XIV 
CONTENTS 
PAGE 
13. Continuation. Equal Roots ....... 342 
14. Small Oscillations of a System with n Degrees of Freedom . . 343 
III. Geometrical Interpretation. Singular Solutions 
15. Meaning of a Differential Equation ...... 345 
16. Continuation. Differential Equations of the Second Order, and 
Higher 348 
17. Singular Solutions .......... 349 
18. Continuation. The General Case ....... 352 
IV. Solution by Series. Integrating Factor 
19. Bessel’s Functions. Zonal Harmonics ...... 353 
20. Integrating Factor .......... 356 
V. Partial Differential Equations 
21. Nature of the Solution 359 
22. Linear Partial Differential Equations of the First Order . . 360 
23. General Partial Differential Equation of the First Order . . 363 
24. Integration by Characteristics ....... 366 
25. Extension to the Case of n -f- 1 Variables ..... 370 
26. The Equations of Dynamics ........ 371 
27. The Partial Differential Equations of Mathematical Physics . 371 
CHAPTER XV 
ELASTIC VIBRATIONS 
1. Simple Harmonic Motion 376 
2. Damping ........... 377 
3. Discussion of the Result ........ 378 
4. Forced Vibrations .......... 379 
5. Integration of the Differential Equation . . . . .381 
6. Discussion of the Result ........ 382 
7. The Differential Equation of the Vibrating String .... 383 
8. Continuation ; the General Case ....... 386 
9. The Differential Equation of the Vibrating Membrane . . . 389 
CHAPTER XVI 
FOURIER'S SERIES AND ORTHOGONAL FUNCTIONS 
1. Formal Development into a Fourier’s Series 391 
2. The General Problem of Development into Series. Power Series . 394 
3. Continuation. Series of Orthogonal Functions .... 397 
4. Approximations according to the Principle of Least Squares . . 399 
5. Zonal Harmonics .......... 400 
6. Bessel’s Functions 402