 
 
 12.0 
 
 P.M. 
 
 18.0 
 
 17.7 
 
 16.1 
 
 13.8 
 
 14.0 
 
 2 
 
 9.7 
 
 9.6 
 
 8.2 
 
 8.5 
 
 9.5 
 
 2 
 
 18.6 
 
 17.3 
 
 15.5 
 
 12.0 
 
 12.0 
 
 4 
 
 9.3 
 
 9.3 
 
 8.4 
 
 8.2 
 
 8.9 
 
 4 
 
 18.8 
 
 16.4 
 
 14.6 
 
 11.8 
 
 11.3 
 
 6 
 
 9.9 
 
 9.7 
 
 9.8 
 
 8.7 
 
 8.6 
 
 6 
 
 18.3 
 
 15.1 
 
 14.2 
 
 11.8 
 
 11.2 
 
 8 
 
 11.0 
 
 11.3 
 
 11.0 
 
 10.3 
 
 10.8 
 
 8 
 
 17.8 
 
 15.0 
 
 14.8 
 
 13.3 
 
 13.3 
 
 10 
 
 12.1 
 
 12.3 
 
 12.0 
 
 12.1 
 
 13.3 
 
 10 
 
 17.4 
 
 15.0 
 
 15.8 
 
 15.4 
 
 15.3 
 
 
 
 
 
 
 
 
 
 
 
 
DIURNAL VARIATIONS OF ELEMENTS 
 TABLE 72 (Continued) 
 
 319 
 
 5. Vapor Pressure e 
 
 6. Electric Potential Fall. 
 
 d 
 
 
 
 Blue Hill (Summer) 
 
 
 Kew 
 
 Kremsmiinster 
 
 Greenwich 
 
 
 000 
 
 200 
 
 400 
 
 1000 
 
 
 Sum. 
 
 Wint. 
 
 Sum. 
 
 Wint. 
 
 Sum. 
 
 Wint. 
 
 A.M. 
 
 11.07 
 
 12.15 
 
 8.78 
 
 7.53 
 
 A.M. 
 
 + 3 
 
 - 8 
 
 -12 
 
 -33 
 
 + 6 
 
 - 1 
 
 2 
 
 10.91 
 
 11.40 
 
 8.41 
 
 6.80 
 
 2 
 
 -14 
 
 -33 
 
 -19 
 
 -51 
 
 + 1 
 
 - 9 
 
 4 
 
 10.75 
 
 11.16 
 
 8.08 
 
 6.44 
 
 4 
 
 -20 
 
 -58 
 
 -24 
 
 -54 
 
 - 3 
 
 -13 
 
 6 
 
 10.88 
 
 10.74 
 
 8.15 
 
 6.44 
 
 6 
 
 00 
 
 -36 
 
 + 2 
 
 -23 
 
 + 1 
 
 -13 
 
 8 
 
 11.25 
 
 10.75 
 
 8.45 6.67 
 
 8 
 
 +23 
 
 + 8 
 
 +16 
 
 + 4 
 
 + 1 
 
 - 6 
 
 10 
 
 11.14 
 
 11.11 
 
 8.92 
 
 7.47 
 
 10 
 
 +14 
 
 +34 
 
 +13 
 
 +22 
 
 + 5 
 
 + 5 
 
 P.M. 
 
 10.67 
 
 11.62 
 
 9.48 
 
 8.19 
 
 P.M. 
 
 -10 
 
 +10 
 
 + 7 
 
 +19 
 
 - 3 
 
 + 4 
 
 2 
 
 10.50 
 
 11.53 
 
 10.08 
 
 8.84 
 
 2 
 
 -24 
 
 - 6 
 
 + 4 
 
 +29 
 
 - 9 
 
 + 3 
 
 4 
 
 10.46 
 
 11.12 
 
 10.27 
 
 9.02 
 
 4 
 
 -21 
 
 + 8 
 
 + 5 
 
 +23 
 
 - 8 
 
 + 6 
 
 6 
 
 10.74 
 
 10.85 
 
 9.72 
 
 8.81 
 
 6 
 
 - 2 
 
 +25 
 
 + 4 
 
 +35 
 
 - 3 
 
 + 8 
 
 8 
 
 11.15 
 
 10.79 
 
 9.15 
 
 8.31 
 
 8 
 
 +24 
 
 +29 
 
 +11 
 
 +32 
 
 + 1 
 
 + 7 
 
 10 
 
 11.17 
 
 11.38 
 
 8.87 
 
 7.81 
 
 10 
 
 +27 
 
 +21 
 
 - 7 
 
 + 3 
 
 + 8 
 
 + 7 
 
 2. The pressure B has semidiurnal waves from the surface 
 upward, diminishing to extinction on the 3,000-meter level. 
 The morning crest of maximum is smaller than that of the 
 afternoon. There is not the least evidence that the semi- 
 diurnal pressure waves embrace an oscillation of the entire 
 atmosphere, as Kelvin's theory of the forced oscillations demands, 
 and therefore several discussions and other inferences depending 
 on that theory are really without proper foundations. 
 
 3. The vapor pressure is subject to this interchange of 
 periods. At Salton Sea, Southern California, Tower No. 1 
 was located in the desert, 1,500 feet from the water, and the 
 semidiurnal period is clearly denned at every stage from the 
 surface to 40 feet. Tower No. 4 was located in the sea, at one 
 mile from the shore, and it was observed, as in section 4, that 
 the diurnal vapor pressure e d near the water converts itself 
 into a semidiurnal wave within 40 feet of the water. At Tower 
 No. 1 the diurnal convection could not obtain vapor from the 
 surface sufficient to fill up the diurnal wave, while at the water 
 this deficiency did not exist near the surface, in consequence of 
 the rapid evaporation. 
 
 Section 5, for the vapor pressure in the free air at considerable 
 heights above the surface of Blue Hill, shows a recombination 
 
320 TERRESTRIAL AND SOLAR RELATIONS 
 
 of the semidiurnal waves at the surface into diurnal waves at 
 less than 1,000 meters above the surface. Here the diurnal 
 convection carries the vapor upward to cooler strata and there 
 concentrates it into a single maximum at about 4 P.M. The 
 details of the physical conditions of these periodic interchanges 
 must be left for more minute researches into the prevailing forces 
 that are at work. 
 
 Section 6 gives some examples of the well-known change 
 from the semidiurnal waves of the electric potential fall, prevail- 
 ing generally in the summer where there is vigorous convection 
 in the lower strata, into the diurnal or approximately diurnal 
 wave which is characteristic of the winter months. Here the 
 positive ions in the atmosphere appear to move up and down, 
 relatively to the surface, upward in the convection of the after- 
 noon, thereby diminishing the potential gradient, and downward 
 at 8 A.M., and 8 P.M., thereby increasing the potential gradients. 
 
 Table 73 contains a collection of the coefficient of dissipation 
 a f , in percentage per minute, from observations made throughout 
 the 24 hours at Daroca, Porta Cceli, Guelma, Stations of the 
 U. S. Eclipse Expedition, 1905, also at Bona, and on the U. S. S. 
 Ccesar during the voyage across the Atlantic Ocean. 
 
 Daroca is on the Aragon plateau; Porta Coeli is near Val- 
 encia; Guelma is in the interior of Algeria and Bona is the port. 
 The observations on the U. S. S. Ccesar were made during the 
 voyage from Gibraltar to the United States: (i) on the forward 
 hatch, and (2) under the shelter of a large iron bulkhead. The 
 values of a' + and a' _ at Guelma have been multiplied by the 
 factor ^2, and those at U. S. S. Ccesar , bulkhead, by the factor 2, 
 in order to reduce them to the scale of the other series in 
 taking the general means. These data are plotted in Fig. 60 in 
 percentage per minute a' . 
 
 Fig. 60 shows that the diurnal variation of the dissipation 
 coefficient in percentage per minute has a maximum at about 3 
 P.M., corresponding with the vertical convection, and secondary 
 maxima at 10 P.M., and 2 to 4 A.M., corresponding with the two 
 descending branches. The minor crests occur at 1 A.M., 4 A.M., 
 8 A.M., P.M., 3 P.M., 5 P.M., and 9 P.M. Each of them corresponds 
 
DIURNAL VARIATIONS OF ELEMENTS 
 
 321 
 
 U 
 
 UK 
 
 1 
 
 
 UK 
 
 O 
 
 
 OOOi-HOOOSOOO 
 
 OOOOSO O>OMC500-#001010-<J<CO-<JI 
 10 10 <0 t-OOOOt-t-00<Ot>t-OOC-<0 
 
 T}< 05 <N 
 
 T* 10 l> T}< > 
 
 D tO ^ O O * 00 O O O rj< O 
 C^OOkO^t~D t-t- > -<t-kOOO 
 
 cgoooor-iaseo oooia>Deo 
 
 CJCO^O^^OCO LClO^UdUD 
 
 otO(N<ocoeo 
 o eo ^ eo o TJI 
 
322 
 
 TERRESTRIAL AND SOLAR RELATIONS 
 
 with some feature of the diurnal circulation, as it transports the 
 ions from one level to another in this complex system of local 
 
 Oa.m. 2 
 
 10 Op.m. 2 
 
 10 a.m. 
 
 1.60 
 
 1.50 
 1.40 
 
 -CtA 
 
 1.60 
 1.50 
 1.40 
 
 3 
 
 1 
 
 FIG. 60. Mean diurnal variation of the coefficient of electrical dissipa- 
 tion in percentage per minute. 
 
 *1 
 
 c 
 AT +1.0 
 
 Temperature o.O 
 
 .m.2 4 6 8 10 Op.m. 2 4 6 8 10 \ 
 
 Cordoba 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 __5 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 \ 
 
 
 
 ^ 
 
 ^~ 
 
 \ 
 
 
 
 
 2 
 
 
 
 \ 
 
 
 
 600 meter level -1.0 
 
 m.m. 
 AB -0.50 
 Pressure o.OO 
 
 
 
 \s 
 
 
 
 4 
 
 
 
 
 
 X^_ 
 
 _s/ 
 
 400-1000 meters 
 Cordoba 
 
 
 
 
 " 
 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 ^ 
 
 N 
 
 
 
 
 
 
 
 
 
 ~7 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 + 0.50 
 
 A m.m. 
 A -1.00 
 Vapor pressure 0.00 
 
 ^ 
 
 "X 
 
 
 
 
 / 
 
 
 
 
 
 \ 
 
 
 Surface 
 Salton Sea 
 
 
 
 ^^^ 
 
 
 
 . 
 
 7 
 
 
 
 
 
 
 
 ~ ^ 
 
 
 
 
 
 
 
 
 / 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 \ 
 
 
 
 
 ^ 
 
 -\ 
 
 
 
 
 / 
 
 
 
 
 
 
 +1.00 
 
 V 
 AV -25 
 Electric Potential 0.0 
 
 _^--' 
 
 
 
 ^^ 
 
 
 ^ 
 
 
 
 
 
 \ 
 
 _-^i 
 
 California 
 
 Grenwich 
 Paris 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s^^ 
 
 X 
 
 
 
 
 
 . 
 
 
 
 
 
 / 
 
 
 
 \ 
 
 
 
 / 
 
 
 \ 
 
 
 
 . 
 
 
 
 
 x 
 
 
 
 + 25 
 
 a' 
 
 + 0.10 
 Electric 
 
 
 
 
 "X. 
 
 
 / 
 
 
 
 
 
 V 
 
 / 
 
 Potsdam 
 Potsdam 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 , 
 
 X. 
 
 
 
 
 
 ^~ 
 
 1 
 
 ^^ 
 
 
 
 / 
 
 
 > 
 
 
 
 
 / 
 
 
 
 ^ 
 
 ^^ 
 
 
 Dissipation -0.10 
 
 
 
 
 ^ 
 
 
 ^ ' 
 
 
 
 
 
 
 
 Daroca 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 FIG. 61. Summary of the various semidiurnal waves. 
 
 circulations. These data should be much more extensively 
 studied. 
 
 It is generally found that the coefficient of dissipation varies 
 
DIURNAL VARIATIONS OF MAGNETIC FIELD 323 
 
 as follows: (1) Greater in clear air and less in cloudy, dusty air; 
 (2) Greater with increase of the wind velocity; (3) Greater 
 with increase of the temperature; (4) Greater with the higher 
 vapor pressure. We may finally compare the diurnal curves 
 of the several elements with the temperature waves in the strata 
 400-1,500 meters above the surface. 
 
 The distribution of the evaporation in the soil from the sur- 
 face to 100 c.m. has been carefully worked out at Cordoba, with 
 the result that the evaporation from the water table in the soil 
 takes place in a diurnal curve-, exactly agreeing with that of 
 the vapor pressure. It is, therefore, thought that A e, A F, a' 
 of Fig. 61 should be inverted and referred to subsurface evap- 
 oration of ground water. 
 
 The Diurnal Variations of the Terrestrial Magnetic Field 
 
 There is another effect of the diurnal circulation in the earth's 
 atmosphere, first, in the generation of electric currents, and 
 
 (-) North Pole 
 
 'Day 
 
 Night / \ I Day 
 
 Night 
 
 (+) South Pole 
 FIG. 62. Scheme of the diurnal circulation in zones. 
 
 secondarily, in the induction of the diurnal magnetic deflecting 
 vectors that cause the variations of the normal magnetic field. 
 
324 
 
 TERRESTRIAL AND SOLAR RELATIONS 
 
 In Bulletin No. 21, U. S. W. B., 1898, were published the results 
 of a computation on the observed elements H. D. V. at 30 
 stations. 
 
 TABLE 74 
 THE MEAN MAGNETIC DEFLECTING VECTORS IN FOUR ZONES 
 
 
 Arctic Zone 
 
 North Temperate 
 
 Tropic Zone 
 
 South Temperate 
 
 
 
 Zone 
 
 
 
 Zone 
 
 Hours of the 
 
 
 
 
 
 
 Observations 
 
 Mag. Lat. 
 
 Mag. Lat. 
 
 Mag. Lat. 
 
 Mag. Lat. 
 
 
 78 to 62 
 
 61 to 28 
 
 + 10 to - 
 
 15 
 
 30 to 55 
 
 
 Stations 7 
 
 Stations 13 
 
 Stations 
 
 5 
 
 Stations 5 
 
 
 s a p 
 
 s a p 
 
 5 a 
 
 f 
 
 sa0 
 
 Midnight 
 
 60 -36 345 
 
 15 -30 111 
 
 20 -33 
 
 5 
 
 19 +27 259 
 
 1 A.M. 
 
 63 -44 355 
 
 14 -35 109 
 
 19 -32 
 
 16 
 
 19 +31 250 
 
 2 
 
 69 -43 5 
 
 14 -32 102 
 
 20 -36 
 
 7 
 
 17 +35 251 
 
 3 
 
 74 -44 16 
 
 14 -33 108 
 
 20 -42 
 
 6 
 
 18 +36 243 
 
 4 
 
 75 -42 25 
 
 15 -35 112 
 
 18 -34 
 
 10 
 
 20 +36 226 
 
 5 
 
 77 -42 30 
 
 17 -33 110 
 
 17 -37 
 
 6 
 
 21 +33 223 
 
 6 
 
 78 -40 32 
 
 20 -31 112 
 
 19 -36 
 
 4 
 
 24 +31 222 
 
 7 
 
 76 -40 36 
 
 22-6 107 
 
 21 -37 
 
 339 
 
 26 +24 235 
 
 8 
 
 65 -37 45 
 
 25+3 99 
 
 24 -30 
 
 297 
 
 28 +28 248 
 
 9 
 
 54 -18 68 
 
 26 +24 66 
 
 26 +23 
 
 228 
 
 28 +33 256 
 
 10 
 
 39 +31 117 
 
 27 +37 49 
 
 35 +25 
 
 210 
 
 26 -27 296 
 
 11 
 
 47 +44 195 
 
 25 +38 312 
 
 43 +22 
 
 204 
 
 25 -37 327 
 
 Noon 
 
 56 +43 200 
 
 33 +35 287 
 
 43 +30 
 
 193 
 
 28 -41 47 
 
 1 P.M. 
 
 64 +42 204 
 
 32 +26 277 
 
 40 +31 
 
 163 
 
 32 -36 53 
 
 2 
 
 73 +41 206 
 
 29 +23 268 
 
 34 +27 
 
 156 
 
 30 -35 72 
 
 3 
 
 83 +39 206 
 
 25 -24 263 
 
 17 +27 
 
 121 
 
 30 +30 82 
 
 4 
 
 89 +39 206 
 
 22 -39 259 
 
 16 -31 
 
 40 
 
 28 +30 85 
 
 5 
 
 87 +36 209 
 
 19 -45 260 
 
 16 -25 
 
 18 
 
 24 +39 84 
 
 6 
 
 78 +37 209 
 
 18 -54 234 
 
 19 -22 
 
 12 
 
 20 +40 84 
 
 7 
 
 62 +34 212 
 
 17 -44 183 
 
 21 -30 
 
 3 
 
 17 +45 78 
 
 8 
 
 54 +32 219 
 
 16 -40 183 
 
 23 -30 
 
 2 
 
 17 +46 108 
 
 9 
 
 51 +11 256 
 
 14 -39 103 
 
 23 -30 
 
 358 
 
 17 +47 283 
 
 10 
 
 50 - 6 279 
 
 15 -36 96 
 
 23 -28 
 
 2 
 
 18 +41 269 
 
 11 
 
 51 -37 336 
 
 13 -33 105 
 
 23 -28 
 
 6 
 
 18 +36 264 
 
 s = the vector in units of the 5th decimal (C. G. S.), o.ooooi dyne. 
 
 a = the vertical angle, positive above the horizon 
 
 P = the azimuth angle, from the South through E. N. W. 
 
 (740) s = (dx*+d y *+ </z 2 )*, o- = (dx z + dy 2 ^, tana = ~ tan/? = ~ 
 
 Table 74 contains a condensed summary of these vectors, s, a, /3, 
 in the four principal zones (Fig. 62), that for the Antarctic being 
 omitted for lack of observations. Fig. 63 contains a diagram 
 
DIURNAL VARIATIONS OF MAGNETIC FIELD 
 
 325 
 
 of the vectors, which illustrates the system to some extent, 
 though they can be properly studied and appreciated only by 
 
 Midnight 
 
 6a.m. 
 
 6 a.m 
 
 6p.m. 
 
 Midnight 
 
 Arrows with for angles above the horizon, + a. 
 Arrows with > for angles below the horizon, a. 
 
 FIG. 63. Scheme of the directions of the deflecting forces causing the 
 diurnal variations of the magnetic field in five principal zones. (Figure 
 on page 90, Bulletin No. 21, U. S. W. B., 1898.) 
 
 reference to the original 30-inch globe model. The reader is 
 directed to Bulletin No. 21 for further discussion of these data. 
 
326 TERRESTRIAL AND SOLAR RELATIONS 
 
 This magnetic system has constituted a difficult problem for 
 solution, as it is necessary to have a simple, world-wide cause 
 capable of producing these diverse effects. 
 
 The most prominent fact is the inversion of vectors as be- 
 tween the two hemispheres, and it is easy to show that the 
 diurnal convection is oppositely directed in reference to the 
 normal magnetic field, positive in the southern hemisphere and 
 negative in the northern hemisphere. 
 
 In the Tropic zone the air rises nearly vertically by day and 
 falls by night; in the Temperate zones it flows toward the poles 
 by day, and toward the equator by night, being oppositely 
 directed in each hemisphere relative to the positive direction 
 of the magnetic field; in the Arctic and Antarctic zones the 
 movement is upward by day toward the sun and downward 
 at night. These five zones of circulation are marked off from 
 each other by the high-pressure belts in latitudes + 30 and 30 
 and by the low-pressure belts in latitudes + 66 and 66. 
 The zones of circulation agree with the zones of magnetic vectors 
 as defined in 1892. 
 
 Fig. 64 contains a scheme of the circulation vectors (black), 
 and the magnetic vectors (dotted), as derived from the two 
 sources indicated. There is remarkable agreement so far as 
 the observational data extend, and the corresponding portions 
 of the circulation adopted by natural inference agree with the 
 parts that are known. It is generally true, (l) that the circula- 
 tion vectors and the magnetic vectors are at right angles to each 
 other, and (2) that the turning points in both systems coincide 
 in all parts of the five zones. The conclusion is almost imperative 
 that the circulating vectors, through the generated ions in 
 streams, induce the observed magnetic deflecting vectors. 
 While there is much to be done by observations fully to verify 
 this theory, it is clear that the main features of both the systems 
 are in remarkable conformity to the known facts of the observa- 
 tions. The horizontal and vertical components of the two sets 
 of vectors in Fig. 64 should be united in one set of spacial vectors, 
 in order that this system may be properly comprehended. The 
 evidence is very strong that the magnetic variations depend 
 
DIURNAL VARIATIONS OF MAGNETIC FIELD 327 
 
 Mn 2 4 6 8 10 Noon 2 4 6 8 10 12 
 
 
 *'** 
 
 S' 
 
 *'"" 
 
 f --- 
 
 X 
 
 r "' 
 
 Arcti 
 
 !\ 
 
 ^> x< 
 
 cZon 
 * 
 
 \ 
 
 i 
 
 e 
 i 
 
 \ 
 
 \ 
 
 k 
 
 \ 
 
 V- 
 
 \ 
 
 \ 
 
 
 *''' 
 
 /x Vertical 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 / 
 
 V 1 
 
 / 
 
 / 
 
 X 
 
 X 
 
 
 
 ^* Planes 
 
 \ 
 
 \ 
 
 A 
 
 -A 
 
 N 
 >*\ 
 
 arth2 
 
 --^ 
 
 remp 
 /\ 
 
 eratu 
 
 reZo 
 
 ^---. 
 
 ne 
 
 /*. 
 
 / ^ 
 
 /\ 
 
 ,'\ 
 
 ^ Horizontal 
 
 
 / 
 
 J 
 
 / 
 
 / 
 
 \ 
 
 V 
 
 /N 
 
 /" N 
 
 ^^ 
 
 2 
 
 
 / 
 
 Planes 
 / Vertical 
 
 \ 
 
 \ 
 
 X 
 
 "* 
 
 
 
 
 
 / 
 
 >.' 
 
 
 N 4. 
 
 s * Planes 
 
 
 ^ 
 
 ^ 
 
 .^ 
 
 *^ > 
 
 \ 
 
 Iropi 
 \ 
 
 cZon 
 
 e 
 
 f 
 
 / 
 
 s 
 
 / 
 
 y 
 
 s Horizontal 
 
 * 
 
 / 
 
 \ 
 
 / 
 
 \ 
 I 
 
 1 
 / 
 
 V 
 / ^ 
 
 ^ 
 
 r^ 
 
 
 ' \ 
 
 \ 
 
 \ 
 </ 
 
 \ 
 / 
 
 \ 
 J 
 
 ^ Planes 
 / Vertical 
 
 \i 
 
 \ 
 
 v x 
 
 ^ 
 
 v y 
 
 / 
 
 
 * 
 
 
 '"-*. 
 
 X A 
 
 % A 
 
 NN A Planes 
 
 > x 
 
 \ 
 
 A, 
 
 / \ 
 
 / \ 
 
 s 
 /\ 
 
 outh 
 
 / N 
 
 Temp 
 
 erati 
 / 
 
 ireZc 
 / 
 
 me 
 i 
 
 / \ 
 
 
 * 
 
 p Horizontal 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 x-' 
 
 'r*. 
 
 J 
 
 N A 
 ft 
 
 x ^ 
 
 ^*. 
 
 SZ 
 
 N 
 
 \ 
 
 Planes 
 Vertical 
 
 *.' 
 
 tf 
 
 *-' 
 
 if 
 
 *.' 
 
 
 
 \ 
 
 ^ i 
 
 "7 
 
 ** / 
 
 -A *' 
 
 >-' 
 
 Planes 
 
 
 / 
 
 / 
 
 / 
 
 / 
 
 A 
 
 / 
 
 a tare 
 f 
 
 ticZo 
 
 ne 
 
 '^i 
 
 4 
 
 /\ 
 
 X 
 
 / i 
 
 ,-V 
 
 / 
 
 ^Vertical 
 ^^ Planes 
 
 ^-.^ 
 
 "*** 
 
 *"*x 
 
 **ik 
 
 *-x 
 
 A 
 
 
 
 
 
 
 "^X 
 
 FIG. 64. The probable types of the diurnal wind vectors as generators 
 of the diurnal magnetic deflecting vectors. Full-lined vectors = the elec- 
 trical currents in the streams of the diurnal convection secondary vectors. 
 Dotted vectors = the induced magnetic deflecting vectors as computed from 
 the observations and given in the U. S. Weather Bureau Bulletin No. 21, page 
 87, 1897. 
 
328 
 
 TERRESTRIAL AND SOLAR RELATIONS 
 
 upon ionization in the lower strata, and not upon any system of 
 ionization currents in the upper strata, as has been claimed. 
 Besides the vector directions of the magnetic forces we may 
 approximately obtain the forces by the formula, 
 (741) A# = 4wen + u + , 
 
 where e = 3.4 X 10~ 10 , n+ = the number of positive ions per 
 cubic centimeter, and u+ the velocity of the circulation in centi- 
 meters per second. The available data for the level 400 meters 
 are probably approximately as follows, in the South Temperate 
 zone: 
 
 TABLE 75 
 
 THE DEFLECTING MAGNETIC VECTORS AS COMPUTED AND OBSERVED 
 A H = 4 TT e n u in C. G. S. units. 
 
 
 
 
 A.M 
 
 
 
 
 Formula 
 
 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 47T 
 
 12 56 
 
 
 
 
 
 
 e 
 
 3.4X10~ 10 
 
 
 
 
 
 
 n+ 
 u + 
 AH 
 Observed 
 
 1045 
 35 
 .00016 
 .00019 
 
 1226 
 38 
 .00020 
 .00017 
 
 1374 
 40 
 .00023 
 .00020 
 
 1284 
 36 
 .00028 
 .00024 
 
 1245 
 34 
 .00018 
 .00028 
 
 1223 
 46 
 .00024 
 .00026 
 
 
 
 
 P.M 
 
 . 
 
 
 
 Formula 
 
 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 47T 
 
 
 
 
 
 
 
 e 
 
 
 
 
 
 
 
 n + 
 u + 
 AH 
 Observed 
 
 1195 
 75 
 .00038 
 .00028 
 
 1232 
 70 
 .00037 
 .00030 
 
 1287 
 55 
 .00030 
 .00028 
 
 1225 
 50 
 .00026 
 .00020 
 
 1242 
 45 
 .00024 
 .00017 
 
 1274 
 38 
 .00021 
 .00018 
 
 The number of ions per cubic centimeter was obtained at 
 the surface at Daroca and Guelma, 1905, and from the other 
 available published data; the velocity of the moving medium 
 in cms/sec, was adopted from the study of the Argentine data. 
 The results are so far in harmony with the observed A H = s of 
 Table 74, South Temperate zone, equivalent approximately to 
 
THE APERIODIC MAGNETIC VECTORS 329 
 
 5 of the North Temperate zone, that we must admit that there 
 is a close causal connection. The magnitude and direction of 
 the deflecting magnetic vectors are so far in harmony with the 
 convectional vectors, in all parts of the earth, that the subject 
 will deserve to be further studied, especially in the determina- 
 tion of the wind vectors in the lower strata of the atmosphere. 
 
 The Aperiodic Magnetic Vectors Along the Meridians 
 
 According to Tables 66, 67, and Fig. 55, there are two principal 
 regions of the absorption of the incoming solar radiation, the 
 cirrus region and the cumulus region, in both of which there 
 is transformation of energy into heat or into electric ions. The 
 consequences of such ionization have been studied in the cumulus 
 region, in the induced periodic diurnal or low-level variations 
 of the magnetic field. It remains to give some account of the 
 effects of the ionization in the cirrus region upon the earth's 
 normal magnetic field. In order to analyze this subject the 
 hourly variations are eliminated by taking the mean daily values 
 of H the horizontal force, D the declination, and V the vertical 
 force, as commonly published. As an example of the world- 
 wide correlation of these daily movements of the magnetic field 
 the horizontal force is transcribed in scale divisions, or units of 
 force, for Greenwich, Toronto, Singapore, St. Helena, Cape of 
 Good Hope, Hobarton, very widely separated in latitude and 
 longitude. It is seen that substantially the same sort of varia- 
 tions, + A H, A H, occur nearly simultaneously all over the 
 earth. Similarly, there are + A D, - A D, + A F, - A V 
 variations occurring from day to day. These rectangular 
 variations must first be all transformed to C. G. S. units d x, 
 dy, dz, and these are to be combined in polar co-ordinates s, 
 a, 8, which give the magnetic deflecting vectors disturbing the 
 normal field of the earth. In computing A H, A D, A F from 
 day to day, since there is an incessant secular or long-period 
 variation of H, D, V, it is necessary to secure a proper base 
 line, with appropriate slope, to which A H, AD, A F may be 
 referred. This is best done by constructing the 10-day con- 
 
330 
 
 TERRESTRIAL AND SOLAR RELATIONS 
 
 secutive means, H 0) D , V for every day of the year, so that 
 the consecutive mean plus the variation is the observed value, 
 H + A H = H , D + A D = D, F + A V = V. In passing 
 the dates of excessive magnetic storms, it is proper to substitute 
 a minimum A H min = 0.00025 C. G. S. It has been proposed 
 to obtain the normal field by taking out the " quiet' 7 days for 
 
 .0370 
 Greenwich .0360 
 Abstract, p.7 .0350 
 
 560 
 Toronto 550 
 p. 395 540 
 
 .1050 
 Singapore .1000 
 p. 15 .0950 
 
 60 
 St.Helena 8 
 p. 81 56 
 
 66 
 
 Cape Good Hope 64 
 p. 161 62 
 
 120 
 ."Hobarton 118 
 p. 371 116 
 
 7 
 
 +20 
 +10 
 Direct Type 
 -10 
 -20 
 
 1 
 
 8 
 
 a 
 
 i 
 
 5 
 
 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 i; 
 
 14 
 
 15 
 
 1C 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 2~ 
 
 23 
 
 24 
 
 25 
 
 2fa 
 
 27 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -^^, 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 s~ 
 
 \ 
 
 / 
 
 
 
 s 
 
 
 
 
 
 S 
 
 
 S 
 
 
 
 
 
 \ 
 
 
 /" 
 
 -N 
 
 
 / 
 
 r 
 
 N 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -^ 
 
 
 
 
 
 / 
 
 
 
 
 
 x 
 
 
 
 ^ 
 
 
 ^\ 
 
 
 
 
 / 
 
 \ 
 
 / 
 
 N 
 
 
 
 
 
 
 - 
 
 
 \ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 ^ 
 
 
 
 N 
 
 
 
 
 s~ 
 
 
 
 \ 
 
 
 S 
 
 -^. 
 
 
 / 
 
 ^ 
 
 
 -- 
 
 
 ^ 
 
 
 V. 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *-* 
 
 
 
 *"' 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^s 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 ^^ 
 
 
 
 
 
 
 
 
 
 
 , 
 
 
 
 
 / 
 
 
 
 
 V 
 
 
 
 
 
 / 
 
 
 V 
 
 ^ 
 
 
 
 
 
 ^^ 
 
 
 
 
 / 
 
 
 "* 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 / 
 
 
 
 
 
 
 
 
 t 
 
 J 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 ^ 
 
 
 
 
 
 / 
 
 
 ^. 
 
 -~N 
 
 
 1 
 
 
 
 / 
 
 
 
 ^s 
 
 / 
 
 
 J 
 
 
 ^^ 
 
 
 
 ~^ 
 
 
 
 V 
 
 ^s 
 
 
 
 
 
 
 
 
 1 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 "^ 
 
 ^s 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f 
 
 
 
 
 / 
 
 
 
 \ 
 
 
 
 
 / 
 
 - 
 
 
 
 
 
 A 
 
 
 
 
 
 V 
 
 
 / 
 
 
 ^_s 
 
 
 
 
 
 
 
 
 
 "^ 
 
 
 
 
 
 
 
 V 
 
 s 
 
 
 ^ 
 
 ' 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -, 
 
 
 
 
 '-. 
 
 t 
 
 , 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -.. 
 
 
 
 
 
 
 
 
 
 
 
 .... 
 
 
 
 
 
 -. 
 
 s 
 
 
 '. 
 
 
 
 
 
 
 
 
 
 
 
 -.. 
 
 
 
 
 
 
 \. 
 
 . 
 
 
 
 
 
 
 
 
 
 
 
 
 ! 
 
 /' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 FIG. 65. Example of the variations of the ^-component of the magnetic 
 field in all latitudes, and showing the derived normal type curve in the 
 26.68-day period (direct), beginning August 3.58, 1845. y = 0.00001 C. G. S. 
 unit. 
 
 each month, and computing the means from these selected 
 days. Unfortunately the fact that a day is "quiet" does not 
 guarantee that the day is near the normal, because "quiet" 
 days are as likely to run on one side of the normal as are the 
 rough or moderately disturbed days. The best daily and monthly 
 means are derived by taking all the observations as they occur, 
 
THE APERIODIC MAGNETIC VECTORS 
 
 331 
 
 except that all variations greater than 0.00025 C. G. S. shall be 
 counted at that value for the sake of taking out the consecutive 
 means. There has been great confusion in the instrumental 
 data, in the manner of discussing the variations, and in the 
 interpretation of the results. 
 
 The computed magnetic vectors have been found to possess 
 
 .0380 
 Greenwich .0370 
 Abstract.p.7 .0360 
 
 620 
 Toronto 610 
 p. 401 600 
 
 .1050 
 Singapore .1000 
 p. 18 .0950 
 
 52 
 St.Helena 50 
 p. 87 48 
 
 M 
 Cape Good Hope 52 
 p. 1C7 50 
 
 114 
 
 Hobarton 112 
 p. 377 110 
 
 y 
 
 + 20 
 +10 
 Inverse Type 
 -10 
 -20 
 
 1 
 
 \ 
 
 
 
 
 
 
 
 
 10 
 
 11 
 
 \-z 
 
 la 
 
 H 
 
 ir> 
 
 10 
 
 17 
 
 18 
 
 lt> 
 
 M 
 
 i\ 
 
 i-i 
 
 i\\ 
 
 ,U 
 
 -> 
 
 M 
 
 27 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 S 
 
 f 
 
 -~^ 
 
 N 
 
 
 
 
 ^ 
 
 , . 
 
 / 
 
 \ 
 
 \ 
 
 
 S 
 
 
 
 N 
 
 
 
 
 /* 
 
 ^ 
 
 
 
 
 
 / 
 
 
 
 
 
 \ 
 
 
 J 
 
 
 
 
 
 
 * 
 
 
 
 
 
 s 
 
 -^ 
 
 ' 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s~ 
 
 ~\ 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 .^ 
 
 
 *\ 
 
 
 
 
 
 
 /~ 
 
 v_ 
 
 / 
 
 
 
 V 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 * 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 . , 
 
 
 
 
 ** 
 
 
 \ 
 
 
 ^^ 
 
 
 
 
 "> 
 
 
 
 
 
 
 / 
 
 " 
 
 
 
 ~x 
 
 
 
 s 
 
 *-" 
 
 
 
 
 3 
 
 
 
 
 
 
 
 " ' 
 
 
 ^ 
 
 
 
 
 ^ 
 
 ^ji 
 
 
 
 
 
 
 V, 
 
 ^s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f~ 
 
 ^_j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 /^ 
 
 ^ 
 
 
 
 ^ 
 
 
 V 
 
 
 
 
 ^~ 
 
 -v, 
 
 Vx 
 
 
 
 
 
 
 
 
 
 
 
 
 s~ 
 
 
 
 y 
 
 
 
 ^^ 
 
 ' 
 
 
 
 
 ^ ^ 
 
 /~\ 
 
 _y 
 
 
 \ 
 
 
 
 
 
 
 / 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 jS- 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f 
 
 > 
 
 
 
 
 
 
 
 ^ 
 
 v^, 
 
 
 
 
 
 
 
 
 v_ 
 
 *s 
 
 
 
 
 
 
 
 
 J 
 
 
 \ 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 v_x 
 
 
 
 
 
 ^ 
 
 -^ 
 
 ~s 
 
 
 \ 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 s- 
 
 ^_ 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 ^ 
 
 */ 
 
 
 
 
 
 s^^ 
 
 ~J~ 
 
 
 
 
 
 
 
 / 
 
 
 
 > 
 
 
 
 
 
 
 . / 
 
 
 ^-^ 
 
 
 
 
 / 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 _^ 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 \ 
 
 
 
 ' 
 
 
 
 
 
 
 
 f 
 
 "^ 
 
 
 s~. 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 " ' 
 
 
 
 \ 
 
 ) 
 
 
 
 
 
 
 
 v> 
 
 
 
 
 
 I 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 */ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 \ 
 
 
 
 
 
 
 C 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 I 
 
 
 
 
 
 
 
 
 
 t 
 
 
 
 
 
 ,* 
 
 N 
 
 
 
 
 
 / 
 
 : 
 
 
 \ 
 
 : 
 
 
 X. 
 
 
 
 
 
 
 
 .-' 
 
 
 
 
 
 
 
 
 
 
 
 .' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ; 
 
 
 
 \ 
 
 / 
 
 : 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 FIG. 66. Example of the variations of the /^-component in the 26.68-day 
 period (inverse), beginning November 18.30, 1845. A careful study of these 
 curves, 1843-1905, shows that this type (reversed) occurs semiannually. 
 Direct type, Feb. 1-April 20 and July 15-Oct. 15; inverse, April 20-July 15 
 and Oct. 15-Feb. 1. 
 
 interesting and important characteristics. Bulletin No. 21, 
 U. S. W. B., 1898, contains a full explanation of these data, to 
 which the reader is referred. (1) These vectors of deflection 
 are generally closely confined to the magnetic meridians, and 
 depend chiefly upon A H, A V. According to the latitude of 
 
332 
 
 TERRESTRIAL AND SOLAR RELATIONS 
 
 the station the vector, s, a, /3 has well-defined lengths and vertical 
 angles, similar to those seen in Figs, 68, 69. Corresponding 
 with the waves in Figs. 65 66 these vectors point first south- 
 
 FIG. 67. Generation of the ions in the cirrus and cumulus regions. They 
 flow alternately toward the north and south poles. 
 
 FIG. 68. Closed circuit for southward magnetic vectors. The + ions flow 
 toward the north pole. 
 
 ward, Fig. 68, and then northward, Fig. 69, alternating about 
 every three days. (2) A very extensive study of these vectors for 
 the years 1843-1910 shows that they have a well-defined period 
 of recurrence, on the average 26.68 days in length, and witji two 
 
THE APERIODIC MAGNETIC VECTORS 
 
 333 
 
 types, the direct as in Fig. 65, and the inverse, Fig. 66, the 
 relative intensity from day to day being shown in the lower 
 section of each figure. The recurrences are complicated with 
 many irregularities, but the periodic action is unmistakable 
 and corresponds with the synodic period of the rotation of the 
 sun on its axis, as observed in the equatorial zone. The inference 
 follows that these magnetic meridian deflecting vectors depend 
 upon certain variations in the solar radiation, distributed in 
 
 FIG. 69. Closed circuit for northward magnetic vectors. The + ions flow 
 toward the south pole. 
 
 solar longitudes in such a manner that certain areas of the solar 
 surface emit stronger radiations than do others in different 
 longitudes. The equatorial period, 26.68 days, is exactly the 
 same as the period determined from numerous direct observations 
 on the sun spots, the faculae, and certain spectrum lines. From 
 a least square solution of the magnetic data, an ephemeris was 
 constructed on the period 26.679 days, and epoch, June 13.72, 
 1887. 
 
 (3) The periodic reversal of the type curve occurs in semi- 
 annual periods, as determined by the records, 1841-1894. Take 
 the successive periods by years and match the type curve with the 
 observations as in Figs. 65 and 66. 
 
334 
 
 TERRESTRIAL AND SOLAR RELATIONS 
 
 TABLE 76 
 
 SOLAR MAGNETIC EPHEMERIS, PERIOD 26.679 DAYS, EPOCH 
 JUNE 13.72, 1887 
 
 1840 
 
 Jan. 16.87 
 
 1870 
 
 Jan. 24.05 
 
 1900 
 
 Jan. 5.55 
 
 41 
 
 Jan. 24.38 
 
 71 
 
 Jan. 5.88 
 
 01 
 
 Jan. 14.06 
 
 42 
 
 Jan. 6.21 
 
 72 
 
 Jan. 14.39 
 
 02 
 
 Jan. 22.57 
 
 43 
 
 Jan. 14.72 
 
 73 
 
 Jan. 21.90 
 
 03 
 
 Jan. 4.40 
 
 44 
 
 Jan. 23.23 
 
 74 
 
 Jan. 3.73 
 
 04 
 
 Jan. 12.91 
 
 45 
 
 Jan. 4.06 
 
 75 
 
 Jan. 12.24 
 
 05 
 
 Jan. 20.42 
 
 46 
 
 Jan. 12.57 
 
 76 
 
 Jan. 20.75 
 
 06 
 
 Jan. 2.25 
 
 47 
 
 Jan. 21.08 
 
 77 
 
 Jan. 1.59 
 
 07 
 
 Jan. 10.76 
 
 48 
 
 Jan. 2.91 
 
 78 
 
 Jan. 10.09 
 
 08 
 
 Jan. 19.27 
 
 49 
 
 Jan. 10.42 
 
 79 
 
 Jan. 18.60 
 
 09 
 
 Jan. 26.78 
 
 1850 
 
 Jan. 18.93 
 
 1880 
 
 Jan. 27.11 
 
 1910 
 
 Jan. 8.61 
 
 51 
 
 Jan. 27.44 
 
 81 
 
 Jan. 7.94 
 
 11 
 
 Jan. 17.12 
 
 52 
 
 Jan. 9.27 
 
 82 
 
 Jan. 16.45 
 
 12 
 
 Jan. 25.63 
 
 53 
 
 Jan. 16.78 
 
 83 
 
 Jan. 24.96 
 
 13 
 
 Jan. 6.46 
 
 54 
 
 Jan. 25.29 
 
 84 
 
 Jan. 6.79 
 
 14 
 
 Jan. 14.97 
 
 55 
 
 Jan. 7.12 
 
 85 
 
 Jan. 14.30 
 
 15 
 
 Jan. 23.48 
 
 56 
 
 Jan. 15.63 
 
 86 
 
 Jan. 22.81 
 
 16 
 
 Jan. 5.31 
 
 57 
 
 Jan. 23.14 
 
 Epoch 87 
 
 Jan. 4.64 
 
 17 
 
 Jan. 12.82 
 
 58 
 
 Jan. 4.97 
 
 88 
 
 Jan. 13 . 15 
 
 18 
 
 Jan. 21.33 
 
 59 
 
 Jan. 13.48 
 
 89 
 
 Jan. 20.66 
 
 19 
 
 Jan. 3.17 
 
 1860 
 
 Jan. 21.99 
 
 1890 
 
 Jan. 2.49 
 
 1920 
 
 Jan. 11.67 
 
 61 
 
 Jan. 2.82 
 
 91 
 
 Jan. 11.00 
 
 21 
 
 Jan. 19.18 
 
 62 
 
 Jan. 11.33 
 
 92 
 
 Jan. 19.51 
 
 
 
 63 
 
 Jan. 19.84 
 
 93 
 
 Jan. 27.02 
 
 
 
 64 
 
 Jan. 1.67 
 
 94 
 
 Jan. 8.85 
 
 
 
 65 
 
 Jan. 9.18 
 
 95 
 
 Jan. 17.36 
 
 
 
 66 
 
 Jan. 17.69 
 
 96 
 
 Jan. 25.87 
 
 
 
 67 
 
 Jan. 26.20 
 
 97 
 
 Jan. 6.70 
 
 
 
 68 
 
 Jan. 8.03 
 
 98 
 
 Jan. 15.21 
 
 
 
 69 
 
 Jan. 15.54 
 
 99 
 
 Jan. 23.72 
 
 
 
 THE SEMIANNUAL REVERSAL OF THE DIRECT AND INVERSE TYPES 
 
 Period 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 D 
 
 17 
 
 32 
 
 42 
 
 41 
 
 18 
 
 7 
 
 13 
 
 33 
 
 43 
 
 41 
 
 30 
 
 16 
 
 10 
 
 12 
 
 I 
 
 37 
 
 22 
 
 12 
 
 13 
 
 36 
 
 47. 
 
 41 
 
 21 
 
 11 
 
 13 
 
 24 
 
 38 
 
 44 
 
 42 
 
 Type 
 
 / 
 
 D W. D 
 
 / /max. / 
 
 D >max. D D 
 
 / /max. / 
 
 The direct type prevails annually, February 1 to April 20. 
 The direct type prevails annually, July 15 to October 15. 
 The inverse type prevails annually, April 20 to July 15. 
 
SYNCHRONOUS ANNUAL VARIATIONS 335 
 
 The inverse type prevails annually, October 15 to February 1. 
 
 These facts of periodic action from the sun in the equatorial 
 period of 26.68 days, together with the semiannual inversion 
 of the type, indicate that the problem of the solar radiation at 
 the sun, and in its effects throughout the earth's atmosphere is 
 an exceedingly complex phenomenon, which will require extensive 
 researches of various kinds. 
 
 By way of suggestion it may be seen on Fig. 67 that if the 
 incoming radiation transforms a part of its energy in the cirrus 
 region into positive (+) and negative ( ) ions, it may be 
 supposed that they seek the poles of the earth in opposite direc- 
 tions, as (+) to the north pole and the ( ) to the south pole, 
 completing their circuit through the outer shell of the earth. 
 This generates the magnetic vector system pointing southward, 
 and the corresponding earth electric currents; at another time the 
 (+) ions seek the south pole and the ( ) ions the north pole, 
 thus producing the northward vectors, and the corresponding 
 earth electric currents. This reversal of direction from time 
 to time depends upon the physical condition of the atmosphere 
 as a conducting medium for the ions, its congestion of ions, its 
 accumulation of ice and vapors, producing the magnetic vectors, 
 auroras, magnetic storms, electric currents, in the well-known 
 conditions as observed. The energy expended at the earth 
 is that transformed from the solar radiation; it is inexhaustible 
 in amount, and depends for the observed aperiodic irregularities 
 upon the prevailing states of the solar and terrestrial atmospheres. 
 
 The Synchronous Annual Variations of the Solar and the Terres- 
 trial Elements 
 
 The possibility of a scientific forecast of the type of weather 
 likely to prevail in a large country as the United States or 
 Argentina, whether the coming year is to be rainy and cool, or 
 dry and warm, depends upon the establishment of the following 
 two propositions: (1) The radiation output of the sun is a 
 variable quantity, as 4 or 5 per cent, each side of the mean; 
 (2) The meteorological elements, temperature, barometric and 
 
336 TERRESTRIAL AND SOLAR RELATIONS 
 
 vapor pressures, and the precipitation synchronize with the 
 solar changes in their annual variations. The evidence at 
 present enables us to affirm that both are true, and that the 
 synchronism exists, though in a very complex form, because the 
 prevailing local conditions depend primarily upon the general 
 circulation, and therefore only indirectly upon the solar varia- 
 tions. It is not possible in this place to do more than summarize 
 the general principles that have been established in a research 
 extending over twenty years, and embracing the available solar 
 and terrestrial data. The first task is to procure homogeneous 
 material of the several observed quantities, extending over a 
 long series of years, sun-spot frequencies, solar -prominence 
 frequencies, amplitudes of the terrestrial magnetic field, baro- 
 metric pressures in all parts of the world, temperatures, and 
 vapor pressures in all countries, precipitation in many districts, 
 direct observations of the solar radiation in calories per square 
 centimeter per minute. Unfortunately the difficulties of secur- 
 ing such homogeneous data of any of these elements is greatly 
 complicated by the irregular and inconsistent methods that 
 have been employed by meteorologists. In consequence of the 
 necessity of substituting a few selected hours of observing for 
 the twenty-four hours of each day, it is necessary to reduce the 
 means from selected hours to the mean of twenty-four hours, which 
 involves a long, special research for each country. The selected 
 hours are different in different countries; the series are broken 
 by changes in the selected hours in consequence of some admin- 
 istrative requirement; the corrections change from place to 
 place when the same hour is made the basis of the work, as 
 where the 75th meridian of the United States is made the hour 
 of observing, which involves a range of three hours locally 
 between the Atlantic and the Pacific States; or where the 
 Greenwich noon is the basis of simultaneous world observations, 
 involving variations up to twelve hours in local conditions; the 
 altitudes and locations of the instruments in great cities have 
 been not infrequently changed, and the instrumental equipment 
 and the methods of computing have never been uniform for 
 the long series. It is necessary to overcome these obstacles by 
 
SYNCHRONOUS ANNUAL VARIATIONS 
 
 337 
 
 setting aside a reasonable number of permanent stations for 
 long series of fundamental work in meteorology, just as astrono- 
 mers dedicate certain observatories to fundamental star places 
 upon which the National Ephemerides are based. Cordoba, in 
 
 1875 
 
 1885 
 
 1900 
 
 1905 
 
 
 
 
 
 "" "^ - 
 
 ~~-^ 
 
 
 / ._ _J 
 
 ^ 
 
 
 / S f_ 
 
 ^ 
 
 11.1 year 
 
 
 \ 2 
 
 
 2 *\ Z 
 
 
 's 
 
 ^ . / s-^^s^ 
 
 ^ ^ ^ 
 
 
 "- 4 f- -i - - - 
 
 7 /^ ~~ / ^ 
 
 
 
 
 
 ' \/ y / 
 
 
 
 _/ r- ' ~f~ "^ 
 
 C 7 \^ 52 ^ 
 
 +400 
 
 
 
 + 200 
 
 
 z V <c 
 
 1 
 
 
 !^/> y^ _^ 1 
 
 
 ^ i 'i E 5 " 
 
 r 7 L 
 
 
 Z. ^^ a 1 
 
 
 -0.50 
 
 
 
 -0.25 / 
 
 5 I Z ^v^ \- 
 
 T "~ ~ T ~/ "^" 
 
 Argentine 0.00 
 
 L- -X ~ \- 4 7 
 
 
 Barometric +0.26 / 
 
 
 
 Pressure - ~? 
 
 /. s _ 
 
 
 + 050 -4 
 
 " 
 
 
 +o'.25 7 
 
 n/N /~\ ^_v ^ 
 
 
 
 
 J Ji X 
 
 Centigrade "O* 25 
 
 ^ Z ^ 
 
 
 
 
 
 
 ' 
 
 T 
 
 +100 /I 
 
 f r 'Sv L 
 
 " x \2. \ 'A 
 
 o.i / 
 
 
 H A Z J Z _ 
 
 
 PL ^2 w S 
 
 / 1 / 
 
 
 \ y v / 
 
 / / 
 
 4.Yft u 
 
 y y .~ 
 
 
 + 05 ./ 
 
 L "u 6 3 /x 
 
 ^^ ^ ^ -/^ 
 
 o.o T 
 
 t * 
 
 , / V 
 
 'United States _ 5 
 
 T \ 7 , } 
 
 7 ^ J 
 
 Fahrenheit 
 
 
 S Z 
 
 
 
 * 
 
 
 
 
 + 15.0 
 
 
 
 + 12.0 
 
 T 
 
 
 + 9.0 
 
 " " " 
 
 
 + 6.0 
 
 
 C- V -^ 
 
 + 3.0 
 
 .-^ ^/ V 
 
 T r 
 
 United States o.O 
 
 / "I \_\ /. 
 
 / ... 
 
 Excess Precipitation _ 3 ' 
 
 r r j^ ^. 
 
 r" 
 
 - 6.0 
 
 r _, 
 
 j 7 7 
 
 -9.0 _ 
 
 
 ^ v/ 
 
 -12.0 
 
 
 \/ 
 
 -15.0 
 
 \i_ \T 
 
 
 
 
 
 FIG. 70. Synchronism between the annual variations of the solar and 
 terrestrial phenomena. 
 
 Argentina, is such a first-class meteorological station, because, 
 since 1870, the instruments have had the same natural exposure, 
 and practically the same apparatus has been used continuously, 
 subjected to numerous tests for normality. There is no station 
 
338 
 
 TERRESTRIAL AND SOLAR RELATIONS 
 
 TABLE 77 
 
 THE SYNCHRONISM BETWEEN THE SOLAR AND THE TERRESTRIAL ANNUAL 
 VARIATIONS OF THE METEOROLOGICAL ELEMENTS 
 
 Year 
 
 Sun-spots 
 
 Promi- 
 nences 
 
 Horizont 
 Mag. Am. 
 
 Argentina 
 
 United States 
 
 CM 
 
 R 
 
 CM 
 
 R 
 
 CM 
 
 R 
 
 AB 
 
 AT 
 
 A.e 
 
 Free. 
 
 &B 
 
 AT 
 
 Free. 
 
 
 
 
 
 
 
 
 mm. 
 
 C 
 
 mm. 
 
 mm. 
 
 inch 
 
 F. 
 
 inch 
 
 1872 
 
 720 
 
 +500 
 
 1800 
 
 +845 
 
 2033 
 
 +679 
 
 -0.94 
 
 +0.21 
 
 +0.29 
 
 - 76 
 
 
 
 - 9.60 
 
 73 
 
 645 
 
 +151 
 
 1700 
 
 +353 
 
 2054 
 
 -251 
 
 .58 
 
 + .15 
 
 + .72 
 
 + 32 
 
 
 
 + 9.35 
 
 74 
 
 578 
 
 - 43 
 
 1569 
 
 -154 
 
 1894 
 
 +168 
 
 + .05 
 
 - .58 
 
 - .48 
 
 + 78 
 
 
 
 - 7.46 
 
 75 
 
 364 
 
 -159 
 
 1188 
 
 -332 
 
 1630 
 
 - 61 
 
 + .47 
 
 + .09 
 
 - .31 
 
 - 50 
 
 
 
 - 0.89 
 
 76 
 
 213 
 
 - 77 
 
 865 
 
 + 11 
 
 1681 
 
 -358 
 
 + .04 
 
 + .04 
 
 + .26 
 
 + 94 
 
 
 
 - 0.99 
 
 77 
 
 120 
 
 + 27 
 
 695 
 
 + 47 
 
 1715 
 
 -320 
 
 - .52 
 
 + .72 
 
 + .70 
 
 -104 
 
 
 
 + 2.65 
 
 78 
 
 177 
 
 -136 
 
 791 
 
 -353 
 
 1948 
 
 +110 
 
 + .25 
 
 - .62 
 
 + .20 
 
 +252 
 
 -0.021 
 
 +0.6 
 
 +22.11 
 
 79 
 
 260 
 
 -188 
 
 1015 
 
 -454 
 
 2256 
 
 - 23 
 
 + .33 
 
 - .14 
 
 - .51 
 
 -327 
 
 + .005 
 
 +0.4 
 
 -15.18 
 
 1880 
 
 373 
 
 + 14 
 
 1332 
 
 + 6 
 
 2729 
 
 + 1 
 
 - .26 
 
 - .22 
 
 - .05 
 
 - 69 
 
 + .023 
 
 -1.4 
 
 - 6.15 
 
 81 
 
 518 
 
 +133 
 
 1064 
 
 +394 
 
 2946 
 
 - 80 
 
 - .41 
 
 + .37 
 
 + .46 
 
 +107 
 
 - .010 
 
 +0.4 
 
 + 4.47 
 
 82 
 
 656 
 
 + 60 
 
 2063 
 
 +264 
 
 8144 
 
 +613 
 
 + .22 
 
 - .18 
 
 - .49 
 
 -108 
 
 + .003 
 
 -0.2 
 
 + 0.60 
 
 83 
 
 704 
 
 + 61 
 
 2252 
 
 -456 
 
 3193 
 
 - 47 
 
 + .20 
 
 - .01 
 
 + -17 
 
 + 73 
 
 + .021 
 
 -0.3 
 
 - 5.39 
 
 84 
 
 634 
 
 +127 
 
 2170 
 
 +686 
 
 3204 
 
 + 19 
 
 - .18 
 
 + .37 
 
 + .36 
 
 - 22 
 
 - .024 
 
 -0.9 
 
 +18.36 
 
 85 
 
 523 
 
 +103 
 
 2080 
 
 +204 
 
 2920 
 
 + 53 
 
 + .14 
 
 - .48 
 
 - .19 
 
 +174 
 
 - .011 
 
 +1.6 
 
 - 0.81 
 
 86 
 
 386 
 
 - 81 
 
 2098 
 
 -513 
 
 2779 
 
 +142 
 
 + .33 
 
 - .25 
 
 - .36 
 
 - 87 
 
 + .009 
 
 -0.3 
 
 -14.84 
 
 87 
 
 249 
 
 - 92 
 
 1671 
 
 +206 
 
 2569 
 
 -233 
 
 - .17 
 
 + .35 
 
 - .11 
 
 -204 
 
 + .010 
 
 -0.6 
 
 - 5.76 
 
 88 
 
 140 
 
 - 59 
 
 1352 
 
 +535 
 
 2298 
 
 +144 
 
 - .43 
 
 + .54 
 
 + .48 
 
 + 83 
 
 - .014 
 
 +0.6 
 
 + 7.59 
 
 89 
 
 165 
 
 - 90 
 
 1402 
 
 -678 
 
 2220 
 
 - 48 
 
 + .11 
 
 - .45 
 
 + .15 
 
 +217 
 
 - .009 
 
 +0.6 
 
 - 0.31 
 
 1890 
 
 309 
 
 -224 
 
 1521 
 
 -836 
 
 2473 
 
 -854 
 
 + .50 
 
 - .26 
 
 - .77 
 
 -186 
 
 + .016 
 
 -0.6 
 
 0.44 
 
 91 
 
 496 
 
 - 69 
 
 1592 
 
 +244 
 
 2613 
 
 - 82 
 
 - .57 
 
 + .29 
 
 + .49 
 
 +108 
 
 - .002 
 
 +0.3 
 
 - 0.51 
 
 92 
 
 669 
 
 +207 
 
 1740 
 
 +732 
 
 2795 
 
 +804 
 
 + .04 
 
 - .10 
 
 + .08 
 
 - 31 
 
 - .003 
 
 +0.4 
 
 + 3.69 
 
 93 
 
 805 
 
 +214 
 
 1931 
 
 +310 
 
 2993 
 
 +151 
 
 + .65 
 
 - .42 
 
 -1.03 
 
 -185 
 
 + .003 
 
 -1.0 
 
 + 0.33 
 
 94 
 
 820 
 
 +116 
 
 1806 
 
 -339 
 
 3073 
 
 + 7 
 
 + .05 
 
 - .43 
 
 - .29 
 
 + 60 
 
 + .006 
 
 -0.5 
 
 - 5.31 
 
 95 
 
 708 
 
 + 60 
 
 1534 
 
 +106 
 
 2806 
 
 -195 
 
 - .47 
 
 + .48 
 
 + .15 
 
 + 35 
 
 .000 
 
 0.0 
 
 - 7.48 
 
 96 
 
 568 
 
 - 67 
 
 1230 
 
 - 192650 
 
 +280 
 
 - .08 
 
 + .91 
 
 + .83 
 
 + 21 
 
 - .007 
 
 +0.7 
 
 + 2.15 
 
 97 
 
 410 
 
 - 95 
 
 1037 
 
 + 76 
 
 2441 
 
 -177 
 
 + .52 
 
 - .14 
 
 + .09 
 
 - 17 
 
 + .005 
 
 +0.2 
 
 + 8.38 
 
 98 
 
 279 
 
 + 42 
 
 817 
 
 - 96 
 
 2236 
 
 +130 
 
 + .09 
 
 -1.10 
 
 - .91 
 
 -112 
 
 + .001 
 
 -0.4 
 
 - 5.39 
 
 99 
 
 185 
 
 - 40 
 
 604 
 
 -106 
 
 1925 
 
 +107 
 
 - .45 
 
 + .21 
 
 + .28 
 
 +129 
 
 - .003 
 
 -0.5 
 
 - 2.80 
 
 1930 
 
 135 
 
 - 21 
 
 398 
 
 +145 
 
 1765 
 
 -176 
 
 - .25 
 
 + .59 
 
 + .61 
 
 +169 
 
 - .002 
 
 +1.0 
 
 + 6.63 
 
 01 
 
 129 
 
 - 96 
 
 295 
 
 -150 
 
 1717 
 
 -343 
 
 + .11 
 
 - .04 
 
 - .67 
 
 -211 
 
 + .008 
 
 +0.3 
 
 - 7.98 
 
 02 
 
 201 
 
 -140 
 
 305 
 
 -222 
 
 1853 
 
 -389 
 
 - .44 
 
 + .16 
 
 + .11 
 
 -134 
 
 - .017 
 
 -0.7 
 
 + 2.16 
 
 03 
 
 330 
 
 - 38 
 
 300 
 
 - 96 
 
 1896 
 
 +230 
 
 + .22 
 
 - .32 
 
 + .17 
 
 + 76 
 
 + .014 
 
 -0.5 
 
 - 0.85 
 
 04 
 
 453 
 
 + 51 
 
 400 
 
 +150 
 
 2000 
 
 -213 
 
 + .01 
 
 - .03J+ .06 
 
 +229 
 
 + .008 
 
 +0.7 
 
 -11.53 
 
 05 
 
 550 
 
 +212 
 
 600 
 
 - 81 
 
 2180 
 
 + 14 
 
 - .08 
 
 - 1 .19 + .12 
 
 + 41 
 
 - .009 
 
 0.0 
 
 + 7.81 
 
 06 
 
 650 
 
 4 
 
 
 
 
 
 .17 
 
 -l_ 74' 01 
 
 197 
 
 
 
 +19.11 
 
 07 
 
 
 
 
 
 
 
 _ .08 .02; + .15 
 
 - fi 
 
 
 
 + 0.26 
 
 08 
 
 
 
 
 
 
 + .41 .26 .17 + 26 
 
 
 
 
 09 
 
 
 
 
 
 
 
 + .20 - .01 - .47! -115 
 
 
 
 
 
 1910 
 
 
 
 
 
 
 .16 + .45 .20 66 
 
 
 
 
 11 
 
 
 
 
 
 
 
 - .11|- .46 - .07 +178 
 
 
 
 
 
 
 
 
 
 
 CM = consecutive mean; R = residual; C M + R = Observed. 
 A B = variation of barometric pressure; A T = variation of temperature. 
 A e = variation of the vapor pressure; Prec. = excess of precipitation. 
 
 in the United States that compares with it, because of changes 
 of one kind or another in the instrumental conditions or the 
 hours of observing. The writer spent many years in adjusting 
 the imperfect observations in the United States, and finally 
 
SYNCHRONOUS ANNUAL VARIATIONS 339 
 
 produced a set of series of Pressure, Temperature, Vapor Pres- 
 sure, and Precipitation, that are fairly homogeneous, and form 
 the fundamental basis from which the annual variations may be 
 computed. Similar reductions to homogeneous data are being 
 made in Argentina, and in other countries, and in time it is 
 hoped that world-wide comparable series of reduced observa- 
 tions may be made accessible to the scientific public. 
 
 Table 77 and Fig. 70 contain a series of examples of the results 
 of such a comparison of the solar and terrestrial annual data, 
 enough to give the reader a fair idea of the possibilities of this 
 important subject. The sun-spots are from Wolfer's data, and 
 the consecutive means C. M. added to the residuals R produce the 
 observed annual means O, C. M. + R = O; the solar prominence 
 frequencies are from the data of the Italian observers; the 
 amplitudes of the horizontal magnetic force were compiled from 
 several European observatories: the Argentine Meteorologica 
 can be found in Bulletin No. 1, Oficina Meteorologica Argentina, 
 1911; the United States data may be found in the Barometry 
 Report, U. S. W. B., 1902; Temperatures and Vapor Pressures 
 in Bulletin S, 1909; Temperature and Precipitation Normals, 
 Bulletin R, 1908; Temperature Departures, Bulletin U, 1911; 
 Climatological Summary in 106 sections, Bulletin W, 1912, all 
 prepared under the writer's supervision. The consecutive 
 means represent a long periodic cycle averaging 11.1 years in 
 duration, but very irregular in length, as from 8 years to 14 years 
 between certain maxima; the residuals represent a short periodic 
 cycle averaging 3.75 years, but ranging between 3 years and 5 
 years. The first curves of Fig. 70 represent the 11.1 years and 
 the 3.75 years cycles between 1875 and 1905. Following them 
 are several curves for Argentina and the United States in the 
 short period cycle. The synchronism in the short period is 
 pronounced, in spite of certain irregularities, demonstrating 
 the general fact that the sun has a variable output of radiation 
 which persistently modifies the earth's circulation and climatic 
 conditions. The barometric pressures and temperatures were 
 studied in all parts of the world and the result summarized in 
 Monthly Weather Review, October and November, 1903. 
 
340 TERRESTRIAL AND SOLAR RELATIONS 
 
 Barometric Pressure. The net work of barometric pressures 
 for the world, taken for the annual variations, shows that the 
 stations must be divided into two classes: (1) Those where the 
 synchronism is direct between the pressure and the prominences, 
 as surrounding the Indian Ocean, and those where the synchron- 
 ism is inverse as in North and South America. Under the 
 external impulse from the sun an increase of the annual radiation 
 accelerates the general circulation in such a way that the pressure 
 is simultaneously higher in certain large regions and lower in 
 others. This is due to the fact that the total pressure of the 
 earth's atmosphere is an invariable constant, so that if the 
 pressure in one region is relatively high, that in another region 
 is relatively low at the same time. The wandering cyclones and 
 anticyclones, added to the more permanent centers of high and 
 low pressures, should sum up to the same constant for the 
 world. The oscillation of regional pressures is, therefore, a 
 fundamental fact leading to an extensive study of the pressure 
 conditions in various localities. 
 
 The Temperatures. Similar studies of the annual tempera- 
 tures divide the stations into two groups, (l) Those in the 
 Tropics with direct synchronism, (2) those in the Temperate 
 zones, on the poleward side of the high-pressure belt, with inverse 
 synchronism. There are many places of mixture or disintegrated 
 effects which it is still difficult to classify. An increase of solar 
 radiation increases the vertical convection of the Tropics, with 
 increase of the surface temperature; this is followed by an in- 
 crease of downflow in the Temperate zones, with an extension 
 of the high areas and cooler temperatures. The temperature 
 integral of the entire earth's atmosphere must be nearly a con- 
 stant, or else the earth's rotation period of twenty-four hours 
 would indicate variations of an astronomical value, which have 
 never been detected in the observations. 
 
 Precipitation. The changes in the general and the local cir- 
 culations, depending upon the solar variations, carry with them 
 the rain-bearing currents, as from the oceans to the continents, 
 and thence the annual amounts of the precipitation in the regions 
 concerned. There are great irregularities in these precipitations 
 
SYNCHRONOUS ANNUAL VARIATIONS 341 
 
 from one region to another, from one year to another, and for 
 the same station. The results from Argentina and the United 
 States indicate clearly that the precipitation synchronizes with 
 the solar variations, and that the variations are of large amounts, 
 ranging through 400-500 millimeters in Argentina at the same 
 station, and several inches in the United States. 
 
 Partial Formations. Fig. 70 shows that the annual crests 
 in certain elements occasionally fail to form completely in the 
 3.75-year period, and for this cause irregularities appear in the 
 series of curves. It is easy to see how this may occur in many 
 cases by a sort of self-contradiction in natural causes and effects. 
 Thus, if in a certain region the excess of solar radiation of the 
 Tropics has produced higher temperatures, this has resulted in 
 spreading a rain and cloud sheet over another region at a distance 
 from it, both due to the same cause. This very cloud sheet, 
 however, acts as a screen upon the surface temperatures, so 
 that lower local temperatures are registered at the surface, while 
 they are really higher above the cloud sheet. The rain currents 
 may precipitate so much aqueous vapor on one side of a mountain 
 range that the overflow on the other side is dryer than usual, so 
 as to give opposite effects for the same efficient increase of 
 circulation, excess in one region, and defect in another region. 
 The observations of the solar prominences depend upon the 
 number of clear days per month. Hence, an increase in solar 
 radiation, following an increase in the frequency of the promi- 
 nences, may locally produce a 'cloud sheet, and hence a lower 
 annual count in the number of the prominences. It is quite 
 irrelevant to attempt to discredit the facts of synchronism, by 
 presenting irregularities or inconsistencies in certain localities, 
 unless the trouble is taken to understand the full series of causes 
 and effects between solar action and final local conditions. 
 Since opposite results, inversions of effects, are inevitable in 
 terrestrial meteorology, from the same solar cause, it will be 
 necessary to study carefully the history of each region, before 
 attempting to arrive at any conclusions. 
 
 The magnetic field presents similar synchronous variations, 
 as may be seen by plotting the amplitude curves. This element 
 
342 TERRESTRIAL AND SOLAR RELATIONS 
 
 is very sensitive to many radiation and ionization influences, 
 and it is our purpose to pursue the research into the function 
 connecting these several elements. 
 
 The radiation in calories per square centimeter per second 
 does not yet present annual variations which seem to be reliable. 
 The cause of this result is seen in the section on radiation, and 
 may be verified by studying the divergent annual values on 
 Table 62. 
 
 The possibility of annual forecasts of the weather conditions 
 is being tested in Argentina by projecting forward the normal 
 3.75-year curve from 1911 to 1915. The results for 1911, 1912, 
 and 1913 are entirely successful, the precipitation being quite the 
 same as indicated in Bulletin No. 1, 0. M. A. It is certainly 
 possible to make similar forecasts for the United States as to 
 precipitation in different districts, wherever the sequence of 
 the rainfall in each district is studied in relation to the funda- 
 mental solar 3.75-year period. Compare Abstract No. 3, U. S. 
 W. B., 1909. with the data of Fig. 70. 
 
 The Aqueous Vapor in the Atmosphere 
 
 It is evident from the discussions on radiation, on cloud 
 formation and precipitation, and on evaporation of aqueous vapor 
 from areas of water, as in lakes and oceans, that the presence of 
 aqueous vapor in the atmosphere is of primary significance. 
 We can compute the number of grams of aqueous vapor per cubic 
 meter of air, or per kilogram of air, according to convenience. 
 
 Grams of Aqueous Vapor in 1 Cubic Meter of Saturated Air 
 (742) ^ = 
 
 The full form most used at all elevations above the sea level. 
 
 By substituting the observed vapor pressure e, the tempera- 
 ture 7", the barometric pressure B, at any other point, the 
 corresponding /* can be computed. Extensive tables have been 
 
AQUEOUS VAPOR IN THE ATMOSPHERE 343 
 
 prepared for // where T ranges from 50 C. to +50 C., and 
 B from 800 mm. to 20 mm. 
 
 Grams of Aqueous Vapor in One Kilogram of Saturated Air. 
 
 (743) ft = 0.622 ^ + 0.235 ^ 
 
 e Q is the saturated vapor pressure for temperatures ranging 
 from - 50 C. to + 50 C. 
 
 When the air is not saturated the following formula serves: 
 e Q = the saturated vapor pressure, / = the dry-bulb temperature, 
 /i = the wet-bulb temperature. 
 
 Vapor Pressure in Millimeters when the Air is not Saturated 
 
 (744) e = e Q - 0.00066 B (t - tj (l + g^) . 
 
 Tables applicable to practical work may be found in Bulletin. 
 No. 2, Oficina Meteorologica Argentina, 1912. 
 
 In the free air the aqueous vapor is distributed approximately 
 by Harm's formula, 
 
 h 
 (745) e = e, 10 6517 , 
 
 where h is the height in meters. 
 
 The Laws of the Evaporation of Water from Lakes, Pans, and 
 Soils with Plants 
 
 The subject of the evaporation of water has been very ex- 
 tensively studied, and there is a large literature on the results. 
 These, however, are unsatisfactory as concerns the terms and 
 the coefficients of the proposed formulas. Another research 
 was undertaken by the writer in 1907 for the U. S. Weather 
 Bureau, at Reno, Nevada, where the proper type of formula was 
 determined; it was continued in 1908 at Indio and Mecca, So. 
 California, and at the Salton Sea, 1909, 1910, in co-operation 
 with numerous stations in various parts of the United States, 
 during which the coefficients were approximately computed; the 
 
344 TERRESTRIAL AND SOLAR RELATIONS 
 
 work was continued in 1911, 1912, at Cordoba, Argentina, and 
 extended to include evaporation from soils, and soils with plants 
 of different kinds, and the final coefficients with the necessary 
 working tables for the computations were constructed. The 
 results of this work are summarized in Bulletin No. 2, Argentine 
 Meteorological Office, 1912. Several special pieces of apparatus 
 have been invented: Bigelow's micrometer hook gage for measur- 
 ing the water height, Bigelow's dial gage for measuring the water 
 height in soil tanks, Wilcken's self -registering apparatus for 
 continuous records of every position of the water surface. The 
 principal difficulty in arriving at conclusions has been due to the 
 necessity of using pans for evaporation, in which case the wind 
 in blowing over the pan greatly complicates the action of the 
 evaporation. Pans of different sizes in the same wind evaporate 
 different amounts during the same interval of time, because the 
 wind carries away the evaporated vapor at different rates, 
 according to the size of the pan, and thus produces a varying 
 mixture of dry air and vapor. A large body of water in a wind, 
 and a small pan in a calm, produce the same effect as an evaporat- 
 ing medium, because the vapor is actually the same in density 
 near the water on a lake in a wind, which merely transports 
 it from place to place without really removing it, as in a calm 
 air over a small pan. The result is that lakes evaporate only at 
 about two-thirds the rate from pans near by in moderate winds. 
 In certain places it was found that a small pan evaporates three 
 times as much water as does a lake in the neighborhood. For 
 example, there were three towers built in the Salton Sea, No. 2 
 near the shore, No. 3 about half-mile from No. 2, and No. 4 
 
 TABLE 78 
 EXAMPLES OF THE ANNUAL EVAPORATION AT THE SALTON SEA 
 
 Tower No. 1. Pan (5), 40 feet above the desert, 195 inches. 
 
 Tower No. 1. Pan (1), on the ground of the desert, 165 
 
 Tower No. 2. Pan (5), 40 feet above the water, 138 
 
 Tower No. 2. Pan (1), 2 feet above the water, 109 
 
 Tower No. 4. Pan (5), 40 feet above the water, 140 
 
 Tower No. 4. Pan (1), 2 feet above the water, 106 
 
 The evaporation from the Salton Sea itself, 72 
 
FORMULA OF EVAPORATION 
 
 345 
 
 about one mile from the shore, while No. 1 was 1,500 feet inland 
 from the sea in the desert. These towers carried pans near 
 the surface of the water and at every 10 feet up to 40 feet above 
 the water. The evaporation for a year was as follows at several 
 pans, as summarized in Table 78. 
 
 The evaporation was registered at other stations from pans 
 of different sizes on the ground, and on a stand 10 feet high, of 
 which annual examples follow. 
 
 EVAPORATION AS RECORDED IN SEVERAL PLACES 
 
 Station 
 
 Indio 
 
 Mecca 
 
 Brawley 
 
 Mammoth 
 
 Height 
 Size of pan 
 Evaporation 
 
 Ground 10 feet 
 6 feet 2 feet 
 119 200 
 
 Ground 10 feet 
 6 feet 2 feet 
 108 170 
 
 Ground 10 feet 
 6 feet 2 feet 
 104 164 
 
 Ground 10 feet 
 6 feet 2 feet 
 126 179 
 
 Station 
 
 N. Yakima 
 
 Cincinnati 
 
 Birmingham 
 
 Lake 
 Tahoe 
 
 Lake 
 Kechess 
 
 Height 
 Size of pan 
 Evaporation 
 
 Ground 10 feet 
 4 feet 3 feet 
 68 86 
 
 Water 10 feet 
 4 feet 3 feet 
 46 62 
 
 Water 10 feet 
 4 feet 2 feet 
 51 64 
 
 2 feet 
 4 feet 
 42 
 
 10 feet 
 3 feet 
 33 
 
 The formulas that have been found to be adequate to follow 
 
 the course of evaporation in all climates, that is, in all conditions 
 
 of temperature, vapor pressure, and wind velocity, are as follows : 
 
 Hours of observation for 4-hour intervals (2, 6, 10) A.M. 
 
 (2, 6, 10) P.M. 
 
 t = the temperature of the dry bulb on the whirling psy- 
 
 chrometer as usually employed. 
 ti = the temperature of the wet-bulb thermometer. 
 e d = the computed vapor pressure at the dew point d. 
 S = the temperature of the water surface. 
 e s = the computed vapor pressure at saturation S. 
 
 de 
 
 -rz = the rate of change of the vapor pressure with the tempera- 
 
 d o 
 
 ture change of the water. 
 
 w = the velocity of the wind in kilometers per hour, derived 
 from the successive anemometer readings. 
 
346 TERRESTRIAL AND SOLAR RELATIONS 
 
 Formula of Evaporation from Large Water Surfaces 
 (746) -T = - 0230 ~ ^ + ' 084 ) ; ( Ar S entine 
 
 anemometer) . 
 
 Formula of Evaporation from Pans of Different Areas 
 = 0-0230 F - l 1 + 0.084 w ). 
 
 F (w) = a factor depending on the area of the pan, which 
 varies with the wind velocity up to about 10 kilometers per 
 hour. FI (w) applies to the Dines' system of wind velocities, 
 used by the Argentine Meteorological Office, and a pan of 1.0 
 meter 2 area; F 2 (w) to the same wind system and a 0.5 m 2 area 
 pan; F (w) to the wind system used in the United States, where- 
 in the same wind velocity is recorded higher in the ratio 1.21 
 to 1.00, and a 1.0 m 2 pan; F 4 (w) to the U. S. wind system and a 
 pan 1.17 m 2 area or 4 feet in diameter; F 5 (w) to the U. S. wind 
 system and a pan 0.29 m 2 area, or 2 feet in diameter. There are 
 two wind systems in use: (1) that based upon the Dines' 
 pressure-velocity, and (2) that based upon the whirling machine 
 velocities. Thus the anemometers by Casella, Negretti, and 
 Zambra, U. S. Weather Bureau Freiz, Richard, are approxi- 
 mately in agreement together, but they are about 20 per cent 
 higher than the Dines, Hess of the Oficina Meteorologica 
 Argentina, Munro, and Tschau system of anemometers. Mar- 
 vin's table of corrections to the Robinson anemometer gives 
 about 20 per cent correction to reduce from the indicated to 
 the true wind velocity, Monthly Weather Review, October, 1906, 
 Table 64, so that the first group becomes equivalent to the 
 second group after making this reduction. Unfortunately, it 
 is customary to omit these reductions, so that the published wind 
 velocities of the United States, and other countries using the 
 above-mentioned anemometers, are about 20 per cent too great. 
 It is indispensable in evaporation reductions that the coefficients 
 should be adjusted to correct wind velocities. For this purpose 
 the following factors F (w) are introduced into the working 
 Tables: 
 
FORMULA OF EVAPORATION 
 
 347 
 
 TABLE 79 
 
 THE FACTORS F (w) FOR ADJUSTING THE EFFECTS OF THE WIND VELOCITIES 
 FOR PANS OF DIFFERENT AREAS 
 
 System 
 
 Argentina 
 
 United States 
 
 Wind 
 
 1.0m 2 
 
 0.5 m 2 
 
 1.0 m2 
 
 1.17 m 2 
 
 0.29 m 2 
 
 Velocity 
 
 Mw) 
 
 F(w) 
 
 F,(w) 
 
 F 4 (w) 
 
 F t 
 
 w = 
 
 1.000 
 
 1.000 
 
 .000 
 
 1.000 
 
 1.000 
 
 1 
 
 1.150 
 
 1.148 
 
 .120 
 
 1.150 
 
 1.160 
 
 2 
 
 1.265 
 
 1.274 
 
 .240 
 
 1.212 
 
 1.290 
 
 3 
 
 .376 
 
 1.392 
 
 .320 
 
 1.289 
 
 1.410 
 
 4 
 
 .463 
 
 1.493 
 
 .400 
 
 1.367 
 
 1.520 
 
 5 
 
 .542 
 
 1.592 
 
 .480 
 
 1.433 
 
 1.630 
 
 6 
 
 .600 
 
 1.667 
 
 .540 
 
 1.480 
 
 1.710 
 
 7 
 
 .617 
 
 1.712 
 
 .590 
 
 1.523 
 
 1.760 
 
 8 
 
 .627 
 
 1.746 
 
 .615 
 
 1.542 
 
 1.810 
 
 9 
 
 .629 
 
 1.762 
 
 .623 
 
 1.552 
 
 1.830 
 
 10 
 
 1.629 
 
 .777 
 
 1.629 
 
 1.561 
 
 1.840 
 
 15 
 
 1.629 
 
 .782 
 
 1.629 
 
 1.561 
 
 1.850 
 
 2Q 
 
 1.629 
 
 .782 
 
 1.629 
 
 1.561 
 
 1.850 
 
 25 
 
 1.629 
 
 .782 
 
 1.629 
 
 1.561 
 
 1.850 
 
 30 
 
 1.629 
 
 .782 
 
 1.629 
 
 1.561 
 
 1.850 
 
 The complete tables for evaporation computations may be 
 found in Bulletin No. 2, Argentine Meteorological Office, 1912. 
 
 It has been found that about 90 per cent of the computed 
 results are less than 0.30 mm. from those as observed. This 
 difference includes the errors of measurement as well as of 
 computation. The computed difference from the observed 
 amounts for entire months in Cordoba is about 4 millimeters, 
 and the total difference for the year on one pan was 3 milli- 
 meters, and on another pan 10 millimeters, the total in the 
 first case being 1,091 millimeters, and in the second case 1,945 
 millimeters. Our experience leads us to conclude that pans 
 need not be employed in work on evaporation, but that compu- 
 tations are quite as accurate provided observations of the water 
 temperature S, the vapor pressure of the air e dj and the velocity 
 of the wind w are made. As it is impossible to float pans on 
 large bodies of water, lakes and reservoirs, except under restricted 
 
348 TERRESTRIAL AND SOLAR RELATIONS 
 
 conditions that injure the observations, it becomes necessary to 
 dispense with pans entirely and depend upon simple computations. 
 There are several methods of abbreviation for computing the 
 mean monthly amounts of the evaporation from lakes and 
 reservoirs, which make the computations an insignificant labor. 
 Studies on evaporation from soil, sand, soil planted with 
 alfalfa, wheat, barley, beans, have been carried on, which 
 analyze successfully the amount of water lost under all conditions 
 throughout the year, from soils by themselves, and from the 
 plants by themselves. Thus, the transpiration of plants is 
 subject to accurate measurement and analysis, and the results, 
 when sufficiently verified, will be of great value to meteorological 
 agriculturalists and botanists. 
 
 The Polarization of Sunlight in the Atmosphere 
 
 Common sunlight vibrates indifferently in every plane 
 perpendicular to its wave front, but when it falls upon any 
 object, large relatively to its wave length, a portion of the light 
 is refracted and a portion reflected in the plane of incidence 
 containing the incident and reflected rays. The vibrations in 
 the reflected ray become at least partially constrained to vibrate 
 parallel to the surface of reflection, and it is plane polarized. 
 The plane of the polarization is at right angles to the plane of 
 vibration, and therefore contains the incident and the reflected 
 rays. If an observer looks at any point in the sky he will receive 
 certain reflected rays that have proceeded from the sun to the 
 reflecting particle and to the eye, this plane being the plane of 
 polarization, and the vibrations are at right angles to it. If the 
 polarization is partial, and the motion circular, elliptical, or of 
 any other figure, components of plane polarized light may be 
 constructed for this plane and another at right angles to it, so 
 that partially polarized plane vibrations in two directions at 
 right angles may more or less neutralize each other between 
 the limits per cent and 100 per cent. A turbid medium, such 
 as air mixed with small solid particles of dust, ice, or even mole- 
 cules, whose diameters are small relative to the wave lengths of 
 light, scatters and polarizes light by Rayleigh's Laws, in which 
 
POLARIZATION OF SUNLIGHT 349 
 
 |8 = the angle of departure from the line of incidence for the 
 reflected ray. 
 
 (748) Intensity of scattering = 1 -f- cos 2 /3. 
 
 (749) Fraction of light polarized = : _ ^. 
 
 Hence the maximum scattering at = 90 from the sun 
 is twice as much as in the direction of the sun, /3 = 0; the 
 amount of polarization is 100 per cent at (3 = 90 from the 
 sun, and it is per cent at /3 = in the direction of the sun. 
 In the atmosphere with the sun on the horizon, as at the equinox 
 in the east, the maximum polarization is in the zenith, and 
 in the vertical plane passing through the zenith and the north 
 and south points. If the solar point is east, the antisolar 
 point is west; as the sun rises the antisolar point sinks below 
 the horizon; as the sun moves to any other usual point the 
 plane of polarization is that which includes the sun, the point 
 of reflection in the sky, and the eye of the observer, the vibrations 
 being generally at right angles to this plane. 
 
 Besides the primary scattering and polarization on the 
 small particles in a turbid atmosphere, it is found that the light 
 is only partially polarized, so that a secondary polarization 
 exists at right angles to the primary, primary and secondary 
 vibrations, and polarizations at right angles to each other, thus 
 tending more or less to complete neutralization of plane polarized 
 light as the primary and secondary components approach 
 equality. There are several such points of neutralization: 
 Babinet's neutral point about 15 to 25 above the solar point, 
 Brews ter's neutral point about the same distance below as the 
 sun rises above the horizon, and Arago's neutral point about 
 15 to 25 above the antisolar point when the sun is on the 
 horizon. The positions of these points vary with the position 
 of the sun in the heavens, and the relative turbidity of the at- 
 mosphere. Since the dust particles accumulate chiefly in the 
 lower atmosphere, in a stratum less than two miles thick, there 
 is an apparent ring of special turbidity close to the horizon, 
 which causes the light to be horizontally polarized within a 
 
350 TERRESTRIAL AND SOLAR RELATIONS 
 
 few degrees of the horizon. Generally, polarization is a maxi- 
 mum in the zenith, and diminishes to the north and south 
 horizon points, and from these to the east point, for a sun in 
 the east and on the horizon. There are numerous variations 
 of these principal results, due to change in the intensity of solar 
 light from radiation, and change in the contents of turbidity 
 in the atmosphere. 
 
 The subject of polarization is discussed fully in "Tatsachen 
 und Theorien der atmospharischen Polarisation," Friedr. Busch 
 and Chr. Jensen, 1911. The literature of the observations and 
 discussions is very extensive during one hundred years. 
 
 The observations are made by a polarimeter, consisting of 
 a grating of parallel bars and spaces, from which the light falls 
 upon a Rochon prism which separates it into the ordinary and 
 extraordinary rays. These fall upon a Nichol, and by its 
 rotation, there is extinction, or flattening of the appearance 
 of the field, at four angles of observation. Thus for four angles 
 of observed extinction, the computation is of the following form. 
 
 Cordoba. (1) (2) (3) (4) (2-1) (4-3) Mean P% 
 
 Feb. 8, 1912, 11.3 94.0 190.0 278.0 82.7 88.0 85.4 73.4% 
 
 A convenient table is prepared for obtaining the mean percent- 
 age of polarization P% from the mean angles (2-1) and (4-3), 
 where (1), (2), (3), (4) are the successive readings. 
 Various relations have been traced out, such as: 
 
 (1) Movement of the altitude of the neutral points with the 
 frequency of the number of the sun-spots. 
 
 (2) Minimum polarization at the time of maximum tempera- 
 ture and maximum convection. 
 
 (3) Maximum polarization in winter rather than summer. 
 
 (4) Water drops have little effect on the polarization at 90 
 from the sun; ice crystals and large particles in that direction 
 decrease the polarization and increase the natural scattered 
 light; light that is reflected from the earth's surface, or from 
 snow areas, diminishes the polarized and increases the reflected 
 common light. 
 
 The relations between these several terms have numerous 
 interesting optical considerations, and they serve to measure 
 
POLARIZATION OF SUNLIGHT 
 
 351 
 
 to some extent the state of turbidity in the atmosphere, and 
 hence have value in connection with the absorption and radiation 
 of solar energy in the atmosphere. 
 
 The polarization at Daroca, Spain, August 19, 26, 1905, was 
 relatively high following rains, but it often fell to 40 per cent or 
 
 Date 
 
 10 11 Noon 1 
 
 August 19 
 
 August 26 
 
 August 30 
 
 lipse 
 
 Aim 
 
 Sept.21-Oct.10 
 U.S.S. Caesar 
 
 FIG. 71. Percentage of polarization of the sky light at Daroca, Spain, 
 during August, and on the U. S. S. Casar, Sept. 21-Oct. 10, 1905. 
 
 50 per cent on account of fine dust in the air; that on August 30 
 shows the effect of the passage of the shadow of the solar eclipse, 
 the sky having been thoroughly cleaned of dust by a rain on 
 August 29; the observations on the U. S. S. Casar during the 
 voyage from Gibraltar to Norfolk, frequently in the clear spaces 
 between cumulus clouds, showed a normal high percentage of 
 polarized light at 90 from the sun. Impurities from solid 
 particles produce natural light by reflection, fine particles and 
 gas molecules produce polarized light. 
 
352 TERRESTRIAL AND SOLAR RELATIONS 
 
 Solar Physics 
 
 It will be possible merely to summarize a few important 
 points in the subject of solar physics, in this connection, because 
 it is very extensive in amount, and in consequence of the fact 
 that much of the theory is in a conjectural stage of development 
 and is still indecisive. 
 
 (1) It is evident that the thermodynamic equations employed 
 in the discussion of the earth's atmosphere are applicable to 
 the sun's atmosphere, by changing the data in a proper manner. 
 Thus, gravity becomes G = g Q X 28.028; the pressure on the 
 photosphere is about five atmospheres, so that P = 5 X G p m E n ] 
 the temperature at the photosphere is apparently 7500 C.; 
 at the top of the chromosphere or lower layers of the inner 
 corona, 10" arc = 7260000 meters, 6900: and top of the inner 
 corona, 35" arc = 25410000 meters, 6500. From these data 
 for a hydrogen atmosphere, or a calcium atmosphere, the various 
 thermodynamic terms can be computed as far as the united 
 terms of the velocities' and radiations' energies. 
 
 (2) The probable velocities in the sun-spots, assuming that 
 they are the stream lines of the funnel-shaped or the dumbbell- 
 shaped vortices on the upper plane of reference, on a level with 
 the layer of the photosphere, can be computed from the general 
 dimensions of the penumbra and umbra, and checked to some 
 extent by the spectroscopic observations on velocities. The 
 vertical and the horizontal velocities in different layers of the 
 sun's atmosphere are being studied with the prospect of ultimate 
 success in a few years. It may be hoped that the radiation 
 output from the sun, computed from such data, may be found to 
 conform to the radiation energy at the earth as derived from the 
 pyrheliometer and the bolometer records, but much research 
 will be required to accomplish this result. 
 
 (3) The rotational velocity of the sun's atmosphere in different 
 latitudes on the level of the photosphere, and in other higher 
 layers as already determined, indicate a very complex kind of 
 circulation, of an entirely different type from that in the earth's 
 
SOLAR PHYSICS 353 
 
 atmosphere. The latter consists of a thin shell heated on the 
 tropics, and acquiring an approximately steady type of equili- 
 brium, as heretofore explained, while the sun has maximum 
 velocity at the equator diminishing to the poles on the level of 
 the photosphere, and increasing upward in all latitudes. Since 
 the integral in every small column along a radius extended must 
 conform to the gravity integral, which is the sum of the pressure, 
 circulation across it, and radiation through it, there is an op- 
 portunity to determine these terms through an approximation 
 by trials. 
 
 (4) Table 80 contains a convenient series of transformations 
 between sidereal and synodic periods. Table 81 contains a 
 collection of the observed synodic periods of rotation in different 
 latitudes. Bigelow's data from the prominences refer to the 
 higher levels of the sun's atmosphere, because they are seen 
 projected above the chromosphere. The acceleration in the 
 polar region over the velocities devised from spectrum dis- 
 placement lines is probably correct, because the spectrum lines 
 are all located at lower levels. Some of the Mt. Wilson data 
 are in conformity with this result. 
 
 (5) The magnetic data at the earth, as already indicated, 
 produce a synodic period of 26.68 days at the sun's equator, 
 conforming closely to the general mean value 872' or 26.58 days 
 from the eight researches quoted. The Zeeman effect has been 
 detected by Professor Hale in the sun-spots, due to the rotation 
 of electric ions in the tube of the vortex. This proves that 
 electric ions in circulation produce magnetic field at solar tem- 
 peratures. Hence, the interior of the sun, if polarized into 
 rotation filaments by its circulation, by rotation on its axis, and 
 processes of radiation, is probably magnetized throughout its 
 mass, in much the same way that the earth carries an internal 
 and external magnetic field, though its interior is at a high tem- 
 perature. There is evidence of such spherical magnetism in the 
 shapes of the polar rays seen in the minimum activity of the 
 coronal formation, where the observed rays from the sun, seen 
 in projection, conform to the lines of force surrounding a sphere, 
 supposing that they are generated chiefly in a polar ring about 
 
354 
 
 TERRESTRIAL AND SOLAR RELATIONS 
 
 TABLE 80 
 
 THE SIDEREAL AND SYNODIC PERIODS OF THE ROTATION OF THE SUN FOR 
 CERTAIN ASSUMED DAILY ANGULAR VELOCITIES. -= = -$= k n. 
 
 1 L O 
 
 Daily 
 Angular 
 Velocity 
 
 X 
 
 Daily 
 Angular 
 Velocity 
 in Degrees 
 
 f 
 
 Sidereal 
 Period 
 in Days 
 
 r 
 
 Angular 
 Velocity of 
 the Sun 
 in Days 
 
 h* 
 
 Angular 
 Velocity of 
 the Earth 
 in Days 
 1 
 ~=n 
 
 Synodic 
 Velocity of 
 the Sun 
 in Days 
 
 j = k-n 
 
 Synodic 
 Period 
 in Days 
 
 S 
 
 1 
 
 700 
 
 o 
 
 11.67 
 
 30.86 
 
 .03241 
 
 .00274 
 
 .02967 
 
 33.70 
 
 705 
 
 11.75 
 
 30.64 
 
 .03264 
 
 " 
 
 .02990 
 
 33.44 
 
 710 
 
 11.83 
 
 30.42 
 
 .03287 
 
 " 
 
 .03013 
 
 33.19 
 
 715 
 
 11.92 
 
 30.21 
 
 .03310 
 
 " 
 
 .03036 
 
 32.94 
 
 720 
 
 12.00 
 
 30.00 
 
 .03333 
 
 " 
 
 .03059 
 
 32.69 
 
 725 
 
 12.08 
 
 29.79 
 
 .03357 
 
 " 
 
 .03083 
 
 32.44 
 
 730 
 
 12.17 
 
 29.59 
 
 .03380 
 
 " 
 
 .03106 
 
 32.20 
 
 735 
 
 12.25 
 
 29.39 
 
 .03403 
 
 " 
 
 .03129 
 
 31.96 
 
 740 
 
 12.33 
 
 29.19 
 
 .03426 
 
 " 
 
 .03152 
 
 31.73 
 
 745 
 
 12.42 
 
 28.99 
 
 .03449 
 
 n 
 
 .03175 
 
 31.50 
 
 750 
 
 12.50 
 
 28.80 
 
 .03472 
 
 " 
 
 .03198 
 
 31.28 
 
 755 
 
 12.58 
 
 28.61 
 
 .03495 
 
 " 
 
 .03221 
 
 31.05 
 
 760 
 
 12.67 
 
 28.42 
 
 .03518 
 
 11 
 
 .03244 
 
 30.83 
 
 765 
 
 12.75 
 
 28.24 
 
 .03542 
 
 " 
 
 .03268 
 
 30.61 
 
 770 
 
 12.83 
 
 28.05 
 
 .03565 
 
 " 
 
 .03291 
 
 30.39 
 
 775 
 
 12.92 
 
 27.87 
 
 .03588 
 
 " 
 
 .03314 
 
 30.18 
 
 780 
 
 13.00 
 
 27.69 
 
 .03611 
 
 " 
 
 .03337 
 
 29.97 
 
 785 
 
 13.08 
 
 27.52 
 
 .03634 
 
 " 
 
 .03360 
 
 29.76 
 
 790 
 
 13.17 
 
 27.34 
 
 .03657 
 
 n 
 
 .03383 
 
 29.56 
 
 795 
 
 13.25 
 
 27.17 
 
 .03681 
 
 11 
 
 .03307 
 
 29.35 
 
 800 
 
 13.33 
 
 27.00 
 
 .03704 
 
 11 
 
 .03430 
 
 29.15 
 
 805 
 
 13.42 
 
 26.83 
 
 .03727 
 
 " 
 
 .03453 
 
 28.96 
 
 810 
 
 13.50 
 
 26.67 
 
 .03750 
 
 " 
 
 .03476 
 
 28.77 
 
 815 
 
 13.58 
 
 26.50 
 
 .03773 
 
 11 
 
 .03499 
 
 28.58 
 
 820 
 
 13.67 
 
 26.34 
 
 .03796 
 
 11 
 
 .03522 
 
 28.39 
 
 825 
 
 13.75 
 
 26.18 
 
 .03819 
 
 " 
 
 .03545 
 
 28.20 
 
 830 
 
 13.83 
 
 26.02 
 
 .03843 
 
 " 
 
 .03569 
 
 28.01 
 
 835 
 
 13.92 
 
 25.87 
 
 .03867 
 
 " 
 
 .03592 
 
 27.83 
 
 840 
 
 14.00 
 
 25.71 
 
 .03889 
 
 "- 
 
 .03615 
 
 27.66 
 
 845 
 
 14.08 
 
 25.56 
 
 .03912 
 
 " 
 
 .03638 
 
 27.49 
 
 850 
 
 14.17 
 
 25.41 
 
 .03935 
 
 11 
 
 .03661 
 
 27.32* 
 
 855 
 
 14.25 
 
 25.26 
 
 .03958 
 
 11 
 
 .03684 
 
 27.14 
 
 860 
 
 14.33 
 
 25.12 
 
 .03982 
 
 " 
 
 .03708 
 
 26.97 
 
 865 
 
 14.42 
 
 24.97 
 
 .04005 
 
 " 
 
 .03731 
 
 26.80 
 
 870 
 
 14.50 
 
 24.83 
 
 .04028 
 
 " 
 
 .03754 
 
 26.64f 
 
 875 
 
 14.58 
 
 24.69 
 
 .04051 
 
 " 
 
 .03777 
 
 26.48 
 
 880 
 
 14.67 
 
 24.55 
 
 .04074 
 
 it 
 
 .03800 
 
 26.32 
 
 885 
 
 14.7.5 
 
 24 .41 
 
 .04097 
 
 " 
 
 .03823 
 
 26.16 
 
 890 
 
 14.83 
 
 24.27 
 
 .04120 
 
 " 
 
 .03846 
 
 26.00 
 
 895 
 
 14.92 
 
 24.13 
 
 .04144 
 
 it 
 
 .03870 
 
 25.84J 
 
 900 
 
 15.00 
 
 24.00 
 
 .04167 
 
 11 
 
 .03893 
 
 25.69 
 
 930 
 
 15.50 
 
 23.22 
 
 .04306 
 
 n 
 
 .04032 
 
 24.80 
 
 *Lat. .13, spots. f Equator of sun. $ Terrestrial. Hydrogen. 
 
SOLAR PHYSICS 
 
 355 
 
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356 TERRESTRIAL AND SOLAR RELATIONS 
 
 23 degrees distant from the coronal pole. This system has 
 been traced from one epoch to another, through several eclipses 
 from 1878 to 1905 at least, as if the synodic period 26.68 days 
 was also fundamental in producing the aspect they present upon 
 the series of eclipse photographs. A model was constructed 
 of such a magnetic field, and turned by its astronomical co- 
 ordinates into the required positions on the days of the several 
 eclipses. The coincidence in position between the pole of the 
 sun, pole of the earth, pole of the corona, and its stream lines as 
 parts of a spherical magnetic field, are too striking to be over- 
 looked. Superposed upon this deep-seated magnetic field, em- 
 bracing the entire interior of the sun, is a strong electrostatic 
 surface field with its rays in normal directions, and in many dis- 
 torted positions. These two fields interplay among the forces of 
 circulation and radiation to produce the numerous fantastic forms 
 seen on the edge of the sun. Astron. Soc. Pac. No. 27, 1891. 
 
 The Spherical Astronomy of the Sun 
 
 It is necessary to give a brief account of the variable relations 
 due to the rotation of the sun on its axis, and the revolution of 
 the earth in. its orbit about the sun. If the spherical conditions 
 of Fig. 72 be transferred to a small rubber ball, it will greatly 
 facilitate the study of this complicated branch of solar physics. 
 The photographs of the sun give pictures which must be inter- 
 preted in terms of spherical co-ordinates, and this is a great 
 labor of computation, where any large number of points are to 
 be considered. Some mechanical devices have been used for 
 securing heliocentric co-ordinates approximately, but for defini- 
 tive work the micrometer measurements must be employed with 
 accuracy. As the earth passes around the sun the aspect of the 
 disk undergoes an annual periodic change which must be followed, 
 and as the sun rotates on its axis the positions of the spots, 
 faculaa, and prominences change from day to day. The following 
 definitions and formulae can be very readily verified from the 
 diagram, and by Chauvenet's treatise on Spherical Trigonometry. 
 The angle H is the apparent projection of SC K on a plane 
 
SPHERICAL ASTRONOMY OF THE SUN 
 
 357 
 
 perpendicular to the plane of sight, the angle G is the apparent 
 projection of K C E on the same plane, the angle P is the position 
 angle of the spot 2 from the north. E, counted positive eastward, 
 so that X = H + G -f P, the position angle from the sun's pole 
 S. At the same time p is the angular distance of S from C, as 
 measured at the center of the sun. The prime meridian is the 
 central meridian at one adopted epoch, mean midnight, Decem- 
 ber 31, 1853 (Carrington) ; Greenwich mean noon, June, 13.72 
 (1887), Bigelow. The rotation periods of the sun change in 
 latitude, from a maximum, 26.68 days synodic at the equator, 
 to a minimum, about 30.00 days at the poles. Carrington's 
 adopted period of rotation is applicable to latitudes = 12, and 
 
 O=Center of Sun 
 C = Center of Disk 
 2= Position of Spot 
 
 '' (l.d.) Sun's Equator 
 
 Ecliptic 
 
 (cC.5.) Earth's Equator 
 L from N to C >- Center 
 I from N to "2 -* Spot 
 /' from N to M-> Prime Meridian 
 
 FIG. 72. Spherical positions on the sun's equator. 
 
 represents the average period of rotation for the sun-spots in 
 that latitude alone. The radiation effects measured at the earth 
 are for the equatorial period. 
 
 Poles. 5 = sun's equator; E = earth's equator; K = 
 
 Ecliptic. 
 
 Inclinations. 7 = sun's equator to the ecliptic, 7. 15' = KO'S. 
 co = earth's equator to the ecliptic, 23.27' = 
 
 EO' K. 
 
358 TERRESTRIAL AND SOLAR RELATIONS 
 
 H + G = earth's equator to the sun's equator, 26 20' = 
 
 EO'S. 
 H = the projection of S 0' K at the center of the 
 
 disk C. 
 G = the projection of K O' E at the center of the 
 
 disk C. 
 
 P = the position angle of 2 = EC 2. 
 X = H + G + P. 
 
 The positive direction of the angles H, G, P } X is through the 
 east. 
 
 p = the heliocentric angle of 2 from C = C 0' 2 . 
 
 Co-ordinates of the center of the disk C. 
 
 at 
 
 L = longitude from the node N on the sun's equator, 
 
 NO ' L - 
 
 D = latitude from the sun's equator = C. L. 
 
 f E = right ascension from the node ^ on the earth's 
 
 equator, TO' E. 
 [ F = declination from the earth's equator = C E. 
 
 f O = sun's celestial longitude from the node T on the 
 Ecliptic] ecliptic, T O f C. 
 
 (^ ... = the latitude is zero = C C. 
 
 Co-ordinates of the spot or point 2 on the disk. 
 
 c , / = longitude from the node N on the sun's equator, 
 
 NO'L 
 equator . , . , 
 
 ^ a == latitude from the sun's equator = / S. 
 
 . , ( a = right ascension from the node T on the earth's 
 Earth's ^ ^ 
 
 -j equator T a. 
 
 [ 8 = declination from the earth's equator, a S. 
 
 A = heliocentric longitude from the node T on the 
 Ecliptic -j ecliptic, TO r A. 
 
 /? = heliocentric latitude from the ecliptic = A 2. 
 /'= longitude of the prime meridian from N on sun's 
 equator. 
 
SPHERICAL ASTRONOMY OF THE SUN 359 
 
 L-l = heliocentric longitude from the central meridian, 
 or difference of heliographic longitudes of 
 center of disk and spot. 
 
 L-V heliographic longitude of center of disk from the 
 prime meridian. 
 
 /-/' = heliographic longitude of 2 from the prime 
 
 meridian. 
 Reduction of the photographic plate. 
 
 R = solar radius on the photograph. 
 R' = solar radius corrected for distortion on the plate. 
 R" = solar radius as given in the ephemeris. 
 r = the perpendicular distance of S from the line 
 
 COO', = 02. 
 
 r f = the measured distance of the spot from the cen- 
 ter of the sun-picture, corrected for distortion. 
 p' = the angular distance 2 as seen from the earth. 
 
 (750) p' = jr R". ~r = sin 0' 2 2' = sin (p + p'). 
 
 (751) p + P '= sin" 1 -. p = sin' 1 ~- - p'. 
 
 From the right spherical triangle TEC (Chauvenet, p. 
 172, 88), 
 
 (752) tan G = tan co cos O ; from cos O = cot co cot (90-). 
 From the right spherical triangle N L C (Chauvenet, p. 
 
 172, 88), 
 
 (753) tan H = cos (O - AQ tan 7; from cos (O - N) = cot I 
 cot (90-#). 
 
 From the right spherical triangle N L C (Chauvenet, p. 
 171, 86), 
 
 (754) sin D = sin (O - N) sin 7. 
 
 From the right spherical triangle N L C (Chauvenet, p. 
 171, 87), 
 
 (755) tan L = tan (O - N) cos 7. 
 
 From the spherical triangle S C 2, two sides and the included 
 angle known, 90-Z), X, p (Chauvenet, p. 179, M), 
 
360 TERRESTRIAL AND SOLAR RELATIONS 
 
 (756) sin d = cos p sin D + sin p cos DcosX. 
 
 From the spherical triangle S C 2, from the two sides and 
 angle opposite one of them (Chauvenet, p. 193, 148), 
 
 (757) Sin (L - /) = sin X sin p sec d. 
 
 From the right spherical triangle T E C (Chauvenet, p. 171, 
 86, 87), 
 
 (758) sin F = sin O sin co. 
 
 (759) tanE = tan O cos co. 
 
 T = the fraction of a revolution executed by the 
 prime meridian at a given date. 
 
 / = time from the epoch, June 13.72, 1887. 
 
 K = the mean angular velocity of the sun on its axis. 
 
 n = the mean angular velocity of the earth around 
 
 the sun. 
 K n = the synodic angular velocity of the sun. 
 
 m = the complete number of sidereal rotations of the 
 prime meridian M since the epoch. 
 
 (760) T = - m (Bigelow): 
 
 (761) l' = TX = T X 14.4783 (Bigelow). 
 
 To transform (A . /?) to (L . D), compute the auxiliary angle a, 
 
 (762) tan a = sin A cot 0. Then, 
 
 / \ si* 1 (I + 0) 
 
 (763) tan L = - ^ -- tan A. 
 
 sin a 
 
 ( 764 ) si nZ? = 
 
 cos a 
 
 The Magnetic Fields of the Earth and the Sun 
 
 The sun presents many aspects of a magnetized sphere, with 
 the positive pole on the south side of the sun's equator; the earth 
 is a magnetized sphere with its positive pole on the south side 
 of the equator. An isolated (supposed) positive (+) magnetic 
 pole tends from the south side to the north side of the sun's 
 
SPHERICAL ASTRONOMY OF THE SUN 
 
 361 
 
 equator, and from the south side to the north side of the earth's 
 equator. 
 
 (x) Magnetic pole near the south geographical pole. 
 FIG. 73. Magnetic coordinates and component forces. 
 
 The following formulas are convenient for reference: 
 
 (765) Mass = M = - x R*. 
 
 o 
 
 (766) Mass Potential P = y = -g TT R s - 
 
 dP 4 r R* 
 
 (767) Vector Potential V e = - I -^ = -g * I -^ * = 
 
 TT / cos 6 (external). 
 
 (768) 
 
 (internal), 
 
 Exterior Forces 
 
 e , 
 
 (770) X., --.+ */ 
 
 a* 
 
 4 I? 3 
 
 TT -r / . 3 sin e cos 
 
 O IT* 
 
362 
 
 (771) 
 
 TERRESTRIAL AND SOLAR RELATIONS 
 
 dV. 4 R* Zzy 
 
 + s" 71 " 7 ^ '~^~ = 0.since;y = 0. 
 
 dy 3 r 3 r* 
 
 
 (772) Z = F n = Z cos + X sin = - TT 7 
 
 o T" 
 
 (- cos + 3 cos 3 5 + 3 sin 2 cos 0) 
 
 r 
 
 = x / 2 cos 0. 
 
 (773) X = F t = - Z sin + X cos = -| TT 7 ^ 
 
 (sin 0-3 cos 2 sin + 3 sin cos 2 0) 
 
 (774) 
 
 Z 2 = 
 
 r 
 = TT / r . sm 0. 
 
 o f- 5 
 
 = TT 7 (l + 3 cos* 0). 
 
 Surface R = r 
 
 (775) Z e = - TT 7 . 2 cos 
 
 o 
 
 (776) X e = - T 7 . sin ' 
 
 Equator 
 
 8 r 
 
 o- * /. 
 
 Pole - 
 
 'e = 0. 
 
 4 
 
 = 0. 
 
 Interior Forces 
 
 C778) ^ - - 7? T * 
 
 (780) Line of Force 
 
 = constant. 
 
SPHERICAL ASTRONOMY OF THE SUN 363 
 
 (781) Equipotential Surface V = K R 3 <r~ = constant. 
 
 O T 
 
 A magnetized sphere may be induced by two layers of 
 positive and negative masses, circulating surface electric currents, 
 two positive and negative poles placed near the center, and by 
 other physical devices, as rotating Ampere electric currents in 
 the interior around lines parallel to the axis, in which case the 
 sphere is polarized. This is probably the case for the earth and 
 for the sun. For surface exterior currents, as in the earth's 
 atmosphere, it is evident that variation in the east- west direction 
 changes the strength of the existing magnetic field along the lines 
 of force. The vectors of Fig. 68, 69, can only be produced by 
 north and south ionization currents, as there indicated, distinct 
 from east and west currents on the normal field. The asymetric 
 position of the sun's magnetic poles deduced by Bigelow in 
 1891 has been confirmed by direct observation at the Mt. 
 Wilson observatory, 1913. 
 
 Conclusion 
 
 The general functional relations between the incoming solar 
 radiation, the portion of it transformed into ionization currents, 
 the magnetic disturbing vectors depending upon them, all re- 
 main to be discovered. It has been possible to indicate in this 
 Treatise some of the important elements in these problems, and 
 it is hoped that the formulas and methods of discussion here 
 adopted will greatly facilitate the pursuit of such researches by 
 many students. The practical side of the matter consists in 
 the development of the branch of Meteorology and Solar Physics 
 which will culminate in the ability to predict the seasonal climatic 
 conditions likely to prevail during the coming year in the several 
 large agricultural regions of the earth. The extent and scope of 
 these subjects are so great that the co-operation of many institu- 
 tions and national offices will be essential for the success of so 
 important an enterprise. 
 
CHAPTER VII 
 
 Extension of the Thermodynamic Computations to the Top of 
 
 the Atmosphere 
 
 Remarks on the Bouguer Formula 
 
 THE computations on the thermodynamic data of the earth's 
 atmosphere have heretofore in the examples been limited to 
 about 20,000 meters, but it is very desirable to extend them to 
 the vanishing plane of the gaseous envelope. The following 
 summary illustrates a method for accomplishing this purpose, 
 and the results afford approximate material for further important 
 researches. These concern the problems of the solar radiation 
 and its absorption, together with the correlative terrestrial 
 radiation and its absorption. It is especially proposed to 
 test the result that the "solar constant," as derived from 
 
 pyrheliometer observations is about 1.92 -^-~- r-, or whether 
 
 cm. 2 mm.' 
 
 it is about twice as much, in conformity with the bolometer 
 data. The Bouguer formula of reduction, in the preceding 
 notation, 
 
 has been shown to contain two unknown terms /o, mo, which 
 together determine the source of the effective radiation, so soon 
 as their details are understood. It has been customary to 
 identify / with the solar radiation falling on the outer limit of 
 the atmosphere for which it is assumed that m = i and disappears 
 from the equation, enabling it to be solved for 7 . This is, in 
 fact, a special interpretation by assumption, and doubt is thrown 
 upon it by the bolometer ordinates, since these require an energy 
 spectrum curve of about T = 6,900 (3.88 calories), while 
 /o = 1.92 calories is satisfied with 5,800 for the effective solar 
 temperature. It has, furthermore, been shown by the pyrheliom- 
 
 364 
 
FIRST DISTRIBUTION OF TEMPERATURE 365 
 
 eter observations that /o is not a constant quantity, that is, a 
 source of radiation independent of atmospheric conditions as to 
 aqueous vapor, dust and ice contents, not now referring to the 
 small variations of the radiation energy emitted at the sun itself. 
 The evidence is that as the coefficient of transmission p dimin- 
 ishes in value the term 7 is progressively depleted. The extra- 
 polated points /o on the axis of ordinates do not concentrate at 
 /o = 1.92 calories. It follows that /o is a terrestrial variable, 
 and it is the purpose of this chapter to investigate further the 
 properties of the terms, 7 , m Q . 
 
 I. FIRST DISTRIBUTION OF TEMPERATURE 
 
 The balloon ascension, Huron, So. Dak., September 1. 1910 
 
 The balloon ascension, Huron, So. Dak., September 1, 1910, 
 reached a height of 30,000 meters, recording temperatures 7", 
 but without velocities q. These have been provisionally sup- 
 plied by taking the mean values for ten other ascensions in order 
 to balance approximately the terms of the gravity equation, 
 
 (196) g (* - *) - - ^-p^ - J fo 2 - <?o 2 ) - (Qi - Co). 
 
 ^10 
 
 Table 82 contains the adopted T, q, from which all the other 
 terms were computed up to 30,000 meters, and the results were 
 plotted on large diagrams of which Figs. 74, 75, are reduced 
 copies. The method of extending the T data to the higher 
 planes is as follows: On the 30,000-meter plane T = 232.3, 
 P = 1352 (B = 10.14 mm. as computed, while the automatic 
 register recorded B = 10.70 mm.), p = 0.05512, R = 105.61, 
 the check equation P = p R T being perfectly satisfied. At first 
 the P and p curves were extended graphically, and from these 
 values up to 40,000 meters approximate values of T were com- 
 puted, carrying the T curve around the corner of the isothermal 
 region into a very rapid gradient. These temperatures are 
 necessary to continue the P, p, curvatures at that elevation. 
 Then all the other terms were computed directly from T and the 
 diagrams were extended. At 40,000 meters T = 170.5, P = 
 
366 
 
 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 fl 
 
 c/3 
 
 a gs 
 
 M S3 
 
 t* g 
 
 *!? 
 
 II 
 
 u 
 
 I s 
 
 OK 
 5 
 
 11 
 
 O ^ 
 
 I 
 
 M 
 
 to 
 
 Tempe 
 ture 
 
 rococococococococococococoro.cocococococococo 
 
 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
 
 CO 
 
 o T-H 
 
 i i O 
 i-H*O 
 
 i ICOCOO5(NCOO5(M>O 
 
 COO5(M>OOOi IIO 
 I li (C^tNfNCOCO 
 
 *OI i(MOO 
 i IIOO5OO 
 
 COOOiOOi<MOi IT iiOrH 
 
 .T 
 
 OOOOO<NCOOOO 
 
 iOi lOSGOOOOiOi i<N^f>O 
 CO CO C^l C^ C^ Ol CO CO CO CO CO 
 
 ^ 
 
 vsx^f'OiO' !COOiCOTtiCO 
 y\/N c>q-^iu7if>.cOcOOiOt^ 
 
 O51>iOCOCO(NOO 
 COO'-HCOcO' i i I 
 
 rH <N 
 
 X X ^ 
 
 GO 00 i 1 CO l> O5 CO 
 
 rH TH t^ 10 TJH CO CO <M 
 
 
 
FIRST DISTRIBUTION OF TEMPERATURE 367 
 
 OO 00 ^2 * 4 C^ CO 
 
 cococococococococococococococococooocococococococococococccococo 
 
 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 
 
 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 
 
 
 
 CO 
 
 <N 
 
 OiCOOO5C<l 
 
 ^ 
 
 CO 
 
 co <N C5 as t Tt< <M i- o o o o <o co o t * o o 
 
 C^COCOC^C^C^C<lC^C<IC^C<JrHrHrHrHC^C^CO^TjHUOO^ 
 
368 
 
 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 279 (B = 0.002095 mm.), P = 0.01796, R = 91.20, and P = 
 p R T is again exactly satisfied. These data indicate that the 
 vanishing plane of the atmosphere is approaching, and a trial was 
 made of z = 50,000 meters for T = the approximate temper- 
 ature of space. Further experience with the data indicates that 
 
 50000 
 
 40000 
 
 FIG. 74. The dynamic data to the top of the atmosphere at 50,000 meters, 
 from the balloon ascension, Huron, S. D., Sept. I, 1910 
 
 P = Pressure in kilograms per square meter. 
 K = Radiation energy. 
 
 p = Density in kilograms per cubic meter. 
 T= Temperature in absolute degrees. 
 
 r j = Temperature before the formation of the isothermal region. 
 Cp = Specific heat ratio at constant pressure. 
 
 The balloon ascension, Huron, S. D., September I, 1910, reached the 
 elevation 30,000 meters. The lines for P, K, />, Cp, and T (up to the iso- 
 thermal level) converge at a point on the vanishing plane 50,000 meters. By 
 ,a series of trials T was determined such as to make the P, K, p curves run 
 out smoothly to a common vanishing point. 
 
FIRST DISTRIBUTION OF TEMPERATURE 
 
 369 
 
 the vanishing plane is nearer 90,000 meters, but the example is 
 reproduced because several interesting points are illustrated. 
 The temperature T at any level, whether observed or adopted, 
 can always be fully checked by the combination of these two 
 equations. If there is an important residual for g (zi z ) it 
 must be due to the fact that an erroneous T was ascribed to the 
 level. In this case the residuals are large from 41,000 to 50,000 
 meters, but this can be remedied by further trial computations in 
 
 50000 
 
 40000 
 
 30000 
 
 20000 
 
 10000 
 
 000 
 
 (QrQul/v*' 
 
 
 
 (Oi-O 
 
 (u,- 
 
 Uo) 
 
 (W : -W ) 
 (Ui-Uo) 
 
 2000. 4000. 6000. 8000. 10000. 12000. 
 
 3.50 3.&0 3.70 3.80 
 
 FIG. 75. The thermodynamic and radiation data to the top of the atmos- 
 phere at 50,000 meters, from the balloon ascension, Huron, S. D., Sept. 
 I, 1910. 
 
 (Qi Qo) =The free heat transmitted by the solar radiation. 
 (WiWo) =The external work of expansion. 
 (Ui Uo) =The inner energy of molecular motion. 
 
 a=The exponent in the radiation equation. 
 (Ei Eo) =The total incoming free heat. 
 (A i Ao) =The absorbed portion of it. 
 
 (Qi- <2o) = (W, - W,} + ( Ui - C/o) 
 
370 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 connection with other high-level balloon ascensions. It should 
 be noted in this example that the value of R = 268.46 at 
 46,000 meters has resumed nearly its surface value, 287.03. This 
 means that the thermodynamic system has been subjected to 
 great heating influences in the levels above 40,000 meters, and 
 this is easily seen by studying the data of Table 82, and Figs. 
 74, 75. The P, K, p, curves run out smoothly to the vanishing 
 
 plane, but the Cp = R ~^ (Qi - ft), (Wi ~ W ), (U,- Z7 ), 
 
 log c, a, curves all are much distorted in the strata above 40,000 
 meters. This is the thermodynamic result of heating the rarefied 
 gases of the outer levels from T = on the vanishing plane 
 to T = 208.2 at 38,000 meters. The so-called isothermal 
 layer from 11,000 to 38,000 meters is merely that part of the 
 atmosphere where the surviving solar radiation nearly balances 
 the terrestrial radiation. Above 38,000 meters is a region of 
 very powerful absorption of the solar radiation, in which the 
 energy of the short waves of the solar spectrum, * = 0.00 /* 
 to 0.40 M, are absorbed, the energy being used in heating this 
 stratum through about 208 degrees. This is the true albedo 
 of the earth's atmosphere, and it amounts to nearly one-half of 
 the total solar radiation falling on the earth's outermost stratum 
 of atmosphere. If a line be drawn, T 1 , from the lower part to 
 the upper part of the T-curve, the enclosed area between T and 
 T 1 represents the region of absorption, and the isothermal curve 
 to the right of it is one of its three boundaries. This area can 
 be studied for several thermodynamic conditions. 
 
 The divergence between the pressure curve P and the 
 radiation energy curve K is due to the free heat per volume 
 ?i ~ Co) - (Wi - TFo) + (tfi - Z7o), where (ft - ft) repre- 
 sents the excess of the inner energy (Ui U ) over the work of 
 expansion (Wi WQ) . At the points where the P and K curves 
 coincide, as at the surface, there is no free heat at liberty to move. 
 In the case of an adiabatic atmosphere, (ft ft) = 0, and there 
 is no radiation. It should be carefully noted that meteorological 
 discussions and the tables in common use have been heretofore 
 uniformly based upon strictly adiabatic formulas. Since the 
 
FIRST DISTRIBUTION OF TEMPERATURE 371 
 
 adiabatic and non-adiabatic systems are identical at the surface, 
 this distinction is not important there, but because these systems 
 diverge to a large amount in the upper atmosphere the inference 
 follows that all meteorological tables based upon R = constant 
 produce erroneous values of the density above the surface. For 
 example, with R = constant, g (zi Z Q ) = Cp a (T a T ) = 
 
 , and from this comes the adiabatic gradient 77 = 
 
 PaO Cp a 
 
 - ( a ~ T ^ = - 9.870 per 1,000 meters. If, on the other 
 
 (21 - z ) 
 
 hand, the terms for circulation \ (qi 2 q<?) and for radia- 
 tion (Qi QQ) are to be introduced, this can only be done by 
 changing p ao to pio, since the pressure term (Pi P Q ) is fixed 
 by direct measures at heights which do not correspond with any 
 adiabatic temperature gradient. It follows that we have 
 
 another value of the specific heat ratio Cpio which differs from 
 
 p -p 
 
 Cp^ so that Cpio (T a TO) = - and the non- 
 
 |10 
 
 adiabatic pio differs from the adiabatic p a . By subtraction we 
 find that, 
 
 (198) - (Cp a - Cp lQ ) (T a - T ) = -\ (qi 2 - qo 2 ) - (Qi - Qo), 
 
 as heretofore indicated. If for the moment we assume that 
 ~" i (<?i 2 ~~ #o 2 ) = 0, as is the case at the top and at the bottom 
 of the atmosphere, there are two extreme values of (Qi Q ) : 
 
 1 . At the vanishing plane, for Cpio 0, 
 
 - (Qi ~ <2o) = - (Cpa - Cpio) (T a - To) = 993.58 X 9.87 
 = 9806, per 1,000 m. 
 
 2. At the surface, for Cp w = Cp a , 
 
 ~ (Ql - <2o) = - (Cpa ~ Cpio) (T a - T ) = 0. 
 
 The adiabatic formula requires that Cpio = Cp a from the 
 surface to the vanishing plane, which is in disagreement with 
 the observed temperature gradients. 
 
372 
 
 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 Illustrations of the Use of Erroneous Densities 
 
 In order to illustrate the amount of divergence between 
 p a and p, we have for this balloon ascension 
 
 TABLE 83 
 
 THE DIFFERENCE BETWEEN /> (NON-ADIABATIC) AND p a (ADIABATIC) 
 
 DENSITY 
 
 z 
 
 ft 
 
 Pa 
 
 50000 
 
 0.0000 
 
 0.0000 
 
 45000 
 
 0.0035 
 
 0.0033 
 
 40000 
 
 0.0180 
 
 0.0057 
 
 35000 
 
 0.0324 
 
 0.0099 
 
 30000 
 
 0.0551 
 
 0.0203 
 
 25000 
 
 0.0929 
 
 0.0426 
 
 20000 
 
 0.1586 
 
 0.0935 
 
 15000- 
 
 0.2754 
 
 0.2073 
 
 10000 
 
 0.4760 
 
 0.4190 
 
 5000 
 
 0.7730 
 
 0.7151 
 
 surface 
 
 1 . 1485 
 
 1 . 1485 
 
 Since p is greater than p a it follows that, 
 
 (782) 
 
 Pi-Po. Pi - 
 
 is smaller than 
 
 pio 
 
 Pao 
 
 and this is compensated in the gravity equation by the terms 
 for the circulation and radiation. It has been tested by numerous 
 
 p r> 
 
 computations that g (zi z ) = Cp a (T a T ) = 
 
 Pao 
 
 without important residuals from the surface to the vanish- 
 
 -p __ r> 
 
 ing plane, and that - Cpu (T a - T ) = - holds true 
 
 pio 
 
 throughout the atmosphere. No system of meteorology 
 
 k 
 
 founded on R a \ _ ^ = Cp a = constant is applicable in prac- 
 tical studies of atmospheric conditions. 
 
 There is another important matter in which erroneous den- 
 sities have been introduced into studies of radiation phenomena. 
 
ILLUSTRATIONS OF THE USE OF ERRONEOUS DENSITIES 373 
 
 It has been thought proper to modify the Bouguer formula in 
 such a way that the observed variations in I,p, shall be dependent 
 upon the nature of the gaseous path m, so that the integrated 
 amount of aqueous vapor in the form of liquid water may be 
 made a factor attaching to the exponent m. Thus it has been 
 computed that the water mass S/* = 2e Q , where S/* is the integral 
 of the water contents in grammes and CQ the vapor pressure in 
 millimeters of mercury at the surface. The hair hygrometer 
 measures the relative humidity of the air at the different levels, 
 and the corresponding e in mm. is computed. The full formula 
 is easily shown to be, 
 
 (783) M = p f -f - 4 (0.622 -I + 0.235 f). 
 
 It has been customary to assume R = 287 and to omit the 
 
 e 2 
 
 term in -57, so that there remains, 
 &* 
 
 273 6 
 
 (784) M = po-^-X 0.622. 
 
 This formula diverges from the true one in the same way that p 
 differs from p a at the different levels, so that the sum of /*, S/t, 
 is erroneous in the formula 2/z = 2e, being too small. 
 
 Furthermore, the solar radiation suffers two sorts of deple- 
 tion, the first as true absorption upon which p depends, the sec- 
 ond as scattering upon which /o partly depends. We have shown 
 how /o should be modified for vapor contents (7 0.0214 e) 
 before introducing it into the Bouguer Formula. The ice crystals 
 of the upper strata, 8,000 to 25,000 meters, are effective reflec- 
 tors of the radiation energy in these strata. The hair hygrometer 
 takes no account of the ice contents in the (3-stage), nor the 
 water contents (f and ft stages) which may be present, but 
 only of the vapor contents (<*-stage). It follows that since water 
 and ice exist throughout the lower atmosphere up to 25,000 
 meters, but not always in cloud forms, though these have 
 been observed up to 16,000 meters, that the a-stage Formula 
 is inadequate to the purpose which has been imposed upon it. 
 
374 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 The remedy for these difficulties is being considered in a special 
 research. 
 
 The Thermodynamic Terms 
 
 Table 82 and Fig. 75 give the relations of (Qi - Q ) = 
 (Wi - Wo) + (Ui - Uo). (ft - Qo) begins at the surface 
 without value because the external work and the inner energy 
 are in equilibrium; it increases by a ci*rve determining the 
 amount of the lower absorption to a point between the eleva- 
 tions 26,000 and 27,000 meters, this being the level where 
 2 (Qi - Qo) = (Ui - Uo) - (Wi - Wo), which defines the true 
 isothermal level where there is no absorption of solar energy, 
 in the same sense that there is no absorption at the top and at 
 the bottom of the atmosphere. This gives three points on the 
 line of total solar energy, 9,806 on the vanishing plane, about 
 5,780 at 26,800 meters, and at the surface. The (Ei - E ) 
 line of total solar radiation energy on the several levels was 
 drawn by connecting these three points. The area between the 
 axis of ordinates and the (Qi Q Q ) curve represents the free 
 heat of transmission, the area between the (Qi Q ) and the 
 (Ei EQ) curves that of absorption, while (E\ E ) = 
 (Qi Qo) + (Ai AQ), the total area. The data on the several 
 levels are for 1,000-meter areas, except below 3,000 meters, 
 where the vertical height is 500 meters. The sum of these 
 areas from the surface to 50,000 meters apparently represents 
 the true " solar constant" of radiation energy, with the constit- 
 uents of transmission and absorption. 
 
 The (Qi Qo) curve continues to increase in value upward, 
 as (Wi Wo) and (Ui U ) separate, till a maximum value 
 for the free heat of transmission is reached at 6963 on the 
 38,000-meter level. It will be shown that the pyrheliometer 
 receives an amount of heat at the surface such as results from 
 a summation up to about this elevation, as if this were the 
 general efficient source of radiation at about 1.92 calories. 
 The (Qi - Qo) and (Ei - E Q ) curves begin to diverge at 27,000 
 meters, and very rapidly after passing 38,000 meters. (Qi - Qo) 
 falls to a minimum at 48,000, and rises immediately to its 
 
THE THERMODYNAMIC TERMS 375 
 
 primary maximum, 9,806 at 50,000 meters, in this computation. 
 (Wi Wo) and (Ui Z7 ) pass through entirely similar changes 
 above 38,000 meters, and they all correspond to the very rapid 
 increase in the temperature of the gases of the upper absorption 
 level. It is evident that the pyrheliometer measures the trans- 
 mission and the minor absorption up to about 38,000 meters, 
 but is not cognizant of the great absorption area above it where 
 the absorption is nearly complete. The bolometer measures the 
 relative ordinates of the true solar spectrum, unless some of 
 them are absorbed, but for its total energy depends upon the 
 computations and integrations of the spectrum curve at given 
 temperatures. This is shown by the spectrum to have lost 
 nearly half the incoming energy in the short waves having long 
 ordinates, these corresponding with the upper absorption 
 region, besides some minor absorptions attributed to aqueous 
 vapor selective absorption bands belonging to the lower region. 
 The general slope of the (Ei E ) line indicates that the 
 equal terrestrial* radiation is required to balance the solar radi- 
 ation, generally large below but small above, in accordance 
 with the prevailing temperatures and the equation of equilib- 
 rium, as will be explained. 
 
 The corresponding values of K\Q, log c, a, in KIQ = c T a , 
 have been computed. The exponent a = 3.819 at the surface 
 gradually falls to a minimum 3.564 at 38,000 meters; it rises 
 to a maximum 3.905 at 45,000 meters, then falls to about 3,600 
 at the vanishing plane. Further computations will be under- 
 taken in respect to the region above 38,000 meters. (Fig. 79.) 
 
 The Constituents of the Solar and the Terrestrial Radiations in 
 the Earth's Atmosphere 
 
 It will be convenient to transform the thermal data which 
 have been computed in the (Meter, Kilogram-second) (M. K. S.) 
 system into the corresponding values in the (Centimeter-gram- 
 second) (C. G. S.) system, before proceeding to a further 
 discussion of the transmitted and absorbed constituents of the 
 solar and the terrestrial radiations. The term (Qi Q ) in 
 
376 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 o (Zi - 80) = " Cpa (Ta - To) = - ?l ~J* - J fe 2 - g 2 ) - 
 
 (Ci ~ Co) 
 
 must satisfy the dimensions of all these terms, as well as the 
 equations P = P R T, and (ft- g ) = (PFi - Wo) + (/!- tf ). 
 (61 - &) was derived from - - (Cp a - - Cp 10 ) (T a - T ) 
 + i (^i 2 ~~ ?o 2 ) = (Qi Qo) so that it has the same dimensions 
 
 k 1 P Pv 
 
 SLS R = T Q>, and since R = = -~r they are the same 
 
 as, those for work, (Wi W ) = PW (vi V Q ), and for the 
 inner energy (Ui U ) = Cv (T a To). In transformations 
 from the (M. K. S.) to the (C. G. S.)system the dimension factors 
 are, [M] = 1000 = 10 3 , [L] = 100 = 10 2 ; in transformations 
 from the (C. G. S.) to the (M. K. S.) system, [M] = 1Q- 3 , 
 [L] = 10~ 2 . All are in mechanical units. 
 
 Pressure, kilog./met., [P 1 -Po]=ML~i 
 
 _ _ 
 
 Density, kilog./met., [ft ] =ML~M P 10 ~ ' L 
 Velocity square, [gi 2 g 2 ] = i 2 = 10 4 = .................... = 10000 
 
 Specific heat, [Cp.Cv.R.K.] = U = 10 4 = .................... = 10000 
 
 Heat, [<2i - <2o = (C - Cp 10 ) (T a - Tc)] = L 2 = 10 4 = ........ = 10000 
 
 Work, [(TFj - TFo) = P 10 (i - o5] = ML~\ M~ ! L 3 = U = 10 4 = . = 10000 
 Inner Energy, [(Z7 t - Z7 ) = Cy (r - T )] = L 2 = 10 4 = ........ = 10000 
 
 The common factor for reducing all the terms of the gravity 
 and heat equations from (M. K. S.) to (C. G. S.) in mechanical 
 units is 10 4 = 10,000. 
 
 From the general equations (Wi Wo) = PW (VIVQ) and 
 (Qi ~ Qo) = (W 1 - W ) + (/! - J7 ), we have, 
 
 U, - U ft-Qo a 
 
 AIO = " - = - X = C L 
 
 V 
 
 which is the radiation equation employed in the computations. 
 The dimensions of all these terms are the same in reducing from 
 
 10000 
 the (M. K. S.) to the (C. G. S.), the factor being - - = 10, as 
 
 1UUU 
 
SOLAR AND TERRESTRIAL RADIATION 377 
 
 is commonly used in the pressure P 1Q . The reduction from 
 mechanical units to heat units (C. G. S.) is by the factor 
 
 4 1851 X 10 7 ' and the factor for chan S e from second to minute 
 is 60. Hence, to reduce (Qi Q Q ) in M. K. S., mechanical units 
 
 gram calorie , 
 
 into - , the factor is, 
 
 cm. 2 minute 
 
 1000^1851% - Sil 5 =0-^14336; Log.-5.15644. 
 
 Compare the constants, coefficients, and dimensions of Table 95, 
 where the formulas afford many other combinations in conform- 
 ity with the kinetic theory of gases. 
 
 The Total, Transmitted and Absorbed Amounts of the Solar and 
 the Terrestrial Radiations 
 
 Huron, September i, 1910 
 
 The method of separating the total radiation (Ei E ) 
 received at the earth from the sun into the transmitted (Qi Q ) 
 and the absorbed (Ai A ) constituents on each level has been 
 
 explained. The coefficient of transmission is p = ~ ^=, 
 
 L\ ~ L>Q 
 
 and of absorption is k = ~ ^r. The terrestrial radiation 
 
 h,\ Lo 
 
 is obtained as follows : Let JQ c T 4 , the black body radiation 
 of the atmosphere at the point whose temperature is T; let 
 J a = c T a the actual radiation for c and a as computed from 
 the thermodynamic data, so that the actual absorption between 
 two planes is / a -i Ja-o = c\ T\ l CQ T Q a . The general 
 equation of radiation equilibrium is 
 
 2 J = E + D, where D is the total terrestrial radiation. 
 Hence, D = 2 J Q E, and B = D J a , where B is the trans- 
 mitted terrestrial radiation. The coefficient of terrestrial 
 
 7? - - 7? 
 transmission is p 1 = 77 ^, and that of absorption is k 1 = 
 
 U\ L/o 
 
 ~^ pp- . The results of these computations appear on Table 
 
 
 
378 
 
 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 w o 
 
 rt rH 
 
 u 2 
 
 
 
 c/) 
 
 s ^ 
 
 IcS 
 
 <X 
 
 8 I 
 
 <3 O 
 H Jo 
 g ^ 
 
 I I 
 " I 
 
 B 
 
 I 5 
 
 a 
 
 * 2 
 
 '31 
 
 1 
 
 'J3 S 
 
 CO 
 CN 
 
 CO CO CO CO CO CO CO 
 CO(MOOOSCOCOC s Q 
 OOS'*t t t > - OSO5OSO01>-COiOCO 
 CO CO i t rHCOOiOCOcOCOcOcOCOCOcOcOCDcOcO-OiO 
 
 I I I I I I I I I I I I I I I I I I I I I I I 
 
 ^J^ CO 
 
 O : ll> 
 rHOSOO 
 
 rH CO rH OS O CO 
 I I I I I 
 
 CO O Tt< CO O CN 
 *H O 00 - CO 00 
 OS Tt< l> l> l> l> 
 
 00 10 00 1> 
 
 I I I I I I I I I I I I I I I I I I I I 
 
 COCOOOt^OOOSOOCOOOO 
 
 lO(MOSTtlrHI>.COrHlCOb- 
 rHrHrHrHrHOOOOpp 
 
 r r 
 
 rHCOb-OSOTh'OOC^OOrHlOCOOOCOOOrHlOGOCMCO 
 
 I I I I I I I I I I I I I I I I I I I I 
 
 I>-OTH|>.O'~"i 
 
 OSl lO5OOl I 
 
 O^ l CNC^ir-i' I 
 
 rH CO 
 CN rH 
 
 CO *O CO *O CO 
 
 SCOOO5Tt<rHTt<COrHOOSb-TtlGO 
 OS*OGOrHCOTt<lOCOb-b-OOOSOS 
 
 !> b* OO 00 Os Os Os Os Os Os OS Os OS os 
 
 rH OCJ O b- O CO 
 rH rH rH O O O 
 
 O O 
 
 00 00 
 
 I I I I I I I I I I I I I I I I I I I I I I + 
 
 CO <M OS OS CO 
 
 O CO CO 00 00 
 
 IO CO rH OS b- 
 
 OS OS OS 00 00 
 
 o o o o o 
 
 OOOlOCNOI^^fNOSb-TtHrHOSCOT^rHGOCOCOrHOOJOCO 
 tfCOCOCOCO<N<N<NrHrHrHrHOOOOO5O5OSOSOOOOOO 
 ^^1 IrHrHrHrHrHrHrHrHrHrHrHrHrHOOOOOOO 
 
 r r r \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
 
SOLAR AND TERRESTRIAL RADIATION 
 
 379 
 
 CMrHCSt^-tf't^^t^OOiOt^cOr- icDCOfNiO^COCOTttcOC^^iOOO-^ 
 COTfi i OS l> -^ r- (OOiO<NOOiO<NOOcOTficN'-HOO5001>t>'COCN>-i 
 uooO^T^T*Tt<COCOCO<NcN<NrHrHrHrHrHrH 
 
 I I I I I I I I I I I I I I I I I I I I I I I I I I I I 
 
 I I 
 
 ' 
 
 b- OO 
 
 CO rH 
 ^H <M 
 
 OOOOCO rfi 
 rH OS OS i i 
 
 <NOOO5t-ia500l> 
 
 <N CO OS i i > i CO GO 00 
 
 i-H CO 
 
 l>* "^ 
 
 i-H <N . 
 
 CO (M O i-4 i-H 
 CO GO O OS -00 
 
 - 
 
 ^^ rH T-H ^^ i-H T-H T-H C^J rH rH C^ rH CO CO CO "^ CO ^O t^ 00 t^* ^^ OS CO ^ f^ IO ^O 
 
 CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO i"H ^^ TH CO CO CO CO 
 
 I I I I I I I I I I I I I I I I I I I I I I I I I I I I 
 
 ' iO 
 
 : b- 
 
 :s 
 
 +++ i ++++ i i i i i i i i i i i i i + i i i i 
 
 T (Ol>- T tiOO5rHrHOCO 
 OOOOi i i iCOCO^^f 
 
 . CO 
 
 I 
 
 . 00 
 
 ; oo 
 
 01 
 
 rH CO CO O 
 t^ rH l^ rH 
 
 00 00 l> l~~ 
 
 
 8888 
 
 + + I I i 1 I I I I I I I I I I I I I I I I I I I I I I 
 
 CO CO CO 
 
 rt< b CO 00 l^ 00 
 
 rH CO rH CO rH CO 
 
 GO CO Tt< CO CM rH O 
 
 o o o o o o o 
 
 8 
 
 cooooco 
 
 ^f "^ ^t^ CO CO 
 
 r r i i r i i i i i i i i i i i i i i i i i i i i i i i i 
 
 t^-f>-i x *t > r>-iv 
 
 d C^l Ol i-^ ^H i-^ 
 
 CM <N <N <N <N <N CN 
 
 o o 
 
380 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 84, inj^'a^ units. The values of (Q l - Q ) in M. K. S. 
 
 mechanical units are found in the last column. The summa- 
 tions are as follows for the entire depth of the atmosphere. 
 
 Total sum of solar radiation, 2 (Ei - ) = - 3.6942 gr \ Ca .' 
 
 cm. 2 mtn. 
 
 Transmitted solar radiation, 2 (Qi - Q ) = - 2.7177 " 
 Absorbed solar radiation, S (Ai A ) = -0.9101 " 
 Atmospheric (black) radiation, S (/O.I-/O.Q) = - 3.8181 " 
 Total terrestrial radiation, S (A -A) = - 3.9419 " 
 Transmitted terrestrial radi- 
 ation, S (Bi -B ) = - 2.5744 " 
 Absorbed terrestrial radiation 2 (J a >i Ja-o) = 1.3675 
 
 The minus ( ) sign is due to the fact that while the positive (+) 
 direction is along the axis z outward, the temperature and 
 thermal gradients decrease in this direction; a positive (+) 
 sign would indicate an inversion of temperatures and thermal 
 quantities outward. In the case of p, k, the coefficients are all 
 positive; in the case of p l , k 1 they are generally positive in the 
 lower levels, but sometimes (+) and sometimes ( ) occur 
 in the upper levels. Since the original observations of the 
 velocity are lacking, those used being supplied by analogy, it is 
 probable that the data of Huron, September 1, 1910, are inade- 
 quate to produce accurate values of p l , k 1 . 
 
 II. SECOND DISTRIBUTION OF TEMPERATURE 
 The Balloon Ascension, Uccle, November 9, ign 
 As already stated the gravity residuals in the check equation 
 
 (196) g (z, - zo) = - Pl ~ P - (c Pa - C#io) (T a - T ) 
 
 Pio 
 
 + A g ( Zl - z ) 
 
 were quite large above 40,000 meters in the Huron ascension. 
 It was supposed that this was due to the fact that the velocities 
 q were assumed, that g was taken constant in the adiabatic 
 
SECOND DISTRIBUTION OF TEMPERATURE 381 
 
 gradient, and that the formulas might begin to fail when the 
 P, p terms became very small. Accordingly, a new series of 
 computations has been undertaken for Uccle, June 9, Septem- 
 ber 13, November 9, 1911, the necessary data including veloc- 
 ities having been courteously supplied by Dr. Vincent, Directeur 
 de 1'Institut Royal Meteorologique de Belgique. In these, 
 besides using the observed velocities, the gravity ranges with 
 the height. It is known that a temperature T introduced into 
 the series of computations, building upward from the surface 
 from level to level, whether observed or assumed, can always 
 be checked by the control gravity equation. If the series of T 
 is correct, A g (^ z ) vanishes. Hence, it is practical, by a 
 set of trial computations, to arrive at such temperatures through- 
 out the atmosphere as will satisfy the entire group of ther- 
 modynamic equations-. The balloon ascensions to 20,000 or 
 30,000 meters give the necessary foundations upon which the 
 entire structure can be built up to 90,000 meters. Up to 40,000 
 meters the four ascensions give about the same residuals, with 
 occasional wide pairs depending on an erroneous T. The trials 
 to reduce the check residuals were therefore limited to the region 
 above 40,000 meters. It appears that positive residuals are to 
 be diminished by increasing the temperatures on the several 
 levels. The general outcome was to introduce a second tem- 
 perature region beginning at about T = 170 on the 40,000- 
 meter plane and terminating near the 90,000-meter plane. The 
 remarkable result is that the formulas are rigorous, however 
 small the values of P, p, R, may become for the successive T, 
 and that small residuals above 50,000 meters imply rather large 
 changes in the temperatures T. Extending the temperatures 
 from 40,000 meters to levels higher than 50,000 meters, the 
 residuals have the following mean values: 
 
 Huron, Sept. 1, 1910, 40,000 to 55,000, three trials, A g (zi-z ) = +163.0 
 
 Uccle, June 9, 1911, 40,000 to 69,000, four " " = + 17.8 
 
 Uccle, Sept. 13, 1911, 40,000 to 80,000, four " " = + 4.8 
 
 Uccle, Nov. 9, 1911, 40,000 to 90,000, two = + 2.5 
 
 Table 85 contains a summary of the Uccle, Nov. 9, 1911, 
 ascension, where the gravity differences may be examined in the 
 
382 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 I 
 
 f 
 
 1 
 
 !| 
 
 R 1 
 
 g 8 
 
 l< 
 
 S 1 g 
 
 w I I 
 ^ H 
 
 S 
 
 W 
 
 B 2 
 
 fe 
 O a> 
 
 8? 
 
 s ^^ 
 
 ^ ^ 
 
 W o 
 
 Efl 
 
 
 
 
 S 
 
 
 
 .2 * 
 
 11 
 
 Ml* 
 
 II 
 
 <NOasooooooooot-t-i>t-t-t-t-t>t>t-t-t-<o<o<o 
 77 I I I I I I I I I I I I I I I I I 1 | | | 
 
 ^sss'sskssss^S 
 
 xxxxxxxxxxxxxxxxg^ 
 
 rHt-mcooio-'*a>oix>-'*?ooQ'<f-rt<ooc<ioo 
 
 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 i 
 
 imioicioiomiijioioioioicus 
 I I I I I I I I I I I I I I I I I I I I I I I 
 
 O^f^^fCDCiOQiCOi 
 TH(N(N(N(N03eOCO 
 
 f?f?TTTTTTT7TTrrrrrttrT 
 
 ooooooooooooooooooooooo 
 XXXXXXXXXXXXXXXXXXXXXXX 
 
 
 TTTTTTTTTTTTTTTTTTrr 
 oooooooooooooooooooo 
 
 XXXXXXXXXXXXXXXXXXXX! 
 
THE THERMODYNAMIC TERMS 
 
 383 
 
 eoeoeoeocoeocoeoeocoweococoeocoeocoeoOTcoeocoeoeoeococo 
 
 xxxxxxxxxxxxxxxxxxxxxxxxxxxx 
 
 kO t> ai r-i rH N <N ^' 10 t> 00 I-! T-! i-( 
 
 
384 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 sixth column. They are smaller from 90,000 to 65,000 than 
 from 65,000 to 40,000 meters. The inference is that the entire 
 thermodynamic system up to vanishing quantities is reliable for 
 numerous researches depending upon the data. In order to 
 illustrate how small these quantities really are, we have at 
 65,000 meters, B = 0.001225 millimeter of mercury, p = 
 0.00000009629 C.G.S., while hydrogen at normal surface con- 
 ditions is p H = 0.00008924 grJcm 3 . The remarkable precision 
 of the computations is proved by the system of checks from the 
 surface to 90,000 meters. 
 
 These data above 40,000 meters, therefore, modify the data 
 of Table 82, and may be substituted for them. The results 
 of further studies on these computations will be published in 
 Bulletin No. 4 of the Argentine Meteorological Office. It may 
 be here remarked that the thermal efficiency of the atmosphere 
 begins at about 65,000, there being little absorption above it. 
 The shape of the absorbing area above 40,000 meters has been 
 changed by this extension, but the amount is not very different, 
 as will be illustrated. Compare Figs. 74, 75, 79. 
 
 Summary of the Computations for Twenty-one Balloon Ascensions 
 
 In order to improve the data as far as practical, similar 
 computations were extended to twenty other balloon ascensions. 
 
 (10) United States, Omaha, February 21,22,23, 1911; Sep- 
 tember 28, 1909. Huron, September 1, 4, 7, 16, 1910. In- 
 dianapolis, October 6, 30, 1909. 
 
 (6) Europe, Lindenburg, July 27, 1908; April 27, May 5, 6, 
 September 2, 1909. Milan, September 7, 1907. 
 
 (5) Atlantic Tropics, Otaria, June 19, 1906; July 29, August 
 29, September 9, 25, 1907. The mean values for the several 
 quantities appear on Table 86, where the data can be conveniently 
 examined. 
 
 (Ei Eo). The same values were adopted throughout, 
 but since balloon ascensions now reach 28,000 meters the critical 
 value at that elevation can be further examined. 
 
 (61 - 60), (Ai - 4 ). These areas are seen on Fig. 75, and 
 
SUMMARY FOR TWENTY-ONE BALLOON ASCENSIONS 385 
 
 they have been described. It should be noted that the course 
 of (Qi - Qo) in the lower absorption region does not in the 
 least follow the distribution of the aqueous vapor contents, 
 which is at a minimum where (Qi Q ) is at its maximum. This 
 problem is very difficult to solve satisfactorily. 
 
 p increases rapidly up to 2,000 meters, then more slowly 
 up to p = 1.000 in the true isothermal level at 27,000 meters; 
 above that level p falls to a small value p = 0.053 at 46,000 
 meters, and then rises to about unity on the vanishing plane. 
 
 k passes through inverse relations in respect of p. 
 
 J Q has a maximum value at 4,000 meters, and proceeds 
 irregularly to zero at the top. Its value can always be recovered 
 from J = $ (D + E) in the tables. 
 
 D has a large negative maximum at the surface, passes 
 through zero with change of sign near 18,000 meters, and gradu- 
 ally increases to its maximum + 0.1406 on the vanishing plane. 
 
 B falls from a negative maximum at the surface to zero at 
 the 13,000-meter level, at the bottom of the so-called isothermal 
 layer, and then increases irregularly to the top of the atmosphere. 
 
 J a begins with a maximum at the surface and gradually 
 falls to a small quantity, finally vanishing at the top. 
 
 p 1 has a nearly constant value of 0.800 in the convectional 
 region and probably about 0.500 above it. k l is correlative to p 1 . 
 
 Uccle, November 9, 1911 
 
 The data are fully computed for the Uccle ascensions, as 
 extended to 90,000 meters, and the results above 40,000 meters 
 will be briefly summarized. The total atmospheric radiation 
 energy, 
 
 I Jo = 2 ( Cl TV - c TV) 
 
 from the surface to 90,000 meters, is equivalent to the "solar 
 constant" at the distance of the earth, because it represents the 
 amount of heat required to maintain the existing temperature 
 distribution in equilibrium with the incoming and outgoing 
 radiations that are in operation day and night. Summarizing 
 the Jo data, we have, 
 
386 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 e C 
 
 3 % 
 
 ri o 
 
 CQ <; 73 
 
 O Tj-OOO 
 t^Ti-(N Os 
 
 t^oo O> O Oirft^O* 
 
 O> O O\ rf t- 
 oo M 1-1 M\O 
 MrO'<frOr-i 
 
 88 
 f I 
 
 O\t^OOOOOOOOOOO f*J fO O MOO N 
 OOO ^t O\ M Tf IH t^w O O O\ O> O\ 
 
 <fr o 00 O vO <N 
 
 row O\OO 
 
 s^ 
 
 'OO Ov 
 100 i^ 
 
 O O O\O\OvO\OO\OiOOoO t^t^OO >O 10 
 
 O\tO*frfOOOMOO>OO < <tNt- 
 
 Ol^OOt^i/5^t>-cO\OO>O/jTt 
 
 MWMWHMMOOOOO 
 
 ooooooooo 
 
 Ssssssssf Illi 
 
 I I I I I I I I I f f I I 
 
 S G 
 
 a c 
 
 ION H t^rorooo OvrO" O\O VO\O 
 
 OOOOOo"o'oMwwi-ii-ii-iMM)-i 
 OOOOOOOOOOOOOOOOO 
 
 I I I I I I I I I I I I I I I I I I I I I I I I I I I 
 
 o o 
 
 r r f r f r f r f f f f 
 
 888 
 i f f 
 
 :.:.!": :.'LJ:::1 
 
 888 
 i i i 
 
 I I I 
 
 O M 
 
 r^t^t^-vOOO ioio^t"T}--, 
 
 oooooooooooooooooooooo 
 
 I I I I I I I I I I I I I f I I I I f f I I 
 
 1 I I I I I 
 
 TjMoo ION O>\O roo r^Tj-M 
 TT^J-rOrOPONNNNWMM 
 
 oooooooooooo 
 
 ooo' v o v o v o o'o'o'o' 
 
 I I I I I I I I I I I I I I I I I I I I III I 
 
 I I I I I 
 
 80800^ 
 
 ^OO^OOLOs 
 
SUMMARY FOR TWENTY-ONE BALLOON ASCENSIONS 387 
 
 ^ I O\ O 00 O\ O 00 00 00 00 00 Oi O O\ 00 t^ O O 00 1/5 
 
 < I ::::::::: : ? *!". ^ 1 ".^ ".". ".". 
 
 i i i i 
 
 \O PO >/5 00 N OlOO /5 1/5 Ov 
 
 I ! ! I I '. . '. \ot^t^oot^oooooooooooo 
 
 oooot^i^oo 
 
 '. '. ' lSt^MOOi5o-"i-i 
 . . . .t^^Ot^Ot-OOOOOO 
 
 O O 
 O>M 
 t-OO 
 
 HI roroN>O t^oooo o> Ovoooo lO'tfO'O oi 
 
 ooooooooooooooooooo 
 
 r i* r r r r r r i" f r r \ \ \ \ r r i 
 
 fOt^-t^t^ooOvOvt^-O TtOOON 
 OOOOOOOOOOOOi-iOOOOO 
 
 i' i' \ \ \ \ \ \ \ \ \ \ \ \ i i i \ 
 
 MMNMMMMMNMT-'v^OO 
 OOOOOOOOOOOOOOOOOOMMMNWMOMMO 
 
 r r r r r r r r r r r r \ \ \ \ r \ \ \ \ \ \ \ \ \ \ \ \ 
 
 r^^*OoQWC*OM^' N OO^"OvOOQf^iot > -oo 
 
 : : : : : : : : : :?yo$l^|*5-t'S5 
 
 : r r r r r r r r r r r r r .r r r i* r i 
 
 ^^^J^^-r^M M N o 2 Tj-oo r^M POOv^-w Ttoo fOOO N O roo rt-N 
 OOOOOOOOOOOOOOwwMNro^tWfOfO^'-J 1 ^ 1 ^^ 1 ^ 
 
 ' r r f f r r r r i' \ \ \ \ \ \ \ \ r r 
 
 M \O *-" O\ 
 t^-oOOO 
 
388 
 
 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 ' 66,000 
 
 ' 76,000 
 
 " 56,000 
 
 ' 66,000 
 
 " 46,000 
 
 1 56,000 
 
 " 36,000 
 
 ' 46,000 
 
 " 26,000 
 
 ' 36,000 
 
 " 16,000 
 
 ' 26,000 ' 
 
 " 6,000 
 
 ' 16,000 ' 
 
 " Surface 
 
 ' 6,000 
 
 From the 76,000 to 90,000 meters, 2 J = -0.0001 
 
 -0.0032 
 -0.0372 
 -0.1981 
 -0.2952 
 -0.2914 
 -0.3275 
 -1.0151 
 - 1 . 5658 
 
 Total atmospheric radiation energy 3.7336 
 
 From the Huron ascension we obtained 3.8181. It is 
 apparently a question of distribution rather than the amount 
 that is concerned in these two temperature systems. The total 
 " solar constant" derived from the black atmospheric radiation 
 is about twice the amount measured by the pyrheliometer. More 
 data on this important subject will be accumulated. 
 
 First Method of Computing the Pyrheliometric Data. 
 The Effective Energy of Radiation and the Solar Constant. 
 
 When the preceding Chapter V, on radiation, was written it 
 was supposed that the extension of these computations would 
 be limited to the lower levels in the atmosphere where direct 
 temperatures were obtained in balloon ascensions. But 'it is 
 now evident that temperatures may be measured, or assumed 
 by trial, and they will be correct provided they produce such 
 values of P. p. R. T. in successive stages as will satisfy the 
 gravity equation with small residuals, 
 
 Pi -Po 
 
 (196) g (zi - ZQ) = - - J (qi 2 - 2 ) - (Qi ~ Co). 
 
 Pio 
 
 Hence, by a series of trial computations, it has become 
 possible to extend the data up to 90,000 meters, and four balloon 
 ascensions have been computed up to 70,000 or 90,000 meters. 
 We thus obtain by (335), (333) 
 (Qi ~ Co) = (Wi - W ) + (Ui - U ) = P 10 (vi -%) + (tfi- Z/o), 
 
 = K 
 
 -Co 
 
 = Ja 
 
 for selective radiation, and 
 
THE CONSTANTS AND COEFFICIENTS FOR DRY AIR 
 
 389 
 
 (344) 
 
 J = ci 2V - c TV 
 
 for black-body radiation, in each 1,000-meter stratum from the 
 surface upward, and thence the total thermodynamic and 
 radiation energies throughout the atmosphere by taking the 
 summations for all of the strata. The results are summarized 
 in Tables 82, 85, and Figs. 75, 79. Similarly, the computations 
 include the data of the Kinetic Theory of Gases, (H 9 U) external 
 and internal energies; (q. f) velocities; (n. N) number of mole- 
 cules, l m ax free path length, y number of collisions, W H mass of 
 the hydrogen atom, e^ the negative ion charge, in all strata to 
 90,000 meters. Tables 96, 97. See BuUetin No. 4, O. M. A. 
 
 TABLE 87 
 
 SUMMARY OF THE RESULTS OP THE COMPUTATIONS ON THREE BALLOON 
 ASCENSIONS FOR THE VALUES OF THE THERMODYNAMIC AND RADIATION 
 ENERGIES 
 
 Station and Date of the Observations 
 Height in Meters of Temperature 
 
 Uccle 
 June 9, 1911 
 70,000 
 
 Uccle 
 Sept. 13, 1911 
 80,000 
 
 Uccle 
 Nov. 9, 1911 
 90,000 
 
 2%(zi 20)] gravity acceleration .... 
 2[ (<?i 2 <7o 2 )] kinetic energy of 
 circulation ... 
 
 9.5792 
 0.0000 
 5.4250 
 
 4.1542 
 4.1146 
 
 3.9690 
 
 11.0895 
 -0.0006 
 6.6676 
 
 4.4219 
 4.1608 
 
 3.9261 
 
 12.4565 
 0.0002 
 8.6590 
 
 3.7975 
 3.9386 
 
 3.5373 
 
 2[ (Qi Qo)] free heat (non-adia- 
 batic) 
 
 S[0(2i-2o) +M?i 2 -?o 2 ) -f- 
 (Qi-Qo)] summary 
 
 L PlQ J " 
 
 S[/o = CiTV - CoTV] black body ra- 
 diation 
 
 2 [K i KO] radiation energy . . 
 
 1.4528 
 
 0.0000 
 1.4602 
 
 1.4603 
 
 1.4672 
 
 -0.0087 
 1.4404 
 
 1.4600 
 
 1.4802 
 
 0.0352 
 1.4085 
 
 1.4395 
 
 t 
 
 2 rQi Qo~| free heat per volume 
 l_ v\. #o J change 
 
 2 [Pi Po] pressure differences 
 2[/ = CiTi al - c 7> ] selective ra- 
 diation 
 
 
 General mean of the thermodynamic data 
 General mean of th&black body radiation data / 
 General mean of the radiation and pressure data . 
 General mean of the selective radiation data J a . 
 
 4.0979 
 3.9476 
 1.4516 
 1.4536 
 
390 
 
 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 Table 87 shows that the thermodynamic state of the earth's 
 atmosphere is such that about 4.00 gr. cal./cm. 2 min. is required 
 to raise it from the frozen and solid state of air at A into 
 the expanded state it now has with its P. p. R. T. in all strata. 
 This is, also, shown to be equal to the black body radiation J Q 
 as derived from the thermodynamic data. The selective absorp- 
 tion J a in a state for continuous emission is about 1.46 calories, 
 and this is equal to the temporary amount of energy in the 
 (K! - KQ) and (Pi - Po) terms. 
 
 Table 88 contains a summary of the terms in the equation of 
 equilibrium, (785) S = /i + B + J a and h = R. 
 
 TABLE 88 
 
 THE TERMS IN THE "EFFECTIVE" RADIATION FORMULA, 
 
 WHERE h = R 
 
 R J a 
 
 z 
 
 5 
 
 Ja 
 
 h =R 
 
 P 
 
 Pw 
 
 Wo 
 
 90000 
 
 4.00 
 
 0.000 
 
 2.000 
 
 1.000 
 
 
 14000 
 
 76000 
 
 4.00 
 
 0.000 
 
 2.000 
 
 1.000 
 
 
 10000 
 
 66000 
 
 4.00 
 
 0.000 
 
 2.000 
 
 1.000 
 
 
 10000 
 
 56000 
 
 4.00 
 
 0.000 
 
 1.999 
 
 0.999 
 
 
 10000 
 
 46000 
 
 4.00 
 
 0.007 
 
 .993 
 
 0.998 
 
 
 10000 
 
 36000 
 
 4.00 
 
 0.034 
 
 .983 
 
 0.996 
 
 
 6000 
 
 36000 
 
 4.00 
 
 0.070 
 
 .967 
 
 0.994 
 
 . . . 
 
 3000 
 
 27000 
 
 4.00 
 
 0.100 
 
 .950 
 
 0.992 
 
 . 
 
 3000 
 
 24000 
 
 4.00 
 
 0.130 
 
 .935 
 
 0.991 
 
 
 3000 
 
 21000 
 
 4.00 
 
 0.170 
 
 .915 
 
 0.988 
 
 
 3000 
 
 18000 
 
 4.00 
 
 0.220 
 
 .890 
 
 0.984 
 
 
 3000 
 
 15000 
 
 4.00 
 
 0.295 
 
 .853 
 
 0.975 
 
 
 3000 
 
 12000 
 
 4.00 
 
 0.410 
 
 .795 
 
 0.961 
 
 
 3000 
 
 9000 
 
 4.00 
 
 0.580 
 
 .710 
 
 0.950 
 
 
 3000 
 
 6000 
 
 4.00 
 
 0.776 
 
 .617 
 
 0.929 
 
 0.920 
 
 3000 
 
 3000 
 
 4.00 
 
 1.050 
 
 .475 
 
 0.887 
 
 0.890 
 
 3000 
 
 000 
 
 4.00 
 
 1.461 
 
 .270 
 
 0.840 
 
 0.840 
 
 
 /i = the "effective" radiation = 2.00 at 90,000 meters. 
 
 S = the solar constant = 4.00 calories. 
 
 R = the " reflected" radiation = 2.00 at 90,000 meters, and 
 
 neutralizes one-half the solar constant, /i = R on all 
 
 levels, except during changes in T. 
 J a = the absorbed radiation in the lower levels, small at 
 
 40,000 meters, and increasing by variable values of p 
 
THE EFFECTIVE ENERGY OF RADIATION 391 
 
 in the Bouguer Formula, for mo about 3,000 meters, 
 
 / = /.#-. 
 
 p = the values of the coefficient of transmission formed by 
 
 p = using the values of 7 for different 3,000-meters. 
 
 lo 
 
 p w the general observed values of p at the sea level, at 3,000 
 meters as La Quiaca, with its probable value at 6,000 
 meters. 
 m = the adopted depth of the stratum in the zenith. 
 
 It is evident that p is not constant throughout the atmos- 
 phere, and that mo is not to be taken as unity for the 90,000 
 meters. On the other hand, 
 
 7T ^ m > ^ m2 A. m * x 1* n 
 
 = /O pl p2 p3 p n , 
 
 for variable (p. m.). 
 
 /i= the effective radiation as observed by the pyrheliometer. 
 
 /o = 2.00 cal. at 90,000 meters, 
 /o penetrates to 30,000 meters undepleted, and then 
 
 diminishes by a variable p, constant through m = 3,000 
 
 meters, in the Bouguer Formula, to 1.46 calories at the 
 
 surface. 
 R = /i at every level. This return current neutralizes one-half 
 
 of the solar constant S = 4.00 calories. 
 J a = the absorbed energy in the lower levels, and together 
 
 with R neutralizes J S. 
 
 The Bolometer measures the transmitted parts of S. 
 The Pyrheliometer measures the transmitted parts of 7 , 
 that is, one-half the "solar constant." 
 
 Figure 76 shows the relations of these terms throughout the 
 atmosphere. The effective radiation /o proceeds downward 
 till it is deflected by J a acting outward, arriving at the surface 
 as 1 1. Simultaneously, the returning radiation R = I\ in every 
 level, during temperature equilibrium, reaches the top of the 
 atmosphere, with the addition of J a in the lower levels, 
 having the value R = 2.00 cal. which neutralizes one-half 
 of the incoming S = 4.00. It is easily seen that the pyrheli- 
 
392 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 E = 1.950 S = 3.950 
 
 
 I 0.000 
 
 0.500 1.000 1.500 
 
 Scale for the density p 1.200 
 
 2.000 
 0.960 
 
 2.500 
 0.720 
 
 3.000 
 0.480 
 
 3.500 
 0.210 
 
 4.000 
 0.000 
 
 FIG. 76. Two methods of discussing the pyrheliometer observations. 
 
 p 
 
 A 
 
 B 
 
 Area 
 
 Area 
 
 Area 
 
 Ja 
 
 J 
 
 = the solar constant 3.95 gr. cal./cm. 2 min. 
 
 = the radiation energy absorbed in producing temperature T. 
 
 = the density curve. 
 
 = the probable pyrheliometer curve. 
 
 = the probable bolometer curve. 
 
 S B = the temperature radiation. 
 
 B A = the scattered radiation. 
 
 A = the free-heat radiation. 
 
 absorbed energy in 1000 meters. 
 
 black body energy in 1000 meters. 
 
 intensity of zenith sun. 
 
 = I \ / pa- 
 
THE EFFECTIVE FORCE OF RADIATION 
 
 393 
 
 ometer measures only the effective radiation 7 = J S, while 
 the bolometer measures such ordinates as are necessary for the 
 construction of a 4.00 calories curve at solar temperature be- 
 tween 6,900 and 7,000. 
 
 If we supply by simple interpolation the missing ordinates 
 on the energy spectrum curve for 6,900, so that the wave-length 
 interval is A A = 0.05 /*, and similarly fill out Abbot's ordinates 
 for Washington, D. C. (34 m), Mt. Wilson (1,780 m), Mt. 
 Whitney (4,420 m), (Bui. No. 3, O. M. A. Tables XXVI, 
 XXVII), we have the following results for the sum of the ob- 
 served ordinates, that is, the relative areas of transmission. 
 
 TABLE 89 
 
 COMPARISON OF THE ABSORBED ENERGY BY THE BOLOMETER AND THERMO- 
 
 DYNAMIC DATA 
 
 Energy Spectrum 
 
 2J 
 
 D 
 
 t 
 
 Ratio 
 
 Ja 
 
 fci 
 Ratio 
 
 6900 
 
 80.295 
 
 
 
 4.00 
 
 
 Washington 
 
 51.19 
 
 29.105 
 
 0.363 
 
 1.45 
 
 0.363 
 
 (34) 
 Mt. Wilson 
 
 59.55 
 
 20.745 
 
 0.258 
 
 1.15 
 
 0.287 
 
 (1780) 
 Mt. Whitney 
 (4420) 
 Final curve . . . 
 
 61.38 
 67.26 
 
 18.915 
 13.035 
 
 0.236 
 0.162 
 
 0.90 
 0.65 
 
 0.225 
 0.163 
 
 (8000) 
 
 
 
 
 
 
 S / is the sum of the ordinates of transmission at each station. 
 
 D is the absorbed part =27 (6,900) - S / (station). 
 
 k is the ratio of absorbed part at each station to J (6,900). 
 
 J a is the computed thermodynamic absorption at same levels. 
 
 k 1 is the ratio of the absorbed part at each station to 4.00. 
 
 It is seen that these two ratios (k, k 1 ) are substantially the 
 same, except for minor variations. 
 
 It must be concluded that the amount of absorption in the 
 lower atmosphere J a is about the same as the amount of deple- 
 tion in the bolometric energy spectrum, referred to a 6,900 
 curve, or 4.00 calories for the solar constant. This analysis 
 points to a very different method of discussing pyrheliometer 
 
394 THE EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 observations from that commonly practised, and it promises to 
 harmonize the several branches of this hitherto conflicting 
 subject. 
 
 SUMMARY 
 
 1. The bolometer curves as observed are best satisfied by 
 an energy spectrum of 6,900 or a solar constant of 4.00 gr. cal./ 
 cm. 2 min. 
 
 2. The thermodynamics of the atmosphere requires an ex- 
 penditure of 4.00 calories to produce and maintain the existing 
 pressures, densities, temperatures, and thermodynamic values 
 up to 90,000 m. 
 
 3. The pyrheliometer measures only one-half the solar 
 constant, that is, the efficient energy 2.00 calories, because one- 
 half of the incoming radiation is neutralized by the returning 
 energy stream of 2.00 calories. 
 
 4. The amounts of absorbed radiations, as measured by the 
 bolometer ordinates or by the thermodynamic conditions, agree 
 in giving about the same ratios (k, k 1 ). 
 
 5. The conclusions derived from the extensive computations 
 summarized in this section verify, generally, the text in 
 Chapter V. 
 
 Change of Theory 
 
 The foregoing analysis is based upon the view that the 
 pyrheliometer measures the efficient incoming solar radiation at 
 2.00 calories, and upon the formula that half the incoming ray 
 advances to the surface while half of it is scattered back to space. 
 Further experience brings both these ideas into doubt, and we 
 proceed to give evidence that the I curve of Fig. 76 should 
 be made the I a curve, thus enclosing the scattering of radia- 
 tion between the curves A and B. 
 
 The Second Method of Discussing Pyrheliometric Data 
 
 The discussion of pyrheliometer observations begun in 
 Chapter V has been continued by the development of a new 
 
A NEW METHOD OF DISCUSSING PYRHELIOMETRIC DATA 395 
 
 method of computation, which will be briefly described. Com- 
 pute from the observed pyrheliometer readings the values 
 of the intensity /i, for the sun in the zenith, and with p w) 
 the unconnected coefficient of transmission, extrapolate I the 
 intensity reduced to the mean solar distance. Abbot multiplies 
 I by a bolometer factor, 1.123 for Washington, and 1.094 for 
 Mt. Wilson on the average, to which a small correction is then 
 added for the effect of the aqueous vapor pressure e. We pro- 
 ceed to develop the data in another way. Collect the individual 
 observations in convenient groups, according to the observed 
 values of p w , as (0.900 - 0.880), (0.880 - 0.860), ... for 
 I OJ /i, 6j and take the mean values, such as appear in Table 90 
 for Washington, D. C., at 34 meters, Bassour at 1,160 meters, 
 La Quiaca at 3,465 meters. Plot on diagrams with p w for 
 abscissas, and values of 7 , /i, in gr. cal./cm. 2 min. for ordinates. 
 It is seen that in the case of Washington, Fig. 77, that I is a line 
 sloping downward to meet h sloping upward in the contact point 
 on the ordinate I a = 1.528; in the case of Bassour I is a hori- 
 zontal line meeting the upward sloping h in the contact point 
 1.680; in the case of La Quiaca I is an upward sloping line to 
 meet /i in the contact point 2.010. In each case the contact 
 point is on the ordinate axis to which corresponds the coefficient 
 of transmission p w = 1.000, which is that for perfect transmission. 
 Whatever may be the physical cause of the sloping of the I 
 lines, below or above the horizontal, the contact point has 
 eliminated that cause from the system. In Case III, Wash- 
 ington, it is necessary to depress the sloping line I into a hori- 
 zontal position I a ') in Case II, Bassour, this is already done; in 
 Case I, La Quiaca, the sloping line must be raised to the horizontal 
 position 7 a . In Case III, if Ii/p w = I , then Ii/p a = /a, 
 where p a is larger than p w ; in Case I, if h/p w = I , then 
 I\/pa = fa, where p a is smaller than p w . It is necessary to 
 determine an equation for each station depending upon the 
 aqueous vapor pressure e which will convert p w into the required 
 p a . This is done in Section 2 of Table 90. Assume p l w at con- 
 venient intervals 1.000, 0.980, 0.960, . . . ; take the contact 
 point I 1 from the diagram, and 7 1 ! from the mean line on the 
 
396 
 
 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
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COMPUTATION OF P a AND I a 
 
 397 
 
 diagram; then compute p a = y. Plot down e on an auxiliary 
 
 lo 
 
 ordinate scale for the same axis of abscissas, and draw the mean 
 e. Now it is evident that if the aqueous vapor effect is to be elimi- 
 nated the line e should pass through the origin where p w = 1.000. 
 In order to do this, draw a line through the origin parallel to e, 
 that is (e + 2.4) mm. for Washington, (e 7.5) mm. for Bassour, 
 and (e + 5.0) mm. for La Quiaca. Take p a p w = F (e + B\ 
 where F = the coefficient, in fact the ratio of the ordinate of e 
 to the ordinate of h counted from the horizontal line 7 a . Hence, 
 we find the equation, 
 
 where B is the vertical ordinate between the two vapor-pressure 
 lines. In type II, p a = p w ; in type III, p a = p w + F (e + B) ; 
 in type I, p a = p w F (e + B). At each station there is to be 
 computed such a value of the coefficient of transmission p a as 
 will make the values of the extrapolated /i fall upon the hori- 
 zontal line / a , instead of upon the sloping line I . If they do 
 fall upon a sloping line, it follows that the resulting mean values 
 of a series, winter series, for example, will differ radically from the 
 summer series, because the means will pertain to different groups 
 of p w . Take the mean values of each half of the groups in 
 Section 1: 
 
 Station 
 
 Pw 
 
 /o 
 
 Pw 
 
 /o 
 
 Difference 
 
 Washington 
 Bassour 
 
 0.854 
 0.869 
 
 1.633 
 1.676 
 
 0.772 
 0.763 
 
 1.687 
 1.674 
 
 + 0.054 Type III 
 -0.002 Type II 
 
 La Quiaca 
 
 914 
 
 1 893 
 
 876 
 
 1 819 
 
 -0 074 Type I 
 
 
 
 
 
 
 
 This is evidently one cause for the incessant variations that 
 characterize pyrheliometer mean values. It is plain that many 
 such fluctuations as have been attributed to solar action are, in 
 fact, due to the imperfect elimination of the terrestrial effects of 
 aqueous vapor and dust as well as density from the intensity 
 of radiation at the station. 
 
398 
 
 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 Summary of the Correction Equations 
 
 Calories 
 
 Washington D.O., Height 34m. 
 
 Ba 88 ourAlgeria,HeightllCOm. ^ Quiaca Height 
 
 2.000 
 
 Type III 
 
 Type II 
 
 Type I 
 
 1.900 
 1.800 
 
 1.700 
 1.600 
 1.500 
 
 1.400 
 1.300 
 
 1.200 
 1.100 
 1.000 
 
 O O 00 
 
 e'=(e- 
 
 7.5) = O 
 
 pw. 1.000 0.900 
 
 0.800 
 
 1.000 
 
 0.900 
 
 0.800 
 
 1.000 0.900 
 
 FIG. 77. Three types of pyrheliometric data 
 
 Type I. 
 
 La Quiaca, height 3,465 meters, p a = p w - 0.0080 (e + 5.0). 
 Mt. Wilson, " 1,780 " p a = p w - 0.0020 (e + 2.0). 
 La Confianza (4,483), Mt. Whitney (4,420), Humahuaca 
 (2,939), Maimara (2,384), have similar equations, but the 
 series of observations is too short to determine the constants 
 
 Type II. 
 
 . Bassour, height 1,160 meters, p a = #,. 
 Mt. Weather, " 526 " p a = p w . 
 There is some uncertainty about the equation for Mt. Weather 
 on account of the indecisive data of I . 
 
 Type III. 
 
 Jujuy, height 1,302 meters, p a = p w + 0.0060 (e - 6.0). 
 Cordoba, " 438 " p a = p w + 0.0087 (e - 2.6). 
 Pilar, " 340 " p a = p w + 0.0087 (e - 2.6). 
 
 Potsdam, " 89 " p a = p w + 0.0090 (e + 0.0). 
 Washington" 34 " p a = p w + 0.0100 (e + 2.4). 
 
 The coefficient F diminishes from + 0.0100 at Washington 
 
REDUCTIONS TO SEA LEVEL 
 
 399 
 
 (34) through 0.000 at Bassour (1,160), to - 0.0080 at La Quiaca 
 (3,465); the amount of the B is very variable with the local 
 conditions of the vapor-pressure e at the station. 
 
 Apply these equations to the values of /i and compute 7 fl . 
 
 TABLE 91 
 WASHINGTON, D. C., STATION EQUATION, p a 
 
 p w + 0.0100 (eX2A) 
 
 No. 
 
 Pw 
 
 e 
 
 e +2.4 
 
 A 
 
 Pa 
 
 * 
 
 la 
 
 Isea 
 
 A/ 
 
 
 7 
 
 .884 
 
 3.78 
 
 6.2 
 
 .062 
 
 .944 
 
 1.410 
 
 1.497 
 
 5 
 
 - 
 
 
 16 
 
 .874 
 
 3.00 
 
 5.4 
 
 .054 
 
 .928 
 
 1.402 
 
 1.511 
 
 *t 
 
 Js 
 
 
 20 
 
 .865 
 
 2.09 
 
 4.5 
 
 .045 
 
 .910 
 
 1.409 
 
 1.548 
 
 rH 
 
 ^ 
 
 
 16 
 
 .854 
 
 2.60 
 
 5.0 
 
 .050 
 
 .904 
 
 1.392 
 
 1.540 
 
 *0> 
 
 <fl 
 
 
 22 
 
 .846 
 
 3.34 
 
 5.7 
 
 .057 
 
 .903 
 
 1.386 
 
 1.535 
 
 > 
 
 B 
 
 
 24 
 
 .834 
 
 4.06 
 
 6.5 
 
 .065 
 
 .899 
 
 1.410 
 
 1.568 
 
 "J 
 
 fi 
 
 
 17 
 
 .824 
 
 3.80 
 
 6.2 
 
 .062 
 
 .886 
 
 1.353 
 
 1.527 
 
 
 
 '~ 
 
 rt 
 
 
 17 
 
 .815 
 
 3.69 
 
 6.1 
 
 .061 
 
 .876 
 
 1.418 
 
 1.619 
 
 ti 
 
 
 
 
 10 
 
 .804 
 
 4.30 
 
 6.7 
 
 .067 
 
 .871 
 
 1.334 
 
 1.532 
 
 ^ 
 
 JS 
 
 mean 1.542 
 
 14 
 
 .795 
 
 5.02 
 
 7.4 
 
 .074 
 
 .869 
 
 1.349 
 
 1.553 
 
 *-, 
 
 ** 
 
 
 18 
 
 .784 
 
 5.10 
 
 7.5 
 
 .075 
 
 .859 
 
 .316 
 
 1.532 
 
 0) 
 
 1 
 
 
 8 
 
 .773 
 
 5.10 
 
 7.5 
 
 .075 
 
 .848. 
 
 .303 
 
 1.537 
 
 
 
 
 
 9 
 
 .764 
 
 5.19 
 
 7.6 
 
 .076 
 
 .840 
 
 .335 
 
 1.590 
 
 > 
 
 d 
 o 
 
 
 11 
 
 .756 
 
 6.10 
 
 8.5 
 
 .085 
 
 .841 
 
 .272 
 
 1.512 
 
 bit) 
 
 ' 
 
 
 7 
 
 .743 
 
 7.03 
 
 9.4 
 
 .094 
 
 .837 
 
 .260 
 
 1.506 
 
 g 
 
 E 
 
 
 5 
 
 .736 
 
 7.69 
 
 10.1 
 
 .101 
 
 .837 
 
 .262 
 
 .508 
 
 
 8 
 
 
 5 
 
 .714 
 
 8.85 
 
 11.3 
 
 .113 
 
 .827 
 
 .252 
 
 .514 
 
 <u 
 
 o 
 
 
 3 
 
 .703 
 
 7.40 
 
 9.8 
 
 .098 
 
 .801 
 
 .170 
 
 .461 
 
 H 
 
 H 
 
 
 4 
 
 .696 
 
 6.06 
 
 8.5 
 
 .085 
 
 .781 
 
 .212 
 
 .552 
 
 
 
 mean 1.527 
 
 
 
 
 
 
 
 
 .534 
 
 1.525 
 
 -0.009 
 
 
 The computed values of I a from p a , in place of I from /?,, 
 plot near the dotted line in the diagram. The mean of the first 
 nine values is 1.542, and of the last ten values 1.527, showing 
 that the variations in p w , and the effect of the vapor-pressure e, 
 are quite well eliminated. A longer series would probably prove 
 that the elimination is complete. 
 
 Table 92 contains the summary of the computed I a for nine 
 stations and their mean values. If these mean values of I a are 
 plotted, for the height z on the axis of ordinates and calories 
 on the axis of abscissas, the points fall nearly on a straight line, 
 so that I a diminishes in proportion to the height. This line cuts 
 the sea level at 1.525 calories, and the differences which corre- 
 
400 
 
 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
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ANNUAL MEAN VARIATIONS 401 
 
 spond to the station height are readily computed. It is equally 
 easy to determine the normal value of I a at any station, and 
 several examples are added for stations having only short series. 
 
 The Annual Mean Variations 
 
 The station equations can be applied to the individual p w 
 groups, or to individual observations, during different years, 
 and their mean sea-level values are collected in Table 93; also, 
 the annual means are taken for all stations which were observed 
 during the same year. We have such means extending from 
 1903 to 1914, and they are plotted on the upper curve of Fig. 78, 
 giving minima in 1903, 1907, 1912, 1914. The 1907 and 1912 
 minima are already well known. In the second curve of Fig. 78 
 is reproduced the adopted mean meteorological curve from 
 Bulletin No. 1, Oficina Meteorologica Argentina, written in 
 1910, which contained a summary of data till 1910, and a forecast 
 from 1910 to 1915. It should be noted that the forecast for 1911, 
 1912, 1913, was very well verified, as it has actually been in all 
 other of the Argentine meteorological data. In 1914 the mini- 
 mum, which the forecast placed in 1915, seems to have come in 
 1914, but this is very unexpected, because there is an interval of 
 only two years following 1912, whereas the ordinary periodic 
 interval is 3.75 years. Another forecast is added with maxi- 
 mum in 1917 and minimum in 1919. It is necessary to maintain 
 suitable solar observations, in order to study the causes of 
 such irregular fluctuations in the output of the solar radiation, 
 and several solar physics observatories, adapted to meteorolog- 
 ical purposes, are indispensable in the interests of long-range 
 forecasts. 
 
 The observatories at Pilar and La Quiaca, Argentina, are 
 well adapted to supplement the work of Washington and Mt. 
 Wilson in the United States. 
 
 The General Summary 
 
 It has been shown that the mean values of I a plot along a 
 straight line with a given slope. On Fig. 76 is plotted S = 3.95 
 the solar constant; S J a the curve of the absorbed radiation from 
 
402 
 
 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 
 s 
 
 i : ill : 
 
 
 
 
 
 
 
 
 rH 
 
 rH i H rH rH 
 
 
 
 
 
 
 
 
 
 CO 
 
 : i : : : 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 a 
 s 
 
 ^4 C^ tO t^ t^* " CO 
 
 rH rH rH O rH rH 
 
 o to to o o to 
 
 
 
 
 
 
 
 
 
 rH . 1 , 1 r ( rH 
 
 
 
 
 
 
 
 
 iH 
 
 s 
 
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 rH rH rH rH 
 
 
 
 
 
 
 
 
 O 
 
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 s 
 
 tij tij 41^ 
 
 
 
 
 
 
 
 
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GENERAL SUMMARY 
 
 403 
 
 the outer limit to the station level, or it may be taken as (S 
 2 J a ) = B y plotting S J a from the ordinate S; the density 
 curve p is plotted with the abscissa as indicated; the pyrheliom- 
 eter curve I a = A is plotted up to 4,500 meters, and its course 
 follows the p curve so closely as to suggest that these two func- 
 tions run out together, that is, at the solar constant 3,950 calories. 
 If Abbot's ordinates for the bolometer are filled up by proper 
 interpolations for the several intermediate wave lengths, and 
 the sums taken, we shall have good relative values of the several 
 bolometer energy areas. 
 
 TABLE 94 
 SUMMARY OF THE BOLOMETER DATA 
 
 Bolometer Data 
 
 Abbot's 
 Scale 
 Sum. 
 
 Calories 
 
 Factor 
 of 
 Reduction 
 
 Calories 
 B 
 
 Height 
 
 z 
 
 Black body 6900 A 
 Abbot's extrapolated 
 
 80.30 
 67.26 
 
 3.876 
 
 20.7 
 
 3.876 
 3.25 
 
 23000 
 6500 
 
 Mt. Whitney 
 
 61 38 
 
 
 
 2 97 
 
 4420 
 
 Mt. Wilson 
 
 59 55 
 
 
 
 2 88 
 
 1780 
 
 Washington, D. C 
 
 51.19 
 
 
 
 
 2.47 
 
 34 
 
 At the solar temperature 6900 A, for black body radiation, 
 the value at the outer limit of the earth's atmosphere is 3.876, 
 and corresponding to this the sum of Abbot's ordinates on his 
 arbitrary scale is 80.30. The factor of reduction is taken 20.7, 
 and the calories B for the several stations at the height z follow. 
 Plot these on Fig. 76, and they seem to belong to the S J a curve. 
 We conclude that the pyrheliometer data approximate the p 
 density curve, and the bolometer data represent the selective 
 radiation curve S/ a . The total solar radiation now divides 
 itself into three parts: 
 
 (1) Radiation intensity S J a used in producing temperature. . 1.46 
 
 (2) Radiation energy scattered back to space 1.00 
 
 (3) Free radiation energy penetrating to the sea level , . 1.52 
 
 Total or solar constant energy in calories 3.98 
 
 The corresponding temperature would be 6930 A. 
 The 7o curve of the first method has been changed into 
 the I a - curve by the second method of computation. 
 
404 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 GENERAL REMARKS 
 
 On Fig. 76 are drawn the typical curves from the data of 
 Chapter VII: J a the selective radiation absorbed in each 1,000- 
 meter stratum, and 2 J a the total amount absorbed down to 
 the given level; J the black body radiation in each 1,000-meter 
 stratum/ anpl S/ the total amount down to the given level; 
 /i the computed radiation for the zenith sun on each level, I 
 the computed Ii/p w , and I a = Ii/p a ; the density curve p; the 
 solar constant S = 3.950, and Abbot's solar constant = 1.950. 
 
 (1) Abbot multiplies I by a small bolometer factor, and adds 
 a small function of e to produce E = 1.950 on each level. 
 Bigelow's curve I a , derived from the contact points for p w = 
 1.000 at perfect transmission, crosses I at an angle, because 
 Abbot's mean values of I are too large at low-level stations, and 
 too small at high-level stations, both systems agreeing for p w = 
 pa at about 1,200 meters, the level of the cumulus cloud base. 
 The pyrheliometer curve A, following closely upon the density 
 curve p, will run out at about 3.950 colories, in conformity with 
 the S/ = 3.950 calories. 
 
 (2) The bolometer ordinates indicate an amount of energy 
 well represented by the curve S/ or B, so that Abbot's small 
 bolometer factor should be greatly increased, up to 2.00 in the 
 higher levels. The theory which led to his small bolometer 
 factor, namely, that the pyrheliometer registers the same amount 
 of heat as that indicated by the bolometer, when the small 
 corrections have been applied, is erroneous. The curve I 
 does not represent the true pyrheliometer intensities, which are 
 7 fl , nor does it take any account of the energy scattered in the 
 area between A and B, nor of the heat absorbed in making the 
 temperatures T in the area between B and S. 
 
 (3) t The pyrheliometer does not directly measure the solar 
 constant at low levels, and the Bouguer formula is incapable 
 of making the complete extrapolations; the bolometer does not 
 measure the solar constant except by an approximate inference; 
 the thermodynamics of the atmosphere affords data for com- 
 
GENERAL REMARKS 405 
 
 puting S/a and S/ in all levels, and the final sum for S/ at 
 the sea level is the solar constant, about 3.950 calories. 
 
 (4) The thermodynamic computations are in accord with this 
 interpretation of the pyrheliometric data and the bolometric 
 data, hi requiring about 3.950 calories to do the work actually 
 measured in the atmosphere, and it would be impossible to 
 maintain the existing temperature, pressure, and density distri- 
 butions by expending only 1.950 calories. The pyrheliometer is 
 useful in determining the relative annual variations of the solar 
 radiation at any station; the bolometer is necessary in studying 
 the relative absorptions of different wave lengths in the atmos- 
 pheres of the sun and of the earth; but the thermodynamic 
 data are required for the mutual interpretation of these two 
 types of data. 
 
 This subject is treated at greater length in Bulletin No. 4, 
 Oficina Meteorologica Argentina, 1914. 
 
406 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 The Constants and Coefficients of Dry Air in the Kinetic Theory 
 of Gases throughout the Atmosphere 
 
 It facilitates the discussion of the problems of the upper 
 layers of the earth's atmosphere to have the data of the Kinetic 
 Theory of Gases for Dry Air in a form for reference. Such a 
 table of constants and coefficients for the (M. K. S.) and the 
 (C. G. S.) systems is given in Table 95, and the results of their ap- 
 plication from the bottom to the top of the earth's atmosphere 
 as computed in Tables 96, 97. The fundamental data for N = 
 6.062 X 10 23 ,e = 4.774 X 10 ~ 10 , conform to the summary of con- 
 stants by R. A. Millikan, Physical Review, August, 1913, the 
 values for n, EQ, e, k, h, cz, differing a little in consequence of 
 slight variations in the adopted meteorological data for P, p, R. 
 It is interesting to note that these fundamental physical quan- 
 tities, and many others, can be computed, not only at the surface 
 in standard conditions, but throughout the atmosphere. Sim- 
 ilarly, the sun's data can be computed provided P, T, p, R } can 
 be secured at the required points in the solar atmosphere. The 
 formulas of Table 95 indicate the method of the computations, 
 and numerous subordinate formulas can be derived from them. 
 The transformation factors between the (M. K. S.) and the 
 (C. G. S.) systems have been checked throughout this group of 
 Formulas. In making transformations from (M. K. S.) to 
 (C. G. S.) M = 10 3 , L = 10 2 ; and from (C. G. S.) to 
 (M. K. S.) M = 10~ 3 , L = 10- 2 . 
 
 In Table 96, the kinetic energy per unit volume H y and U 
 the inner energy per unit volume, decrease in a curve similar 
 to those of P and K to vanishing values; q, the mean square 
 velocity, and T, the arithmetical mean velocity, diminish till 
 they disappear at the outer limit; N, the number of molecules 
 per kilogram-molecule, and n, the number of molecules per 
 cubic meter, diminish slowly, but do not vanish. If the term N 
 ought to remain constant, this diminution may go back to some 
 additional physical forces not included in these simple formulas. 
 Compare Fig. 70 for typical curves of the Dynamic, Thermo- 
 
THE CONSTANTS AND COEFFICIENTS FOR DRY AIR 407 
 
 dynamic, and the Kinetic Theory data, throughout the atmos- 
 phere. 
 
 A second more refined computation, with improved tempera- 
 tures, has been undertaken. The data of Table 97 should be 
 substituted for those of Table 96 above 40,000 meters, and 
 the following remarks are based upon it. l max , Maxwell's free 
 path, increases from the surface upward, rapidly above 45,000 
 meters ; v, the number of collisions per second, diminishes with 
 the height. The supposed height of the atmosphere, as derived 
 from meteors at 100 kilometers, finds complete verification in 
 these computations. m Hj the mass of the hydrogen atom, as 
 computed, shows a small increase in value, probably due to 
 some defects in our adopted simple formulas. Similarly, e_, the 
 elementary negative electric charge, assumed in this formula 
 to hold a constant ratio to m H , 3.4554 X 10~ 13 (M. K. S.), 
 follows the fortunes of m B . 
 
408 
 
 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 aS 
 
 a -*-> 
 H 
 
 (/3 
 
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 c/5 
 
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 c^SSoo 
 
 CO (N i I O 
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 i-H rH 1-1 T-H rH 
 
 /\ /\ s\ 
 
 I><NaO(N 
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 X X 
 
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 05 ^H 
 
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 XX X X X X 
 
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 5 
 
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 liiili 
 
THE CONSTANTS AND COEFFICIENTS FOR DRY AIR 
 
 409 
 
 
 
 t>- O5 
 
 % % 
 
 05 00 
 
 CO rH 
 
 10 d 
 
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 I 
 
 (N rt< O5 (M 
 
 rH <N T^ O 
 
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 O5O51O I s - H CO lO CM 
 
 I rH (M CO rH rH 
 
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 5 - 
 
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 ass hydrog 
 ass electric a 
 
 
410 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 & . ail 
 
 S 
 
 m H 
 ass of 
 ydrogen 
 Atom 
 
 
 Freep 
 Leng 
 Maxw 
 
 S O 
 
 3 !_ 
 
 K S 
 
 X X X 
 
 THrl<r}<T-lt-Tj<eOCO(MC^NNNN 
 
 ns s f, 
 
 ii i 
 
 rH iH r-l r-t l-i 
 
 XxXX X 
 
 r i- Tj C< T- i- r-( O> 00 00 t- t- t- <O > tf> kO US 
 
 000 
 
 X X XxX 
 
 t*OOOCOOO 
 
 
 00 11 1 
 
 r-H iH T-l T-H tH 
 
 XX XX X 
 
 CO 00 t- N * T- 10 CO d rH T- r T- tH OS 00 t- <0 >0 ' 
 
 XXXX 
 
 T-<0t-C<|T}lOt-C5HT-lT-lT-lr-lT-lT-lT-lT-ll-lT-ll-e 
 
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 xxxxx x 
 
 <NI 
 
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 11 
 
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 r-l T-H T-l 
 
 XXX 
 
 nt-^Tfcococowco 
 
THE KINETIC THEORY OF DRY AIR SYSTEM OF UNITS 411 
 
 wi to T* -<t <* ri * eo eo 
 
 T "I 
 
 X X 
 
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 -^' CO 09 fri C3 M' fri -i r-l r-l i-l i-l 00 
 
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 fcS8S88Sfe8?SSSS 
 
 ^< O N O . o O I- O T* iH 
 
 g2 
 
412 
 
 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 I 
 
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THE KINETIC THEORY OF GASES 
 
 413 
 
 CDlOWOT^^^C^-OOOCOOrHi-HO^U^ 1 
 
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 05t-t-0000000>0500r-ltH(M(MeOCO^<T)< 
 
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 ooit-eo<NOicon<ocoiMNU5cococDio ~ 
 
414 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 Summary of the Dimensions with Special Reference to the 
 Equivalents 
 
 In transformations from the (M. K. 5.) to the (C. G. S.) 
 systems, the following dimensions should be observed: 
 
 Mass = M=10 3 . Length = L = 10 2 . Time=r=l second. 
 Velocity = L T~ l . Momentum = M.L T~ l . Acceleration = 
 
 LT.~ l T.~\ 
 Force = M.L T ~ l T~ l = M L T~ 2 . Impulse = M.L T~ l . 
 
 Force per unit mass or force of acceleration . = L T~ 2 
 Force per unit volume = L T~ 2 . M L~* . . = M L~ 2 T~ 2 
 
 I Specific Volume = Volume per unit mass . . = M~ l L 3 
 I Specific Density = Mass per unit volume . . = M L~ 3 
 
 Work = force X length = M L T~ 2 . L . . = M L 2 T~ 2 
 Kinetic Energy = Mass times velocity squared = M L 2 T~ 2 
 Increase of kinetic or potential energy = work 
 
 done ..... ........ = ML 2 T~ 2 
 
 I Kinetic Energy per unit mass ...... = L 2 T~ 2 
 
 | Gravity potential = work per unit mass . . = L 2 T~ 2 
 
 Kinetic Energy per unit volume = L 2 T~ 2 . M L~* = M L~ l T~ 2 
 Pressure = force per unit area =M L T~ 2 . L~~ 2 = M L~ l T~ 2 
 Pressure = work per unit volume =ML 2 7^ 2 . IT 3 =M L~ l T~ 2 
 
 In the equations of Condition, we have for T = 1, 
 p _ p 
 
 , g ( Zl -z )= - -% ( qi * - g 2) - (Q l - Qo), s 
 
 Pio 
 
 L.L = M L~\ M~ 1 L 3 = L 2 = L 2 = 10 4 = 10,000. 
 
 i =L 2 .M L-*= 
 
 Pressure has two definitions: (1) force per unit area, as Kilo- 
 grams per square meter, or grams per square centimeter; (2) work 
 per unit volume, as free heat per unit volume, hydrostatic pres- 
 
SUMMARY OF DIMENSIONS 415 
 
 sure, inner energy per unit volume, radiation energy, and these 
 all have the same dimension, M L~ l . Since the Erg and the 
 Joule are units of work they must refer to the unit volume and 
 not to the unit area. 
 
 Hence = ML i T~\ L~* = M ZT 1 T~* = 10 3 X 1(T 2 = 10, 
 
 While -~ = ML* T~\ L~* = M T~* = 10 3 = 1,000 
 is distinctly erroneous. 
 
 Summary of Dimensions 
 
 In the conversion from (M. K. S.) mechanical units into 
 gr. cal./cm. 2 minute, the correct factor becomes: 
 
 (M.K.S.) X 10 X 60/4.1851 X 10 7 = 0.000014336 
 
 On the other hand, should the area L? be placed in the denomi- 
 nator under Joule, instead of the volume Z, 3 , the factor (M. K. S.) 
 X 1000 X 60/4.1851 X 10 7 = 0.0014336 would be incorrect, in- 
 
 asmuch as the - - is the work energy per unit volume and 
 
 not per unit area. 
 
 The equations of condition assume two typical forms: 
 
 (a) - (Pi - Po) - = g (zi - z ) fto-j, 
 
 M L~ l = L.L. M L~* = M L~ l = 10 
 , M (Pi - Po) 60 60 
 
 (b) -^r"^ = g(Si " 2o) ^ 
 
 M L~ l . M~ l D = L . L = D = 10 4 
 For the equation (a), from (M. K. S.) Mech. through (C. G. S.) 
 
 Mech., (M. K. S.) X 10 X 60/4.1851 X 10 7 = 0.000014336 
 For the equation (a), from (M. K. S.) Mech. through Kil. Cal./ 
 
 m. 2 min., (M. K. S.) X 60/4.1851 X 10 3 X 10 3 = 0.000014336 
 For the equation (b), from (M. K. S.) Mech. through (C. G. S.) 
 
 Mech., (M. K. S.) X 10 4 X 60/4.1851 X 10 10 = 0.000014336 
 For the equation (b), from (M . K. S.) Mech. through Kil. Cal ./M 2 
 
 min., (M. K. S.) X 60/4.1851 X 10 3 X 10 3 = 0.000014336 
 
416 EXTENSION OF THERMODYNAMIC COMPUTATIONS 
 
 TABLE 98 SUMMARY OF DIMENSIONS 
 
 Formulas 
 
 Dimensions 
 
 M. K. S. 
 
 C. G. S. 
 
 Work 
 
 
 ,_kilog.m 2 
 
 gr. cm. 2 
 
 P pR T 
 
 ML 1 M 
 
 kilog. Joule 
 
 sec. 2 
 gr. erg 
 
 p (m) = mol. wt. 
 U. 
 
 T 2 L 2 L T 2 
 
 ~L? 
 L 3 
 
 m. sec. 2 m 3 
 kilog. 
 m 3 
 
 m* 
 
 cm. sec. 2 cm 3 . 
 gr. 
 cm. 3 
 
 cm. 3 
 
 (M) = M (m) 
 R Cp Cu 
 
 M 
 L* 
 
 kilog. 
 
 gr. 
 cm. 2 
 
 K = R(m) 
 
 T^deg. 
 L 2 (m) 
 
 sec. Meg. 
 m 2 (m) 
 
 sec. 2 deg. 
 cm. 2 (wi) 
 
 A = mech. eq. H 
 
 T 2 deg. 
 LML 
 
 sec. 2 deg. 
 
 4 13*51 V 1fV kilo &- m * 
 
 sec. 2 deg. 
 
 4 IS'JI V 1010 /oor 2 
 
 Watt 
 
 T^cal 
 
 (M) 
 
 sec. 2 
 kilog. Joule 
 
 1 Watt. 107 erg 
 
 L 2 T 
 
 N 1 
 
 r 3 deg. 
 
 1 
 
 sec. 3 deg. 4 m 2 sec. deg. 4 
 1 
 
 cm. 2 deg. 4 cm. 2 sec. deg. 4 
 1 
 
 n 
 
 M 
 
 I 
 
 mass hydrogen atom 
 1 
 
 mass hydrogen atom 
 
 1 
 
 
 L ML-i (m)U (m)L 2 
 
 m 3 (mol. wt.) 
 (mol. wt.) m 2 
 
 cm. 3 (mol. wt.) 
 
 Pv RT 
 
 T 2 M T 2 
 M L~i L 3 L 2 
 
 sec. 2 
 m 2 
 
 sec. 2 
 cm 2 . 
 
 2 3P0 
 
 N 
 
 T 2 M~ T 2 
 
 sec. 2 
 
 , , m* 
 
 M-mol. ; 
 sec. 2 
 
 m 2 
 
 sec. 2 
 
 M-mol. ' 
 sec. 2 
 
 cm. 2 
 
 2 V- 
 77 3 ^ 
 
 T 2 
 
 sec. 2 
 
 sec. 2 
 
 2 n 
 
 (M)L 2 
 
 m sec. 2 
 
 sec. 2 
 (M) cm. 2 
 
 t ** 
 
 T 2 deg. 
 
 sec. 2 deg. 
 kilog. m 2 Joule 
 
 sec. 2 deg. 
 gr. cm. 2 erg. 
 
 AT 
 
 L 
 
 T 2 deg. 
 ML 2 ML* 
 
 sec. 2 deg. deg. 
 kilog. m 2 
 
 sec. 2 deg. deg. 
 gr. cm. 2 
 
 Ci = Sirch 
 
 f y2 
 
 ML? 
 
 _, J ouie sec. 
 sec. 
 
 kilog. m 3 
 
 sec. 
 gr. cm. 3 
 
 C2=ch/k 
 
 Ldeg. 
 ."z. r~2 , s/ / i y 2 
 
 sec. 2 Joule meter 
 meter deg. 
 kilog. Joule 
 
 sec. 2 
 cm. deg. 
 gr. erg 
 
 H 2 P 
 
 TJ C n T 
 
 pi M L. 1 - 
 
 L 2 M , M 
 
 met. sec. 2 m 3 
 kilog. Joule 
 
 cm. sec. 2 cm. 3 
 gr. erg 
 
 C 
 g 
 
 a = 
 
 T 2 deg.' L 3 deg> ~ L T 2 
 M L\ 1 M 
 
 m sec. 2 m 3 
 w/sec. 
 w/sec. 2 
 kilog. Joule 
 
 cm. sec. 2 cm. s 
 cm./sec. 
 cm./sec. 2 
 gr. erg 
 
 a c 
 
 T 2 L 4 deg. 4 LT 2 deg. 4 
 M L M 
 
 m. sec. 2 deg. 4 m 3 deg. 4 
 kilog. Joule 
 
 cm. sec. 2 deg. 4 cm. 3 deg. 4 
 gr. erg 
 
 4 
 
 cr 
 
 L r 2 deg. 4 ' T r 3 deg. 4 
 M T 2 cal. M cal. 
 
 sec. 3 deg. 4 m. 2 sec. deg. 4 
 kilog. cal. 
 
 sec. 3 deg. 4 cm. 2 sec. deg. 4 
 gr. cal. 
 
 ^ A A 
 <r 
 
 r'deg. 4 L 2 " L 2 r d eg 4 
 
 m 2 sec. deg. 4 
 kilog. Joule 
 
 cm. 2 sec. deg. 4 
 gr. ere; 
 
 ^ 
 
 T 3 deg. 4 
 
 sec. 3 deg. 4 m 2 sec. deg. 4 
 
 sec. 3 deg. 4 cm. 2 sec. deg. 4 
 
GENERAL PROBLEMS 417 
 
 It should be noted that a X j, = cr, so that a is the work 
 
 energy per unit volume. When this gains the velocity L Ti, 
 it becomes cr or radiation through the unit area. The equation 
 (b), page 414, KIQ = cT a gives c of the same dimension as a, 
 and it must be multiplied by a velocity to become comparable 
 with o- in the Stefan Law. The computations for the radiation 
 coefficients conform to these dimensions. 
 
 Similar computations have been successfully made for 
 hydrogen and calcium vapor in the atmosphere of the sun, and 
 the results are in conformity with the indications of the well- 
 known observations. This knowledge of P.p.R.T. in the 
 superficial layers of the sun's envelope is of great value in in- 
 terpreting the lines of the visible and radiation spectra. Indeed, 
 the thermodynamic data are so complex that it will be im- 
 practical to determine them by direct observations. The change 
 of R = constant to R = variable, that is, the transition from 
 adiabatic to non-adiabatic conditions, converts the so-called 
 constants k, h, c, c 2} a, a into a series of coefficients which change 
 from level to level. It follows that the laws of radiation for 
 each chemical element are much more complicated than is 
 implied in the use of the Stefan or Wein-Planck Laws with 
 simple constants. This subject is of such far-reaching im- 
 portance that it is reserved for further study and research. 
 
 General Problems in Atmospheric and Solar Physics 
 
 The purpose of the development of non-adiabatic meteorology 
 has been to discover a method of computations applicable to all 
 atmospheres, as those of the earth and the sun. The process 
 consists in fixing an initial P , /o , Ro t T on some level z 0j and then, 
 by assuming 7\ on the level 2i, determine the corresponding PI, 
 Pi, TI on that level, all to be checked by the gravity equation. 
 From these values of T the entire system can be obtained. In 
 the case of the sun the initial data can be fixed approximately 
 as follows: 
 
 If the Boyle- Gay Lussac Law in the atmosphere of the earth 
 
418 
 
 GENERAL PROBLEMS 
 
 is, P U = R T, this can be converted to the sun's atmosphere 
 by multiplying each term with 28.028 = G/g = T, the ratio 
 of the surface gravity on the sun to that on the earth, so that, 
 PT X UT = RT X Tr. The corresponding factor for all other 
 terms follows very readily, and the entire thermodynamic 
 system can be developed for each element by the method of 
 trials. The check is found in the gravity equation. 
 
 g 
 
 p _ p 
 
 - ZQ) = - l ~ (C p a - C #10) (T a ~ T ). 
 
 The analogy between the physical condition in the atmosphere 
 of the sun just above the photosphere, and in that of the earth 
 between the levels 37,000 and 44,000 meters, is very interesting. 
 Compare Fig. 79. In both there is apparently a very rapid 
 change of temperature with the height, and a transition from ex- 
 cessively rarefied media to the tru egaseous condition. In the 
 earth's atmosphere there is passage through such conditions as : 
 
 z in meters 
 
 37,000 
 
 44,000 
 
 
 T (absolute) 
 P 
 B (mercury in M M) 
 
 211 
 437 
 3.3 
 
 124 
 98 
 0.7 
 
 M. K. S. Sys- 
 tem of units 
 
 P 
 
 0.025 
 
 0.0087 
 
 
 R 
 
 82.77 
 
 91.49 
 
 
 The physical transformations in this transition at 44,000 
 meters correspond with those sensitive states in the Geissler 
 Tubes at which the electric and electro-magnetic phenomena 
 are especially active. From this it is inferred that the origin of 
 the auroral atmospheric electricity consists in the transformation 
 of a portion of the incoming solar radiation of short wave lengths 
 into ions, or free electric charges, and that these pursue their 
 paths to the polar regions, as indicated in Fig. 69, generating 
 the corresponding magnetic deflection vectors. In Table 95 the 
 
 mass of the hydrogen atom is m H = -r = 1.6496 X 10~ 27 , and 
 
GENERAL PROBLEMS 419 
 
 if this is a constant, N = 6.062 X 10 26 should, also, be a con- 
 stant throughout the atmosphere. But by Tables 96, 97, N 
 diminishes with the height, especially above 40 km. 
 
 z = 90,000 N = 1.665 X 10 20 
 
 80,000 1.064 X 10 24 
 
 70,000 7.023 X 10 24 
 
 60,000 2.216 X 10 26 
 
 50,000 5.364 X 10 25 
 
 40,000 1.092 X 10 26 
 
 30,000 1.730 X 10 26 
 
 20,000 2.671 X 10 26 
 
 10,000 4.161 X 10 26 
 
 000 No = 6.147 X 10 26 
 
 The differences between No and N may be taken as the 
 number of transformed ions on the several levels, to be used in 
 generating the auroral electric currents. Furthermore, the region 
 above 44,000 meters in the earth's atmosphere, with its excessively 
 rarefied media, in electric sensitiveness under the impact of the in- 
 coming solar radiation of short wave lengths, is distinctly coronal in 
 its physical conditions, and it corresponds with the deep corona 
 that exists above the sun's chromosphere. The rapid changes of 
 temperature near the sun's photosphere produce a layer which 
 is the source of great radiation energy outward, while the rapid 
 temperature increase in the levels 44 to 37 km. of the earth's 
 atmosphere is the evidence of an important absorption of radia- 
 tion energy. Hence, the sharpness of the sun's visible disk is a 
 phenomenon depending upon the thermodynamic conditions 
 existing between the rapid decrease of temperature and the in- 
 crease of radiation outward to space. The coronal and the pho- 
 tospheric regions in the atmospheres of the sun and the earth 
 are, therefore, the physical counterparts of each other. 
 
 The laws of the absorption of the incoming radiation in the 
 earth's atmosphere need further study. In place of interpreting 
 the bolometer data in terms of the pyrheliometer data, and 
 ascribing the great depletion of 4.00 gr. cal./cm. 2 min. down to 
 only 2.00 calories, as the result of a very imperfect action of the 
 sun as a black body radiator, the result of the examination of 
 the pyrheliometer and thermodynamic data seems to indicate 
 
420 GENERAL PROBLEMS. 
 
 that the pyrheliometer data should be interpreted in terms of 
 the data of the bolometer, so that the "Solar constant" is 
 about 4.00 calories, with the sun's effective radiation tempera- 
 ture 6950 A. In the Bouguer formula, the coefficients of trans- 
 mission p w as directly observed, and p a as computed, have an 
 interesting relation to the wave lengths in the spectrum. Abbot's 
 values of p w at different wave lengths were grouped together, 
 at Mt. Wilson and at Washington, and diagrams of the mean 
 values of p w , as 0.936, 0.925, 0.915 . . . were plotted on the 
 abscissas, 0.20 M, 0.30 ju ... 1.60 M. If the values of p w and 
 p a are plotted on these curves, they intersect at about wave 
 length 0.57 M. If 0.57 M on p w = 1.00 is taken as the maximum 
 ordinate on the energy curve, this corresponds with about 
 5000 and 1.07 gr. cal./cm. 2 min. at the distance of the earth. 
 Now, this is the average direct reading of the pyrheliometer at 
 sea level, so that we infer that not only the aqueous vapor and 
 dust, but also the density of the atmosphere, must be removed in 
 discussing the coefficients of transmission. This conforms to the 
 result shown in Fig. 76, where the second method of computa- 
 tion identifies the depletion of I a = I\/p a with the density 
 P of the atmosphere. This research is not yet ready for 
 final statement. It is very desirable that the excellent condi- 
 tions prevailing at La Quiaca, at the elevation 3465 meters 
 (11,037 feet) on the Bolivian Plateau, should be utilized for a 
 solar physics observatory, equipped with a bolometer, spectro- 
 heliograph, and direct-vision spectroscope for prominences, to 
 be operated in cooperation with Mt. Wilson and Washington, 
 D. C. The conditions of living in La Quiaca are comparatively 
 easy for observers, and there are complete railroad facilities for 
 transportation from Buenos Aires. La Confianza, at 4483 me- 
 ters (14,070 feet), can, also, be occupied the year around, as the 
 village of San Vincente and the neighboring mining camp pro- 
 vide the necessary accommodations. It is important that the 
 bolometric data should be obtained at these high levels, and 
 there is an excellent opportunity for an expedition to make the 
 necessary observations at those stations. 
 
 An inspection of the data of Chapter VII suggests that there 
 
GENERAL PROBLEMS 421 
 
 are a very large number of problems in general physics that can 
 be studied to advantage with the data which have been already 
 developed. The laboratory is quite incapable of furnishing the 
 fundamental relations that are shown to exist above the 45,000- 
 meter level, where the pressure is a small fraction of one milli- 
 meter of mercury. Fortunately, the balloon ascensions to 30,000 
 meters cover a very interesting region, the so-called " isothermal 
 layer," and they provide quite accurate observations of the 
 temperature. The most important outstanding problem is to 
 find a thermodynamic function for the kinetic energy of circula- 
 tion, ]4 (<?i 2 <?o 2 ), in order to separate it from the free heat 
 
 - (ft - 90- 
 
 It is hardly possible, in opening up so much new research 
 material in the physics of the atmosphere, to have escaped 
 imperfections and even errors, but it is thought desirable to indi- 
 cate to meteorologists and astro-physicists some of the possible 
 channels of investigation that appear to be accessible to such 
 studies as are here illustrated. 
 
 
 
INDEX 
 
 Adiabatic atmosphere, V. Bjerknes' system, 51 
 
 Margules' system, iii 
 
 Adiabatic and non-adiabatic systems, differences, 51 
 Angular velocities, cylindrical, 140, 142 
 
 earth's, 160 
 
 polar, 140, 142 
 
 rectangular, 140, 142 
 Aqueous vapor in the atmosphere, grams per cubic meter, 342 
 
 grams per kilogram, 343 
 
 Atlantic Ocean, meteorological observations, 56 
 Atmosphere, homogeneous, 2 
 Avogadro's Law, number of atoms, 28 
 Azimuth, coordinates, 7 
 
 B 
 
 Balloon ascensions, results, 115 
 
 Huron, So. Dak., Sept. 1, 1910, 365, 377, 381, 410 
 
 Twenty-one, 384 
 
 Uccle, June 9, Sept. 13, Nov. 9, 1911, 380, 385, 412 
 Barometer corrections, for local effects, 45 
 
 for mean column temperature, 43 
 
 for plateau temperature, 45 
 
 for removal of station, 42 
 
 for standard gravity, 42 
 
 for standard temperature, 41 
 
 for vapor pressure, 45 
 
 reduction tables, 46 
 
 to normal instrument, 42 
 Barometric formulas, general, 39 
 formulas, 23 
 
 pressure, annual variations, 338 
 
 transformations, 38 
 
 V. Bjerknes' paper on Dynamic Meteorology, 133 
 Bolometer, absorbed energy of, 393 
 energy spectrum, 278, 283 
 instruments, 277 
 observed ordinates, 284 
 Bouguer formula, application in atmosphere, 270 
 
 indeterminate development, 267 
 
 remarks, 364 
 
 423 
 
424 INDEX 
 
 Boyle-Gay Lussac law, P =P R T, 1, 28 
 Boyles' law, Mariotte, 28 
 
 Calories, large and small units, 15 
 Carnot's function, law, 71 
 cyclic process, 72, 73 
 
 examples, 77 
 
 Circulation, direction, and velocity on the earth, 190 
 general in the atmosphere, 189 
 transformation of energy in 
 
 Case I, change of position of two layers, 180 
 Case II, adiabatic and non-adiabatic mixture, 184 
 Case III, overturn of strata, 185 
 
 Case IV, masses of different temperatures on the same level, 187 
 Case V, discontinuous temperatures, 188 
 work of, 153 
 
 Clausius' law of kinetic energy, 28 
 Coefficient of Conductivity, discrepancy of results, 304 
 examples, 305 
 formulas, 295 
 
 (c), in formulas of the planetary circulation, 126 
 Components, analytic construction, 229 
 general and local, 220 
 graphic construction, 232 
 Concentration, in mixtures, 30 
 Consecutive means, 338 
 Constants, of dynamic meteorology, 9 
 of static meteorology, 9 
 of thermodynamic meteorology, 10 
 three systems, 13 
 
 (M. K. S.) and (C. G. S.), 406, 408 
 
 Continuity, equation of in rectangular, cylindrical, and polar coordinates, 145 
 Convection, diurnal in lower strata, 98 
 
 semi-diurnal waves, 99 
 Co-ordinate accelerations, 136 
 
 cylindrical on the rotating earth, 148 
 forces, 137 
 
 polar on the rotating earth, 149 
 velocities, 136 
 Co-ordinates, rectangular, cylindrical, polar, 135 
 
 systems of, 8 
 
 Coronas, solar and terrestrial, 419 
 Current function, in angular velocity, 171 
 
 in the total pressure, 171 
 Cyclone, example of the evaluation of the several terms, 254 
 
INDEX 425 
 
 Dal ton's law, of partial pressures, 28 
 Data, Thermodynamic, 389, 393 
 Deflecting force of the earth's rotation, 178 
 Density, in mixtures, 20 
 
 in the isothermal region, 96 
 
 in the planetary circulation, 118 
 
 of the atmosphere, 13 
 
 use of erroneous, 372 
 
 variations of, 17 
 
 Differential, total of -j- f 146 
 
 Dimensions, Conversion Factors, 376 
 
 Summary of equivalent, 414 
 Diurnal convection, five types in the atmosphere, 312 
 
 examples, 313 
 
 magnetic vectors, compared with convection vectors, 328 
 variations, electric dissipation, 321 
 magnetic field, 323 
 meteorological elements, 318 
 Dumb-bell-shaped vortex, equations of, 177 
 formulas collected, 250 
 tangential angle i, 251 
 Dyne, unit of force, 14 
 
 Eastward drift, 227 
 
 Ebert ion counter, formulas, 299 
 
 example, 299, 301 
 
 velocity formulas, 300 
 Electric potential of the atmosphere, formulas, 309 
 
 examples, 310 
 
 Elster and Geitel dissipation apparatus, formulas, 297 
 Entropy, (S\ So), in mixtures, 30 
 in the isothermal region, 93 
 in the planetary circulation, 124 
 Ephemeris, solar magnetic, 334 
 Equations, general, 6 
 
 of motion, cylindrical, 147 
 
 in cylindrical vortices, 168 
 
 in Ferrel's local cyclone, 164 
 
 in the German local cyclone, 166 
 
 in thermal energy, 151 
 
 First form, 147 
 
 Second form, 149, 150 
 
 Third form, 152 
 
 Fourth form, 161 
 Erg, unit of work, 14 
 
426 INDEX 
 
 Evaluation, of -jg, 159 
 
 of f in terms of temperature, 154 
 
 for standard density, 157 
 
 for the non-adiabatic atmosphere, 156 
 
 through the Boyle-Gay Lussac Law, 156 
 
 183 
 
 Evaporation of water, laws, 343 
 examples, 344, 345 
 factors for pan effect, 347 
 formulas, 346 
 from soil and plants, 348 
 Expansion heat, coefficient of, 69 
 Exponent (a), in the planetary circulation, 126 
 
 F 
 
 Forces, of expansion and contraction, 137 
 of inertia, 137 
 of rotation, 139 
 
 Formula (172), differentiation of, 58 
 
 Free energy, of thermodynamic potential at constant volume and pressure, 67 
 heat, (Qi Qo), calories as unit of, 15 
 applications of formulas, 80 
 evaluation in balloon ascensions, 61 
 Friction, force of, 179 
 Funnel-shaped vortex, equations of, 175 
 
 G 
 
 Gas coefficient R, as a constant, 1 
 
 as a variable, 16 
 
 in a mixture, 29 
 
 in the planetary circulation, 118 
 Gerdien apparatus, for number and velocity of ions; formulas, 302 
 
 examples, 303 
 
 Gold's paper, on radiation and absorption, 134 
 Gradient ratio n, 5 
 
 variable values of, 57 
 Gradients, adiabatic and non-adiabatic, 4 
 Gravitation, force of, in altitude, 19 
 
 in latitude, 18 
 
 in metric and English measures, 20 
 Gravity potential, 67 
 
 H 
 
 Heat, evaluation through the entropy, 69 
 in anticyclones, 109 
 in cyclones, 108 
 
INDEX 427 
 
 Heat energy, application of formulas, 77 
 
 formulas, 66 
 
 in the isothermal region, 92-96 
 inner energy of, 67, 69 
 losses in the convection region, 103 
 mechanical equivalent of, 15 
 
 Historical review of the theories of cyclones and anticyclones: 
 Bigelow's asymmetric cyclonic system, 218 
 Ferrel's cold-center anticyclone, 216 
 Ferrel's warm-center cyclone, 216 
 Hann's dynamic cyclone and anticyclone, 217 
 Horizontal magnetic force, semiannual variations, 333-335 
 
 I 
 Inner energy, applications of formulas, 77 
 
 bound energy, 67 
 
 formulas, 66 
 
 in mixtures, 29 
 
 in the isothermal region, 93-95 
 
 in the planetary circulation, 124 
 
 kinetic, 26 
 
 potential, 26 
 
 total heat and work, 26 
 Integral mean, of temperature, 36 
 
 '1L37 
 T 
 
 of pressures, 22 
 lonization, in atmosphere, 292 
 
 conduction currents, 296 
 
 electrostatic relations, 295 
 
 notation and formulas, 293 
 Isobars, observed and local components, 224 
 Isotherms, normal and local, 235 
 
 observed and local, 239 
 Isothermal region, in Europe, 91, 94 
 in the Tropics, 95 
 
 K 
 Kinetic energy, of motion, 67 
 
 in planetary circulation, 120 
 theory of gases, for the atmosphere, 30 
 constants and formulas, 31 
 Huron, Sept. 1, 1910, 410 
 Uccle, Nov. 9, 1911, 412 
 
 L 
 
 Land cyclone, observed temperatures, 255 
 observed velocities, 255 
 
428 INDEX 
 
 Latent heats, definitions, 69 
 
 examples, 78-80 
 
 second form of the equations, 75 
 Leakage currents, Bigelow's, 219 
 Lindenburg observations in balloon ascensions, 54-64 
 Local circulations, general equations applied, 163 
 
 M 
 
 Magnetic aperiodic vectors along the meridians, 329 
 
 semiannual inversions, 334 
 field, coordinates and forces, 360 
 horizontal force, observed variations, 330-331 
 sphere, theory, 363 
 
 Margules' paper, on the theory of storms, 133 
 Mass, in mixtures, 29 
 Meteorological elements, annual variations in Argentina, 338 
 
 in the United States, 338 
 Meteorology, constants of the static, dynamic and thermodynamic, 9-10 
 
 status of, 1 
 Mixture of gases, formulas, 23-29 
 
 N 
 
 Non-adiabatic atmosphere, application of formulas, 77 
 general formulas, 50 
 working equations, 53 
 
 O 
 
 Operator V 2 , in rectangular, cylindrical, polar co-ordinates, 146 
 
 P 
 
 Partial formations, in the annual curves, 341 
 Physics, fundamental laws of, 28 
 Polarization, of sunlight, description, 348 
 examples, 350 
 percentage observed, 351 
 Potential gradient, definition, 144 
 Precipitation, annual variations, 337 
 Pressure, compared with temperature observations, 103 
 
 dynamic and barometric units, 112 
 
 gradients, rectangular, cylindrical, polar, 143 
 evaluations, 157 
 
 in anticyclones and cyclones, 107, 108 
 
 in isothermal region, 92-95 
 
 in mixtures of gases, 28 
 
 semi-diurnal waves, 100 
 
 three types of units, P, p, B, 2-3 
 
 variations of, 17 
 
INDEX 429 
 
 Problems, general, in physics, 417 
 Processes, adiabatic, 68 
 
 Carnot's cyclic, 72 
 
 cycle for vapors, 73 
 
 irreversible and reversible, 70 
 
 isenergetic, 68 
 
 isodynamic, 66, 68 
 
 isoelastic, 66, 69 
 
 isopiestic, 66, 69 
 
 Prominences on the sun, annual variations, 337 
 Pyrheliometer, instrument, 262 
 
 annual mean variations, 402 
 
 example, 264 
 
 first method of discussion, 364 
 
 reduction to sea level, 399 
 
 second method of discussion, 395 
 
 theory, 263 
 
 Radiation, calories of, 288 
 
 co-efficients and exponents, 82-90 
 
 constituents, 375 
 
 depletion from the cirrus maximum, 274 
 
 determination of the intensity, 265 
 
 effective, 388 
 
 energy, in the planetary circulation, 126 
 
 formula, -*, 290 
 
 mean observed values, 291 
 
 function, K = c T, 1 
 formulas, 82 
 
 intensity at different elevations, 271 
 
 intensity at the earth, 280 
 
 in the isothermal region, 86 
 
 in the planetary circulation, 124, 129 
 
 maximum value, 274 
 
 mean annual values, 272 
 
 relative absorption by aqueous vapor, 275-276 
 
 selective and black, 388 
 
 solar and terrestrial, 375 
 
 total, transmitted, absorbed, 377 
 Relative humidity, observed in the atmosphere, 33 
 Remarks, general, 404 
 Residuals, 339 
 
 Rocky Mountain Plateau, temperature gradients, 34 
 Rotating earth, equations of motion, cylindrical, 148 
 equations of motion, polar, 149 
 
430 INDEX 
 
 Solar constant, derived from the bolometer, 291 
 
 derived from the pyrheliometer, 276 
 
 derived from thermodynamics, 292 
 physics, magnetic data at the earth, 353 
 
 siderial and synodic periods, 354 
 
 thermodynamic equations, 352 
 
 velocities of motions in sun-spots, 352 
 
 velocity of rotation in latitude, 352 
 Specific heat, at constant pressure, 1 
 
 examples, 77-79 
 
 in isothermal region, 93-95 
 
 in terms of latent heat, 77 
 
 variations, 38 
 
 variations in monatomic gases, 27 
 Spherical astronomy, application to the sun, 356 
 
 formulas, 357 
 
 positions on the sun's equator, 357 
 Stefan's radiation formula, 279 
 Stoke's current function, Eq. (494), (495), for funnel-shaped vortex, 172 
 
 for dumb-bell-shaped vortex, 172 
 Strata, interflow of warm and cold, 248 
 Summary, 394 
 general, 403 
 of dimensions, 408 
 Sun-spots, annual variations, 337 
 Synchronism, solar and terrestrial phenomena, 338 
 Synchronous variations of the elements, 335 
 
 Tangential angle i, in circulations, 251 
 Temperature, annual variations, 338 
 
 first distribution, 365 
 
 gradients, observed in the atmosphere, 32 
 computed for the plateau, 34 
 
 in cyclones and anticyclones, 105, 237 
 
 in semi-diurnal waves, 100 
 
 in the isothermal region, 92-95, 115 
 
 in the planetary circulation, 118 
 
 of the sun, probable values, 281-285, 403 
 
 second distribution, 380 
 
 standard, 3 
 
 variations of, 17, 37 
 
 virtual, 37 
 
 Theory, change of, 394 
 Therm, unit of heat, 16 
 Thermodynamics, first law, 65 
 
INDEX 431 
 
 Thermodynamics, second law, 70 
 
 terms, 374 
 Transmission of energy, in the isothermal and convectional region, 287-292 
 
 U 
 
 Units, equivalent values, 1 
 of heat and work, 14-1G 
 
 systems of: (M.K.S.), (C.G.S.), (F.P.S.), 2, 408-409 
 transformations of, 12-14 
 
 Variations of P, p, R, T, 17 
 
 Vector, definitions of velocity and direction of motion, 7 
 
 Velocities, in cyclones and anticyclones, 106-109 
 
 in different latitudes (velocity function), 123 
 
 in the planetary circulation, 120 
 
 linear absolute and relative, 161 
 
 normal and local in cyclones and anticyclones, 240 
 
 normal and local in storms, 221 
 
 observed in different levels, 226 
 Viriol, of inner energy, 26 
 Vortex formulas, dumb-bell-shaped tube, 172 
 funnel-shaped tube, 172 
 general formulas, 174 
 tubes, successive, 173 
 Vortices, composition of, 212 
 
 connection between the funnel and dumb-bell, 252 
 
 cylindrical equations, 168, 175 
 
 evaluation of equations, 256-259 
 
 examples of the structure in the atmosphere, 196 
 
 examples of (1) Cottage City waterspout, 199 
 
 examples of (2) St. Louis tornado, 202 
 
 examples of (3) DeWitte typhoon, 205 
 
 examples of (4) Ocean cyclone, 209 
 
 general equations, 213 
 
 reversed dumb-bell, 213 
 
 W 
 
 Westward drift, 224-229 
 
 Wien displacement formula, 280 
 
 Wien-Planck formula, of the spectrum energy, 278 
 
 evaluations, 281 
 Work, energy, formulas, 66 
 applications, 81 
 
 erg as unit of, 15 
 
 heat equivalent of, 16 
 
 in the isothermal region, 93-95 
 
 in the planetary circulation, 124 
 
 total work done, 27 
 
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