VIII 



CONTENTS. 



CHAPTER IV. 

 Principle of Least Action. Generalized Equations of Motion. 



Art. Page 



34. Hamilton's Principle ... 97 



35. Principle of Least Action . 99 



36. Generalized Coordinates. La- 

 grange's Equations .... 108 



37. Lagrange's Equations by di- 

 rect Transformation. Various 

 Reactions 115 



Art. 



38. Equation of Activity. Integral 

 of Energy 



39. Hamilton's Canonical Equat. 

 39a.Varying Constraint 



40. Hamilton's Principle the most 

 general dynamical principle 



41. Principle of Varying Action 



CHAPTER V. 

 Oscillations and Cyclic Motions. 



42. Tautochrone for Gravity . . 144 



43. Damped Oscillations . . . 148 



44. Forced Vibrations. Resonance 152 



45. General Theory of small 

 Oscillations ...... 157 



Vibration of a String of Beads. 

 Continuous String . . . . 164 



Forced Vibrations of General 

 System ....... ".173 



Cyclic Motions. Ignoration 



of Coordinates 175 



of 



46 



47 



48 



49. Example. Three Degrees 

 Freedom. General Case . . 



50. Effect of Linear Terms in 

 Kinetic Potential. Gyroscopic 

 Forces 



51. Cyclic Systems 



52. Properties of Cyclic Systems. 

 Reciprocal Relations . . . 



53. Work done by the Cyclic and 

 Positional Forces . . . . 



54. Examples of Cyclic Systems 



Page 



125 

 126 



129 



130 

 131 



181 



184 

 188 



190 



192 

 193 



PART II. 



DYNAMICS OF RIGID BODIES. 



Systems of Vectors. 



CHAPTER VI. 

 Distribution of Mass. 



Instantaneous Motion. 



199 



55. Translations and Rotations 



56. Rotations about two Parallel 

 Axes 201 



57. Rotations about Intersecting 

 Axes. Infinitesimal Rotations 202 



58. Vector -couples 204 



59. Statics of a Rigid Body . . 205 

 59 a. Parallel Forces. Force- 

 couples 205 



60. Reduction of Groups of Forces. 

 Dualism 209 



61. Variation of the Elements of 

 the Reduction. Central Axis. 

 Null- System 209 



62. Vector- cross . . ... . 212 



63. Complex of Double -lines. . 214 



64. Composition of Screws . . 216 



65. Work of Wrench in Produ- 

 cing a Twist 220 



66. Analytical Representation. 

 Line Coordinates 221 



67 



Momentum Screw. Dyna- 

 mics 224 



68. Momentum of Rigid Body . 225 



69. Centrifugal Forces .... 228 

 Moments of Inertia. Parallel 



Axes 229 



Moments of Inertia at a Point. 

 Ellipsoid of Inertia .... 229 



72. Ellipsoid of Gyration ... 232 



73. Ellipsoidal Coordinates . . 234 

 Axes of Inertia at Various 



Points 237 



Calculation of Moments of 



Inertia . . 240 



Analytical Treatment of Kine- 

 matics of a Rigid System. 



Moving Axes 243 



Relative Motion 247 



78. Angular Acceleration . . . 248 



79. Kinetic Energy and Momen- 

 tum due to Rotation . 249 



70 



71 



74 



75 



76 



77 



