48 DYNAMICAL THEOEY OF SOUND 



where A (p 2 ) is the determinant on the left-hand side of 16 

 (3), with p 2 written for n 2 . The general conclusion is that when 

 a periodic force of simple-harmonic type acts on any part of 

 the system, every part will execute a simple-harmonic vibration 

 of the same period, with synchronism of phase, but the 

 amplitude will of course be different in different parts. When 

 the period of the forced vibration nearly coincides with that 

 of one of the free modes, an abnormal amplitude of forced 

 vibration will in general result, owing to the smallness of the 

 denominator in the formulae (2). For a complete account of 

 this matter we should have to take dissipative forces into 

 consideration, as in 12. 



A remarkable theorem of reciprocity, first proved by Helmholtz 

 for aerial vibrations, and afterwards greatly extended by Lord 

 Rayleigh, follows from (2). If we imagine a second case of 

 forced vibration (distinguished by accents) in which Q/ = 

 whilst Q 2 ' varies as cos pt, we shall have 



Comparing with (2), we see that 



,:ft = 9,':ft'. ..................... (4) 



The interpretation is most easily expressed when the "forces" 

 Q 1 and Q 2 ' are of the same character, e.g. both ordinary statical 

 forces, or both couples, in which case we may put Qi = Q 2 ', and 

 obtain q 2 = qi'. In words: The vibration of type 2 due to a 

 given periodic force of type 1 agrees in amplitude and phase 

 with the vibration of type 1 due to an equal force of type 2. 

 An example from the theory of strings will be found in 28. 

 The above proof is easily extended to the general case of 

 ra degrees of freedom. 



18. Composition of Simple-Harmonic Vibrations in 

 Different Directions. 



We recur to the subject of composition of simple-harmonic 

 vibrations which, though not so important as in Optics, claims a 

 little further attention. If in a freely vibrating system we fix 

 our attention on a particular particle, the directions in which it 



