494 Mr. R, II. M. Bosanquet on the Beats 



.r, wc Lave an example of a law of force expressed by such a 

 series, which for small displacements coincides with the pen- 

 dulum law, while for displacements of but moderate extent the 

 higher terms rapidly become prominent. 



62. Assuming the existence of a law of this kind in the 

 transmitting mechanism of the ear, we should have for the 

 force corresponding to displacement u such an expression as 



— [iru + air + $u % + 7?i 4 + ...). 



Suppose the system acted upon by two harmonic forces, 



E cos pt, F cos (qt — e), 



and the mass = unity, or included in the coefficients ; the equa- 

 tion of motion is 



— +ri 2 u= —uu 2 -~/3ii:>—'yif. . , + E cos pt F cos (qt — e) 



(following the notation of Lord Rayleigh on Sound, i. p. 65). 

 For the first approximation we neglect powers of u above 

 the first ; then 



u= e cos pt +f cos (qt — e) , 

 where 



E F 



e ~ n 2- p v J- n'-q 2 ' 



63. "We may here mention that, in the present case, n is 

 negligible. This is easily seen, since, if n had any value 

 corresponding to a frequency within the limits of the ordinary 

 range of hearing, there would be a series of notes strengthened 

 by the correspondence. But the only notes thus strengthened 

 are those which are supposed to correspond to the ear-cavity. 

 They are so high in the scale that the connexions of the 

 internal ear would require to be nearly as rigid as brass or 

 steel to produce them. A further reason for n not being large 

 will be arrived at in speaking of combination-tones. And we 

 shall assume that n is smaller than any values of p or q which 

 occur in practice. 



64. The first approximations to the subsequent terms may 

 be now all made by substituting in them the value of u above 

 obtained. The process to be followed for u 2 coincides with 

 that commonly adopted ; and the result is given in Lord Ray- 

 leigh's book, i. p. 66. 



In the cases of u 6 and higher powers the process is simpler 

 than that which has been previously indicated. 



