787 



O)' — nox 



^ / (2/4-l).-Ta Y 



The frequencies arrived ut are those of odd order. aUered by 

 current and field. For large values of R an approximate value of 



n can easily he expressed in the form /?^ -| . From (B) follows 



'iPvi, ice 



^ ()R H.J I 

 s being an odd number. / being taken zero, while for to and ??, their 

 values for R = oc must be put. Taking x = 0, i.e. neglectijig the 

 air-damping in comparison with the electrical damping, we tind 



jt sa 4H^U 



In the solution, therefore, there is a damping factor of the form 



The influence of the damping is the less, the greater the value of 

 s is. This is directly evident, for if .s- is great, the string vibrates in 

 a great number of parts with opposite motion. The electromotive 

 force generated by those parts Therefore is annulled. 



In case R is small, the i-oots of the equation (B) are those of the 



transcendental equation 



?il 2a . nl 



I cos sin — = 



2a n 2a " 



or 



2a nl 



~7*^w- = ^ (10)^) 



nl 2a 



nl -r 



The quantity — approaches to odd multiples of -. For small values 



of R an approximate form n^-\-aR can be easily indicated. Taking 

 again L=0 and x = 0, we tind 



2aRQ , 



Us = /is- H /. 



where iis is an arbitrary root of (10). In case the resistance is small, 



all vibrations suffer the same damping. 



For ff we find 



nl . n (I — x) . nx 



sin sm — — — sin — 



2 a a 



(p — . 



nl 



sin — 



(I 



1) Compare for instance Riemann-Weber, Partielle-Differeulial Gleichungen, II, 

 p. 129. 



