196 Prof. A. Steinhauser on the Theory 



since ^> 1 =90°— w 1} and <£ 2 =90°— w 2 , we have 



cos <f>i = sin Wi = sin (a. + ft) cos 7, 

 cos <£ 2 = sin w 2 = sin (/3 — a) cos 7 ; 



and the areas of the cross sections of the parallel pencils of rays 

 of sound which reach the left and right pinnae, respectively, 

 are 



/i=/i cos £i=/i sin (« + P) cos 7, 

 /a =/a cos <£ 2 =/ 2 sin (/3-a) cos 7. 



But as these cross sections constitute, as has already been 

 shown, the measures by which the rays of sound which respec- 

 tively reach the left and right pinnae are to be reckoned, we 

 may draw the following conclusions: — 



(1) That, with respect to the angle of altitude 7, the smaller 

 this angle becomes the better will the hearing be, since in the 

 sum 



f\ +/s =/i sin (a + ft) cos 7 +/ 2 sin (ft - a) cos 7, 



which must be a maximum for the condition of best hearing, 

 both terms have maximum value when 7=0. 



We hear therefore binaurally the best, relatively, when the 

 source of sound is situated in that plane in which are 

 situated the line of sight and the line joining the middle points 

 of the pinnce (" plane of best hearing "), and the best, absolutely, 

 when it is situated in the line of sight. 



(2) That the intensities i x and i 2 with which a sound is per- 

 ceived in the two ears, for any values of 7 whatever, always 

 remain equal so long as a = 0, since then 



f\ =/i sin ft cos 7 =/ 2 sin /3 cos y, 



and, because f 1 =f 2 =f, 



f f 1 =fsin /3 cos 7, and /' 2 =/sin /3 cos y. 



(3) That the intensities i\ and i f 2 , with which a sound 

 coming from above or below (in the latter case 7 is negative) 

 is perceived in the two ears, are in the ratio f\ :f 2 , or as 



sin(a + /3): sin (ft— a). 



And since also in this case the proportion 



i x : i 2 =. sin (a + /3) : sin {ft — a) 



holds good, which was deduced for the case in which the source 

 of sound was "situated in the plane of best hearing, it follows, 

 either by calculation from equation (2), that 



tan a = i — ^ tan ft, 



^l + ^2 



