Vibrations by Forces of Double Frequency. 



155 



In the former paper some examples were given drawn from 

 ordinary mechanics and acoustics. To these may be added 

 the case of a stretched wire, whose tension is rendered periodi- 

 cally variable by the passage through it of an intermittent 

 electric current. It is probable that an illustration might be 

 arranged in which the vibrations are themselves electrical. 

 © would then represent the stiffness of a condenser, ^ re- 

 sistance, and <3> self-induction. The most practicable way of 

 introducing the periodic term would be by rendering the self- 

 induction variable with the time (^j). This could be effected 

 by the rotation of a coil forming part of the circuit. 



The discrimination of the real and imaginary values of c is 

 of so much importance, that it is desirable to pursue the ap- 

 proximation beyond the point attained in (26). From (11) 

 we find 



S)(l) = l+COS(7T C ) 



Sy(l) 1+ COS (77V ©o)' ' ' ' ^ } 



from which, or directly, we see that if c = l, corresponding to 

 the transition case between real and imaginary values, 



2>(i)=o. 



(59) 



If, as we shall now suppose, © 2 , ®3 • • • vanish, (59) may be 

 written in the form 



=0, . (60) 



where 



a r 



©o-l 

 ©! ; 



© -9 



©! 



«3 = 



©o-25 



©! ' 



(61) 



The first approximation, equivalent to (26), is found by 

 considering merely the central determinant of the second 

 order involving only % ; thus, 



af-l^O. ...... (62) 



The second approximation is 



(63) 



