DAVIS. — LONGITUDINAL VIBRATIONS OF A RUBBED STRING. 709 



of the wire there is always a very distinct spot of light at each end of 

 the line, and k— 2 other spots equally spaced between them, correspond- 

 ing to the k horizontal portions of the vibration curve ; but in all other 

 cases the line is of nearly uniform intensity, even at the ends. 



The portion of the envelope between x = l/k and x = 2 l/k is given by 



T k \k I T k V I h 



= 4tf|* + * 



¥ ' k 







and is a straight line intersecting the parabola 



r x I — x 



h = ±H 



I I 



at the points (l/k, AH (k — 1) /k 2 ) and (2 l/k, 8H (k — 2) /P), as was 

 to be proved. 



Similar reasoning can be applied to a point in the third Xth, or, in 

 general, to any point x, where 



J* 0'+ 1)* * . , 



j < a < k » andj <k; 



but it is necessary to distinguish between a case in which j is even and 

 one in which it is odd. The resulting formulae are given in Table II. 

 In each case the envelope is a straight line intersecting the above men- 

 tioned parabola at the points [j l/k, \Hj (k — j)\ k 2 ] and [(_/ + l)^/&, 

 4 B(j + 1) (k —j—Y) j h"\ and the second part of the proof is completed. 



The geometry of Figure 1, which was obtained experimentally for 

 longitudinal vibrations, is, therefore, a verification of the established 

 theory for the transverse vibrations of a string bowed at an aliquot 

 point. 



It is easy to construct from the data of Table II either the curve 

 y= 4> (0 f° r a given x, or the curve y = i[s (x) for a given t, in any 

 desired aliquot case. The results for k = 2, 3, and 4 are given in Figures 

 3, 4, and 5. From them one can get a reasonably definite idea of the 

 general case. It is interesting to notice that the vibration of a string 

 bowed in the middle is identical with that of a string plucked in 

 the middle, although under other circumstances the corresponding 

 modes of vibration of a bowed and of a plucked string are wholly 

 different. 



