Periodic Motion by Solid Friction. 165 



proportional to the product o£ the speed of the bow and the 

 reciprocal of the smaller segment of the string, provided 

 this product is not too large. 



If the bow is transferred from a more important node, 

 such as that at a trisection of the string, to one at a distance 

 of only a small fraction of the length, the missing com- 

 ponents are restored without other appreciable change, since 

 the components dropped are relatively insignificant. Equa- 

 tions (1) and (2) hold good so long as tcu/sin roc is small, and 

 the node harmonics are therefore brought to normal ampli- 

 tude when the fractional displacement is large compared with 

 ■k/o; where a is the lowest of the frequencies involved. The 

 rapid restoration of missing components in passing from a 

 node has been observed aurally. 



3. The motions examined experimentally have been found 

 to be of the type (4) : others, however, might occur. These 

 are to be sought for among the known group of undamped 

 motions. A sufficient illustration is afforded by the simple 

 case when the bow is applied at a point one fourth of the 

 length from an end. There are then three undamped motions, 

 and of the two in which the ratio of the intervals is equal to 

 that of the segments, under air resistance only that is possible 

 in which the motion with the bow occupies the longer in- 

 terval. In the third the intervals are equal and 



. 4zV 1 . 7T 



A r = n sin r — 



it v I 



so that the series in (3) is equal to Akttv from to 7r/2, and 

 to —4cK7rv between 7r/2 and it. The motion is therefore 

 steady if the pressure is not less than 8/c7rvp/(iui—fji'). 

 In the ordinary case the pressure is not less than 



16K7TVp/[/UL— ///), 



so that for the range of pressure between these limits only 

 that motion is maintained in which the times to and fro 

 are equal : 



w °° n C i 



# = 4^/2- X (-1) 1 sinl + 4ii#sinl + 4ra* 



— /;> . v 2 sin 3 -f ±n x sin 3 + 4n t \> 



a vibration distinguished by the absence of all the odd 

 octaves. 



Whether any possible steady motion is set up by suitable 

 bowing remains a matter for practical trial. 



October, 1910. 



