of Binaural Audition. 263 



pinnae ; lastly, let the angles which the line of sight makes 

 with the wall and with the reflected rays be respectively 

 and a r . 



We have, then, the following relations, amongst others, 

 between these quantities : — 



a = + ^ i/r = a — 0; 

 also 



u = cc r + 2f = ec r + 2a-2(j>, 

 and 



a r =2cf) — u. 



And, according to what has preceded, the ratio between the 

 intensities produced by the direct action of the rays of sound 

 when they make an angle u with the line of sight is 



i x '. i 2 — m : n = (tan a + tan /3) : (tan /3 — tan a). 



Hence, further, it follows that by analogy the ratio between 

 the intensities produced indirectly by reflexion, when the 

 reflected rays of sound make the angle a r with the line of sight, 

 must be 



Pi' P2= (tan« r + tan/3) : (tan /3 — tana,.), 



p i:/O2 = [tan(20-«)-tan/3] : [tan/3- tan (20 -a)]. 



That these ratios shall be equal requires that p x : p 2 = h : *2> 

 or that 



[tan (20— a) + tan /3] : [tan /3- tan (2$-u)~] 



— (tan a + tan /3) : (tan /3— tan a), 

 or that 



2tan/3: 2 tan (20 -a) = 2 tan /3 : 2 tana; 

 from which it follows that 



tan (20 — a) = tan a. 



But since tan (20 — a) may equal tana either when 20 — a=a 

 or when 20 — a = 180° + a, we have two values for 0, 

 namely = a and = 90° + « respectively ; but of these the 

 first alone has any meaning in the case under consideration, 

 since, as we see from simple inspection of figure 7, cannot 

 have a value greater than a if a reflexion is to be possible. 



Hence the estimate of direction luill be unaffected by the re- 

 flexion at a single vertical ivall only if that wall runs parallel 

 to the direction of the rays of sound. 



The direct rays reach both ears so long as a ^ (3. Similarly 

 also the indirect rays reflected at the wall W will reach both 

 ears so long as a ? . ~ /3. For, since ct = u r + 2i/r, as may be seen 

 from figure 7, it follows that a r < a always ; and hence a,. < @ 

 as long as «<|3. 



