450 Dr. C. V. Burton on some 



6. If we put, then, #=B cosjerf, 

 the expression for the impressed force is 



(p2 v 



cos pt-{ : 2 cos*ptj; 



so that 



F ( 9 ./- 3B 2 \) .. . ..B* , 



t5 = < — mp l + hi 1 + j -g J V cos ^ — ^/} sin pt + i'* — ^ cos 6 'pt. 



Let F x denote the first harmonic term of F (that is, the sum 

 of the terms in cos pt and sinjt?£), while F 3 denotes the term 

 in cos opt. Then, since 



oc= — B/? sin^t, 

 we have 



F?(max.)_ 1 f .,.,/, ,.3B*\\« 



{-^+*(l+|5)}*+JF. 



x 2 (max.) j? 2 



When wij A:, /i, a, and B are given, the minimum value of the 

 right-hand side corresponds to 



^ 2=/i ( 1+ i5) >/( - 



Also 



Fornax.) = _1_ h2 B A 

 i ,2 (max.) ltip 2 a 4 ' 



which diminishes continually as p 2 increases. Hence in 

 general terms we may say that to produce a given amplitude 

 of velocity requires the least amplitude of impressed periodic 

 force when the periodic time of the force is somewhat shorter 

 than would be the natural period of the system for infinitesimal 

 vibrations if the frictional term were abolished. And return- 

 ing to the case of our strings, we may infer that a similar 

 result will hold good. Hence a force of given finite ampli- 

 tude will excite in a given string the greatest amplitude of 

 velocity when the period of the force is somewhat shorter 

 than the " natural " period of the string, and hence when a 

 periodic disturbance of finite amplitude reaches the ear, the 

 string whose resonance is excited most strongly will have a 

 " natural" period longer than the period of the disturbance. 



7. For example, let the note c be sounded with very small 

 intensity, and let that part of the basilar membrane which 

 responds most strongly be called the c-string. If then the 

 same note c is made to sound close to the ear with considerable 

 intensity, the strongest resonance will be excited no longer 

 in the c-string, but in some string of longer natural period, 

 such as the B^-string. Now according to Helmholtz's theory 



