78Ö 



fa hl a f )il\ \ 



v' C = p Asiu +-Bil— cos -f C l . . (4) 



\n a n \ a J J 



whereas, because of the bouiidar}- conditions, we must liave 



A + 6'=i:0 



nl nl ^ 



A cos ^ 4- B siu - + C= 0. 

 a ' a 



TJiis gives for the frequency the transcendental equation 



j 1 — cos — 



7}. z= p I srn -j- I 



n a n . td 



sin — 



or 



nl ( . nl a / )i I 



v"^ sm — = /' ^ '"^ '^1 ^ '~ '-'^'^ ~ 



a \ a n \ a 



From this it appears immediately that we must have 



nl 

 siji — = (5) 



2 a 



or 



nl / )il 2a . nl\ 



n^ cos — ^ p { I cos sin — 1 (b) 



2a '^ V ^« " 2«/ 



(5) can be satisfied by 



~ = k^ (7) 



2a ^ ' 



or, hence 



CO" — itoy. 



I 



As is immediately to be seen, these are the damped vibrations of 

 even order, which the string' can perforin in the absence of the 

 current. It is evident that the presence of current and field have 

 no influence on the vibrations of even order. If the resistance is 

 infinitely great, the constant /> in the e<(uation (6) is zero. In this 

 case the equations can be satisfied by yi = 0. or tu = 0, i.e. the 

 string is at rest ; and further by 



Hence 



nl 

 cos — 3= 0. 

 2a 



nl ÜT 



01' 



