﻿Periodic Field of Force. 25 



on the latter for any given field strength at the poles is 

 obviously a periodic function o£ n0 which vanishes when 



6=~ — , and also when = — , where r is anv 



integer. 



We therefore write 



Couple = Field strength x [<*] sin ?i#+a 2 sin 2m0 4 o^sin 3n04&c] 



= Field strength xf(nd) say, 



where the terms a 1? a 2 , a 3 , &c. rapidly diminish in amplitude. 

 It will be seen that the cosine terms are absent. Since the 

 field strength is periodic, we may write the expression for 

 the couple acting on the wheel thus 



Couple =.\jf(nd) [^sin (pt + e x ) + b 2 sin (2p*4e 2 )4 &c] 

 = L/>0)F(*), say. 



The work done by the couple in any number of revolutions 



= (~Lf{n0)Y{t)dt. 



*j 



It is obvious that this integral after any number of complete 

 revolutions is zero, except in any of the following cases, 

 when it has a finite value proportional to and increasing 

 w T ith t ; i. e. when 



n0 =pt or 2pt or 3j)t or Apt and so on, 



or when 



2n0=pt or 2pt or Zpt or Apt and so on, 



or when 



3n0=pt or 2pt and so on. 



It is therefore a necessary but not, of course, always a 

 sufficient condition for uniform rotation to be possible that 

 one or more of the above relations should be satisfied. The 

 first series corresponds to the synchronous speed and 

 multiples of the synchronous speed. These have been ob- 

 served experimentally by me up to the fifth at least. The 

 second series includes the above and also the half -synchronous 

 speed and odd multiples of the same. These latter have also 

 been observed by me up to the fifth odd multiple. Since a a 

 is much smaller than a l5 the relative feebleness o( the main- 

 tenance of the half-speeds observed in experiment will readily 

 be understood. 



The third series has not so far been noticed in experiment. 



