176 On the Measure of the Intensity of Sound. 



Taking logarithms in (ii), 



log.AX=log.A+log,(l+^) 



ultimately, when —^ is small. If then we put 

 A<7= log e AX and k= 1, 



ACT=-J-, (ill) 



and the differences of the new measure of sensation (<r) and 

 the mechanical intensity vanish together. This equation (iii) 

 is the basis of Fechner's law. 



Eesuming the argument. The hypothesis that differences 

 of a measure of sensation are proportional to the ratios of 

 mechanical intensity is impossible, totidem verbis ; but the dif- 

 ferences of the measure in question are the logarithms of the 

 differences of another measure, which satisfies the conditions 

 of observation, and leads to Fechner's law. 



A third assumption, which may perhaps be stated separately, 

 is that the ratio of the sensations is proportional to the ratio 

 of the intensities — that is, 



and, with the same notation as before, 



Ar 



i+t-H 1 ^ 



If the differences are to vanish simultaneously, 



and 

 where 



Whence it follows that the differences of sensation (AX) are 

 proportional to the differences of mechanical intensity (A I) ; 

 but this was shown to be untrue in connexion with the first 

 hypothesis. This hypothesis is therefore really identical with 

 that first made, and falls with it. 



Of these, the simplest hypotheses that can be made on this 

 subject, the only one which satisfies the elementary conditions, 



