186 Prof. A. Steinhauser on the Theory 



We have then : — 



m=/isin(a + /3), 

 w=/ 2 sin (/3 — a); 



consequently if, as is previously admitted, f x =f 2 =f, we have 



the equation 



m _ sin (a -f- /3) 



n ~~ sin (/3 — a)* 

 Developing the sines 



m _ sin a cos /3 + cos a sin /3 ^ 

 ^ "~ sin/3cosa — cos /3 sin a. 5 

 and dividing numerator and denominator of right-hand mem- 

 ber by cos a cos /3, 



m __ tan a + tan j3 # 

 n tan/3 — tana 

 from which we may further obtain 



m + n tan /3 4 



m— n tana ' 



whence , m—n. n /1X 



tana = — — - tan/3 (1) 



m + n 



Further, let i± and i 2 be the intensities with which the sound 

 which comes in the direction S is perceived in the left and 

 right ears respectively ; then the principal considerations are 

 expressed in the following equations : — 



m _ i t m 

 n ~~ i 2 ' 



and m— n _ ii~i 2 



m + n ii+i%' 

 Substituting this expression in equation (1), we obtain 



tan a =r^tan/3; (2) 



from which observe that the direction in which a source of 

 sound is situated may be estimated by the different intensities with 

 which a sound is perceived in the two ears. 



That which is here deduced from calculation our ears ac- 

 quire by practice in the course of time. That practice is ac- 

 quired in the following way : — The first lesson which the ear 

 learns, using the eye continually as its instructor, is to dis- 

 tinguish the different kinds of sources of sound, as, for example, 

 the clattering wagon, the clanking chain, the barking dog, &c, 

 Conversely the ear thereby acquires the power of recognizing 

 the source of sound, from the nature of the sound-wave. That 



