£48 The Intensity of Periodic Fields of Force. 



The methods of ensuring steadiness and their significance 

 may be arrived at theoretically. We assume that there is no 

 independent driving, and that the field is of impulsive type, 

 being constant through an interval which is small compared 

 with the period. The equation of motion is 



+ 3*0+/ (nt) I a 2r+l sin (2r + l) — = 0, 

 o 6 



where e is the angle between consecutive positions of stable 

 equilibrium, and f(nt) is an even periodic function of period 

 27r/n, equal to c 2 between and t, and zero between t 

 and Trjn, 



. 2-7T 2-7T 



Putting — 6 = nt — a-\ (f>, we have 



° e e T 



</>' + 2k</>+ — 2 {(2r + l)a 2r +i cos (2r + 1) a} f (nt) . 



= 2 {a 2r +i sin (2r + !)«}/(>£) — K n^- 



€ 



where $ is small. The mean value of the right-hand side 

 is zero if %a 2r +i sin (2r 4- l)a = *;e/2c 2 T, which therefore 

 gives the phase of the steady motion. The right now re- 

 duces to the even function equal to «e/2r between and t 

 and —nKe\2iT between t and ir\n ; this gives the small 

 periodic variation in the angular velocity due to the inter- 

 mittency of the force. The steadiness of the motion depends 

 upon the free oscillation of <j>. From the concluding para- 

 graph of § 3 it is evident that there is cumulative effect if 

 the impulsive spring is of sufficient magnitude. For a given 

 phase, a, stability is assured either by the introduction of 



frictional resistance, proportional to 0, to absorb the energy 

 communicated, as in the device cited above, or by a reduction 

 in the spring to destroy the isochronism. The latter result 

 may be attained by an increase in the radius of gyration. 



For a specified motional resistance and strength of current 

 there is a limit of frequency below which there is instability, 

 and the greater the radius of gyration the slower the speed 

 at which the instrument can be used. 



If the field is not impulsive, but acts during an appreciable 

 interval, e. g. half the period, steadiness might be reached 

 in any case of instability, simply by a suitable increase of 

 current. 



July 1910. 



