 <h4>FUNDAMENTAL FORMULAS OF PHYSICS
In Two Volumes </h4>
 <h5>Edited By Donald H. Menzel</h5>
 <h6>Äâà òîìà ñëèòû â îäíó êíèãó </h6>
 <h4>TABLE OF CONTENTS</h4>
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Preface 5  
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VOLUME I Fundamental Formulas Of Physics
ÁËÎÊ_× Chapter 1 : BASIC MATHEMATICAL FORMULAS 1
ÁËÎÊ_Ã 1 Algebra 1
1.1. Quadratic equations 1
1.2. Logarithms 2
1.3. Binomial theorem 2
1.4. Multinomial theorem 2
1.5. Proportion 2
1.6. Progressions 3
1.7. Algebraic equations 3
1.8. Determinants 4
1.9. Linear equations 4
ÁËÎÊ_Ã 2 Trigonometry 5
2.1. Angles 5 
2.2. Trigonometric functions 5 
2.3. Functions of sums and differences 5 
2.4. Addition theorems 5 
2.5. Multiple angles 6 
2.6. Direction cosines 6 
2.7. Plane right triangle 7 
2.8. Amplitude and phase 7 
2.9. Plane oblique triangle 7 
2.10. Spherical right triangle 8 
2.11. Spherical oblique triangle 8 
2.12. Hyperbolic functions 8
2.13. Functions of sums and differences 9
2.14. Multiple arguments 9
2.15. Sine,cosine,and complex exponential function 9
2.16. Trigonometric and hyperbolic functions 9
2.17. Sine and cosine of complex arguments 10
2.18. Inverse functions and logarithms 10
ÁËÎÊ_Ã 3 Differential Calculus 10
3.1. The derivative 10
3.2. Higher derivatives 10
3.3. Partial derivatives 10
3.4. Derivatives of functions 11
3.5. Products 11
3.6. Powers and quotients 12
3.7. Logarithmic differentiaion 12
3.8. Polynomials 12
3.9. Exponentials and logarithms 12
3.10. Trigonometric functions 12
3.11. Inverse trigonometric functions 12
3.12. Hyperbolic functions 13
3.13. Inverse hyperbolic functions 13
3.14. Differential 13
3.15. Total differential 13
3.16. Exact differential 13
3.17. Maximum and minimum values 14
3.18. Points of inflection 14
3.19. Increasing absolute value 14
3.20. Arc length 14
3.21. Curvature 15
3.22. Acceleration in plane motion 15
3.23. Theorem of the mean 15
3.24. Indeterminate forms 15
3.25. Taylor's theorem 16
3.26. Differentiation of integrals 17
ÁËÎÊ_Ã 4 Integral Calculus 17
4.1. Indefinite integral 17
4.2. Indefinite integrals of functions 18
4.3. Polynomials 18
4.4. Simple rational fractions 18
4.5. Rational functions 18
4.6. Trigonometric functions 19
4.7. Exponential and hyperbolic functions 20
4.8. Radicals 20
4.9. Products 21
4.10. Trigonometric or exponential integrands 21
4.11. Algebraic integrands 22
4.12. Definite integral 22
4.13. Approximation rules 23
4.14. Linearity properties 24
4.15. Mean values 24
4.16. Inequalities 24
4.17. Improper integrals 25
4.18. Definite integrals of functions 25
4.19. Plane area 25
4.20. Length of arc 25
4.21. Volumes 25
4.22. Curves and surfaces in space 26
4.23. Change of variables in multiple integrals 26
4.24. Mass and density 26
4.25. Moment and center of gravity 27
4.26. Moment of inertia and radius of gyration 27
ÁËÎÊ_Ã 5 Differential Equations 28
5.1. Classification 28
5.2. Solutions 28
5.3. First order and first degree 28
5.4. Variables separable 28
5.5. Linear in y 29
5.6. Reducible to linear 29
5.7. Homogeneous 30
5.8. Exact equations 30
5.9. First order and higher degree : 30
5.10. Equations solvable for jfr 30
5.11. Clairaut's form 30
5.12. Second order 31
5.13. Linear equations 32
5.14. Constant coefficients 33
5.15. Undetermined coefficients 34
5.16. Variation of parameters 35
5.17. The Cauchy - Euler "homogeneous linear equation" 36
5.18. Simultaneous differential equations 36
5.19. First order partial differential equations 37
5.20. Second order partial differential equations 37
5.21. Runge-Kutta method of finding numerical solutions 39
ÁËÎÊ_Ã 6 Vector Analysis 39
6.1. Scalars 39
6.2. Vectors 39
6.3. Components 39
6.4. Sums and products by scalars 40
6.5. The scalar or dot product, V, ' V2 40
6.6. The vector or cross product, vx x v2 40
6.7. The triple scalar product 41
6.8. The derivative 41
6.9. The Frenet formulas 41
6.10. Curves with parameter t 42
6.11. Relative motion 42
6.12. The symbolic vector del 43
6.13. The divergence theorem 44
6.14. Green's theorem in a plane 44
6.15. Stokes's theorem 44
6.16. Curvilinear coordinates 45
6.17. Cylindrical coordinates 45
6.18. Spherical coordinates 45
6.19. Parabolic coordinates 45
ÁËÎÊ_Ã 7 Tensors 46
7.1. Tensors of the second rank 46
7.2. Summation convention 46
7.3. Transformation of components 46
7.4. Matrix notation „ 46
7.5. Matrix products 46
7.6. Linear vector operation 47
7.7. Combined operators. 47
7.8. Tensors from vectors 47
7.9. Dyadics 47
7.10. Conjugate tensor. Symmetry 48
7.11. Unit, orthogonal, unitary 48
7.12. Principal axes of a symmetric tensor 49
7.13. Tensors in w-dimensions 49
7.14. Tensors of any rank 50
7.15. The fundamental tensor 50
7.16. Christoffel three-index symbols 50
7.17. Curvature tensor 51
ÁËÎÊ_Ã 8 Spherical Harmonics 51
8.1. Zonal harmonics 51
8.2. Legendre polynomials 51
8.3. Rodrigues's formula 52
8.4. Particular values 52
8.5. Trigonometric polynomials 52
8.6. Generating functions 53
8.7. Recursion formula and orthogonality 53
8.8. Laplace's integral 53
8.9. Asymptotic expression" 53
8.10. Tesseral harmonics. 53 
8.1 L Legendre's associated functions 54
8.12. Particular values 54
8.13. Recursion formulas 54
8.14. Asymptotic expression 54
8.15. Addition theorem 55
8.16. Orthogonality 55
ÁËÎÊ_Ã 9 Bessel Functions 55
9.1. Cylindrical harmonics 55
9.2. Bessel functions of the first kind 55
9.3. Bessel functions of the second kind 56
9.4. Hankel functions 56
9.5. Bessel's differential equation 57
9.6. Equation reducible to Bessel's 57
9.7. Asymptotic expressions 58
9.8. Order half an odd integer 58 
9.9. Integral representation. 59
9.10. Recursion formula  59
9.11. Derivatives  59
9.12. Generating function  59
9.13. Indefinite integrals  59
9.14. Modified Bessel   59
10. The Hypergeometric Function 60
10.1. The hypergeometric equation 60
10.2. The hypergeometric series 60
10.3. Contiguous functions 61
10.4. Elementary functions 62
10.5. Other functions 62
10.6. Special relations 62
10.7. Jacobi polynomials or hypergeometric polynomials 63
10.8. Generalized hypergeometric functions 63
10.9. The confluent hypergeometric function 64
11. Laguerre Functions 64
11.1. Laguerre polynomials 64
11.2. Generating function 64
11.3. Recursion formula 64
11.4. Laguerre functions 65
11.5. Associated Laguerre polynomials 65
11.6. Generating function 65
11.7. Associated Laguerre functions 65
12. Hermite Functions 65
12.1. Hermite polynomials 65
12.2. Generating function 66
12.3. Recursion formula 66
12.4. Hermite functions 66
13. Miscellaneous Functions 66
13.1. The gamma function 66
13.2. Functional equations 67
13.3. Special values 67
13.4. Logarithmic derivative 67
13.5. Asymptotic expressions 67
13.6. Stirling's formula 68
13.7. The beta function 68
13.8. Integrals 68
13.9. The error integral 69
13.10. The Riemann zeta function 69
14. Series 69
14.1. Bernoulli numbers 69
14.2. Positive powers 70
14.3. Negative powers 70
14.4. Euler - Maclaurin sum formula 70
14.5. Power series : 70
14.6. Elementary functions 71
14.7. Integrals 72
14.8. Expansions in rational fractions 72
14.9. Infinite products for the sine and cosine 73
14.10. Fourier's theorem for periodic functions 73
14.11. Fourier series on an interval 74
14.12. Half-range Fourier series 74
14.13. Particular Fourier series 74
14.14. Complex Fourier series 76
14.15. The Fourier integral theorem 76
14.16. Fourier transforms 77
14.17. Laplace transforms 77
14.18. Poisson's formula 77
14.19. Orthogonal functions. 78
14.20. Weight functions 78
15. Asymptotic Expansions 79
15.1. Asymptotic expansion. 79
15.2. Borel's expansion 79
15.3. Steepest descent 80
16. Least Squares 80
16.1. Principle of least squares 80
16.2. Weights 81
16.3. Direct observations 81
16.4. Linear equations 81
16.5. Curve fitting 81
16.6. Nonlinear equations 82
17. Statistics 82
17.1. Average 82
17.2. Median 82
17.3. Derived averages 83
17.4. Deviations 8?
17.5. Normal law ' 83
17.6. Standard deviation 83
17.7. Mean absolute error 84
17.8. Probable error 84
17.9. Measure of dispersion. 84
17.10. Poisson's distribution. 84
17.11. Correlation coefficient 85
18. Matrices 85
18.1. Matrix 85
18.2. Addition 86
18.3. Multiplication 86
18.4. Linear transformations 86
18.5. Transposed matrix 87
18.6. Inverse matrix 87
18.7. Symmetry 87
18.8. Linear equations 87
18.9. Rank 88
18.10. Diagonalization of matrices 88
19. Group Theory 89
19.1. Group 89
19.2. Quotients 90
19.3. Order 90
19.4. Abelian group 90
19.5. Isomorphy 90
19.6. Subgroup 90
19.7. Normal divisor 90
19.8. Representation 91
19.9. Three-dimensional rotation group 91
20. Analytic Functions 93
20.1. Definitions 93
20.2. Properties 94
20.3. Integrals 94
20.4. Laurent expansion 94
20.5. Laurent exp ansion about infinity 95
20.6. Residues 95
21. Integral Equations 95
21.1. Fredholm integral equations 96
21.2. Symmetric kernel 97
21.3. Volterra integral equations 98
21.4. The Abel integral equation 99
21.5. Green's function 99
21.6. The Sturm - Liouville differential equations 100
21.7. Examples of Green's function 102
ÁËÎÊ_× Chapter 2 : STATISTICS 107
ÁËÎÊ_Ã 1 Introduction 107
1.1. Characteristics of a measurement process 107
1.2. Statistical estimation 108
1.3. Notation 108
ÁËÎÊ_Ã 2 Standard Distributions 109
2.1. The normal distribution 109
2.2. Additive property 109
2.3. The logarithmic-normal 
distribution 110
2.4. Rectangular distribution 110
2.5. The x2 distribution 110
2.6. Student's t distribution. Ill
2.7. The F distribution Ill
2.8. Binomial distribution Ill
2.9. Poisson distribution 112
ÁËÎÊ_Ã 3 Estimators of the Limiting Mean 112
3.1. The average or arithmetic mean 112
3.2. The weighted average 113
3.3. The median 113
ÁËÎÊ_Ã 4 Measures of Dispersion 113
4.1. The standard deviation 113
4.2. The variance s2 =
S (Xi — x)2l(n — 1) 115
4.3. Average deviation 115
4.4. Range : difference between largest and smallest value in a set of observations, #,,=#,, 116
ÁËÎÊ_Ã 5 The Fitting of Straight Lines 116
5.1. Introduction 116
5.2. The case of the underlying physical law 116
ÁËÎÊ_Ã 6 Linear Regression 120
6.1. Linear regression 120
ÁËÎÊ_Ã 7 The Fitting of Polynomials 121
7.1. Unequal intervals between the x's 121
7.2. The case of equal intervals between the x's — the method of orthogonal polynomials 123
7.3. Fitting the coefficients of a function of several variables 124
7.4. Multiple regression 124
ÁËÎÊ_Ã 8 Enumerative Statistics 125
8.1. Estimator of parameter of binomial distribution 125
8.2. Estimator of parameter of Poisson distribution 125 
8.3. Rank correlation coefficient 126
ÁËÎÊ_Ã 9 Interval Estimation 126
9.1. Confidence interval for parameters 126
9.2. Confidence interval for the mean of a normal distribution 127
9.3. Confidence interval for the standard deviation of a normal distributoin 127
9.4. Confidence interval for slope of straight line 128
9.5. Confidence interval for intercept of a straight line 128
9.6. Tolerance limits 128
10. Statistical Tests of Hypothesis 129
10.1. Introduction 129
10.2. Test of whether the mean of a normal distribution is greater than a specified value 129
10.3. Test of whether the mean of a normal distribution is different from some specified value 130
10.4. Test of whether the mean of one normal distribution is greater than the mean of another normal distribution 130
10.5. Test of whether the means of two normal distributions differ 131
10.6. Tests concerning the parameters of a linear law 131
10.7. Test of the homogeneity of a set of variances 132
10.8. Test of homogeneity of a set of averages 132
10.9. Test of whether a correlation coefficient is different from zero 133
10.10. Test of whether the • correlation coefficient p is equal to a specified value 133
11. Analysis of Variance 134
11.1. Analysis of variance 134
12. Design of Experiments 137
12.1. Design of experiments. 137
13. Precision and Accuracy 137
13.1. Introduction 137
13.2. Measure of precision 137
13.3. Measurement of accuracy 138
14. Law of Propagation of Error 138
14.1. Introduction 138
14.2. Standard deviation of a
ratio of normally distributed variables 140
14.3. Standard deviation of a product of normally distributed variables 140
ÁËÎÊ_× Chapter 3 : NOMOGRAMS 141
ÁËÎÊ_Ã 1 Nomographic Solutions 141
1.1. A nomogram (or nomograph) 141
ÁËÎÊ_× Chapter 4: PHYSICAL CONSTANTS 145
ÁËÎÊ_Ã 1 Constants and Conversion Factors ( of Atomic and Nuclear Physics 145
ÁËÎÊ_Ã 2 Table of Least-Squares-Adjusted Output Values 148
ÁËÎÊ_× Chapter 5 : CLASSICAL MECHANICS 155
ÁËÎÊ_Ã 1 Mechanics of a Single Mass Point and a System of Mass Points 155
1.1. Newton's laws of motion and fundamental motions 155
1.2. Special cases 157
1.3. Conservation laws 159
1.4. Lagrange equations of the second kind for arbitrary curvilinear coordinates 160
1.5. The canonical equations of motion 161
1.6. Poisson brackets 162
1.7. Variational principles 163
1.8. Canonical transformations 164
1.9. Infinitesimal contact transformations 167
1.10. Cyclic variables 168
1.11. Transition to wave mechanics. The optical-mechanical analogy. 171
1.12. The Lagrangian and Hamiltonian formalism for continuous systems and fields 174
ÁËÎÊ_× Chapter 6 : SPECIAL THEORY OF RELATIVITY 178
ÁËÎÊ_Ã 1 The Kinematics of the Space-Time Continuum 178
1.1. The Minkowski "world" 178
1.2. The Lorentz transformation 179
1.3. Kinematic consequences of the Lorentz transformation 181
ÁËÎÊ_Ã 2 Dynamics 186
2.1. Conservation laws 186
2.2. Dynamics of a free mass point 188
2.3. Relativistic force 189
2.4. Relativistic dynamics 189
2.5. Gauge invariance 191
2.6. Transformation laws for the field strengths 192
2.7. Electrodynamics in moving, isotropic ponderable media (Minkowski's equation) 192
2.8. Field of a uniformly moving point charge in empty space. Force between point charges moving with the same constant velocity 194 
2.9. The stress energy tensor and its relation to the conservation laws 196
ÁËÎÊ_Ã 3 Miscellaneous Applications ' 197
3.1. The ponderomotiveequation 197
3.2. Application to electron optics 198
3.3. Sommerfeld's theory of the hydrogen fine structure 200
ÁËÎÊ_Ã 4 Spinor Calculus 201
4.1. Algebraic properties 201
4.2. Connection between spinors and world tensors 203
4.3. Transformation laws for mixed spinors of second rank. Relation between spinor and Lorentz transformations 204
4.4. Dual tensors 206
4.5. Electrodynamics of empty space in spinor form 208
ÁËÎÊ_Ã 5 Fundamental Relativistic Invariants 209
ÁËÎÊ_× Chapter 7 : THE GENERAL THEORY OF RELATIVITY 210
ÁËÎÊ_Ã 1 Mathematical Basis of General Relativity 210
1.1. Mathematical introduction 210
1.2. The field equations 211
1.3. The variational principle 212
1.4. The ponderomotive law 213
1.5. The Schwarzschild solution 214
1.6. The three "Einstein effects" 214
ÁËÎÊ_× Chapter 8 : HYDRODYNAMICS AND AERODYNAMICS 218
ÁËÎÊ_Ã 1 Assumptions and Definitions 218
ÁËÎÊ_Ã 2 Hydrostatics 219
ÁËÎÊ_Ã 3 Kinematics 220
ÁËÎÊ_Ã 4 Thermodynamics 223
ÁËÎÊ_Ã 5 Forces and Stresses 223
ÁËÎÊ_Ã 6 Dynamic Equations 224
ÁËÎÊ_Ã 7 Equations of Continuity for Steady Potential Flow of Nonviscous Fluids 225
ÁËÎÊ_Ã 8 Particular Solutions of Laplace's Equation 227
ÁËÎÊ_Ã 9 Apparent Additional Mass 228
10. Airship Theory 230
11. Wing Profile Contours; Two-Dimensional Flow with Circulation 231
12. Airfoils in Three Dimensions 232
13. Theory of a Uniformly Loaded Propeller Disk 233
14. Free Surfaces 233
15. Vortex Motion 234
16. Waves 235
17. Model Rules 237
18. Viscosity 238
19. Gas Flow, One- and Two-Dimensional 239
20. Gas Flows, Three Dimensional 239
21. Hypothetical Gases 239
22. Shockwaves 240
23. Cooling 241
24. Boundary Layers 242
ÁËÎÊ_× Chapter 9: BOUNDARY VALUE PROBLEMS IN MATHEMATICAL PHYSICS 244
ÁËÎÊ_Ã 1 The Significance of the Boundary 244
1.1. Introductory remarks 244
1.2. The Laplace equation 244
1.3. Method of separation of variables 246
1.4. Method of integral equations 249
1.5. Method of Green's function 249
1.6. Additional remarks about the two - dimensional area 251
1.7. The one - dimensional wave equation 253
1.8. The general eigenvalue problem and the higher-dimensional wave equation 255
1.9. Heat conduction equation 261
1.10. Inhomogeneous differential equations 263
ÁËÎÊ_× Chapter 10 : HEAT AND THERMODYNAMICS 264
ÁËÎÊ_Ã 1 Formulas of Thermodynamics 264
1.1. Introduction 264
1.2. The laws of thermodynamics 265
1.3. The variables 266
1.4. One-component systems 266
1.5. One-component, usually two-phase systems 268
1.6. Transitions of higher orders 269
1.7. Equations of state 270
1.8. One - component, twovariable systems, with dW = Xdy 271
1.9. One - component, twovariable systems, with dW = Xdy 271
1.10. One-component, multivariable systems, with dW = S X4yi 271
1.11. Multicomponent, multivariable systems, dW = pdv 271
1.12. Homogeneous systems 273
1.13. Heterogeneous systems 275
ÁËÎÊ_× Chapter 11 : STATISTICAL MECHANICS 277
ÁËÎÊ_Ã 1 Statistics of Molecular Assemblies 277
1.1. Partition functions 277
1.2. Equations of state 280
1.3. Energies and specific heats of a one-component assembly 281
1.4. Adiabatic processes 281
1.5. Maxwell's and Boltzmann's laws 282
1.6. Compound and dissociating assemblies 282
1.7. Vapor pressure 283
1.8. Convergence of partition functions 284
1.9. Fermi-Dirac and Bose-Einstein statistics 284
1.10. Relativistic degeneracy 286
1.11. Dissociation law for new statistics 287
1.12. Pressure of a degenerate gas 288
1.13. Statistics of light quanta 289
ÁËÎÊ_× Chapter 12: KINETIC THEORY OF GASES : VISCOSITY,
THERMAL CONDUCTION, AND DIFFUSION 290 
ÁËÎÊ_Ã 1 Preliminary Definitions and Equations for a Mixed Gas, Not in Equilibrium 290
1.1. Definition of r, x, y, z, t, dr 290
1.2. Definition of m, n, p, w10,fii2, m0 290
1.3. The external forces F, XyYyZ^ 291
1.4. Definition of c, u, vy w, dc 291
1.5. The velocity distribution function/ 291
1.6. Mean values of velocity functions 291
1.7. The mean mass velocity Co) u0,v0,w0 291
1.8. The random velocity C; U, Vy W; dC 292
1.9. The "heat" energy of a molecule E: its mean value E 292
1.10. The molecular weight W and the constants mu, jvl 292
1.11. The constants J,k, R 292
1.12. The kinetic theory temperature T 293
1.13. The symbols cv, Cv 293
1.14. The stress distribution, 293
1.15. The hydrostatic pressure p; the partial pressures p^Pi 294
1.16. Boltzmann's equation for/ 294
1.17. Summational invariants 294
1.18. Boltzmann's H theorem 294
ÁËÎÊ_Ã 2 Results for a Gas in Equilibrium 295
2.1. Maxwell's steady-state solutions 295
2.2. Mean values when / is Maxwellian 296
2.3. The equation of state for a perfect gas 296
2.4. Specific heats 297
2.5. Equation of state for an imperfect gas 297
2.6. The free path, collision frequency, collision interval, and collision energy (perfect gas) 298
ÁËÎÊ_Ã 3 Nonuniform Gas 299
3.1. The second approximation to/ 299
3.2. The stress distribution 300
3.3. Diffusion 300
3.4. Thermal conduction 301
ÁËÎÊ_Ã 4 The Gas Coefficients for Particular Molecular Models 301
4.1. Models a to d 301
4.2. Viscosity fi and thermal
conductivity A for a simple gas 302
4.3. The first approximation to D 12' 303
4.4. Thermal diffusion 304
ÁËÎÊ_Ã 5 Electrical Conductivity in a Neutral Ionized Gas with or without a Magnetic Field 304
5.1. Definitions and symbols 304
5.2. The diameters as 305
5.3. Slightly ionized gas 305
5.4. Strongly ionized gas 306
ÁËÎÊ_× Chapter 13 : ELECTROMAGNETIC THEORY 307
ÁËÎÊ_Ã 1 Definitions and Fundamental Laws 307
1.1. Primary definitions 307
1.2. Conductors 308
1.3. Ferromagnetic materials 309
1.4. Fundamental laws 309
1.5. Boundarv conditions 309
1.6. Vector and scalar potentials 310
1.7. Bound charge 310
1.8. Amperian currents 311
2, Electrostatics 311
2.1. Fundamental laws 311
2.2. Fields of some simple charge distributions 312
2.3. Electric multipoles; the double layer 312
2.4. Electrostatic boundary value problems 314
2.5. Solutions of simple electrostatic boundary value problems 314
2.6. The method of images 315
2.7. Capacitors 318
2.8. Normal stress on a conductor 319
2.9. Energy density of the electrostatic field 319
ÁËÎÊ_Ã 3 Magnetostatics 320
3.1. Fundamental laws 320
3.2. Fields of some simple current distributions 321
3.3. Scalar potential for magnetostatics; the magnetic dipole !. 322
3.4. Magnetic multiples 323
3.5. Magnetostatic boundaryvalue problems 325
3.6. Inductance 326
3.7: Magnetostatic energy density 327
ÁËÎÊ_Ã 4 Electric Circuits 328
4.1. The quasi - stationary approximation 328
4.2. Voltage and impedance 328
4.3. Resistors and capacitors in series and parallel connection 328
4.4. KirchhofFs rules 328
4.5. Alternating current circuits 329
ÁËÎÊ_Ã 5 Electromagnetic Radiation 330
5.1. Poynting's theorem 330
5.2. Electromagnetic stress and momentum 331
5.3. The Hertz vector; electromagnetic waves 332
5.4. Plane waves 335
5.5. Cylindrical waves 336
5.6. Spherical waves 337
5.7. Radiation of electromagnetic waves; the oscillating dipole 339
5.8. Huygen's principle 340
5.9. Electromagnetic waves at boundaries in dielectric media 341
5.10. Propagation of electromagnetic radiation in wave guides 342
5.11. The retarded potentials; the Lienard-Wiechert potentials; the self-force of electric charge 344
ÁËÎÊ_× Chapter 14: ELECTRONICS 350
ÁËÎÊ_Ã 1 Electron Ballistics 350
1.1. Current 350
1.2. Forces on electrons 350
1.3. Energy of electron 351
1.4. Electron orbit 351
ÁËÎÊ_Ã 2 Space Charge 352
2.1. Infinite parallel planes 352
2.2. Cylindrical electrodes 352
ÁËÎÊ_Ã 3 Emission of Electrodes 353
3.1. Thermionic emission 353
3.2. Photoelectric emission 353
ÁËÎÊ_Ã 4 Fluctuation Effects 354
4.1. Thermal noise 354
4.2. Shot noise 354
ÁËÎÊ_× Chapter 15 : SOUND AND ACOUSTICS 355
ÁËÎÊ_Ã 1 Sound and Acoustics 355
1.1. Wave equation, definition 355
1.2. Energy, intensity 356
1.3. Plane wave of sound 356
1.4. Acoustical constants for various media 357
1.5. Vibrations of sound producers; simple oscillator 357
1.6. Flexible string under tension 357
1.7. Circular membrane under tension 358
1.8. Reflection of plane sound waves, acoustical impedance 359
1.9. Sound transmission through ducts 359
1.10. Transmission through long horn 360
1.11. Acoustical circuits 360
1.12. Radiation of sound from a vibrating cylinder 361
1.13. Radiation from a simple source 361
1.14. Radiation from a dipole source 362
1.15. Radiation from a piston in a wall 362
1.16. Scattering of sound from a cylinder 362
1.17. Scattering of sound from a sphere 363
1.18. Room acoustics 363
VOLUME II Fundamental Formulas Of Physics
ÁËÎÊ_× Chapter 16 : GEOMETRICAL OPTICS 365
ÁËÎÊ_Ã 1 General Considerations 365
1.1. Geometrical optics and wave optics 365
1.2. Media 365
1.3. Index of refraction 365
1.4. Interfaces 366
1.5. Refraction and reflection.The Fresnel formulas 366
1.6. Optical path and optical length 367
1.7. Fermat's principle 368
1.8. Cartesian surfaces and the theorem of Malus 368
1.9. Laws of reflection 369
1.10. Laws of refraction 369
1.11. The fundamental laws of geometrical optics 370
1.12. Corollaries of the laws of reflection and refraction 370
1.13. Internal reflection and Snell's law 370
1.14. Dispersion at a refraction 371
1.15. Deviation 371
ÁËÎÊ_Ã 2 The Characteristic Function of Hamilton (Eikonal of Bruns) 372
2.1. The point characteristic, V 372
2.2. The mixed characteristic, W 374
2.3. The angle characteristic, T 374
2.4. The sine condition of Abbe 376
2.5. Clausius' equation 377
2.6. Heterogeneous isotropic media 378
2.7. Collineation 379
ÁËÎÊ_Ã 3 First Order Relationships 380
3.1. Conventions 380
3.2. Refraction at a single surface 381
3.3. Focal points and focal lengths 381
3.4. Image formation 382
3.5. Lagrange's law 382
3.6. Principal planes 383
3.7. Nodal points 384 
3.8. Cardinal points 385 
3.9. The thin lens 386 
3.10. The thick lens 387 
3.11. Separated thin lenses 387 
3.12. Chromatic aberration 388 
3.13. Secondary spectrum 391 
3.14. Dispersion formulas 391 
ÁËÎÊ_Ã 4 Oblique Refraction 395
4.1. First-order theory 395
4.2. Oblique refraction of elementary pencils 397
4.3. The Seidel aberrations 399
4.4. The Seidel third-order expressions 401
4.5. Seidel's conditions in the Schwarzschild-Kohlschiitter form. 403
ÁËÎÊ_Ã 5 Ray-Tracing Equations 405
ÁËÎÊ_× Chapter 17 : PHYSICAL OPTICS 409
ÁËÎÊ_Ã 1 Propagation of Light in Free Space 410
1.1. Wave equation 410
1.2. Plane-polarized wave 411
1.3. Elliptically polarized wave 411
1.4. Poynting vector 411
1.5. Intensity 411
1.6. Partially polarized light 412
1.7. Light quant 412
ÁËÎÊ_Ã 2 Interference 412
2.1. Two beams of light 412
2.2. Double-source experiments 412
2.3. Fringes of equal inclination 413
2.4. Fringes of equal thickness 413
2.5. Michelson interferometer 413
2.6. Fabry-Perot interferometer 413
2.7. Lummer-Gehrcke plate 414
2.8. Diffraction grating 414
2.9. Echelon grating 415
2.10. Low-reflection coatings 415
ÁËÎÊ_Ã 3 Diffraction 416
3.1. Fraunhofer diffraction by
a rectangular aperture 416
3.2. Chromatic resolving power of prisms and gratings 416
3.3. Fraunhofer diffraction by
a circular aperture 416
3.4. Resolving power of a telescope 416
3.5. Resolving power of a microscope 417
3.6. Fraunhofer diffraction by N equidistant slits 417
3.7. Diffraction of x rays by crystals 417
3.8. Kirchhoff's formulation of Huygens' principle 417
3.9. Fresnel half-period zones 418
3.10. Fresnel integrals 418
ÁËÎÊ_Ã 4 Emission and Absorption 418
4.1. Kirchhoff's law of radiation 418
4.2. Blackbody radiation laws 418
4.3. Exponential law of absorption 419
4.4. Bohr's frequency condition 419
4.5. Intensities of spectral lines 419
ÁËÎÊ_Ã 5 Reflection 419
5.1. Fresnel's equations 419
5.2. Stokes' amplitude relations 420
5.3. Reflectance of dielectrics 420
5.4. Azimuth of reflected plane-polarized light 420
5.5. Transmittance of dielectrics 420
5.6. Polarization by a pitexpf plates 421
5.7. Phase change at total internal reflection 421
5.1. Meridional rays 405
5.2. Skew rays 406
5.8. Fresnel's rhomb 421
5.9. Penetration into the rare medium in total reflection 421
5.10. Electrical and optical constants of metals 422
5.11. Reflectance of metals 422
5.12. Phase changes and azimuth for metals 423
5.13. Determination of the optical constants 423
ÁËÎÊ_Ã 6 Scattering and Dispersion 424
6.1. Dipole scattering 424
6.2. Rayleigh scattering formula 424
6.3. Thomson scattering formula 424
6.4. Scattering by dielectric spheres 424
6.5. Scattering by absorbing spheres 425
6.6. Scattering and refractive index 425
6.7. Refractivity 425
6.8. Dispersion of gases 425
6.9. Dispersion of solids and liquids 426
6.10. Dispersion of metals 426
6.11. Quantum theory of dispersion 426
ÁËÎÊ_Ã 7 Crystal Optics 427
7.1. Principal dielectric constants and refractive indices 427
7.2. Normal ellipsoid 427
7.3. Normal velocity surface 427
7.4. Ray velocity surface 427
7.5. Directions of the axes 427
7.6. Production and analysis of ellipticaily polarized light 427
7.7. Interference of polarized light 428
7.8. Rotation of the plane of polarization 428
ÁËÎÊ_Ã 8 Magneto-optics and Electro-optics 428
8.1. Normal Zeeman effect. 428
8.2. Anomalous Zeeman effect 429
8.3. Quadratic Zeeman effect 429
8.4. Faraday effect 430
8.5. Cotton-Mouton effect 430
8.6. Stark effect 430
8.7. Kerr electro-optic effect 430
ÁËÎÊ_Ã 9 Optics of Moving Bodies 431
9.1. Doppler effect 431
9.2. Astronomical aberration 431
9.3. Fresnel dragging coefficient 431
9.4. Michelson - Morley experiment 431
ÁËÎÊ_× Chapter 18 : ELECTRON OPTICS 433
ÁËÎÊ_Ã 1 General Laws of Electron Optics 436
1.1. Fermat's principle for electron optics 436
1.2. Index of refraction of electron optics 436
1.3. Law of Helmholtz-Lagrange for axially symmetric fields 436
1.4. Upper limit to the current density 7 in a beam cross section at potential CD and with aperture angle a 436
1.5. General lens equation 436
ÁËÎÊ_Ã 2 Axially Symmetric Fields 437
ÁËÎÊ_Ã 6 Electron Paths in Uniform Fields 440
2.1. Differential equations of the axially symmetric fields in free space 437
2.2. Potential distribution in axially symmetric electric field 437
2.3. Behavior of equipotential surfaces on axis 437
2.4. Magnetic vector potential in axially symmetric field 437
2.5. Field distribution in axially symmetric magnetic field 437
ÁËÎÊ_Ã 3 Specific Axially Symmetric Fields 438
3.1. Electric field 438
3.2. Electric field 438
3.3. Magnetic field 438
3.4. Magnetic field 438
ÁËÎÊ_Ã 4 Path Equation in Axially Symmetric Field 438
4.1. General path equation in axially symmetric field 438
ÁËÎÊ_Ã 5 Paraxial Path Equations 439
5.1. General paraxial path equation 439
5.2. Azimuth of electron 439
5.3. Paraxial path equation for path crossing axis 439
5.4. Paraxial ray equation for variable R = r®l/i 439
S S ParnYifil rnv pnnatinn in electric field for variable c = —r'lr 439
5.6. Paraxial ray equation in electric field for variable b = —r'\r + 1 (2*) 440
5.7. Paraxial ray equation in electric field for arbitrarily high voltage 440
6.1. Path in uniform electrostatic field—<£' parallel to z axis 440
6.2. Path in uniform magnetic field 440
6.3. Path in crossed electric and magnetic field. 441
ÁËÎÊ_Ã 7 Focal Lengths of Weak Lenses 441
7.1. General formula for focal length of a weak lens 441
7.2. Focal length of aperture lens 441
7.3. Focal length of electric field between coaxial cylinders 441
7.4. Focal length of magnetic field of single wire loop 441
7.5. Focal length of magnetic gap lens 442
7.6. Focal length of lens consisting of two apertures at potential <P0 and CE>j, separated by a distance d (radius of apertures « d) 442
ÁËÎÊ_Ã 8 Cardinal Points of Strong Lenses 442
8.1. Strong lens 442
8.2. Uniform magnetic field,cut off sharply at z = ± d 443
8.3. "Bell-shaped" magnetic field 443
8.4. Electric field <X> = CD gW**)1* arc tan z/d aa*s 9. Electron Mirrors 444
9.1. Paraxial ray equations 444
9.2. Displacement of electron 444
9.3. Approximate formula for focal length of an electron mirror 444
10. Aberrations 444
10.1. Geometric aberrations of the third order 444
10.2. Chromatic aberrations. 445
10.3. General formula for aperture defect 445
10.4. Aperture defect of weak lens 445
10.5. Aperture defect of bellshaped magnetic field 446
10.6. Aperture defect of uniform magnetic and electric field 446
10.7. Aperture defect of uniform electric field of length / 446
10.8. Chromatic aberration of weak unipotential electrostatic lens 446
10.9. Chromatic aberration of a magnetic lens for large magnification 446
10.10. Chromatic aberration of uniform magnetic and electric field 446
10.11. Relativistic aberration of weak electrostatic unipotential lens. 446
11. Symmetrical Two-Dimensional Fields 447
11.1. Field distributions 447
11.2. Paraxial path equation in electric field 447
11.3. Paraxial path equations in magnetic field 447
11.4. Focal length of weak electric cylinder lens 447
11.5. Focal length of weak slit lens 448
11.6. Focal length and displacement of focal point in z direction for weak magnetic cylinder lens 448
12. Deflecting Fields 448
12.1. Field distribution in two-dimensional deflecting fields 448
12.2. Deflection by electric field for electron incident in midplane 448 
12.3. Deflection by magnetic field of length 449
ÁËÎÊ_× Chapter 19 : ATOMIC SPECTRA 451
ÁËÎÊ_Ã 1 The Bohr Frequency Relation 451
1.1. Basic combination principle 451
ÁËÎÊ_Ã 2 Series Formulas 451
2.1. The Rydberg equation 451
2.2. The Ritz combination principle 452
2.3. The Ritz formula 453
2.4. The Hicks formula 453
ÁËÎÊ_Ã 3 The Sommerfeld Fine Structure Constant for Hydrogen-Like Spectra 454
3.1. Enerffv states 454
ÁËÎÊ_Ã 4 Coupling 455
4.1. LS or Russel-Saunders coupling 455
ÁËÎÊ_Ã 5 Line Intensities 456
5.1. Doublets 456
ÁËÎÊ_Ã 6 Theoretical Zeeman Patterns 459
6.1. Lande splitting factor. 459
6.2. The Paschen-Back effect 459
6.3. Pauli's g-sum rule 460
ÁËÎÊ_Ã 7 Nuclear Magnetic Moments 460
7.1. Hyperfine structure 460
ÁËÎÊ_Ã 8 Formulas for the Refraction and Dispersion of Air for the Visible Spectrum 461
8.1. Meggers' and Peters' formula 461
8.2. Perard's equation 461
8.3. The formula of Barrell and Sears 462
ÁËÎÊ_× Chapter 20 : MOLECULAR SPECTRA 465
ÁËÎÊ_Ã 1 General Remarks 465
ÁËÎÊ_Ã 2 Rotation and Rotation Spectra 465
2.1. Diatomic and linear polyatomic molecules 465
2.2. Symmetric top molecules 469
2.3. Spherical top molecules 472
2.4. Asymmetric top molecules 474
2.5. Effect of external fields 476
2.6. Hyperfine structure 478
ÁËÎÊ_Ã 3 Vibration and Vibration Spectra 480
3.1. Diatomic molecules 480
3.2. Polyatomic molecules 483
ÁËÎÊ_Ã 4 Interaction of Rotation and Vibration : Rotation-Vibration Spectra 489
4.1. Diatomic molecules 489
4.2. Linear polyatomic mocules 490
4.3. Symmetric top molecules 492
4.4. Spherical top molecules 496
4.5. Asymmetric top molecules 497
4.6. Molecules with internal rotation 497
ÁËÎÊ_Ã 5 Electronic States and Electronic Transitions 499
5.1. Total energy and electronic energy 499
5.2. Interaction of rotation and electronic motion in diatomic and linear polyatomic molecules 500 
5.3. Selection rules and spectrum 502
ÁËÎÊ_× Chapter 21 : QUANTUM MECHANICS 505
ÁËÎÊ_Ã 1 Equations of Quantum Mechanics 505
1.1. Old quantum theory 505
1.2. Uncertainty principle 506
1.3. Schrodinger wave equation 506
1.4. Special solutions of the Schrodinger equation for bound states 509
1.5. Solutions of the Schrodinger equation for collision problems 512
1.6. Perturbation methods 514
1.7. Other approximation methods 516
1.8. Matrices in quantum mechanics 518
1.9. Many-particle systems 520
1.10. Spin angular momentum 520
1.11. Some radiation formulas 521
1.12. Relativistic wave equations 522
ÁËÎÊ_× Chapter 22 : NUCLEAR THEORY 525
ÁËÎÊ_Ã 1 Table of Symbols 525
ÁËÎÊ_Ã 2 Nuclear Theory 528
2.1. Nuclear masses and stability 528
2.2. Stationary state properties 529
2.3. Nuclear interactions 531
2.4. Properties of the deuteron 533
2.5. Potential scattering 534
2.6. Resonance reactions 538
2.7. Beta decay 541
ÁËÎÊ_× Chapter 23: COSMIC RAYS AND HIGH-ENERGY PHENOMENA 544
ÁËÎÊ_Ã 1 Electromagnetic Interactions 544
1.1. Definitions and some natural constants 544
1.2. Cross sections for the collision of charged particles with atomic electrons, considered as free 545
1.3. Energy loss by collision with atomic electrons 546
1.4. Range of heavy particles 546
1.5. Specific ionization 547
1.6. Cross sections for emission of radiation by charged particles 547
1.7. Energy loss of electrons by radiation 549
1.8. Cross sections for scattering of charged particles 550
1.9. Scattering of charged particles in matter. 551
1.10. Compton effect 552
1.11. Pair production 552
ÁËÎÊ_Ã 2 Shower Theory 553
2.1. Definitions 553
2.2. Track lengths 554
2.3. Integral spectrum 555
2.4. Properties of the shower maxima 555
2.5. Stationary solutions 556
2.6. Lateral and angular spread of showers 556
ÁËÎÊ_Ã 3 Nuclear Interactions 557
3.1. Nuclear radius and transparency 557
3.2. Altitude variation of nuclear interactions : gross transformation 557
ÁËÎÊ_Ã 4 Meson Production 557
4.1. Threshold energies 557
4.2. Relativity transformations 558
ÁËÎÊ_Ã 5 Meson Decay 558
5.1. Distance of flight 558
5.2. Energy distribution of decay products 559
5.3. Angular distribution in two-photon decay 559
ÁËÎÊ_Ã 6 Geomagnetic Effects 559
6.1. Motion in static magnetic fields 559
6.2. Flux of particles in static magnetic fields 560
6.3. Limiting momenta on the earth's surface 560
ÁËÎÊ_× Chapter 24 : PARTICLE ACCELERATORS 563
ÁËÎÊ_Ã 1 General Description and Classification of High-Energy Particle Accelerators 563
1.1. General description 563
1.2. Classification according to particle accelerated 563
1.3. Classification according to particle trajectories 564
1.4. Designation of accelerators 504
1.5. Basic components 564
ÁËÎÊ_Ã 2 Dynamic Relations for Accelerated Particles 566
2.1. Fundamental relativistic relations 566
2.2. Derived relations 566
2.3. Nonrelativistic relations 566
2.4. Units 566
ÁËÎÊ_Ã 3 Magnetic Guiding Fields 567
3.1. Specification of magnetic guiding fields 567
3.2. Force on a charged particle in a magnetic field 568
3.3. Equations of motion of a charged particle in a magnetic guiding field 568
3.4. Equilibrium orbit 569
3.5. Stability of motion in the equilibrium orbit 570
3.6. Oscillations about the equilibrium orbit 570
3.7. Coupling of oscillations about the equilibrium orbit 570
3.8. Damping of radial and vertical oscillations 571
ÁËÎÊ_Ã 4 Particle Acceleration 571
4.1. Electrostatic and quasielectrostatic acceleration 571
4.2. Induction acceleration. 571
4.3. Traveling wave acceleration 572
4.4. Impulsive acceleration. 572
ÁËÎÊ_Ã 5 Phase Stability and Phase Oscillations 573
5.1. Phase stability 573
5.2. Phase oscillations in circular accelerators 574
5.3. Phase motion in linear accelerators 576
ÁËÎÊ_Ã 6 Injection and Focusing 578
ÁËÎÊ_Ã 7 Additional Remarks about Special Accelerators 578
7.1. The conventional cyclotron 578
7.2. The betatron 578
7.3. The synchrotron 579
7.4. The synchrotron or frequency modulated cyclotron 579
7.5. Linear accelerators 579
ÁËÎÊ_× Chapter 25 : SOLID STATE 581
ÁËÎÊ_Ã 1 Introduction : Crystal Mathematics 581
1.1. Translations 581
1.2. The unit cell and the s sphere 582
1.3. The reciprocal lattice 583
1.4. Periodic boundary conditions 583
ÁËÎÊ_Ã 2 Elastic Constants 584
2.1. Stress and strain components 584
2.2. Elastic constants and moduli 584
2.3. Forms of c{i or s,7- for 13'13 some common crystal classes 585
2.4. Relation of elastic constants and moduli. 585
2.5. Forms taken by the condition of positive de-finiteness, for some common crystal classes 586
2.6. Relation of c^ and s{j to other elastic constants 586
2.7. Thermodynamic relations 586
ÁËÎÊ_Ã 3 Dielectrics and Piezoelectricity 587
3.1. Piezoelectric constants 587
3.2. Dielectric constants 588
3.3. Pyroelectricity and the electrocaloric effect. 589
3.4. Elastic constants of piezoelectric crystals 589
3.5. Relations of adiabatic and isothermal piezoelectric and dielectric constants 590
ÁËÎÊ_Ã 4 Conduction and Thermoelectricity 591
4.1. Conductivity tensor of a crystal 591
4.2. Matthiessen's rule 591
4.3. Thomson effect 591
4.4. Seebeck effect 591
4.5. Peltier effect 592
4.6. Entropy flow and Bridgman effect 592
4.7. Galvanomagnetic and thermomagnetic effects 593
ÁËÎÊ_Ã 5 Superconductivity 593
5.1. The London equations 593
5.2. Field distribution in a steady state 594
5.3. The energy equation 594
5.4. Critical field and its relation to entropy and specific heat 595
5.5. Equilibrium of normal and superconducting phases for systems of small dimensions 595
5.6. Multiply connected superconductors 596
5.7. General properties of time-dependent disturbances in superconductors 597
5.8. A-c resistance of superconductors 597
5.9. Optical constants of superconductors 597
ÁËÎÊ_Ã 6 Electrostatics of Ionic Lattices 598
6.1. Potential at a general point of space, by the method of Ewald 598
6.2. Potential acting on an ion, by the method of Ewald 599
6.3. Potential due to an infinite linear array, by the method of Madelung 600
6.4. Potential acting on an ion in a linear array, by the method of Ma-delung 600
6.5. Potential due to a plane array, by the method of Madelung 600
ÁËÎÊ_Ã 7 Thermal Vibrations 601
7.1. Normal modes of a crystal " 601
7.2. Thermodynamic functions, general case 602
7.3. Thermodynamic functions at high temperatures 603
7.4. Thermodynamic functions at low temperatures 603
7.5. Debye approximation 604
7.6. Equation of state for a crystal 605
7.7. Long wavelength optical modes of polar crystals 605
7.8. Residual rays 606
ÁËÎÊ_Ã 8 Dislocation Theory 607
8.1. Characterization of dislocations 607
8.2. Force on a dislocation 607
8.3. Elastic field of a dislocation in an isotropic medium 608
ÁËÎÊ_Ã 9 Semiconductors 609
9.1. Bands and effective masses 609
9.2. Density of states 610
9.3. Traps, donors, and * acceptors 610
9.4. The Fermi-Dirac distribution 610
9.5. Density of mobile charges 611
9.6. Fermi level and density of mobile charges, intrinsic case 612
9.7. Fermi level and density of mobile charges, extrinsic case 612
9.8. Mobility, conductivity, and diffusion 613
9.9. Hall effect 614
9.10. Thermoelectric effects. 614
9.11. Mean free time and mean free path 615
9.12. The space charge layer near a surface 616
9.13. Contact rectification 617
9.14. Differential capacity of a metal-semiconductor contact 618
9.15. D-c behavior of p-n junctions 619
9.16. A-c behavior of p-n junctions 620
10. Electron Theory of Metals 621
10.1. The Fermi-Dirac distribution 621
10.2. Averages of functions of the energy 622
10.3. Energy and electronic specific heat 622
10.4. Spin paramagnetism 623
10.5. Bloch waves 624
10.6. Velocity and acceleration 624
10.7. Energy levels of almost free electrons 624
10.8. Coulomb energy 625
10.9. Exchange energy 625
10.10. Electrical and thermal conduction 626
10.11. Orbital diamagnetism 627
10.12. Optical constants 627
11. Miscellaneous 628
11.1. Specific heats at constant stress and strain 628
11.2. Magnetocaloric effect and magnetic cooling 629
11.3. The Cauchy relations 629
11.4. The Brillouin and Langevin functions 630
11.5. Relation of thermal release to capture of mobile charges by traps 630
ÁËÎÊ_× Chapter 26 : THE THEORY OF MAGNETISM 633
ÁËÎÊ_Ã 1 Paramagnetism 633
l.l. Classical theory 633
1.2. Quantum theory 634
ÁËÎÊ_Ã 2 Ferromagnetism 636
2.1. Classical theory 636
2.2. Quantum theory 637
2.3. Anisotropic effects 638
ÁËÎÊ_Ã 3 Diamagnetism and Feeble Paramagnetism 639
3.1. Classical theory of diamagnetism 639
3.2. Quantum theory of diamagnetism 639
3.3. Feeble paramagnetism 640
ÁËÎÊ_× Chapter 27 : PHYSICAL CHEMISTRY 641
ÁËÎÊ_Ã 1 Chemical Equilibrium 641
1.1. Equilibrium constant or "mass action law" 641
1.2. Equilibrium constant from calorimetric data 642
1.3. Equilibrium constant from electric cell voltages 642
1.4. Pressure dependence of the equilibrium constant 643
1.5. Temperature dependence of the equilibrium constant 643
ÁËÎÊ_Ã 2 Activity Coefficients 644
2.1. The "thermodynamic" equilibrium constant 644
2.2. Thermodynamic interpretation of the activity coefficient 644
2.3. Activity coefficients of gases 645
2.4. Activity coefficient from the "law of corresponding states" 646
2.5. Activity coefficients of nonelectrolytes in solution 646
2.6. The Gibbs-Duhem equation 647
2.7. The enthalpy of nonideal solutions 647
2.8. The entropy of nonideal solutions 648
2.9. The activity coefficients of aqueous electrolytes 648
2.10. The Debye-Huckel equation 649
ÁËÎÊ_Ã 3 Changes of State 650
3.1. Phase rule 650
3.2. One component, solidsolid and solid-liquid transitions 650
3.3. One component, solidgas and liquid-gas transitions 651
3.4. Two components, solidliquid transition 651
3.5. Two components, liquid vapor transition 652
3.6. Liquid transition 652
3.7. Osmotic pressure 653
3.8. Gibbs-Donnan membrane equilibrium. 653
ÁËÎÊ_Ã 4 Surface Phenomena 653
4.1. Surface tension 653
4.2. Experimental measurement of surface tension 654
4.3. Kelvin equation 654
4.4. Temperature dependence of surface tension 655
4.5. Insoluble films on liquids 655
4.6. Adsorption on solids. 655
4.7. Excess concentration at the surface 656
4.8. Surface tension of aqueous electrolytes 656
4.9. Surface tension of binary solutions 656
ÁËÎÊ_Ã 5 Reaction Kinetics 657
5.1. The rate law of a reaction 657
5.2. Integrated forms of the rate law 657
5.3. Half-lives 658
5.4. Integrated form of rate law with several factors 658
5.5. Consecutive reactions 658
5.6. Multiple-hit processes. 659
5.7. Reversible reactions 659
5.8. The specific rate : collision theory 659
5.9. The specific rate : activated complex theory 660
5.10. Activity coefficients in reaction kinetics 660
5.11. Heterogeneous catalysis 661
5.12. Enzymatic reactions 661
5.13. Photochemistry 661
5.14. Photochemistry in intermittent light 662
ÁËÎÊ_Ã 6 Transport Phenomena in the Liquid Phase 662
6.1. Viscosity : definition and measurement 662
6.2. Diffusion : definition and measurement 663
6.3. Equivalent conductivity : definition and measurement 664
6.4. Viscosity of mixtures. 665
6.5. Diffusion coefficient of mixtures 666
6.6. Dependence of conductivity on concentration 666
6.7. Temperature dependence of viscosity, diffusion and conductivity 667
ÁËÎÊ_× Chapter 28 : BASIC FORMULAS OF ASTROPHYSICS 668
ÁËÎÊ_Ã 1 Formulas Derived from Statistical Mechanics 668
1.1. Boltzmann formula 668
1.2. Ionization formula 668
1.3. Combined ionization and Boltzmann formula. 669 
1.4. Dissociation equation for diatomic molecules 669
ÁËÎÊ_Ã 2 Formulas Connected with Absorption and Emission of Radiation 670
2.1. Definitions 670
2.2. Specific intensity 670
2.3. Einstein's coefficients. 670
2.4. Oscillator strength 671
2.5. Absorption coefficients. 671
2.6. Line strengths 672
2.7. Definition of/-values for the continuum 673
ÁËÎÊ_Ã 3 Relation between Mass, Luminosity, Radii, and Temperature of Stars 673
3.1. Absolute magnitude 673
3.2. Color index 673
3.3. Mass-luminosity law 674
3.4. The equation of transfer for gray material 675
3.5. Non-gray material 675
3.6. Model atmosphere in hydrostatic equilibrium 676
3.7. Formation of absorption lines 676
3.8. Curve of growth 677
3.9. Equations governing the equilibrium of a star 678
3.10. Boundary conditions 679
3.11. Theoretical form of mass-luminosity law 679
ÁËÎÊ_× Chapter 29 : CELESTIAL MECHANICS 680
ÁËÎÊ_Ã 1 Gravitational Forces 680
ÁËÎÊ_Ã 2 Undisturbed Motion 682
2.1. Elliptic motion 684
2.2. Parabolic motion 686
2.3. Hyperbolic and nearly parabolic motion 686
2.4. Relativity correction. 686
ÁËÎÊ_Ã 3 Disturbed Motion 687
3.1. The disturbing function 687
3.2. Variations of the elements 688
3.3. Perturbations of the coordinates 690
3.4. Mean orbit of the earth 690
3.5. Mean orbit of the moon 692
3.6. Mass of a planet from the mean orbit of a satellite 692
ÁËÎÊ_Ã 4 The Rotation of the Earth 693
4.1. Poisson's equations 693
4.2. The Eulerian nutation. 694
ÁËÎÊ_× Chapter 30 : METEOROLOGY 697
ÁËÎÊ_Ã 1 Basic Equations for Large-Scale Flow 697
1.1. The hydrodynamic equation of motion 697
1.2. Conservation of mass 698
1.3. Equation of state 698
1.4. First law of thermodynamics 699
ÁËÎÊ_Ã 2 Derived Equations 699
2.1. Geostrophic wind 700
2.2. Hydrostatic equation. 700
2.3. Adiabatic lapse rate 701
2.4. The circulation theorem 701
2.5. The vorticity theorem. 702
2.6. The energy equation. 702
2.7. The tendency equation 702
2.8. Atmospheric turbulence 703
ÁËÎÊ_× Chapter 31 : BIOPHYSICS 705
ÁËÎÊ_Ã 1 Introduction : Energy Relations 705
ÁËÎÊ_Ã 2 Kinetics of Enzyme Catalyzed Reactions 708
2.1. Simple reactions 708
2.2. Inhibitors 709
ÁËÎÊ_Ã 3 The Cell 711
3.1. Metabolism and concentration distributions 711
3.2. Diffusion forces and cell division 714
3.3. Cell polarity and its maintenance 719
3.4. Cell permeability 720
ÁËÎÊ_Ã 4 The Neurone and Behavior 722
4.1. Excitation and conduction in the neurone 722
4.2. Behavior and structure of central nervous system 725
ÁËÎÊ_Ã 5 The Evolution and Interaction of Populations 733
5.1. The general laws of populations 733
5.2. Equations of biological populations 734
5.3. Simple populations; effect of wastes, nutriment, and space 736
5.4. Interaction of two species 738
5.5. Embryonic growth 739
// @
INDEX 743