of Binaural Audition. 189 



Now, in order to learn whether in the case of a single sen- 

 sation of sound the ratio between the intensities of the sensations 

 in the two ears is the same for both individuals, we have ob- 

 viously only to determine whether 



m i _ m 2 

 n x ~~ n 2 

 or whether 



sin (a + ft) _ sin (a + ft) 



sin (ft — «) sin (ft — « ) ' 



since, according to what has already preceded, those intensi- 

 ties are in the proportion of each m to its respective n. 

 Developing the sines, we get 



sin a cos ft + cos a sin ft _ sin a cos ft + cos a sin ft 

 sin ft cos u — cos ft sin a sin ft cos a — cos ft sin a 



Hence, by algebraic proportion, 



whence 



2 cos u sin ft _ 2 cos a sin & 

 2 sin a cos ft 2 sin a cos ft 



sin ft __ sin ft 

 cos ft ~~ cos ft 



tan ft = tan ft, 



which can only be true in the case proposed if ft = ft. 



The premised condition, that the ratio between the intensi- 

 ties with which the two ears of an observer perceive a sound 

 is the same for two observers for whom the angle j3 differs 

 while other conditions remain the same, is therefore inexact ; 

 whence the standard of each individual for the perception of the 

 direction of sounds is dependent upon the angle {3 included be- 

 tween the effective surfaces of the pinna of his ears. Suppose, 

 for example, that in the observer A, ft = 25°, and in the ob- 

 server B, ft = 30°, and that the angle a which the sound-rays 

 make with the line of sight of each of the observers is 30° ; 

 then : — 



For observer A: 



771!=/ sin (ex. + ft) =/sin 45°, 

 n x =/sin (ft— a)=/sin 5° ; 



consequently the ratio of the intensities is 



m 1 : n 1= sin 45° : sin5°=0'707 : O087 = 8'l : 1. 



