Vibrations of twitched and stroked Strings. 219 



3ries can be integrated term by term ; 1 



Q 2 Cf .la \^ (A+B)LT . nirx 



Both series can be integrated term by term ; hence we 

 have 



or 



_ALT . nirx 



according as n is or is not divisible by m. 



The representation of y by a Fourrier series gives therefore 



LT ^ A n . nirx . 2mrt 

 y= — is- 2, -o-sm -*- -sin— „=— , 

 u it 2 fl ^ 2 L T 7 



in which, when n is not divisible by m, 



Arc = A ; 



and, when w is divisible by m, 



Av-A + B. 



ChristofFel's conditions of discontinuity are spontaneously 

 fulfilled by the equations formed in no. 17; hence the quanti- 

 ties A and B are at first arbitrary. Observations have never- 

 theless shown that the upper tones corresponding to the places 



x= —, , .... are not present in the sound of the string 



m m 7 l °» 



B must therefore be = — A. 



Consequently according to no. 16, for a point between x=0 



and x , the velocity of the ascending motion is equal to 



A , therefore independent ofx ; but that of the descending 



motion fluctuates between the values 



0, -A-, 0, -A -,.•■• 



7 m 7 7 m 



The curve of the velocities will therefore consist of a rectili- 

 near ascending * and a scalariform descending portion. 



This is in accordance with Helmholtz's figures (I. c. p. 144). 

 In the declinations published by Neumann (I. c.) the descend- 

 ing portion is not scalariform, but slightly rippled, and thus 

 the equation A + B = is not perfectly fulfilled. Indeed all 

 the upper tones are absent only when the bow is drawn very 

 uniformly across the string, and touches it very accurately at 

 the wished-for node-point. The number of the ripples which 



* This is confirmed also bv Clifton's observations : cf, Donkin, I. c. 

 p. 136. 



