Another method of finding e/m for the negative ion which is applicable in many cases to which the preceding one is not suitable, is as follows: Let us suppose that the ion starts from rest and moves in a field where the electric and magnetic forces are both uniform, the electric force X being parallel to the axis of x, and the magnetic force Z parallel to the axis of z; then if x, y, are the co-ordinates of the ion at the time t, the equations of motion of the ion are-- d²x dy m --- = Xe - He -- , dt² dt d²y dx m --- = He --. dt² dt The solution of these equations, if x, y, dx/dt, dy/dt all vanish when t = 0, is Xm / / e \ \ x = --- {1 - cos( -- Ht ) } eH² \ \ m / / Xm /e / e \ \ y = --- {-- Ht - sin( -- Ht ) }. eH² \m \ m / / These equations show that the path of the ion is a cycloid, the generating circle of which has a diameter equal to 2Xm/eH², and rolls on the line x = 0. Suppose now that we have a number of ions starting from the plane x = 0, and moving towards the plane x = a. The particles starting from x = 0 describe cycloids, and the greatest distance they can get from the plane is equal to the diameter of the generating circle of the cycloid, i.e. to 2Xm/eH². (After reaching this distance they begin to approach the plane.) Hence if a is less than the diameter of the generating circle, all the particles starting from x = 0 will reach the plane x = a, if this is unlimited in extent; while if a is greater than the diameter of the generating circle none of the particles which start from x = 0 will reach the plane x = a. Thus, if x = 0 is a plane illuminated by ultra-violet light, and consequently the seat of a supply of negative ions, and x = a a plane connected with an electrometer, then if a definite electric intensity is established between the planes, i.e. if X be fixed, so that the rate of emission of negative ions from the illuminated plate is given, and if a is less than 2Xm/eH², all the ions which start from x = 0 will reach x = a. That is, the rate at which this plane receives an electric charge will be the same whether there is a magnetic field between the plate or not, but if a is greater than 2Xm/eH², then no particle which starts from the plate x = 0 will reach the plate x = a, and this plate will receive no charge. Thus the supply of electricity to the plate has been entirely stopped by the magnetic field. Thus, on this theory, if the distance between the plates is less than a certain value, the magnetic force should produce no effect on the rate at which the electrometer plate receives a charge, while if the distance is greater than this value the magnetic force would completely stop the supply of electricity to the plate. The actual phenomena are not so abrupt as this theory indicates.