192 Prof. A. Steinhauser on the Theory 



From which it follows that 



x cos a cos 3 — x sin u sin /3= cos /3 cos a + sin /3 sin «, 

 and, further, that 



(x— 1) cos a cos /3 = (^ + 1) sin a sin /3, 

 whence by algebraic transformation, 



tan a = cot (3. 



x+1 



Tan a and also angle a increase in value as the proper frac- 

 tion T increases : but this fraction becomes greater as x 



x+1 & 



increases in value, and x increases as the degree of deafness 

 of one ear increases. It follows, therefore, that the person hard 

 of hearing must, in order to hear best, turn to the speaker 

 his better-hearing ear to a larger degree in proportion as the 

 hearing of the other ear is relatively worse. 



In the case of complete deafness of one ear, x—^o , and our 

 formula becomes 



i-i 



x 

 tan a = = cot /3, 



1-- 



x 



or 



tan a = cot /3, 

 whence 



a = 90-£, 



which is the case when the rays of sound are perpendicular 

 to the surface of the hearing ear, and that ear is turned straight 

 towards the speaker. 



In the case of two different individuals the angle /3 is gene- 

 rally different. Suppose two individuals hard of hearing in 

 one ear to an equal degree. Since tana and angle a are 

 greater in proportion as cot fi is greater and as /3 is less, it fol- 

 lows that persons equally hard of hearing in one ear must, in 

 order to hear as well as possible, turn the better-hearing ear 

 the more towards the speaker in proportion as the angle which 

 the surfaces of their pinnae make with the line of sight is less. 

 The cases of hardness of hearing in both ears may be separated 

 into two: (1) the cases in which the two ears are equally hard 

 of hearing ; (2) the cases in which they are unequally hard 

 of hearing. 



In the first case suppose, for example, that the left ear 



