ACTION OF A FIELD ON A CONDUCTING NEEDLE. 173 



The components of the force which are exerted on the element du 

 are 



X = KtluAx = KJuAa cos 0, 

 Y = - Kduty = - KduBa sin 0, 



and the tangential component 



T = aKdu(A + B) sin cos 0. 



The condition of equilibrium is thus 



sin cos = 0, 



which gives two directions, = and = -, the first corresponding to 



stable, and the second to unstable, equilibrium. 



For a very small deviation dO from the position of stable equi- 

 librium, the tangential component is 



If the volume-element is isolated, the duration of the oscillation is 

 given by the formula 



KA + B' 



it is thus seen to be independent of a, and to depend only on the 

 density of the substance, on its electrical susceptibility, and on the 

 law of the variation of the field. 



If we collect on a straight line a series of similar particles, and 

 if we suppose that they exert no influence on each other, each of 

 them would act as if it were alone, and the oscillation of the whole 

 needle would take place in the same time as each of its parts ; the 

 duration of the oscillation of a needle for the state of the field in 

 question is therefore independent of its length. 



185. ACTION OF A FIELD ON A CONDUCTING NEEDLE. A 

 conductor behaves in a totally different manner. Consider, for 

 instance, an infinitely small needle, so that the electrification may 

 be supposed identical with that which would be produced in a 

 uniform field. 



