Maintenance of Vibration by Periodic Field of Force. 137 



only the assumed disturbance of frequency 1/T, but also 

 other subsiding components whose frequencies are multiples 

 of 1/T and play a part in the magnification or maintenance 

 of the vibration of that frequency. If, in the distribution 

 of the field F(y ), only the first correction term by is taken 

 account of, the analysis proceeds practically on the same 

 lines as that contained in the Phil. Mag. paper of Jan. 1915, 

 except that, instead of the equations of motion for one 

 degree of freedom, the general formulae for the normal 

 coordinates in the forced vibrations of the wire would have 

 to be used. The same general result would be obtained, 

 that the components in the motion having the same fre- 

 quency as the field or any multiple of it, would not play any 

 part in the maintenance of the motion of the kind now 

 considered. We have already seen how this indication of 

 theory may be verified experimentally. 



When, however, the correction terms of higher order, 

 that is, cy (J 2 , &c, are considered, some modification of this 

 general statement might become necessaiy. 



Summary and Conclusion. 



The present paper considers experimentally, and theoreti- 

 cally, a case of vibrations maintained by a non-uniform 

 periodic field of force which is of some practical importance. 

 It is shown that when a wire divides up into two or more 

 segments and vibrates under the transverse attraction of 

 an electromagnet, the motion which has the same frequency 

 as the field may be rendered unstable by the non-uniformity 

 of the field and then passes over into one, the frequency of 

 which is a submultiple of the frequency of the field. 

 Photographic records illustrating the first six cases of the 

 kind are presented with the paper. 



It is also shown that a motion of this type may be set up 

 and maintained even when the attracted point on the wire is a 

 node and the ordinary forced vibration is therefore absent. 

 The effects of the non-uniformity and of the periodic varia- 

 tion of the field on the ordinary forced vibration are also 

 considered in detail and the mathematical theory of certain 

 effects noticed by Klinkert is set out. 



The experiments and observations described in the present 

 paper were made in the Laboratory of the Indian Association 

 for the Cultivation of Science. 



Calcutta, 

 February 2nd, 1917. 



