239 



cut by the string during its movement, the form taken by the string 

 in its deflected condition is of the greatest significance as well as 

 the strength of the magnetic field. A string under tension deflected 

 in a homogeneous magnetic field takes the form of a parabola. In 

 that case the damping factor is 



w . 10 

 Whenever the string would take another form vvliilst being deflected, 

 the factor f would fake another value. This factor represenis the 

 mean deflection taken over the whole string as compared with the 

 maximum deflection. A perfectly relaxed string, clamped at the ends 

 and uniformly loaded takes the form given by the formula 



in which ?/ gives the deflection for a point at distance a; from the 

 end. If we put x =z U we get the deflection for the middle of the 

 string, which has already been given in formula (2). In order to 

 find the mean deflection we integrate (7) over the whole length : 



I 



r _ p I* 







Comparing (2) and (8) we find 



/ 



y Idw/y 



"""-15 







so that we may state 



S HW 

 X>= — (9) 



15 tvAO' ^^ 



Taking 3f as the mass of the string, we can always represent 

 the time of vibration as 



1 t /'M 



if K be a lateral force and if we suppose the damping to be very 

 slight. If the string should make critically damped vibrations, the 

 damping would be 



D=\/IlifK . (11) 



Eliminating K from (10) and (11) we get, in connection with (9) 



8 HH' 



4:jtMN= — (12) 



15 tvAO' ^ ^ 



