208 M. F. Lindemann on the Forms of the 



pass through the position of equilibrium ; for the moment 

 £=0 selected in equations (18) appears essentially dependent 

 on x. In consequence of this new determination equations 

 (18) change into : — 



y=at forOf^fr^; j 



y=bt+f„ |^ST-f; 



y=at+g„ T-|<*<T. 



(21) 



a, h,f, g are complete linear functions, to be determined, of 

 x. Between them and the function X a series of relations 

 subsist. 



At the time T, namely, every point must again be in the 

 position of equilibrium ; therefore 



0=aT+g (22) 



Further, we have 



aX=bX + 2f, (23) 



b(2T-Z) + 2f=a(2T-Z) + 2g, . . . (24) 

 and thence 



aT=-6(T-£) (25) 



This last equation affirms that the path travelled in ascend- 

 ing differs from that travelled in descending only in its sign, 

 as must be the case in a periodic vibration. 



11. Kespecting X, according to no. 9 two different assump- 

 tions can be made : either X increases when x increases ; or 

 X diminishes as x increases. In both cases for x= and x= L, 

 i. e. for the extremities of the string, must 



£=0mod. T. 



In the former case, therefore, we can take £=0 for x=0, 

 and consequently (since no nodes are supposed to be pre- 

 sent, and since X increases simultaneously with x increas- 

 ing) X = T for x = L; that is, the points in the vicinity of 

 #=0 rise very quickly and descend very slowly, while the 

 neighbouring points to # = L descend rapidly and ascend very 

 slowly. In the other case the behaviour of the two ends of 

 the string is exactly the reverse. In the following we make 

 the former assumption; the other would be decided simply by 

 the exchange ofx with L— x. 



