164 Mr. A. Stephenson on the Maintenance of 



— oLv/(ir — a) for intervals —a to ex. and a to 2ir — a, so that 



2r 1 . 



A r = ,sin ru, 



and the right hand side o£ (3) becomes 



Akttv ^ 1 



Z cos rt. 



it — « sm r« 



In the simpler cases a = 7r/2, 2tt/3, and 37r/4, it is readily 

 seen that this represents a constant between a and 27T — a ; 

 and in any example, since the cosecant coefficients of cos rtlr 

 recur and all lie below a certain limit numerically, a similar 

 result as to the constancy during the shorter interval may be 

 verified arithmetically. Let this constant value be k *, and 

 the greatest value during the interval —a to a, h' . Then 

 the force maintaining the motion is not greater than Y+pk' 

 during the forward swing, and is equal to ¥+pk in the 

 backward swing, p being the tension of the string. 



If the pressure is F, /n the coefficient of statical friction, 

 and fu,' the coefficient at relative speed 7rr/(7r— a), then 



P+^- = F^ and F+pk'$Ffi. 



The equation determines the mean position of the string, and 

 from the inequality it is evident that the pressure must not 

 be less than p(k' — &)/(/x — /u'). 



We have shown then that the bow when applied at a node 

 maintains that steady free motion which does not contain 

 the components corresponding to the node and in which the 

 point of bowing vibrates with constant speeds during intervals 

 proportional to the segments of the string, travelling with the 

 bow during the longer interval without slipping : and the 

 only condition for the possibility of the motion is that the 

 pressure exceed a certain limit. 



All further properties belong to the free vibration defined 

 by these facts : the equation of the motion, in accordance - 

 with (1) and (2) ; is 



— — S -3 sin rx sin rt. .... (4) 



where the summation includes all except the node com- 

 ponents. 



Thus the amplitude of any component, when it occurs, is- 



x^ sin ?'^ n 2 — l7r a -, ., . 



* If p4-q = n, 2* -s— = = 5— tt when t=a, so that 



1 \ 1 > —1 r 2 sin ra n 2 £j 



li: 



2n 2 -l / tt 



3 71* \7T — a 



