Vibrations of twitched and stroked Strings. 205 



Hence the validity of the differential equation (2) is not im- 

 paired by the discontinuities of the differential quotients, accord- 

 ing to x, of series (1), which occur at the angles of the string. 



The stroked String. 



8. The notation introduced in no. 2 shall stand also for the 

 violin-string, to be treated of in the following. We base the 

 mathematical treatment of this upon the following proposition, 

 to which direct observations have conducted*, and which re- 

 presents with approximate accuracy the motion of the several 

 points of the string {conf. infra, no. 16) : — 



Each point in a stroked string vibrates hither and thither with 

 constant velocity ; this velocity is different for the different points 

 of the string. 



It ivill be shown that by this the motion of the string is com- 

 pletely determined, if, in addition, the amplitude of any single 

 point of the string is given, and excluding the occurrence of 

 nodes. 



"We shall find that the velocity of a point of the string (with 

 the exception of the central point) when ascending is neces- 

 sarily different from the velocity when descending. At first 

 w r e make no hypothesis about this ; we will only suppose the 

 velocity in each direction separately to be constant. 



Let us consider a point x of the string, which at the time 

 2 = commences its ascending motion. The total period of a 

 vibration may be denoted by T, the time of the ascent by X, 

 therefore that of the descent by T— 2. We can then put 



y=at + h for 0<*<£, 



y — bt — k „ X^t 



;*} ■ ■ • • m 



Since for all values of t the second differential quotient ac- 

 cording to t of y is =0, this must also be the case with the 

 second differential equation according to x of y, by which 

 equation (2) is satisfied ; a, b, h, k are consequently complete 

 linear functions of x. They are subsequently to be more pre- 

 cisely determined. X is also a function of x, to be determined, 

 while T is a constant. 



In the determination of these functions the occurrence of 

 node-points shall be excluded. 



9. As velocities of opposite motions, a and b are quantities 

 different from one another when they are not both =0. 



* Conf. Hehnholtz I.e. and Clem. Neumann, a Beobachtungen iiber 

 die Schwingimgen gestrichener Saiten," Sitzungshericlite der Wiener Aka- 

 dmie 1870, math.-naturw. Classe, Band lxi. Abth. 2, p. 89. 



Phil. Mag. S. 5. Vol. 9. No. 55. March 1880. Q 



