1.	Fundamental Concepts		1
	2.	Series of Constants		6
	3.	Series of Positive Terms		9
	4.	Alternating Series		15
	5.	Series of Positive and Negative Terms		16
	6	Algebra of Series	.	21
	7.	Continuity of Functions Uniform Convergence		23
	8.	Properties of Uniformly Convergent Senes		28
	9.	Power Series		30
	10.	Properties of Power Series		33
	11.	Expansion of Functions in Power Series		35
	12.	Application of Taylor’s Formula		41
	13	Evaluation of Definite Integrals by Means of Power Series ...		43
	14	Rectification of Ellipse. Elliptic. Integrals		47
	15	Discussion of Elliptic Integrals	.	48
	16	Approximate Formulas in Applied Mathematics		55
	17	Preliminary Remarks		63
	18	Dinchlet Conditions. Derivation fourier coeficients		65
	19	Expansion of Functions in Fourier series		67
	20	Sine and Cosine Series		73
	21.	Extension of Interval of Expansion		76
	22.	Complex Form of Fourier Senes		78
	23.	Differentiation and Integration of Fourier Scries		80
	24.	Orthogonal Functions		81
	25.	Graphical Solutions		83
	26.	Algebraic Solution of Cubic		86
	27.	Some Algebraic Theorems		92
	28.	Horner’s Method		95
	29.	Newton’s Method . .		97
	31.	Determinants of the nth Order.		100
	30.	Determinants of the Second and Third Order		102
	31.	Determinants of the nth Order.		106
	32.	Properties of Determinants . . .		107
	33.	Minors .		110
	34.	Matrices and Linear Dependence		114
	35	Consistent and Inconsistent Systems of Equations		117
	36	Functions of Several Variables		123
	37	Partial Derivatives		125
	38	Total Differential		127
	40.	Euler's Formula		130
	41.	Differentiation of Implicit Functions .		137
	42.	Directional Derivatives		143
	43.	Tangent Plane and Normal Line to a Surface		146
	44.	Space Curves		149
	45.	Directional Derivatives in Space		151
	46.	Higher Partial Derivatives		153
	47.	Taylor’s Scries for Functions of Two Variables		155
	48.	Maxima and Minima of Functions of One Variable		158
	49.	Maxima and Minima of Functions of Several variables		160
	50.	Constrained Maxima and Minima		163
	51.	Differentiation under the Integral Sign		167
	52.	Definition and Evaluation of the Double Integral		173
	53.	Geometric Interpretation of the Double Integral		177
	54.	Triple Integrals		179
	55.	Jacobians. Change of Variable		183
	56.	Spherical and Cylindrical coordinates		185
	57.	Surface Integrals		188
	58.	Green’s Theorem in Space		191
	59.	Symmetrical Form of Green’s Theorem		194
	60.	Definition of Line Integral		197
	61.	Area of a Closed Curve		199
	62.	Green’s Theorem for the Plane		202
	63.	Properties of Line Integrals		206
	64.	Multiply Connected Regions		212
	65.	Line Integrals in Space		215
	67.	Preliminary Remarks		225
	68.	Remarks on Solutions		227
	69.	Newtonian Laws		231
	70.	Simple Harmonic Motion		233
	71	Simple Pendulum	. .	234
	72	Further Examples of Derivation of Differential Equations		239
	73	Hyperbolic Functions		247
	74	First-order Differential Equations		256
		Equations with Separable Variables		257
		Homogeneous Differential equations		259
	77	Exact Differential Equations		262
	78	Integrating Factors		265
		Not Occur Explicitly		267
	80.	Differential Equations of the Second Order		269
	81.	Gamma Functions		272
	82.	Orthogonal Trajectories		277
	83	Singular Solutions	...	279
	84.	Linear Differential equations of the first order		283
	85.	Linear Equations of the First Order . .		284
		Equation)	.	286
	87	Linear Differential Equations of the nth Order		287
	88	Some General Theorems		291
				295
	90.	Oscillation of a Spring and Discharge of a Condenser		299
	91.	Viscous Damping		302
	92.	Forced Vibrations .		308
	93.	Resonance		310
	94.	Simultaneous Differential equations		312
	95	linear Equations with Variable Coefficients .		315
	96.	Variation of Parameters		318
	97	97. The Euler Equation		322
				322
	98.	Solution in Series		325
	99.	Existence of Power Series Solutions		329
	100.	Bessel’s Equation		332
	101.	Expansion in Series of Bessel Functions		339
	102.	Legendre’s Equation		342
	103.	Numerical Solution of Differential Equations		346
	104.	Preliminary Remarks .		350
	106.	Integration of Partial Differential Equations.		353
	107.	Linear Partial Differential Equations with Constant Coefficients		357
				357
	108.	Transverse Vibration of Elastic String		361
	109.	Fourier Series Solution .		364
	110.	Heat Conduction		367
	111.	Steady Heat Flow . .		369
	112.	Variable Heat Flow ...		373
	113.	Vibration of a Membrane		377
	114.	Laplace’s Equation		382
	115.	Flow of Electricity in a Cable . .		386
	116.	Scalars and Vectors		392
	117.	Addition and Subtraction of Vectors		393
	118.	Decomposition of Vectors. Base Vectors		396
	119.	Multiplication of Vectors		399
	120.	Relations between Scalar and Vector Products		402
	121.	Applications of Scalar and Vector Products		404
	122.	Differential Operators		406
	123.	Vector Fields		409
	124.	Divergence of a Vector		411
	125.	Divergence Theorem		415
	126.	Curl of a Vector		418
	127.	Stokes’s Theorem		421
	128.	Two Important Theorems		422
	129	129 Physical Interpretation of Divergence and Curl		423
	130.	Equation of Heat Flow		425
	131.	Equations of Hydrodynamics		428
	132.	Curvilinear Coordinates		433
	133.	Complex Numbers .		440
	134.	Elementary Functions of a Complex Variable .		444
	135.	Properties of Functions of a Complex Variable		448
	136.	Integration of Complex Functions ....		453
	137.	Cauchy’s Integral Theorem		455
	138.	Extension of Cauchy’s Theorem		455
	137.	Cauchy’s Integral Theorem		455
	138.	Extension of Cauchy’s Theorem		455
	139.	The Fundamental Theorem of Integral Calculus		457
	140.	Cauchy’s Integral Formula . .		461
	141.	Taylor’s Expansion. .		464
	142.	Conformal Mapping ....		465
	143.	Method of Conjugate Functions . .		467
	144.	Problems Solvable by Conjugate Functions		470
	145	Examples of Conformal Maps .		471
		Applications of Conformal Representation.	. .	479
	147.	Fundamental Notions		492
	147	Fundamental Notions		492
	149.	Mutually Exclusive Events		497
	150.	Expectation. .		500
	151.	Repeated and Independent Trials		501
	151.	Repeated and Independent Trials		501
	152.	Distribution Curve		504
	152.	Distribution Curve		504
	154.	Probability of the Most Probable Number .		511
	154.	Probability of the Most Probable Number .		511
	155.	Approximations to Binomial Law		512
	156.	Tho Error Function ...		516
	157.	* Precision Constant. Probable Error		521
	158.	Graphical Method .		525
	159.	Differences		527
	161.	Constants Determined by Method of averages		534
	162.	Method of Least Squares		536
	163.	Method of Moments		544
	164.	Harmonic Analysis		545
	165.	Interpolation Formulas		550
	166.	Lagrange’s Interpolation Formula		552
	167.	Numerical Integration		554
	168.	A More General Formula		558
		Answers	. .	561
		Index	.	575
		Index		575
	107.	Linear Partial Differential Equations with Constant coefficients
	14	Rectification of Ellipse. Elliptic Integrals		.47
		Preface		v
	105.	Elimination of Arbitrary Functions.		.
	108.	Transverse Vibration of Elastic String .		.
	94.	Simultaneous Differential Equations		. . 312
	112.	Variable Heat Flow ...		. . 373
	15	Discussion of Elliptic Integrals	.	. . 48
	37	Partial Derivatives		. 125
	39	Total Derivatives		. 130
	66.	Illustrations of the Application of the Line Integrals .		. 217
	75	equations with Separable Variables		. 257
	76	Homogeneous Differential Equations		. 259
		Equation)	.	. 286
	96	96. Variation of Parameters	.	. 318
	105	105. Elimination of Arbitrary Functions.	.	....
	95.	Linear Equations with Variable Coefficients		.... 315
	57.	Surface Integrals		.188
	88	Some General Theorems	. .	.291
	38	Total Differential		i27
		Chapter II FOURIER SERIES
		Chapter III
		SOLUTION OF EQUATIONS
		Chapter IV
		PARTIAL DIFFERENTIATION
		Chapter V
		MULTIPLE INTEGRALS
		Chapter VI LINE INTEGRAL
	66.	Illustrations of the Application of the Line Integrals
		Section
		Chapter VII
		ORDINARY DIFFERENTIAL EQUATIONS
	79	Equations of the First Order m Which One of the Variables Does
	86.	A Non-linear Equation Reducible to Linear Form (Bernoulli’s
	89	The Meaning of the Operator
		Chapter VIII
		PARTIAL DIFFERENTIAL EQUATIONS
	Section	Page
		Chapter IX VECTOR ANALYSIS
		Chapter X
		COMPLEX VARIABLE
		xi
		Sbction
		Chapter XI PROBABILITY
		Chapter XII
		EMPIRICAL FORMULAS AND CURVE FITTING
		section
		Chapter XI PROBABILITY
		Chapter XII
		EMPIRICAL FORMULAS AND CURVE FITTING
		06a
		A
		Absolute convergence of series, lb, 17, 20, 21
		Absolute value of complex number, 441
	441	Addition, of series, 21 of vectors,
	393	parallelogram law of,
	394	Adiabiftic process, 224 Aerodynamics, 133, 431 Algebra, fundamental theorem of, 92 Algebraic theorems, 92-94 Alternating series, 15 am i/, 51
		Addition, of series, 21 of vectors, 393 parallelogram law of, 394 Adiabiftic process, 224 Aerodynamics, 133, 431 Algebra, fundamental theorem of, 92 Algebraic theorems, 92-94 Alternating series, 15 am i/, 51
		Amplitude of complex number, 441 Amplitude function, 51 Analysis, harmonic, 545 Analytic functions, 451-491 Angle, as a line integral, 195 direction, 146, 398 of lap, 240 of twist, 485 solid, 195
		Angular velocity, 61, 191, 236, 404, 424
		Applications, of conformal representation, 479-491 of line integrals, 217-224 of scalar arid vector products, 404-406
		Approximate formula, for n!, 509 for probability of most probable number, 511
		in applied mathematics, 55 Approximation, Laplace’s or normal, 515
		Approximations to binomial law, 512 Arc length, 143 of ellipse, 47
		06b
		Arc length, of sinusoid, 55 Area, 172
		as a double integral, 178 as a line integral, 190-202 element of, 183, 184, 190, 437 positive and negative, 200 surface, 188-196
		Argument of complex number, 441 Associative law, for series, 18 for vectors, 394
		Asymptotic formula for a1, 509 Asymptotic senes, 524 Atmosphere, thickness of, 61 Attraction, law of, 218, 232 motion under, 58, 218 of cone, 196 of cylinder, 196 of sphere, 196, 232 Augmented matrix, 118 Auxiliary equation, 292 Averages, method of, 534 Axes, right- or left-handed, 397
		B
		Base vectors, 396 Beam, 240-242, 307 Belt on pulley, slipping of, 239 Bending moment, 241 Bernoulli-Euler law, 241, 307 Bernoulli’s equation, 286 Bessel functions, 273, 336, 381 expansion in, 339 Bessel's equation, 332, 380 Beta function, 27*6 Binomial law, 502
		approximations to, 512 Binomial series, 40 Biot and Savart, law of, 52 Boundary conditions, 242, 351, 363, 370
		Buckling, 299
		07a
		Lagrange’s interpolation formula, 552
		Lagrange’s method of multipliers, 163-167
		Lamellar field, 423 Laplace’s approximation, 515 Laplace’s equation, 195, 369, 382, 385, 386, 439, 451, 470, 481 Law, Bernoulli-Euler, 241 binomial, 502, 512 of attraction, 218 of conservation of matter, 429 of cooling, 254 of dynamics, 231 of error, 520, 536 of gravitation, 232 of small numbers, 512 Least squares, method of, 536 theory of, 521
		Legendre polynomials, 344, 384 expansion in, 346 • Legendre’s equation, 342, 384 Leibnitz’s rule (see Differentiation, under integral sign)
		Leibnitz’s test (see Test, for alternating series) length, of arc, 143 of ellipse, 47 of sine curve, 55 Level surface, 406 Limit, 2, 124, 454 Line, contour, 144 coordinate, 434
		direction cosines of, 146, 147, 151 normal, 144, 146-149 of equal potential, 277 of flow, 475 stream, 277, 432, 467 tangent, 143, 147, 151 vector equation of, 395 Line integrals, 197-224, 410, 421, 454
		07b
		Line integrals, applic&ions of, 217-224
		around a closed curve, 202, 206, 216, 421
		definition of, 197, 454 evaluation of, 202-206, 458 for angle, 195 for area, 201 for work, 217 in space, 215, 410, 421 properties of, 206-217 transformation of, 202, 421 Linear dependence or independence, 116, 317
		Linear differential equations, 283-349, 357
		with constant coefficients, 287 367 with variable coefficients, 284, 315-349
		Linear differential operator, 287-299 Ivog z, 446
		logarithmic paper, 526
		M
		M test, 27
		Maclaurin formula, 36 Maclaurin’s scries, 37, 249 Magnitude of a vector, 393 Map, geographic, 479 of a curve, 466 Mapping functions, 467 Matrix, 114-122 augmented, 118 determinants of, 115 rank of, 115
		Maxima and minima, constrained, 163
		for functions of one variable, 158 for functions of several variables, 160
		Mean error, 516, 522 Mean-value theorems, 210n. Measure numbers, 397 Mechanical quadrature, 554 Membrane, vibration of, 377 Mercator’s projection, 479 Metric coefficients, 437
		08a
		Cable, flexible, 244 flow of electricity in, 386 supporting horizontal roadway, 242
		Cartography, 479 Catenary, 247, 252.
		Cauchy- Riemann equations, 221, 450, 455
		Cauchy’s equation, 322n.
		Cauchy’s integral formula, 401 Cauchy’s integral test, 12 Cauchy’s integral theorem, 455 Center of gravity, 177, 182, 183, 187, 190, 191, 196, 522
		Change of variables, in derivatives, 154
		in integrals, 183-188 Characteristic equation, 292 Charge, distribution of, 487 Charts, distribution, 506 Chemical reaction, 258 Circular functions, 247 Circulation, of a liquid, 475, 477 of a vector, 418, 419 Closed curve, area of, 199-201 direction around, 200 integral around, 201, 203, 206, 216, 421 simple, 200 cn w, 51
		Coefficients, Fourier, 65 metric, 437 Cofactor, 111, 112 Combinatory analysis, fundamental principle of, 493
		Commutative law, 394, 399, 400 Comparison test for series, 9 Complementary function, 290, 292 Complete elliptic integrals, 48 Complex number, 440 absolute value of, 441 argument of, 441 conjugate of, 444, 488 vector representation of, 440 Complex roots of unity, 87
		08b
		Complex variable, 440-491 functions of, 444-491 analytic, 451-491 derivative of, 449 integration of, 453 line integral of, 454 Taylor’s expansion for, 464 Components of force, 217 Composite function, 134, 137 Condenser, 283, 299, 305, 308, 387 Conditionally convergent series, 16, 17, 21
		Conditions, Cauchy-Riemann, 221, 450, 455 Dirichlet, 65
		for exact differential, 212, 216 Conductivity, 367, 426 Conductor, 486, 489 Conformal mapping, 465, 471 Conformal representation, applications of, 479-491 Conformal transformation, 467 Conjugate of a complex number, 444, 488
		Conjugate functions, 468, 470 Conservation of matter, law of, 429 Conservative field of force, 219, 411 Consistent systems of equations, 117-122
		Continuity, equations of, 221, 429, 481
		of functions, 23, 28, 124, 448 Contour line, 144
		Convergence, absolute, 16, 17, 20, 21, 33
		conditional, 16, 17, 21 interval of, 31, 33 of series, 4, 7
		tests for, 9, 11, 12, 15, 20, 27, 31, 33
		radius of, 31, 33 uniform, 23-30, 33 Cooling, law of, 254 Coordinate lines, 434 Coordinate surfaces, 434 Coordinates, curvilinear, 433-439 cylindrical, 152, 185, 190, 191, 378, 386, 434, 438
		09a
		Coordinates, ellipsoidal, 433 parabolic, 439
		polar, 183, 184, 276, 279, 386, 438 spherical, 152, 185, 382, 386, 434, 439
		cos x, 46, 250 cosh, 247
		Cosine, hyperbolic, 247 power series for, 38, 40 Cosine series, 73
		Cosines, direction, 146, 147, 151, 188, 194, 398 coth, 249 Cramer’s rule, 113 Cross product, 400 Cubic equation, algebraic solution of,#86
		graphical solution of, 83 Curl, 418, 422, 423, 438 Current, 386, 427 Curve, distribution, 504, 516 clastic, 240, 307 map of, 466 Curve fitting, 525-560 Curves, integral, 226, 228, 279 orthogonal, 277, 468 Curvilinear coordinates, 433-439 Cylinder functions (see Bessel functions)
		Cylindrical coordinates, 152, 185, 190, 191, 378, 386, 434, 438
		D
		Dam, gravity, 483 Damping, viscous, 302*
		Dead-beat motion, 304 Decomposition of vectors, 396 Definite integrals, 172 change of variable m, 183-188 evaluation of, 172 mean-value theorem for, 21071. Deflection, 299
		Degree of differential equation, 225 Del, V (see Nabla)
		Delta amplitude, dn, 51 De Moivre’s theorem, 90, 442
		09b
		Dependence, functional, 2 linear, 116
		Dependent events, 495 Derivation of differential equations, 231-247 Derivative, 125 directional, 143, 151, 219 normal, 144, 146, 152 of functions of a complex variable, 449, 452, 463
		of hyperbolic functions, 255 of series, 29, 33 partial, 125-143, 153 total, 130-143 Descartes’s rule of signs, 94 Determinants, 102-114 cofactors of, 111 expansion of, 106rc., Ill functional or Jacobian, 183 Laplace development of, 111 minors of, 110 of matrix, 115 product of, 110 properties of, 107-112 solution of equations by, 102-114 Wronskian, 317 Deviation, standard, 523 Diagonal term of determinant, 107 Diagram, pv, 223 Differences, 527
		Differential, exact, 211, 212, 216, 222, 224, 262, 411, 418, 420 of area, 184, 190 of volume, 185, 187, 190 partial, 128-143 total, 127-143
		Differential equations, 225-391 Bernoulli’s, 286 Bessel’s, 332, 380 Cauchy-Ricmann, 221, 450, 455 Cauchy’s, 322n. definition of, 225 degree of, 225 derivation of, 231-247 Euler’s, 322, 430 exact, 262 first order, 256, 267 Fourier, 425
		10a
		Differential equations, general solution of, 230, 290, 292, 350, 358 homogeneous, 259, 2G1 homogeneous linear, 290 integral curve of, 22(5, 228 integrating factor of, 205 Laplace’s, 309, 382, 385, 386, 439, 451, 470, 481 Legendre’s, 342, 384 linear, 226, 283-349, 357 numerical solution of, 346 of electric circuits, 301, 305, 386 of heat conduction, 367 of membrane, 377 of vibrating spring, 308 of vibrating string, 361 order of, 225 ordinary, 225-349 partial, 225, 350-391 particular integral of, 290, 292, 297, 318, 359 particular solution of, 230 second order, 269, 295 separation of variables m, 257 simultaneous, 312-315 singular solution of, 279 solution m series, 228, 325, 349, 364
		solution of, 226
		with constant coefficients, 287-315, 357
		with variable coefficients, 284, 315-349
		Differential expression, 225 Differential operators, 287-299, 357, 406
		Differentiation, of implicit functions, 132-142
		of series, 29, 33, 34, 80 partial, 123-171 term by term, 33, 34, 80 under integral sign, 167 Diffusion, 369, 427 Diffusivity, 368w.
		Direction angles, 146, 398 Direction components, 146 Direction cosines, 146, 147, 151, 188, 194, 398
		10b
		Direction ratios, 150, 151 Directional derivative, 143, 151, 219 (See also Gradient)
		Dirichlet conditions, 65 Discharge of condenser, 299 Discontinuity, finite, 64 Discriminant of cubic, 89 Distance, element of, 435 Distribution of charge, 487 Distribution charts, 506 Distribution turve, 504, 516 Distributive law, 399, 400 Divergence, of senes, 5, 8, 20 / of a vector, 411, 423, 438 Divergence theorem, 191, 415, 425, 428 dn uf 51
		Dot product, 399
		Double integrals, 173, 192, 202, 275 Drying of porous solids, 369 Dynamics, laws of, 231
		E
		c, 42 c'xy 250
		Effects, superposition of, 129, 223 E{k, <p), 48-51, 54 Elastic curve, 240, 307 Elasticity, 241, 422, 484-486 Electrodynamics, 422, 423n. Electron, 315
		Electrostatic field, 475, 477, 479 Electrostatic force, 487 Electrostatic potential, 487 Electrostatics, 486-491 Element, of arc, 467 of area, 184, 190, 437 of distance, 435 of volume, 185, 187, 190, 437 Elementary functions, 315 expansion of, 35-46, 65-82, 465 Ellipse, area of, 177, 202 center of gravity of, 177 length of arc of, 47 Ellipsoidal coordinates, 433 Elliptic functions, 51
		11a
		Minima (see Maxima and minima) Minimax, 162
		Modulus, of complex number, 441, 442
		of elliptic function, fc, 51 Moment, bending, 241 Moment of inertia, 177, 180, 182, 183, 187, 190, 191, 196, 241 Moments, method of, 544 Most probable value, 505 approximation for probability of, 511
		Motion, dead-beat, 304 fluid, 220 1aws of, 231, 234 of a membrane, 377 oscijlatory, 304 pendulum, 48, 234 simple harmonic, 233, 301, 314, 380
		under gravity, 232 Multiple integrals, 172-196 definition and evaluation of, 173, 179
		geometric interpretation of, 177 Multiplication, of complex numbers, 442
		of determinants, 110 of series, 21 of vectors, 399 Multiplicity of root, 93, 294 Multiplier, Lagrangian, 165 Multiply connected region, 205, 212, 455
		Mutually exclusive events, 497 N
		Nabla, or del, V, 194, 195, 407, 414, 422
		Newtonian potential, 196 Newton’s law, of attraction, 218 of cooling, 254 of dynamics, first law, 231 second law, 231, 272, 363 third law, 231, 234 of gravitation, 232 Newton’s method of solution, 97
		11b
		Normal, to a curve, 144 to a plane, 146, 147 to a surface, 147, 188, 407 Normal approximation, 515 Normal derivative, 144, 146, 152 (See also Gradient)
		Normal distribution curve, 516 Normal equations, 537, 540 Normal form, 146
		Normal law (see Gaussian law of error)
		Normal line, 144, 146-149 Normal orthogonal functions, 81 Numbers, complex, 440 measure, 397
		Numerical integration, 554-560 Numerical solution of differential equations, 346
		O
		Odd function, 68 Operator, 528
		differential, 287-299, 357, 406 vector (see Curl; Divergence;
		Gradient; Nabla)
		Order of differential equation, 225 Ordinary differential equations, 225-* ' 349
		(See also Differential equations) Ordinary discontinuity, 64 Origin of a vector, 393 Orthogonal curves, 277, 468 Orthogonal functions, 81, 339, 345 Orthogonal systems, 434 ^Orthogonal trajectories, 277-279 Orthogonal vectors, 398 Oscillation of a spring, 299 Oscillatory motion, 304 Overdamped, 303
		P
		p series, 10 Parabola, 244 Parabolic coordinates, 439 Paraboloid, hyperbolic, 162 Parachute, 253, 255
		12a
		Parallelogram law of addition, 394 Parameters, 277, 280 integrals containing, 167 variation of, 318
		Parametric equations, 143, 149, 150, 199, 215, 247
		Partial derivatives, 125-143, 153 Partial differential equation, 350-391
		derivation of, 351
		Fourier, 425
		integration of, 353
		Laplace's, 369, 382, 385, 386, 439
		linear, 357
		of elastic membrane, 377 of electric circuits, 386 of heat conduction, 367, 425 of vibrating string, 361 Partial differentials, 128-143 Partial differentiation, 123-171 Partial fractions, method of, 297 Partial sum, 4
		Particular integral, 290, 292, 297, 318, 359
		Particular solution, 230 Path, integrals independent of, 208, 216, 452, 455
		Pendulum, simple, 44, 234-238, 306 Periodic function, 64 Picard’s method, 347 Plane, equation of, 147 inclined, 280, 282, 306 normal form for, 146 tangent, 146-149 Point, of inflection, 159 singular, 451 Poisson formula, 512 Polar coordinates, 183,184, 276, 279, 386, 438
		Polygon, rectilinear, 478, 485 Polynomials, Legendre, 344, 384 Porous solids, drying of, 369 Potential, electrostatic, 487 gravitational, 219, 408 lines of equal, 277 Newtonian, 196
		velocity, 221, 222, 277, 430, 432, 453, 467, 480
		12b
		Potential function, 219, 411 Power series, 30-62 differentiation of, 33, 34 evaluation of integrals by, 43-46 expansion in, 35-^16 functions defined by, 33 integration of, 33, 34 interval of convergence of, 31, 33 operations on, 33-35 theorems on, 31-35 uniform convergence of, 33 uniqueness of expansion in, 38 whose terms are infinite series, 40 Power series solutions of differential equations, 325-346 Precision constant, 520, 521 Pressure on dam, 484 Primitive, 458
		Principal part of increment, 128 Probability, 492-524 Probability curve, 521 Probable error, 521 Probable value, most, 505 probability of, 511 Product, of determinants, 110 scalar, 399 vector, 400
		Projection, Mercator’s, 479 stereographic, 479 Pulley, slipping of belt on, 239 pv diagram, 223
		Q
		Quadrature, mechanical, 554 Quotient, of complex numbers, 444 of power series, 40
		R
		Radius of convergence, 31, 33 Radius vector, 195 Rank of matrix, 115 Ratio test, 11, 20, 31 Reaction, chemical, 258 Rearrangement of series, 17 Rectilinear polygon, 478, 485 Recursion formula, 273, 328, 331, 334
		13a
		Region, multiply connected, 205, 212, 455
		of integration, 173 simply connected, 205 Regula falsi, 101 Regular functions, 451 Remainder in Taylor’s series, 36-37 Remainder theorem, 92 Repeated trials, 501 Representation, applications of conformal, 479-491 Residuals, 534, 537 Resonance, 310 Riemann surface, 473 Right-handed system of axes, 397 Rod, flow of heat in, 373 vibitltions of, 366, 367 Roots, of equations, 83-102 isolation of, 92 theorems on, 92-94 of unity, w, o>2, 87 Rot (see Curl)
		Rotational field, 418 Rule, Cramer’s, 113 Simpson’s, 556 trapezoidal, J>56
		S
		Scalar field, 406, 408 Scalar point function, 406, 418 Scalar product, 399 application of, 404 Scalars, 392
		Schwartz transformation, 478, 485, 491
		Seepage flow, 483 Separation of variables, 257 Sequences, 2
		fundamental principle of, 6 limit of, 3
		Series, asymptotic, 524 binomial, 40
		evaluation of integrals by, 43-46 Fourier, 63-82 infinite, 1-62
		integration and differentiation of, 29, 33, 34
		13b
		Series, of constants, 6-22 of functions, 23-62 power, 30-62
		solution of differential equations by, 228, 325-346
		Taylor’s and Maclaurin’s, 37, 155, 228, 249, 464, 539 tests for convergence of, 9, 11, 12, 15, 20, 27, 31, 33
		theorems on, 17, 21, 27, 28, 29, 31, 33, 34, 36, 38
		uniform convergence of, 23-30 Shearing stresses, 485 Simple dosed curve, 200 Simple harmonic motion, 233, 301, 314, 380 equation of, 234 period of, 234
		Simple pendulum, 44, 234-238, 306 Simply connected region, 205 Simpson’s rule, 556 Simultaneous differential equations, 312-315
		Simultaneous equations, 102-122, 139-141 sin xt 41, 250 sin-1 x, 46
		Sine, hyperbolic, 247 length of curve, 55 power series for, 40, 41 Sme series, 73 Singular point, 451 Singular solution, 279 Singularities of function, 222 smh x, 247
		Sink {see Source and sink) Six-ordinate scheme, 548 Slipping of belt on pulley, 239 Small numbers, law of, 512
		sn u, 51
		Solenoidal field, 423 Solid angle, 195 Solids, drying of porous, 369 Solution, of cubic, 86-91
		of differential equations, 226, 228, 325
		general, 230, 290, 292, 350, 358 particular, 230
		14a
		Elliptic integrals, 47-55 complete, 48
		first kind, F(k, *>), 48-55, 238 second kind, E(k, <*>), 48-54 third kind, II(n, k} <p), 50 Empirical formulas, 525-500 Entropy, 224 Envelope, 279 Equation, auxiliary, 292 Bernoulli’s, 286 Bessel’s, 332, 380 characteristic, 292 cubic, 80 Euler, 322 Fourier, 425 indicia!, 334 integral, 347
		Laplace’s, 195, 309, 382, 385, 380, 439, 451, 470, 481 Legendre’s, 342, 384 of continuity, 221, 429, 481 of plane, 147 wave, 432
		Equations, Cauchy-Uiemann, 221, 450, 455
		consistent, 117-122 dependent, 105 differential, 225-391 Euler’s, 430
		inconsistent, 105, 117-122 normal, 537, 540
		parametric, 143, 149, 150, 199, 215 representing special types of data, 528
		simultaneous, 102-122, 139-141 solution of, 83-122 systems of, 102-122
		hofnogeneous linear, 119-122 non-homogeneous linear, 113-119
		Error, Gaussian law of, 520, 536 mean, 516 mean absolute, 522 mean square, 522 of observation, 516 probable, 521 small, 56
		Error function, 516
		14b
		Euler equation, 322 Euler formulas, 78, 251 Euler’s equations, 430 Euler’s theorem, 136 Evaluation of integrals, by differentiation, 169 m series, 43-46 Even function, 68 Events, dependent, 495 independent, 495 mutually exclusive, 497 Exact differential, 211, 212, 216, 222, 224, 262, 411, 418, 420 Exact differential equation, 262 Expansion, in Bessel functions, 339 in Fourier series, 65-82 m Legendre polynomials, 346 in Maclaunil’s series, 37 in power series, 37-46 in Taylor’s scries, 37 m trigonometric series, 65 uniqueness of, 38 Expectation, 500
		Expected number of successes, 508 Exponential form for trigonometric functions, 78, 251, 446, 447 Exponential function, expansion for, 42, 446
		Extremal values, 164 Extremum, 164
		F
		F(k, <?), 48-55 Factor, integrating, 265 Factor theorem, 92 Factonal, n!, approximation for, 509 (See also Gamma functions) Falling body, 58, 232 Field, 406
		conservative, 411 electrostatic, 475, 477, 479 irrotational, 418 Finite discontinuity, 64 Fitting, curve, 525-560 Flexure, 298
		Flow, of a liquid, 220, 424, 428, 477,
		. 478, 480-484
		15
		Section
		Chapter VII
		ORDINARY DIFFERENTIAL EQUATIONS
	79.	Equations of the First Order in Which One of the Variables Does
	86.	A Non-linear Equation Reducible to Linear Form (Bernoulli’s
	89.	'Hie Meaning of the Operator
		Chapter VIII
		PARTIAL DIFFERENTIAL EQUATIONS
		16
		Chapter IV
		PARTIAL DIFFERENTIATION
		Chapter V
		MULTIPLK INTEGRALS
		Chapter VI LINE INTEGRAL
		17
		Pao*
		Preface
		Chapter 1
		INFINITE SERIES
	13	J.3. Evaluation of Definite Integrals by Means of Power Series ... 43
		Chapter II FOURIER SERIES
	18.	Dinchlct Conditions. Derivation of Fourier Coefficients .... 65
		Chapter III
		SOLUTION OF EQUATIONS
		vii
		18
		Chapter IX VECTOR ANALYSIS
		Chapter X
		COMPLEX VARIABLE
		Section
		Chapter XI PROBABILITY
		Chapter XII
		EMPIRICAL FORMULAS AND CURVE FITTING