788 

 hence for y 



nl , n [l — x) . no; 



si7i sin sin — 



,v,/ a a a 



y=' -nl ■ • • • <"' 



sin — 

 a 



The real and imaginary part of this expression satisfy the equations 



and the boundary conditions. A sum of solutions for different values 



dy 

 of to satisfies the equation. If y and — are aiven for ^ = 0, we can 

 ^ ^ dt 



with the aid of the given functions find the solution. The found 



proper functions are not orthogonal, but by an appropriate linear 



substitution orthogonal functions can be obtained. If y is known, I 



can be calculated from (^3). 



§ 3. It is useful to work out the problem. Using the assumption 



(2) of CRpmoRE, we obtain for (fs the following set of equations 



(taking k and / zero) : 



4.HJ 

 (ps + ns^ ffs = — (12) 



and 



RJ^ JS-^^O (13) 



jr s 



where 



sjta 



Here .s^ is an odd number; for even values the second member 

 of (12) is zero, and the even vibrations are therefore unchanged. 

 Now putting 



and 



4.HI 1 



SÜÏQ lis to 



we find 



2/ H^io) ^tts 

 RI -}-— ^- = 0. 



jr s 



The frequency-equation therefore is 



8Z HHoi _ 1 ^ 

 R-\ — 2 = {Qa) 



This frequency-equation has the same roots as equation (6), which 

 if x and / have been taken 0, takes the form 



