Periodic Motion by Solid Friction. 163 



If the bow is applied at a node dividing the length in the 

 ratio p/q where p and q are integers with no common 

 measure, there are p + q — 1 free motions in which the point 

 has constant speeds to and fro, the ratio of the forth and 

 back intervals being r/s where r and s are any integers such 

 that r + s=p + q. Whatever the position of the node, there- 

 fore, two in general distinct motions are always possible, and 

 in these the intervals are proportional to the segments of the 

 string. 



When the string is subject to slight air resistance any 

 possible steady motion must approximate to the undamped 

 kind, and we shall prove that with the bowed point at a node 

 that motion is possible in which the ratio of the forward 

 interval to the backward is equal to the ratio of the greater 

 segment to the less. 



Let the length of the string be 7r, and the point of bowing 

 at distance a, >7r/2, from the end (1) ; also let y l and y 2 be 

 the displacements from the mean positions at distances x l 

 and x 2 from the ends, both measured positive in the direction 

 of bowing. 



If the maintained motion of the bowed point is of free type 



y = XA r sin rt, 



where the summation necessarily does not include com- 

 ponents having a node at the bowed point, 



L— 2— \ sin r Xi ( sin rt + /ca cot r a co^ r^—tcx^ cos rxi cos rt\ . ■ • (1) 



sinm 1 ' J 



A 



j=2 — j— \\sinrxc, (sin rt-\-Kir — a cot rir — a cos rt) — kx 2 cos rx 2 cosrt\ 



. ■ (2) 

 terms of order tc 2 being neglected. 

 At the bowed point, 



p, + p- = 2 K Tr2,rJ±- C o$n (3) 



ax 1 ax 2 sm'ra. 



The quantity on the right with the addition of soms constant 

 represents the force necessary to maintain the motion. The 

 condition for a steady motion is that this be equal to the 

 frictional force so long as there is relative motion, and be 

 not greater than the statical friction when there is no relative 

 motion. 



In the case under examination the velocity is v and 



M2 



