Intensity of Periodic Fields of Force. 847 



negative portions of the spring, the condition for stability is, 

 as before, that the impulsive spring be less than four times 

 the reciprocal of its period. 



4. When the period of the field is made up of intervals of 

 constant strengths, the periodic motions which distinguish 

 the limits of the ranges may be found directly. If, for 

 example, the spring is alternately equal to (an) 2 and —(an) 2 

 during intervals 7r/n, the limits of the odd ranges are given 

 b 7 



e- a7r =tan^(2a-l), e air = tan ^(2a-l), 



and of the even ranges by 



-e a7r = tan J (2a- 1), -e~ a7r = tan J (2a- 1). 



The ranges of cumulative effect lie about the integral 

 values of a, and those of stability about the values differing 

 from an integer by a half; the latter become small without 

 limit as a is taken large. In the case of instability the value 

 of p, equation (1), is 



n , cos (2e + air) 

 log, 



r here 



4tt * e cos(2 e -a7r)' 

 + 1 



cos 2e = sin air 



>2air — J[ 



When the spring is alternately (an) 2 and zero for periods 

 7r/n, the ranges are given by cot air /2=air/2 and cos airj2 = 0, 

 the even by tana7r/2 = — aw/2 and sina7r/2 = 0. Here the 

 integral values of a give the upper limits of the ranges. 



5. The problem under consideration is of importance in 

 connexion with those motions which are maintained by a 

 periodic field of force. Take, for example, the 'phonic 

 wheel,' in which " the essential feature is the approximate 

 closing of the magnetic circuit of an electromagnet, fed with 

 an intermittent current, by one or more soft iron armatures 

 carried by the wheel and disposed symmetrically round the 

 circumference "*. "In some cases the oscillations of the 

 motion about the phase into which it should settle down are 

 very persistent and interfere with the application of the 

 instrument. A remedy may be found in a ring containing 

 water, or mercury, revolving concentrically." 



* Rayleig V Sound,' i. § 63 « 



