216 M. F. Lindemann on the Forms of the 



This system of three equations is to be formed successively 

 for i= 1, 3, 5, ... m— 1 when m is even, 

 for i = l, 3, 5, ... m — 2 when m is odd. 

 In the former case the operation finishes with the last of these 

 three equations, formed for i=m—l; in the other case, with 

 the first, formed for i=m. Therefore, when m is even, the last 

 equation reads 



*-Kf-<h*(^- , )l 



tor " ~2mL =T = ~2L~ ; 



and when m is odd, 



p 2(m — 1)L— mx < t < 21i-—x 

 t0r 2mL = T = "2L"' 



The equation 



^-A(L-.)(T-0-Bg-,)g-;) 



for^=^<^<l . (36) 



must be superadded, in order to have the motion in the in- 

 terval of time from t=0 to t=T represented for all the points 



between #=0 and #= — . From (35) and (36) it follows that 



the ascending motion, for the part of the string considered, 

 proceeds with the constant velocity 



: A<L-*)+Bg-«)j 



while the velocity of the descending motion fluctuates between 

 the values 



-(A+B>, -A*+b(£-*), -(A+B)*,... 



17. The general formula can be constructed in an analogous 

 manner. We have, when 



2i T < ^2^ + l T 

 m — — m 



y= A(L-^ + B(^L^)(^^T) 



for ti < 1 < ^a:-2(i-j + l)L m 

 IOr m =T= 2mL 



