Tones of Bowed Stringed Instruments. 575 



with relatively slio-ht modifications of the principal (Helm- 

 holtzian) type; and the general character of the vibration is 

 determined by the movement, to and fro, on the string, of 

 one large discontinuous change of velocity, and of a number 

 of minor discontinuous fluctuations of velocity. The positions 

 -and magnitudes of these minor fluctuations must be such 

 that in the vibration-curve at the bowed point, we have a 

 uniform gradual rise followed by a steep and generally non- 

 uniform fall, the ratio of the time-intervals occupied by the 

 two stages being the same as in the simple Helmholtzian 

 type. In my monograph I have shown by several examples, 

 that the same dispositions give for the vibration-curve of a 

 point close to the end of the string, a "fluttering " or irregular 

 rise, followed by a steep and uniform jail. 



3. Analysis of the Transitional Modes. 

 The general expression for a discontinuous vibration is 



^ . nirx [ . 2mrt 7 2mrt~\ 



ij—2^ sm -j— a n sin —^- + b n cos -yp I ' 



where 1 r nirC x 7 nirQ 2 1 



a »~ "~" 2 2T "1 C0S — T ^""2 COS , \-kc > 



1 |~J . n7T ^ , 7 ' W ^C 3 , o 1 



~2T x Sin — T — ~*~ 2 S1U r + & c * ' 



di, d 2 , &c. being the magnitudes of the discontinuous changes 

 of velocity, and C 1? 2 , &c. their positions measured from the 

 origin at time t = 0. If a discontinuity is moving with the 

 positive wave, its position is given by its ^-coordinate ; 

 but if it is moving with the negative wave, its position is 

 given by (21 — x). From this expression, the amplitudes 

 {a n -\-b n y and the phases tan -1 b n \a n of the harmonics may 

 be readily calculated, when the magnitudes and positions of 

 the discontinuities are known. In finding the phases, it is 

 convenient to choose the origin of time in such a manner 

 that the phase angle of the fundamental component of the 

 vibration is zero. 



When we have only one discontinuity, we may write 

 d 2 = d s =d 4 ifec. =0; and taking the origin of time such that 

 the discontinuity d 1 is initially at the end of the string, (\ = L 

 we have 



7 ^(-l)"" 1 • mrx . 2imt 

 y = d 1 Z ~ n 2Ti sln i sin T ' 



ir"7T"L Ij I 



This is the simplest Helmholtzian type of vibration including 



and ; 1 f.7 __-_wCi . .7 _• nirCz 



