Vibrations of tivitched and stroked Strings. 209 



According to this, x can always be chosen so small that, 

 for a given value of t. 



XX 



2= = 2 



For points of the string which are situated sufficiently near the 

 end x = 0, therefore, the middle equation of (21) always holds 

 good ; so that for them y = bt +/. For x = must y = ; con- 

 sequently we can put 



b=Bx, /=F#, 



where B and F denote constants. 



The function X reaches its greatest value for^ = L. There- 

 fore, if 2£<T, then the first of the equations (21) holds good 

 in the vicinity of the end x=L ; it must give y = for #=L ; 

 we thus get 



a=A(L — x), 



where A denotes a constant. 



It then follows, further, from (22), that 



g=-AT(L—x). 



In fact, g must contain the factor L— x. since for <#=L and 

 2£>T the third of the equations (21) is valid. 



The function X is now determined from (25); we get 



y _ -bT__ -BTx 



a-b A(L-a)-B« S 



lastly, /is found from (23) — namely, 



/=F f ^J(a-&)=-|BTtf. 



12. The constants A and B, by which all is expressed, can 

 be reduced to a single constant by means of the discontinuity- 

 conditions (19) and (20). In correspondence with the repla- 

 cing of equations (18) by (21), £ must previously be replaced 

 in these relations by \X or by T— \X. 



For the place of discontinuity t — \X both sides of equation 

 (19) become, by virtue of the values just found for a, b,f } g } 

 and X } equal to 



_ ATyr A(L-^) 2 + B^ 2 

 2[A(L-x)-BxY 



This relation is therefore spontaneously fulfilled. 

 Equation (20), on the contrary, gives for the same place of 

 discontinuity : — 



