Vibrations of tivitched and stroked Strings. 

 These equations can only exist for 



05*< 



L-, 



201 



(7) 



L 2' 



for if t exceeded the given upper limit, y f would become >L. 

 From (6) result then 



2a(L — ay' J L — y 

 Consequently, so long as the inequalities (7) subsist, accord- 

 ing to (4) we have : — 



y =&£for0<£<£-|; 



L(aT-2LQ + (L--2a> 

 y 2a(L-a)T 



„ a 2t ^x < a 2t 

 or L"T = L=L + T ; 



y = b — - for T- + 7n f r = l« 



(8) 



L-a L ' T=L 



Of the three straight lines, therefore, which make up the 

 figure of the string in the period under consideration, one only 

 changes with the time, and, indeed, that which does not pass 

 through an extremity of the string ; it always remains parallel 

 to the straight line 



b(L-2a)x ( q^ 



V- 2a(L-a) {) 



At the close of the time-interval (7), i. e. for 

 L-q T 

 L ' 2> 



t= 



(10) 



y becomes = L, 8 r = ; the string now consists of two straight 

 lines, namelv 



j x 

 y = b — 



for 0<a<2a-L, 



_ 6(2a-L)(L-fl) 



(11) 



~ 2a(L-a) » 2a-L^*<L., 



The second straight line is here again parallel to (9). 

 5. In the following section of time, defined by 



L-a T oT 



L ' 2 < * < 2L> • • • 



(12) 



