of Binaural Audition. 267 



be shown, the change in the total intensity (^ + i 2 ) caused by 

 the change in direction of the rays of sound is extremely small — 

 then, by what has preceded, we may conclude from the new 

 ratio between the intensities i± and i 2 what the new direction 

 is in which the source of sound is situated, or, in other words, 

 what the motion of the source of sound to right or left may be; 

 and we may also conclude by the change of the total inten- 

 sity what the change of distance is, or, in other words, what 

 the motion of the source of sound backwards or forwards may 

 be. It follows that every change of 'place of a source of sound 

 situated ivithin the region of direct binaural hearing may be 

 known as to its nature by the hearing of the two ears. 



For, to recapitulate, the change of place of the source of 

 sound is known in respect of right and left from the new ratio 

 i 2 : 4, and in respect of forward or backward by the new sum 

 h + 4- 



From figure 3 we see that the total number of rays of sound 

 reaching both ears may be measured by the length of the 

 straight line de. Now this is longest when the direction of 

 the rays of sound coincides with the line of sight, indicated 

 in figure 8 as Si; but it is shortest when the rays of sound take 

 the direction of the line S 2 (fig. 8). Hence it follows that 

 the ratio between the greatest and least of the possible values 

 of the total intensity, namely J : J z , must be equal to the 

 ratio between the straight lines de and gk. But since de = bc 

 and gk = bv, we have the proportion 



J g :J l ^ be. bv, 

 or, lastly, since bv — bc . cos /3, \ 



J g : J L = 1 : cos /3. 



If we take for ft its greatest possible value, namely about 

 30°, in order that the difference between the two members of 

 the ratio may become a maximum, it follows that 



J^:J, =1:0-866, 



whence we infer that the change in the total intensity with which 

 a sound is perceived occasioned by a change of the position of 

 the source of sound (supposed always to be situated within the 

 region of direct binaural hearing) is very small. 



2. The Theory of Indirect Binaural Audition. 



8. If a source of sound is situated in the region of indirect 

 binaural hearing, no ray of sound can reach the surface of 

 either of the pinnae directly ; and the sound produced by the 

 sonorous body can evoke a sensation^ only as the result of 



