Dr. A. M. Mayer's Researches in Acoustics. 287 



are the absolute energies of the sound-vibrations, we get for 

 the ratio of their corresponding intensities of sensations 

 1 : 2*6 ; but if the absolute energies of the sounds are 10 and 

 40 (and their ratio is also 1:4), we get for their relative 

 sensations 1 : 1*48. Or, what is the same, on the curve 

 expressing the law of Weber, or of Fechner, the ratio of the 

 sensations of two sounds as given by their corresponding 

 ordinates depends on the number of units in the abscissas 

 forming the ratio of the energies of these sounds. 



Professors Cattell and Fuller ton, from extended experi- 

 ments " On the Perception of Small Differences " *, very 

 carefully made and skilfully reduced, have formed the opinion 

 that neither the law of Weber nor Fechner's formula is 

 correct, and a priori considerations lead them to 'the opinion 

 that it is probable that the sensation is directly as the stimulus. 

 If the sensation increases directly as the stimulus, then we 

 can obtain the relative sensations of two sounds whose relative 

 energies are known. Adopting this relation, we have 1 : 4 

 as the ratio of the maximum sensation in the periods taken 



If we take for the relative intensities of the sound-giving vibrations the 

 ratio of the squares of the amplitudes of the forks, the least perceptible 

 difference in sensation corresponding to the differences in the energies 

 of the sounds may be computed. As example, suppose the second 

 fork has —$ of the amplitude of vibration of the first. Then the 

 energy of the maximum sounds of the beats will be 20+ 1 2 =441, and 



the energy of the minimum sound of the beating will be 20— 1 2 = 361, 



441 



and o7rj-=the ratio of the stimuli giving the least perceptible difference 



in sensation. Sound-vibrations of different amplitudes and of different 

 pitch will have to be experimented with, and the fork giving the greater 

 amplitude of vibration should, in successive experiments, be lower in 

 pitch, and then higher in pitch, than the fork giving the lesser amplitude 

 of vibration, for reasons set forth in my research (1) " On the Oblitera- 

 tion of the Sensation of one Sound by the simultaneous action on the 

 ear of another more intense and lower sound ; (2) On the Discovery of 

 the Fact that a Sound even when intense cannot obliterate the sensation 

 of another Sound lower than it in pitch " (Phil. Mag. Dec. 1876 ; 

 ' Nature,' Aug. 10, 1876). Such a research will be difficult and tedious, 

 and will require many precautions in arranging the experiments. 



Any one may readily observe the phenomena described by sounding a 

 fork with long amplitude of vibration, and, gradually bringing up to the 

 ear a second fork with a small amplitude of vibration, giving with the first 

 three beats per second. As the latter fork gradually approaches the ear 

 the beats become stronger, reaching a maximum of intensity, and then 

 diminishing till, at a certain distance of the fork from the ear, they vanish 

 in the more intense sensation of the more intense sound, to reappear when 

 the faintly vibrating fork has been brought closer to the ear. 



* ' Publications of the University of Pennsylvania,' Philosophical 

 Series, No. 2, May 1892. 



