﻿n ^k 



Periodic Field of Force. 21 



We then obtain the following set of relations by substi- 

 tution : 



- (b — P 2 ) Aj + kpBi = — h A 2 + «iB 2 + h A 4 — a 2 B 4 - &c. 



- (^o—^ 2 )B 1 —^A 1 = a 1 A 2 + 6^2 + ^4 + 6^4+ &c. 

 -(6 1 -4p 2 )A 2 + 2%?B 2 =-6 1 A 1 + a 1 B 1 + 5iA 5 -a 1 B 5 +&c. 



- (& -4p 2 )B 2 - 2kpA 2 =a 1 A 1 + b^ + a : A 5 + ^B 5 + &c. (8) 



and so on. 



It must be remembered that these relations are all only 

 approximate, as F(U) in general contains powers of U higher 

 than the first which we have neglected, and which no doubt 

 must be taken into account in framing a more complete 

 theory. The general remarks made above with reference to 

 equation (6) apply here also. 



The exact character of the vibratory motion maintained by 

 the periodic field of force in any case, depends upon the form 

 of the functions F(U) and f(t) which determine respectively 

 the disposition of the field and its variability with respect to 

 time. One very simple and important form of f(t) is that 

 in which the field is of an impulsive character, in other 

 words is of great strength for a very short interval of time 

 comprised in its period of variation, and during the rest of 

 the period is zero or nearly zero. Such a type of variation 

 is not merely a mathematical possibility. In actual experi- 

 ment, when a fork-interrupter is used to render the current 

 passing through the electromagnet intermittent, the magneti- 

 zation of the latter subsists only during the small fraction of 

 the period during which the current flows and at other times 

 is practically zero. When the current is flowing the accele- 

 ration is considerable : at other times, the acceleration is 

 nearly zero, and the velocity practically constant. These 

 features are distinctly shown in all the vibration-curves 

 (except those of the first type) reproduced in PL II., the 

 sudden bends in the curves corresponding roughly to the 

 extreme outward swings of the fork, i. e. to the instants when 

 the magnetizing current was a maximum. It seems possible 

 that a simpler mathematical treatment than that given above 

 might be sufficient to discuss the phenomena of the main- 

 tenance of vibrations by a periodic field of force when the 

 periodicity of the field is of the "impulsive" type ; in other 

 words, when the dynamical system is subject to periodic 

 impulsive " springs/' one, two, three or more of which occur 

 at regular intervals during each complete period of the 

 vibration of the system. 



These experiments on vibrations maintained by a periodic' 



