172 PARTICULAR CASES OF EQUILIBRIUM. 



For the entire needle, the moment of the resultant in reference 

 to the axis will be 



If p is the density of the substance in question, the time /, of 

 infinitely small oscillations, is given by the equation 



I 



cfidu 



If the needle is rectilinear, and of the length 2/, we have 



cfidu 



i 



which gives finally 



Thus, in a variable field, a dielectric needle able to turn about its 

 centre of gravity is directed, not along the line of force, but along 

 the line of greatest variation of the force; the square of the time 

 of oscillations is not simply in the inverse ratio of the force of the 

 field, as it depends on the coefficients of the variation, and, other 

 things being equal, is proportional to the length of the needle. 



184. The phenomenon is particularly useful to consider when 

 the field is symmetrical in reference to the point at which is the 

 needle. Suppose, for instance, the needle to be placed at the 

 middle point O of the distance AA' (Fig. 34) between two equal 

 masses of opposite signs, and that it can oscillate about a right 

 line perpendicular to AA'. In the plane of oscillation, the force 

 at the point O is a minimum for the direction A'A or Ox, and 

 a maximum for the perpendicular direction Oy. For a point 

 adjacent to O we may therefore write 



