Vibrations of twitched and stroked Strings. 199 



At the time £=0 the peg is suddenly withdrawn. The initial 

 velocity for every point of the string is then =0 ; and at the 

 time £ = the string has the form of an angularly bent line, 

 whose vertex lies in the point x=a, y=b. We have therefore, 

 for *=0:— 



y=&- for O^x^a; 



If it be admitted that y can be developed as a function of x 

 in a Fourier series of the form 



2 (A n si 



2njrt , -r, . 2mrt\ . mrx 

 -7^— ( + B n cosin —^- J sin -j-> 



we get, in the known manner, series (1), found also by Helm- 

 holtz. For the reason mentioned in no. 1, the admissibility 

 of the latter must, in the present case, be more closely inves- 

 tigated. 



3. From D'Alembert's solution of equation (2) it is easily 

 proved that the form of the twitched string consists, at any 

 moment, of three straight lines, apart from single moments in 

 which it is composed of two or only of one line*, Conse- 



quently, for all points of the string, ^-j = 0; and, consequently 



a l so ^=0, by which (2) is fulfilled. We must therefore 



ot 

 inquire whether the function (1) is a corresponding linear func- 

 tion of x and t. 



Of the three right lines, one passes through the initial point, 

 another through the point x=L, y=0. Let these two be cut 

 by the third in the points 7, S and y f ,% respectively; then is 



J 7 



y-- — ;_;, ; «.t2*s/; 



L— x 



y= h 'r^±7 „y^x<l. 



7 



(4) 



* Compare Thomas Young's a Experiments and Inquiries respecting 

 Sound and Light/' Phil. Trans. 1800 ; part i. p. 135 ; Monge, Journal de 

 VEcole Pohjtechnique, t. viii. p. 118 (1809) ; and Helmholtz I. c. : 



