DAVIS. —^LONGITUDINAL VIBRATIONS OF A RUBBED STRING. 707 



less steep descents connecting k horizontal stretches. A similar proposi- 

 tion is true for points in the last fa\\ of the string. 13 



The portion of the envelope between x = and x = Ijk is given by 



(\Hk-\\ ( x\ *— lac 



and is a straight line intersecting the parabola 



/ A TT xI ~ x 



; i = 4H 7 - r 



at the points (0, 0) and (l/k, 4 H (k — l)/k' 2 ), which agrees with what 

 was to be proved. 



In the second kt\\ of the string, a point under the influence of the 

 minor vibration alone, would, at the time t = Q, be descending through 

 equilibrium with a velocity 



. \H< x -V ±H ( x 1 



9% T , v T 



and this descent would last until the time 



t = T> - t 2 = T> - T' X -^- = T (- - j\ 



It would be followed by an ascent during an interval centred at t = T/k 

 with a velocity 



/• - t> ' ~T~ ~~ T \k~Tj 



The whole duration of this interval would be 



'■'-"■(r-i)' 



13 This work shows that the figure in Helmholtz's 1860 paper is incorrect. It is 

 true that the differential equation in question has a solution corresponding to the 

 function y = f(t), which his figure represents, and that this solution satisfies the 

 boundary conditions and both of Helmholtz's laws, as can easily be proved by 

 the method of section 5 of this paper. But it does not satisfy Young's law, the 

 6th, 18th, 30th, etc., partials being present, and it could not, so far as our present 

 knowledge of the violin string goes, be actually set up by bowing a string at the 

 point 1/6. 



For some theorems which are very useful in this connection, see Wedmore, 

 Journ. Inst, of El. Engg., 25, 224 (1896) ; and Lyle, Phil. Mag., 6, 549 (1903) ; ibid., 

 11, 25 (1906). 



