194 Prof. A. Steinhauser on the Theory 



But if this is so, 



1 > 2 sin ft 

 or 



J > sin /3 ; 

 and in this case, 



/3<30°. 



Now, since it is the fact that for most persons /3<30°, this 

 explains the well-known position assumed by the listener who 

 turns one ear towards the source of sound. 



5. In the theory of Binaural Audition developed in the pre- 

 ceding paragraphs, it has been tacitly assumed throughout 

 that the source of sound should be situated in that plane (usu- 

 ally horizontal) in which the line of sight and the line joining 

 the middle points of the two pinnas are situated. This is not, 

 however, always the case ; for while the line of sight may be 

 horizontal, the source of sound may be situated above or below 

 the imaginary horizontal plane. Hence it is necessary to 

 enlarge the theory of binaural audition in this respect. In 

 figure 6, which is drawn in so-called parallel perspective, let 

 A A! be the horizontal line of sight; then/2 or adg c, and f x 

 or a deb are the effective surfaces of the pinnae, each enclo- 

 sing the angle /3 between it and the line of sight. Let the 

 plane d k be a prolongation of one of these planes, / 2 . Lastly, 

 let a M be the direction of any ray of sound (the position being 

 a general one) whose projections are M." a upon the vertical 

 planed, and Wa upon the horizontal plane a q. Then to 

 reckon the rays of sound (assumed parallel to one another) 

 which meet the surfaces /i and/ 2 in the given direction de- 

 fined now by the angle 7 vertically, and by the angle u hori- 

 zontally, we must take as measures of the number of those 

 rays the cross sections which can be led orthogonally through 

 the two quadrangular prisms which have respectively for their 

 bases the surfaces/*! and/ 2 , and whose long edges are parallel 

 to the direction of the rays of sound a M. 



These cross sections, which may be drawn upon one common 

 plane perpendicular to the direction of the rays of sound, may 

 be regarded as the projections of the surfaces f x and f 2 upon 

 that plane. And since the area of the projection of a plane 

 figure is equal to the product of the area of that figure into 

 the cosine of the angle included between the plane of the figure 

 and the plane of projection, we shall be able to find the areas 

 of the cross sections, provided we first know the angles w\ and 

 w 2 which the direction of the rays of sound makes with the 

 surfaces /x and/ 2 respectively. Now the plane taken normally 

 to the direction of the rays of sound is the plane in which the 



