268 TIDAL WAVES. [CHAP. VIII 



In the present application to infinitely small motions, these 

 take the form 



a>8qs + c s qs = Q s (6). 



It is easily seen from this that the dynamical characteristics of the 

 normal coordinates are (1) that an impulse of any normal type 

 produces an initial motion of that type only, and (2) that a steady 

 extraneous force of any type maintains a displacement of that 

 type only. 



To obtain the free motions of the system we put Q s in (6). 

 Solving we find 



q s = A s cos (o- s t + e s ) (7), 



where o- 8 = (c 8 /a s )^ (8)*, 



and A 8) e 8 are arbitrary constants. Hence a mode of free motion 

 is possible in which any normal coordinate q 8 varies alone, and 

 the motion of any particle of the system, since it depends 

 linearly on q s , will be simple-harmonic, of period 27r/cr s , and 

 every particle will pass simultaneously through its equilibrium 

 position. The several modes of this character are called the 

 ' normal modes ' of vibration of the system ; their number is equal 

 to that of the degrees of freedom, and any free motion whatever 

 of the system may be obtained from them by superposition, with 

 a proper choice of the ' amplitudes ' (A s ) and ' epochs ' (e s ). 



In certain cases, viz. when two or more of the free periods 

 (27T/0-) of the system are equal, the normal coordinates are to a 

 certain extent indeterminate, i.e. they can be chosen in an infinite 

 number of ways. An instance of this is the spherical pendulum. 

 Other examples will present themselves later; see Arts. 187, 191. 



If two (or more) normal modes have the same period, then 

 by compounding them, with arbitrary amplitudes and epochs, we 

 obtain a small oscillation in which the motion of each particle is 

 the resultant of simple-harmonic vibrations in different directions, 

 and is therefore, in general, elliptic-harmonic, with the same 

 period. This is exemplified in the conical pendulum ; an im- 

 portant instance in our own subject is that of progressive waves in 

 deep water (Chap. IX.). 



* The ratio <r/27r measures the ' frequency' of the oscillation. It is convenient, 

 however, to have a name for the quantity a itself; the term 4 speed ' has been used in 

 this sense by Lord Kelvin and Prof. G. H. Darwin in their researches on the Tides. 



