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charges. Per unit oharoe tliesc forcos are givoii liy the voetor jtroiluft 

 of the velocity and the inaaiietio fmci' in the lieM. 



Let there be a mode of motion A. with freqiKMicy ", before theie 

 is any magnetic force, and let F be the (ïlectromagnetic forces arising 

 from this motion as soon as the field is produced. The direction of 

 these forces will obviously chanj^e witli tiie frequency -', and to deter- 

 mine their action on the system is a pioblem of „resonance" or of 

 „forced vibrations". In general, tlie system will respond to the forces 

 F by a motion in several of its other fundamental modes. In fact, 

 anv particular motion B will ceitainly be excited if only the forces 

 -F do a positive or negative work in an infinitely small displacement 

 corresponding to that mode B. 



Since the electromagnetic forces are perpendicular to the velo- 

 cities, the forces F will do no work if the infinitely small displa- 

 cement belong to the mode A itself. A direct influence of the forces 

 F on the motion A which gave lise to them is thereby excluded. 



As to the other modes, all depends on their frequency. If the 

 frequency n' of a motion B be considerably different from «, the 

 forced vibration B, if it exist at all, will be very insignificant, for 

 experience shows the forces F to be very feeble as compared with 

 the other forces acting in the system. As well as the forces F 

 themselves, the amplitude of the forced vibrations B will be pro- 

 portional to the strength H of the field. Hence, the electromagnetic 

 forces F'j which exist in consequence of the vibration B, will be 

 of the order JJ^, and it will be permitted to neglect their reaction 

 on the original motion. 



The case is quite different as soon as the frequency of B is 

 equal to that of A. The amplitude of the new motion B will then 

 rise to a much higher value; as may be deduced from the equa- 

 tions of the problem, it will reach the same order of magnitude as 

 the amplitude of A itself. The influence of the forces F' on the ori- 

 ginal motion will likewise be much greater than in the former case. 



One mav see by a simple reasoning that this influence will 

 consist in a modification of the period. Since the forces F have 

 the same phase as the velocities in the motion A, there will be a 

 difi'erence of phase of ^/4 period between them and the displacements 



A. On the other hand, the displacements in the motion B have 

 the same or the opposite phase as the forces F, and the phase of 

 the forces F' will differ by V4 period from that of the displacements 



B. These latter forces will therefore have the same or the oppo- 

 site phase as the displacements A, and this is precisely what is 

 required, if they are to change the frequency of A. 



