i 5 4 HERMANN VON HELMHOLTZ 



mathematical deduction that the different simple oscillatory 

 motions of an elastic body are superposed without disturbance, 

 as long as the amplitude of the oscillations is so small that 

 the motive forces excited by the displacements are sensibly 

 proportional to the latter. When, however, the amplitude of the 

 vibrations is so great that the squares of the displacements exert 

 a perceptible influence on the magnitude of the motive forces, 

 new systems of simple oscillatory motions arise, the vibration- 

 period of which corresponds with that of the combinational 

 tones. The vibratory motions of the air, produced by various 

 sources of sound in simultaneous action, correspond with the 

 exact sum of the motions produced by the separate sources 

 of sound only when the vibrations are infinitesimal, i. e. when 

 the alterations of density are so small that they do not come 

 into play as compared with the total density, and when there- 

 fore the displacements of the oscillating particles are vanish- 

 ingly small in comparison with the dimensions of the entire 

 mass; if the law does not hold good, combination tones are 

 produced. It followed from these considerations that the 

 origin of combinational tones is not necessarily dependent on 

 the sensations of the auditory nerve. With two simultaneous 

 tones of the right strength, the combination tones may corre- 

 spond with actual vibrations of the tympanum and auditory 

 ossicles, received by the nervous apparatus in the usual way. 

 But, as Helmholtz pointed out, conditions similar to those 

 affecting the movements of the apparatus of the tympanic 

 cavity may also occur outside the ear, so that vibrations 

 corresponding to combination tones may be produced quite 

 independent of the human ear, and external to it, and he made 

 experiments to prove the objective existence of combination 

 tones. The nature of combination tones has therefore nothing 

 to do with the marvellous property by which the ear analyses 

 a confused group of sound-waves into the single tones of which 

 it is made up, and distinguishes the voices of separate indi- 

 viduals, and the quality of the different musical instruments. In 

 regard to the origin of this property of analysing the aerial 

 motion produced by the joint effect of a number of resonant 

 bodies into the elements corresponding with the particular 

 effects, Helmholtz had indeed formed a special hypothesis, 

 but felt it necessary to test it upon the various phenomena. 



