200 M. F. Lindemann on the Forms of the 



From this results 



nir\ 



\y~~ y f L— 7 r L J' 



The function t/, defined by equations (4) in the interval 

 from #=0 to x = L, is therefore represented by the series 



2Ly 1 YyV — <y f 8 . mry 

 7r 2 ri 2 Vy{y—y') L 



-i Tf tyt-^ rf- Sin-f 1 - S11V 



'1 ' 



(L-/)( y - 7 ') ^ LJ S1L L 



The quantities 7, S, y r , S / must now be so determined as 

 functions of t that this series shall become identical with that 

 defined by (1). This latter is equal to 



hi? v l T. (a 2t\ , . (a , 2A1 . nirx 

 ^a(L-a) 2 i? L Sm n,r (l - T J + Sm Hi + TJJ Sm TT- 



If, therefore, fi, v are undetermined positive or negative 

 whole numbers, we obtain the equations : — 



-*(£-! + v), 



y_^/a 2* 



L(S-8') +?$'-•/«= ± 6L (W)(7-Y) , 



za{L—a) 



(5) 



(6) 



The numbers /^, v are to be determined, and their signs 

 chosen, so as to make 7 and y f constantly positive for all values 

 of t between and T, and 7^7 ; ^L. The same sign must 

 be chosen for the first of equations (5) and the first of equa- 

 tions (6), as well as, on the other hand, for the second equation 

 (5) and the second equation (6). These conditions oblige us 

 to distinguish whether «>L-«ora<L — a. We assume the 

 former ; the other case can be decided in a precisely corre- 

 sponding manner, and therefore need not be specially treated. 



4. First, let yw,=v=0 and 



y _ a_ 2t i__a_ %t 

 L~~L t' L~L + T' 



