DAVIS. — LONGITUDINAL VIBRATIONS OF A RUBBED STRING. 695 



phase changes suddenly by tt when the bowing point passes through the 

 node. 



A notable advance was made by Helmholtz, 4 who, with the aid of 

 Lissajous's vibration microscope, proved 



Helmholtz's Law : When a string is bowed at an aliquot point [1/k], 

 the part of the string immediately under the boiv moves to and fro with 

 constant velocities, whose ratio is equal to the ratio [l/(k — 1)] of the 

 segments into which the string is divided by the point in question. 5 



He also formulated the following 



Velocity Law : The smaller of these two velocities has the same direc- 

 tion as that of the bow and is equal to it. 



In other words, " This point of the string adheres fast to the bow 

 and partakes of its motion . . . then is torn off and jumps back to its first 

 position . . . till the bow again gets hold of it." From these facts he 

 derived his well known solution, 



2 A 



**l n- 



. rnrx . flirt 

 sin — — sin — , 



where / is the length of the string, 2 7 T is its period, and A is a constant 

 depending on the velocity of the bow. This single expression satisfies 

 the necessary conditions for any aliquot case, and may be called Helm- 

 holtz's major solution. In any particular case it must be modified, in 

 accordance with Young's law, by the omission of the kth, 2kth, etc., 

 terms ; that is, by subtracting the expression 



2 — 



^ k- m 2 



. km-rrx . km-n-t 

 sin — sin — ^ : 



which may be called the minor solution for the case in hand. If the 

 point bowed is near one end of the string, as is always true in musical 

 practice, the minor solution is relatively unimportant and may usually be 

 neglected altogether. This work of Helmholtz's solves the problem for 



4 " On the motion of the strings of a violin," Proc. Glasgow Phil. Soc, Dec. 19, 

 1800 ; reprinted in Phil. Mag., Ser. 4. 21, 393 ( 1861 ) . See also Helmholtz, Sensations 

 of Tone, etc., 3d Eng. ed., 80-88, 384-387; Rayleigh, Theory of Sound, 2d ed., 

 208 et seq. ; Donkin, Acoustics, 2d ed., 136-144. 



5 An aliquot point is a point whose distance from one of the fixed ends is \/lct\\ 

 of the length of the string, where k is an integer. 



