206 M. F. Lindemann on the Forms of the 



Therefore ^ is discontinuous for t = X ; that is, at the time 



£ = X the point x receives an impulse. Accordingly the for- 

 mulae of Christoffel already mentioned in no. 7 become appli- 

 cable, viz. 



in which a 2 has its previous meaning, and where the two values 

 of the bracketed quantities on the two sides of the place of 

 discontinuity t = X are distinguished by the signs + and — 

 used as indices. We have 



therefore 



l = dX 

 c dx 



Now a, b, and X must be continuous functions of x, unless 



we suppose the connexion of the string broken. In like 



02/ 

 manner h and k are continuous ; for otherwise ^~, and conse- 



quently also =^, would be discontinuous at other times as well. 



Even -=r- is a continuous function of x j for since all the points 



will not simultaneously be at the greatest distance from the 

 position of equilibrium, for a variable point x equations (18) 

 are to be replaced by 



y = at + h for X <t<Z + Z, 1 



y = bt + k „ £ + £<^£ + T,J 



when the point x arrives at its extreme position at the time 

 £=Xo, where Xo is a continuous function of x. As the motion 

 is to have the period T, 



aZ + h=b(l Q + T) + L 

 This holds good for every point x of the string. Forming 

 the corresponding equation for x + dx and for x—dx, and dis- 

 tinguishing the differentials taken in the positive and negative 

 directions by the indices + and — , there follows, by subtrac- 

 tion of the two resulting equations: — 



■[ffi).-ffi)J-»[(£H©J- 



