THEOKY OF VIBRATIONS 45 



in any one mode determinate, though usually different for 

 different modes, the only arbitrary elements being the absolute 

 amplitude and the phase-constant. 



The equations of motion being necessarily linear, since 

 products and squares of the coordinates and their differential 

 coefficients with respect to the time are expressly excluded, it 

 follows that the different solutions may be superposed by 

 addition of the corresponding expressions. This has been 

 sufficiently illustrated in the preceding examples. By super- 

 posing in this way the m normal modes, each with its arbitrary 

 amplitude and phase, we obtain a solution involving 2m 

 arbitrary constants, which is exactly the right number to 

 enable us to represent the effect of arbitrary initial values of 

 the coordinates q l} q^, ... q m and velocities q lt q z , ... q m . In 

 other words, the most general free motion of the system about 

 a configuration of stable equilibrium may be regarded as made 

 up of the m normal modes with suitable amplitudes and initial 

 phases. This principle dates from D. Bernoulli* (1741). 



In particular cases it may happen that two (or more) of the 

 natural periods of the system coincide. There is then a corre- 

 sponding degree of indeterminateness in the character of the 

 normal modes. The simplest example is furnished by the 

 spherical pendulum, or by a particle oscillating in a smooth 

 spherical bowl. The normal modes may then be taken to 

 correspond to any two horizontal directions through the position 

 of equilibrium. From the theoretical standpoint such coinci- 

 dences may be regarded as accidental, since they are destroyed 

 by the slightest alteration in the constitution of the system 

 (e.g. if the bowl in the above illustration be in the slightest 

 degree ellipsoidal), but in practice they often lead to interesting 

 results. Cf. 53 below. 



An important characteristic of the normal modes, first 

 pointed out by Lord Rayleigh in 1883, has still to be referred 



* Daniel Bernoulli (17001782), one of the younger members of the 

 distinguished family of Swiss mathematicians. Professor of mathematics at 

 St Petersburg (172533), and of physics at Bale (175082). His chief work 

 was on hydrodynamics, on the theory of vibrating strings, and on the flexure 

 of elastic beams. 



