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-JJ..

m MEMOREAM

George Davidson

1825-1911

/

A SYNOPSIS

OF THE PRINCIPAL

FORMULAE AND RESULTS

OF

purr |lilatftematicÂ«iÂ»

CHARLES BROOKE, M. B.

OF ST. JOHN S COLLEGE.

CAMBRIDGE:

Printed by J. Smith, Printer to the Univerait}-.

SOLD BY J. ct J.J. DEIGHTOX, CAMBRIDGE;

AND C. J. G. I'v: F. RIVINGTON, LONDON.

1829

PREFACE.

The Author of the following pages, having experienced

the want of a compendious collection of results, in common

probably with most mathematical students, endeavoured,

during his undergraduateship, to form such a collection for

his o^^^^ use. This has since that time been arranged, and

considerably enlarged, and is now laid before the Public, in

the hope that it may in some measure facilitate the labour

of the analyst, by enabling him to compare and apply the

results of previous investigation.

It was originally intended to restrict these pages to

a mere collection of formulae, but it is hoped that in adding

an outline of many useful methods of operation, sufficiently

detailed to render them applicable to particular cases, the

increased size of the work will not be deemed an objec-

tion.

The references that have been given will probably be

found sufficiently minute to enable the reader to find without

difficulty any demonstration he may require ; had tlie proof

of each result been separately quoted, the size of the work

must unavoidably have been much increased. Tlie object

has been to refer to works of the greatest authority, and to

those which can most readily be procured.

The principles of Notation that have been adopted in the

following pages are explained in the Appendix ; wliere will

also be found an explanation of the very ingenious and

ivi5133i0

F KEl' ACK.

powerful symbols invented by Mr. Jarrett, of Catharine

Hall ; to whom the Author is indebted for several of the

theorems expressed by those symbols, that have been in-

troduced in the notes.

The great difficulty of printing mathematical works

correctly, may be fairly estimated by the number of errors

that occur in the works of the ablest writers on the subject.

As it is probable that some few may still have escaped the

Author's notice, he will feel much indebted to any of his

readers, who will have the kindness to communicate to the

publishers any errors they may happen to detect.

The Author begs leave to take this opportunity of

expressing his acknowledgements to the Syndics of the

University Press, for the very liberal manner in which they

have contributed to defray the expenses of the work.

St John's College,

May, 1829.

TABLE OF CONTENTS.

Algebra.

Art. ' Page

1^ â€” 3. Definitions and rules 1

4 â€” 6. Powers and roots ib.

7. Surds 2

8, 9 Greatest common divisor 3

10 â€” 12. Fractions ib.

13 â€” 15. Equations, simple and quadratic 4

16, 17- Ratios and proportion 7

18, 19. Arithmetical progression 8

20. Harmonical progression 10

21. Figurate and polygonal numbers ib.

22. 23. Geometrical progression 11

24. Permutations and combinations 13

25. Binomial theorem 14

26. 27. Polynomial theorem 17

28. Indeterminate coefficients 22

29. Logarithms ib.

30. 31. Continued fractions 28

32. General properties of equations 36

S3. Transformation of equations 38

34. Elimination 40

35, 36". Depression of equations 43

37, 38. Symmetrical functions of the roots of equations . . 44

3Q. The equation of differences 47

40. Limits of the roots of equations 48

41. Commensurable roots .30

42. Incommensurable roots 51

43. Quadratic factors 53

44. Impossible roots ib.

45. Application oi" the ihcoi-}- of equations to surd? . . 54

VI CONTENTS.

Art. Pack

46. Cubic equations 55

47. Biquadratic equations 56

48 â€” 50. The equation yÂ» + 2) = 59

51. General solution of equations 62

52 â€” 56. Indeterminate equations of the first degree . ... 63

57. Forms of square numbers 73

58 â€” 64. Indeterminate equations of the second degree ... 77

65. Porms of cubes 86

66. Indeterminate equations of the third degree ... 87

67' Forms of biquadrates 88

68. Indeterminate equations of the fourth degree ... 89

69. Solution of homogeneous indeterminate equations. . 90

70. Solution of the equation x^' â€” b=:ai/ 9I

71. General properties of numbers 92

72. Properties of prime numbers 93

73. Quadratic forms of prime numbers 94

74. Resolution of numbers into squares 95

75. Quadratic divisors 96

76. Ternary divisors 97

77. Scales of notation 99

Trigonometry.

I, 2. Divisions of the circle 100

3 â€” 6. Relations of the trigonometrical lines 101

7 â€” 9- Values of sin a, cos a, and tan a 102

10. Formulae relating to two arcs 104

II. Formulae relating to double arcs 105

12. Values of the sine, cosine, &c. of 30Â°, 45Â°, and 60Â°. IO6

13. Formula? relating to three arcs 107

14 â€” 18. Relations between the sides and angles of plane tri-

angles ib.

19. Solution of right-angled plane triangles IO9 .

20. Solution of oblique-angled plane triangles . . . .110

21. General principles of spherical trigonometry . . .112

22. Relations between the sides and angles of spherical

triangles 113

23. Formulae for the area of a spherical triangle . . . II6

24. Values of the radii of the inscribed and circumscribing

circles .; II7

CONTENTS. Vll

Art. 1'aoi:

25. Solution of riglit-angled spherical triangles . . .117

26. Solution of oblique-angled spherical triangles . . . 121

27. Series for the sine, cosine, &c. in terms of the arc 128

28. Series expressing the inverse circulai* functions . .129

2Q. Series for determining the value of tt 130

30. Formulae involving impossible quantities .... ib.

31 â€” 33. Formulae for the sums of arcs, and multiple arcs . 131

34. Powers of the sine and cosine of an arc . . . .136

35. Sums of trigonometrical series 137

36. Resolution of trigonometrical quantities into factors . 140

37. Approximate solution of triangles 141

38. Solution of triangles by series 142

39 â€” 41. Formula3 peculiar to geodetic operations . . . .143

42 â€” 48. Formulae for the construction of tables 144

49- Formulas for the verification of tables 150

50 â€” 53. Trigonometrical solution of equations 151

54. Properties of a quadrilateral inscribed in a circle . 153

55, 56. Properties of polygons 154

Analytical Geometry.

1 â€” 3 Definitions and principles 155

AnalytiCttl Geometry of two dimensions.

4, 5. The equation of a straight line, and its pi-operties . 155

6, 7- Transformation of co-ordinates 157

8â€”10. The circle 158

11 â€” 13. The parabola I60

â€¢14 â€” 16. The ellipse Ifi3

17â€”20. The hyperbola I67

21, 22. Discussion of lines of the second order 172

23, 24 Summary of equations I74,

Analytical Geometry of three dimensions.

25. The straight line I76

26, 27. The plane 177

28. The orthogonal projection of plane figures . . . .179

29Â« Oblique co-ordinates 180

30 â€” 33. Transformation of co-ordinates 181

34. The sphere . ^ 183

VIU CONTENTS.

Art. Pack

35. The cylinder 184

S6, 37. The cone ib.

38 â€” 42. Surfaces of the second order 185

43 â€” iC). The intersection of a surface of the second order, and

a plane ] 89

47. The tangent plane 19I

Differential CALcirLtis.

1. Differentiation of algebraic functions igS

2. Differentiation of exponential functions 194

3 â€” 6. Successive differentiation ib.

7. Differentiation of functions of superior orders . . . I96

8 â€” 10. Development of functions 197

11. Implicit functions 199

12. Transformation of the independent variable .... ib.

13. Elimination of an arbiti-ary function 200

14. 15. Particular values of the variable ib.

16. Maxima and minima 203

Integral Calculus.

1. Fundamental formula^ 204

2. Inverse circular functions ib.

3. Logarithmic integrals 205

4. 5. Decomposition of rational fractions ib.

6. Formulae for rendering surds rational 210

7 â€” 15. Integration of irrational functions 211

16, 17- Integration of elliptic transcendents 218

18. Integration of exponential and logarithmic functions 222

19 â€” 22. Integration of circular functions 224

23. Approximate values of integrals 228

24, 25. Successive integration 230~

26. Integration of functions of several variables . . . ib.

Differential Equations.

27 â€” 32. Differential equations of the first order, and of one

dimension in d^y 231

33 â€” 34,. The introduction of a factor which renders a differential

equation integrable 233

35. Singular solutions 235

CONTENTS. IX

Art. I'AGK

36, 37. Equations of more than one dimension in d^i/ . . . 236

38 â€” 42, Differential equations of the second order .... 239

43 â€” 45. Linear equations of the 71"' order . 240

4(> â€” 48. Simultaneous equations 24.0

49 â€” 52. Approximate integration of differential equations . . 248

53, 54. Comparison of elliptic transcendents 249

55. Total differential equations of several variables . . .251

56 â€” 61. Partial differential equations of the first order . . . 252

62 â€” 67. Partial differential equations of superior orders . . . 255

68. Integration of partial differential equations by series .261

6g. Singular solutions of partial differential equations . . ib.

Application of the Differential and Integral

Calculus to Geometry.

1 â€” 3. The contact of lines 263

4, 5. Asymptotes 265

6 â€” 8. Singular points 266

9 â€” 11. Curves referred to polar co-ordinates 267

12, 13. Circumscribing figures 268

14, 15. The contact of surfaces 269

16, 17. The contact of curves of double curvature . . . .271

18â€”23. Rectification 272

24â€”26. Quadrature 274

27. Cubature 275

28. Transformation of co-ordinates ib.

29. Conditions which render a curve quadrable .... 276

30. Conditions which render a curve rectifiable .... ib.

31. Trajectories ib.

32 â€” 38. Remarkable algebraical curves 277'

39_48. Remarkable transcendental curves 279

Calculus of Variations 288

Calculus of Finite Differences.

Direct method of . differences:

1. Fundamental formula* 292

2, 3. Successive differences ib.

4. Series involving the differences and differential

coeflicients 29 !â€¢

b

X - CON TK NTS,

AUT, I'AGt

5. Difrcronc'c'?: of functions of two or more variables . . 295

6 â€” f). Intcqjolation of series Of-

10. Differences of the trigonometrical lines 298

11 â€” 14. The variation of triangles 299

14 â€” 18. The construction of logarithmic and trigonometrical

tables 305

Inverse method of differences.

19 â€” 20. Integration of algebraic functions ....... 309

21. The numbers of Bernoulli 310

22. Integration of exponential functions 311

23. 24. Successive integration^ 312

25 â€” 32. Equations of differences 314

33, 34. Equations of mixed differences 319

35 â€” 30. Summation of series 320

40 â€” 42. Theory of generating functions 326

Functional Equations.

1 â€” 3. Reduction of functional equations to equations of

differences 330

4 â€” 9. General solution of functional equations obtained from

a particular solution 332

10, 11. Differential functional equations 339

Appendix 341

The following symbols, although not original, may perhaps

require explanation, as they have not yet been generally

introduced. '

Â«;:}>& is read a is not greater than b,

Â«

m MEMOREAM

George Davidson

1825-1911

/

A SYNOPSIS

OF THE PRINCIPAL

FORMULAE AND RESULTS

OF

purr |lilatftematicÂ«iÂ»

CHARLES BROOKE, M. B.

OF ST. JOHN S COLLEGE.

CAMBRIDGE:

Printed by J. Smith, Printer to the Univerait}-.

SOLD BY J. ct J.J. DEIGHTOX, CAMBRIDGE;

AND C. J. G. I'v: F. RIVINGTON, LONDON.

1829

PREFACE.

The Author of the following pages, having experienced

the want of a compendious collection of results, in common

probably with most mathematical students, endeavoured,

during his undergraduateship, to form such a collection for

his o^^^^ use. This has since that time been arranged, and

considerably enlarged, and is now laid before the Public, in

the hope that it may in some measure facilitate the labour

of the analyst, by enabling him to compare and apply the

results of previous investigation.

It was originally intended to restrict these pages to

a mere collection of formulae, but it is hoped that in adding

an outline of many useful methods of operation, sufficiently

detailed to render them applicable to particular cases, the

increased size of the work will not be deemed an objec-

tion.

The references that have been given will probably be

found sufficiently minute to enable the reader to find without

difficulty any demonstration he may require ; had tlie proof

of each result been separately quoted, the size of the work

must unavoidably have been much increased. Tlie object

has been to refer to works of the greatest authority, and to

those which can most readily be procured.

The principles of Notation that have been adopted in the

following pages are explained in the Appendix ; wliere will

also be found an explanation of the very ingenious and

ivi5133i0

F KEl' ACK.

powerful symbols invented by Mr. Jarrett, of Catharine

Hall ; to whom the Author is indebted for several of the

theorems expressed by those symbols, that have been in-

troduced in the notes.

The great difficulty of printing mathematical works

correctly, may be fairly estimated by the number of errors

that occur in the works of the ablest writers on the subject.

As it is probable that some few may still have escaped the

Author's notice, he will feel much indebted to any of his

readers, who will have the kindness to communicate to the

publishers any errors they may happen to detect.

The Author begs leave to take this opportunity of

expressing his acknowledgements to the Syndics of the

University Press, for the very liberal manner in which they

have contributed to defray the expenses of the work.

St John's College,

May, 1829.

TABLE OF CONTENTS.

Algebra.

Art. ' Page

1^ â€” 3. Definitions and rules 1

4 â€” 6. Powers and roots ib.

7. Surds 2

8, 9 Greatest common divisor 3

10 â€” 12. Fractions ib.

13 â€” 15. Equations, simple and quadratic 4

16, 17- Ratios and proportion 7

18, 19. Arithmetical progression 8

20. Harmonical progression 10

21. Figurate and polygonal numbers ib.

22. 23. Geometrical progression 11

24. Permutations and combinations 13

25. Binomial theorem 14

26. 27. Polynomial theorem 17

28. Indeterminate coefficients 22

29. Logarithms ib.

30. 31. Continued fractions 28

32. General properties of equations 36

S3. Transformation of equations 38

34. Elimination 40

35, 36". Depression of equations 43

37, 38. Symmetrical functions of the roots of equations . . 44

3Q. The equation of differences 47

40. Limits of the roots of equations 48

41. Commensurable roots .30

42. Incommensurable roots 51

43. Quadratic factors 53

44. Impossible roots ib.

45. Application oi" the ihcoi-}- of equations to surd? . . 54

VI CONTENTS.

Art. Pack

46. Cubic equations 55

47. Biquadratic equations 56

48 â€” 50. The equation yÂ» + 2) = 59

51. General solution of equations 62

52 â€” 56. Indeterminate equations of the first degree . ... 63

57. Forms of square numbers 73

58 â€” 64. Indeterminate equations of the second degree ... 77

65. Porms of cubes 86

66. Indeterminate equations of the third degree ... 87

67' Forms of biquadrates 88

68. Indeterminate equations of the fourth degree ... 89

69. Solution of homogeneous indeterminate equations. . 90

70. Solution of the equation x^' â€” b=:ai/ 9I

71. General properties of numbers 92

72. Properties of prime numbers 93

73. Quadratic forms of prime numbers 94

74. Resolution of numbers into squares 95

75. Quadratic divisors 96

76. Ternary divisors 97

77. Scales of notation 99

Trigonometry.

I, 2. Divisions of the circle 100

3 â€” 6. Relations of the trigonometrical lines 101

7 â€” 9- Values of sin a, cos a, and tan a 102

10. Formulae relating to two arcs 104

II. Formulae relating to double arcs 105

12. Values of the sine, cosine, &c. of 30Â°, 45Â°, and 60Â°. IO6

13. Formula? relating to three arcs 107

14 â€” 18. Relations between the sides and angles of plane tri-

angles ib.

19. Solution of right-angled plane triangles IO9 .

20. Solution of oblique-angled plane triangles . . . .110

21. General principles of spherical trigonometry . . .112

22. Relations between the sides and angles of spherical

triangles 113

23. Formulae for the area of a spherical triangle . . . II6

24. Values of the radii of the inscribed and circumscribing

circles .; II7

CONTENTS. Vll

Art. 1'aoi:

25. Solution of riglit-angled spherical triangles . . .117

26. Solution of oblique-angled spherical triangles . . . 121

27. Series for the sine, cosine, &c. in terms of the arc 128

28. Series expressing the inverse circulai* functions . .129

2Q. Series for determining the value of tt 130

30. Formulae involving impossible quantities .... ib.

31 â€” 33. Formulae for the sums of arcs, and multiple arcs . 131

34. Powers of the sine and cosine of an arc . . . .136

35. Sums of trigonometrical series 137

36. Resolution of trigonometrical quantities into factors . 140

37. Approximate solution of triangles 141

38. Solution of triangles by series 142

39 â€” 41. Formula3 peculiar to geodetic operations . . . .143

42 â€” 48. Formulae for the construction of tables 144

49- Formulas for the verification of tables 150

50 â€” 53. Trigonometrical solution of equations 151

54. Properties of a quadrilateral inscribed in a circle . 153

55, 56. Properties of polygons 154

Analytical Geometry.

1 â€” 3 Definitions and principles 155

AnalytiCttl Geometry of two dimensions.

4, 5. The equation of a straight line, and its pi-operties . 155

6, 7- Transformation of co-ordinates 157

8â€”10. The circle 158

11 â€” 13. The parabola I60

â€¢14 â€” 16. The ellipse Ifi3

17â€”20. The hyperbola I67

21, 22. Discussion of lines of the second order 172

23, 24 Summary of equations I74,

Analytical Geometry of three dimensions.

25. The straight line I76

26, 27. The plane 177

28. The orthogonal projection of plane figures . . . .179

29Â« Oblique co-ordinates 180

30 â€” 33. Transformation of co-ordinates 181

34. The sphere . ^ 183

VIU CONTENTS.

Art. Pack

35. The cylinder 184

S6, 37. The cone ib.

38 â€” 42. Surfaces of the second order 185

43 â€” iC). The intersection of a surface of the second order, and

a plane ] 89

47. The tangent plane 19I

Differential CALcirLtis.

1. Differentiation of algebraic functions igS

2. Differentiation of exponential functions 194

3 â€” 6. Successive differentiation ib.

7. Differentiation of functions of superior orders . . . I96

8 â€” 10. Development of functions 197

11. Implicit functions 199

12. Transformation of the independent variable .... ib.

13. Elimination of an arbiti-ary function 200

14. 15. Particular values of the variable ib.

16. Maxima and minima 203

Integral Calculus.

1. Fundamental formula^ 204

2. Inverse circular functions ib.

3. Logarithmic integrals 205

4. 5. Decomposition of rational fractions ib.

6. Formulae for rendering surds rational 210

7 â€” 15. Integration of irrational functions 211

16, 17- Integration of elliptic transcendents 218

18. Integration of exponential and logarithmic functions 222

19 â€” 22. Integration of circular functions 224

23. Approximate values of integrals 228

24, 25. Successive integration 230~

26. Integration of functions of several variables . . . ib.

Differential Equations.

27 â€” 32. Differential equations of the first order, and of one

dimension in d^y 231

33 â€” 34,. The introduction of a factor which renders a differential

equation integrable 233

35. Singular solutions 235

CONTENTS. IX

Art. I'AGK

36, 37. Equations of more than one dimension in d^i/ . . . 236

38 â€” 42, Differential equations of the second order .... 239

43 â€” 45. Linear equations of the 71"' order . 240

4(> â€” 48. Simultaneous equations 24.0

49 â€” 52. Approximate integration of differential equations . . 248

53, 54. Comparison of elliptic transcendents 249

55. Total differential equations of several variables . . .251

56 â€” 61. Partial differential equations of the first order . . . 252

62 â€” 67. Partial differential equations of superior orders . . . 255

68. Integration of partial differential equations by series .261

6g. Singular solutions of partial differential equations . . ib.

Application of the Differential and Integral

Calculus to Geometry.

1 â€” 3. The contact of lines 263

4, 5. Asymptotes 265

6 â€” 8. Singular points 266

9 â€” 11. Curves referred to polar co-ordinates 267

12, 13. Circumscribing figures 268

14, 15. The contact of surfaces 269

16, 17. The contact of curves of double curvature . . . .271

18â€”23. Rectification 272

24â€”26. Quadrature 274

27. Cubature 275

28. Transformation of co-ordinates ib.

29. Conditions which render a curve quadrable .... 276

30. Conditions which render a curve rectifiable .... ib.

31. Trajectories ib.

32 â€” 38. Remarkable algebraical curves 277'

39_48. Remarkable transcendental curves 279

Calculus of Variations 288

Calculus of Finite Differences.

Direct method of . differences:

1. Fundamental formula* 292

2, 3. Successive differences ib.

4. Series involving the differences and differential

coeflicients 29 !â€¢

b

X - CON TK NTS,

AUT, I'AGt

5. Difrcronc'c'?: of functions of two or more variables . . 295

6 â€” f). Intcqjolation of series Of-

10. Differences of the trigonometrical lines 298

11 â€” 14. The variation of triangles 299

14 â€” 18. The construction of logarithmic and trigonometrical

tables 305

Inverse method of differences.

19 â€” 20. Integration of algebraic functions ....... 309

21. The numbers of Bernoulli 310

22. Integration of exponential functions 311

23. 24. Successive integration^ 312

25 â€” 32. Equations of differences 314

33, 34. Equations of mixed differences 319

35 â€” 30. Summation of series 320

40 â€” 42. Theory of generating functions 326

Functional Equations.

1 â€” 3. Reduction of functional equations to equations of

differences 330

4 â€” 9. General solution of functional equations obtained from

a particular solution 332

10, 11. Differential functional equations 339

Appendix 341

The following symbols, although not original, may perhaps

require explanation, as they have not yet been generally

introduced. '

Â«;:}>& is read a is not greater than b,

Â«

Online Library → Charles Brooke → A synopsis of the principal formulae and results of pure mathematics → online text (page 1 of 17)