264 Prof. A. Steinhauser on the Theory 



The rays reflected at the wallW will therefore reach both 

 ears so long as this is the case with the direct rays. 



We have then obtained the following measures for reckon- 

 ing the rays of sound which reach the left and right pinnae 

 respectively : — 



(1) for direct rays, the expressions 



/sin(a + /3) and /sin (/3 — a); 



(2) for indirect rays reflected at the wall W, 



/sin [2</> — « + j3] and /sin [ — 20 + a + /3]. 

 Consequently we have, as a measure for the total number of 

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

 the expressions 



{/sin(* + j3)+/sm[20-* + /3]} 

 and 



{/ s inO3-*)+/sin[-20 + a + /3]}, 



provided the wall W is regarded as a completely reflecting 

 surface. 



Therefore the intensities i\ and i f 2 , with which the sound is 

 perceived in the left and right ears respectively in virtue of 

 the united direct and indirect actions, will be to one another 

 in the ratio of the two expressions just determined, and give 

 the following equations : — 



i\__ /sin(* + /3)+/sin[20 — u + $~] 

 i f 2 ""/sin (/3 - a) +/sin [ — 20 + a + /3] 



sin (a + g) + sin [20 + ((3- «)] m 

 sin(/3-a) + sin [(a + /3)-20]' 

 whence 



-sin[(a + /3)-20] 



i'i — i' 2 _ sin (a. + /3) - sin ( / 3 - a) + sin [20 + (/3 — «)] 



» v i + *'s ~~ sin (a + /3) + sin (|3 — a) + sin [20 + (/3 - «)] 



+ sin [(a + j3)- 20] 

 _ 2 cos /3 sin a + 2 cos /3 sin (20 — a) 

 ~ 2 sin ]3 cos a + 2 sin /3 cos (20 — a) 



__ cos/3[sina+ sin (20 — a) J 

 "" sin /3[cos a + cos (20 — a)] 



= cotan /3 tan 



_ tan 

 ~~ tan /3 



The direction in which, then, the source of sound is esti- 



