FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 107 



nates, and that the vessel and orifice have special forms. Finally, the 

 value of the earlier results relative to vortices was shown for the 

 determination of the motion of fluid particles in discbarge. 



In the early part of his stay at Heidelberg, and while engaged in his 

 hydrodynamic researches, he was also pursuing acoustic and aerody- 

 namic investigations. In his articles, "On combination-tones" and 

 " Upon tbetone color of the vowels," he again adopted the view of other 

 physicists wbich he had abandoned in some of his earlier and less 

 important works, namely, that each sensation, as it is aroused by the 

 atmospheric vibrations going out from a single sounding body, is com- 

 pounded from simple sensations or tones such as are caused by a simple 

 vibratory motion of the air. This hypothesis he formulated mathe- 

 matically, proceeding from the theorem of Fourier for the representa- 

 tion of any periodic motion as the sum of a series of sine-motions. 

 The pitch of a tone was defined as the height of the lowest of its con- 

 stituent tones, which is called the fundamental, the others being dis- 

 tinguished as overtones. Exact experimental investigation showed 

 that the musical tone color depends only on the i>resence and strength, 

 but not on the phase differences, of the overtones which are included 

 in the sound. The sounds as they penetrate the ear could be resolved 

 into their simple factors, and these could be again reunited. The 

 sounds produced by the voice were found to differ from the sounds of 

 most other musical instruments in that the strength of their overtones 

 depends not on the corresponding cardinal numbers, but on their abso- 

 lute pitch. Throughout these researches we may perceive the design 

 to make a sharp distinction between the sensations in so far as they 

 consist in impressions peculiar to our nerve apparatus, such as those 

 due to overtones, and the perceptions which form our ideas of outside 

 objects, as, for example, the ideas of the sound combined from the par- 

 tial tones. 



In all the previous considerations of acoustics the very far-reaching 

 hypothesis had been made that the vibratory motions of the air, and 

 other elastic bodies which are produced by the simultaneous action of 

 several sources of sound, are always the exact sum of the motions due 

 to all the separate sound-sources. Helmholtz, however, showed that 

 this law only holds in strictness when the vibrations are of indefinitely 

 small amplitudes, and therefore the density changes are so slight that 

 they are negligible in comparison with the whole density, and the dis- 

 placements of the vibrating particles are also negligible compared with 

 the whole masses. A distinction was made between cases where this 

 law was followed undisturbed both within and without the ear, but the 

 sensations not accurately combined to form the perception, and cases 

 where the combination tone was different from the summation of its con- 

 stituents from causes operating before the auditory nerves are reached. 



His studies in acoustics were soon pursued much further with the 

 aid of the most refined analysis. In his famous treatise on the "Theory 



