214 M. F. Lindemann on the Forms of the 



In the next interval, where — < x < — . we should have:- 



m — — m 



*- A (* L "S)S- l )<* 



t < mx — 2Jj 

 T = 2L ; 



mx — 2L < t < 4Jj—mx 



2L = T = 2L 

 A /3L \ /m jX 4L— mx ^ t ^ ., 



Lastly, we find generally, w T hen 



m = — m 



mx 



-2iL 



y= A (^L-x)t forO<^< 



^- A r ^ /U 7 " " 2L = T = 2L ' 



And when L < x < — L: — 



m = — m 



t ^ 2iL~mx 



'-f^M 1 for ° = T= 2L 



. /2i T \/T A 2iL-to« . t . m.T-2(i-l)L 



,= -A(^L-,)(T-0 for ^-^-l)L <^< L 

 Since ?/ is a function <p(x) of .# satisfying the conditions 



4>(®+—j=<l>(®)> 



in expanding it into a series of sines we need only make use of 

 equations (32). We find (analogously to no. 14) 



ALT^ 1 . nmirx . 2mrt /ni . 



y= 2- 2 -2 sin -y— sin -7p-. . . . (34) 



16. It shall now be assumed, further, that the several por- 

 tions of the string vibrate by themselves, as in the preceding 



