PROFESSOR AT BONN 151 



a continuous sensation of tone is felt to be a consonance, a 

 discontinuous sensation to be a dissonance. Tell Richelot that 

 I am now endeavouring to establish thorough bass upon an 

 integration of partial differential equations of the second order 

 and second degree. I hope this may interest him more than 

 the subjects of my earlier work/ 



On June 18, he writes to William Thomson : ' I have busied 

 myself with certain observations in acoustics during the winter, 

 especially on combination tones, which have shown me that 

 these tones, which have hitherto always been supposed to 

 originate within the ear, can arise externally to it also, when- 

 ever the vibrations of the air or of any other elastic body, 

 including the tympanum of the ear, are so strong that the 

 second power of the elongation has influence on the motion, so 

 that the law of the superposition of small vibrations ceases 

 to be valid. If m and n are the vibration numbers of two 

 simultaneously sounding tones, I have, in addition to the long- 

 recognized tone of (m n) beats, discovered another tone of 

 (m + n) beats/ 



His paper on Combination Tones appeared the same year 

 in Poggendorff's Annalen. It was known that there is on 

 the one hand undisturbed superposition of various sound- 

 waves in the air, and that on the other the ear, when simul- 

 taneously affected by several such waves of sound, has the 

 power of perceiving and recognizing each of them separately. 

 But in such cases the ear not only hears the different tones 

 excited by the resonant bodies, but other additional, if feebler, 

 tones, the combinational tones, which are not primarily pro- 

 duced by one of the sounding bodies, but are of secondary 

 origin from the concurrence of two primary tones. These 

 were formerly held to be subjective phenomena, dependent on 

 the special nature of the sensation of the vibrations of sound 

 through the auditory nerve. Helmholtz, however, submitted 

 this question, as well as the possible existence of other than 

 the known combinational tones, to a searching examination, 

 supplemented by mathematical analysis. 



He names any oscillation of an elastic body, in which the 

 distance of each vibrating particle from the position of equi- 

 librium can be represented as a simple sine-function, with 

 a constant factor, of a linear expression of time, a simple 



