218 M. F. Lindemann on the Forms of the 



added — namely 



'T A . „/ 2iL\/iw-i + l 



y 



-Hi-'M'-^C^ 1 '-) 



f m,£-|-2(m — 2^— 1)L < ^ < 2L— 5 

 ° r_ "" 2mL =T= 2L * 



Accordingly the velocity of the descending motion fluctuates 

 between the values 



-(A + B> + -BL and - (A + B> + ^Li BL. 



The other half of the ascending motion now remains to be 

 considered. It is represented by the equations 



,=-a(l-,)(t-o-b(^l-,X^^'t-0 



f 2(m+j—l)L—mx < t < 2(m — 2i — 1 +/)L + m^g 

 t0r 2S = T = 2mL ' 



„ 2(?n+j — 2i — iyL + mx < tf < 2(m+j)h--mx 

 tor 2SL = T = 2SL 



These equations are to be formed for^l, 2, . . . . 2i — 1. 

 The first of them holds good also for ^'=21 ; if the latter has 

 been superadded, the entire system concludes with the equa- 

 tion 



,= -A(L-.)(T-0 + B(,-^L)(^ T -,) 



for K™-l)L+mx < £ < i 



2mL = T = i * 



Analogous formulae can be constructed for those parts of 



2^—1 

 the string which are limited by the points #= L and 



#= — L. This, however, shall not here be pursued further. 



18. The function y defined by the foregoing conducts to a 

 very simple Fourrier series. We find,, namely, 



^ « . 2nirt , A 



if 7) and At; have the signification stated in no. 16, so that 77 



is equal to the series given in (31), while, according to (34), 



A BLT _, 1 . nmirx . 2nmirt 



A "=,HV 2 ;? sin -ir sm -f— 



