165] THEORY OF NORMAL MODES. 269 



If any of the coefficients of stability (c s ) be negative, the 

 value of cr s is pure imaginary. The circular function in (7) is then 

 replaced by real exponentials, and an arbitrary displacement 

 will in general increase until the assumptions on which the 

 approximate equation (6) is based become untenable. The un 

 disturbed configuration is then unstable. Hence the necessary 

 and sufficient condition of stability is that the potential energy V 

 should be a minimum in the configuration of equilibrium. 



To find the effect of extraneous forces, it is sufficient to 

 consider the case where Q s varies as a simple-harmonic function of 

 the time, say 



Q s =C 8 coa(at+) ..................... (9), 



where the value of a is now prescribed. Not only is this the 

 most interesting case in itself, but we know from Fourier s 

 Theorem that, whatever the law of variation of Q s with the time, it 

 can be expressed by a series of terms such as (9). A particular 

 integral of (9) is then 



This represents the forced oscillation due to the periodic force 

 Q s . In it the motion of every particle is simple-harmonic, of the 

 prescribed period 27T/0-, and the extreme displacements coincide in 

 time with the maxima and minima of the force. 



A constant force equal to the instantaneous value of the 

 actual force (9) would maintain a displacement 



* 



(11), 



the same, of course, as if the inertia-coefficient a s were null. 

 Hence (10) may be written 



*?tr^Stf* ..................... (12X 



where cr s has the value (8). This very useful formula enables us 

 to write down the effect of a periodic force when we know that of 

 a steady force of the same type. It is to be noticed that q s and Q s 

 have the same or opposite phases according as cr J a s , that is, 

 according as the period of the disturbing force is greater or less 

 than the free period. A simple example of this is furnished by a 

 simple pendulum acted on by a periodic horizontal force. Other 



