of Binaural Audition, 191 



therefore (in + n) will be a maximum when 



/(sin u cos j3 + cos « sin /3 + sin ^ cos a — cos /5 sin a), 

 or 



2/sin /3 cos a 



is a maximum. But in this expression, so long as one person 

 only is concerned, the only variable is a ; consequently 

 2/sin /3 cos a will have its maximum value, namely 2/sin /§, 

 when cos a is a maximum, which occurs when u = 0°. 



We hear best therefore ivith the two ears when the sound reaches 

 us from the front in the line of sight. 



It is in consequence of this that we always turn with our 

 face to the speaker, provided that we possess equally good 

 hearing of the two ears. 



If this be not the case, .and we hear worse (for example) 



with the right ear than with the left, and to such an extent 



that we perceive a sound of intensity ? 2 with only the .^th part 



of that intensity, then to find the direction of best binaural 



audition the problem will be the same as if the right ear were 



n 

 considered to receive only - rays of sound instead of n rays, 



or as if the effective surface of that ear were not / but 



f • ' 



only - . Proceeding on this assumption, we see at once that 



x . „ / 



if, in figure 5, ab=f and ac=--, the straight line de drawn 



x 



at right angles to the direction of the rays of sound S measures 



the sum of the rays of sound which reach both ears, and that 



this straight line is longest when parallel to be. The best 



binaural hearing therefore occurs when the direction of the 



rays of sound is perpendicular to the straight line be. In 



order to find the angle a, which gives the direction of best 



hearing for the case under discussion, we have only to consider 



that, in figure 5, be = cd, also that 



be=fcos (a + /3) 



or 



and that 



or 



whence 



=/cos a cos /3 —/sin a cos /3, 

 cd= -(cos(/3— a) 



f f 



= - cos B cos u -f- — sin B sin a ; 



f f 



/cos a cos /3 —/sin a cos /3 = J - cos /3 cos a + - sin /3 sin a. 



OS X 



