DAVIS. — LONGITUDINAL VIBRATIONS OP A RUBBED STRING. 705 

 n 



Sff^ 1 



x . mrt 



hmirx . krrnrt 



1 — z — sln — W ' 



where H\% a constant. If the first term were the complete solution, the 

 motion of any point x of the string would be an ascent from h units 

 below equilibrium to h units above it, with a constant velocity f, during 

 an interval of length 2r, followed by a return to the original position, 

 with another velocity g, during an interval of length 2 (T — t), where 



x I — x 



h = ±H 



I I 



UJl-x 



J ~ ' rp 



T I 



9 = 



t= 7 



AH x 

 ~T 1 



x 



Suppose any one point to be going up through its position of equilibrium 

 at the time t = 0. Then all points would go up through their positions 

 of equilibrium at the times 0, 2 T, 4 T, etc., and would go down through 

 those positions at the times T, 3 T, 5 T, etc. ; and each of these times 

 would be for each point the middle of an interval, during which its ve- 

 locity is constant. The first of the above equations shows the envelope 

 of this major vibration to be a parabola, whose vertex is at the point 



m H )- 12 



The case in hand is modified by the presence of a term 





km-rrx . Icmirt 



sin ; — sin 



Pffl 2 / T 



8 H ^S 1 . trnrx . m tt t 



= ^¥^Tm 2 sm 1/k sin YJk ' 



The motion corresponding to this term alone would be exactly similar to 

 that described above, except that the constants /, T, and H would be 



12 For a proof of these facts see any of the references mentioned in the note on 

 page 695. 



vol. xli. — 45 



