294 DYNAMICAL THEORY OF SOUND 



existence has even been denied. It has however been objectively 

 demonstrated by Riicker and Edser*, by its effect on a tuning 

 fork of the same frequency. 



Difference-tones due to the causes just considered are most 

 easily perceptible where we have a mass of air which is subject 

 to the joint and vigorous action of the primary vibrations, as 

 in the harmonium and the siren; they can then, like other 

 tones, be reinforced by suitable resonators. 



There is however a way in which combination tones may 

 conceivably be originated in the ear itself. To explain this 

 it is necessary briefly to consider the forced vibrations of an 

 unsymmetrical system. When a particle, or any system having 

 virtually one degree of freedom, receives a displacement x, the 

 force (intrinsic to the system) which tends to restore equilibrium 

 is a function of x, and may be supposed expressed, for small 

 values of x, by a series 



An example is furnished by the common pendulum, where 

 the force of restitution is proportional to g sin 6, or 



but here, on account of the symmetry with respect to the 



vertical, the force changes sign with 6, so that 



only odd powers of 6 occur. The correction for 



small finite amplitudes depends therefore on the 



term of the third order in 6. But if the system 



be unsymmetrical, as in the case of a pendulum 



hanging from the circumference of a horizontal 



cylinderf, the term of the second order comes in, 



and the correction is more important. Helmholtz Fi 



lays stress on the fact that in the slightly 



* Phil. Mag. (5), vol. xxxix. (1895). 



f If a be the radius, and I the length of the free portion of the string when 

 vertical, the potential energy is 







where .9 is the arc described by the bob from the lowest position. The restoring 

 force is therefore 



dV_mg a lmga^ t 



