508 Mr. J. Larmor on the Theory of 



series of double lines in a spectrum, with equidistant fre- 

 quencies, might be originated. Suppose, as a very rough 

 illustration, that a polar molecule is constituted of a system 

 of positive electrons around one pole and a system of negative 

 ones around the other, the two systems being so far apart as 

 to have practically separate sets of periods for their orbital 

 motions, each of course disturbed by the presence of the other. 

 Each of these systems moves in the magnetic field, more or 

 less constant, arising from the other ; and the effect of this 

 disturbing field will, as above, be to duplicate all the periods 

 of that set in the above regular way. 



9. It is desirable to precisely formulate the relation 

 between the motions of the electrons and the radiation 

 emitted by them, which has been tacitly employed in the 

 foregoing discussion. The specification of that radiation 

 may be readily assigned by a summation over the different 

 elements of the paths of the oscillating ions. Suppose that an 

 ion e is at A and after a time 8t is at B, where AB = v8t, 

 v being its velocity ; the effect of its displacement is the 

 same as that of the creation of an electric doublet AB of 

 moment evSt ; thus we have only to find the influence 

 propagated from such a doublet, and then integrate along the 

 paths of the ions of the molecule. 



Consider now such a doublet at the origin, lying along 

 the axis of z ; for it, or indeed for any distribution sym- 

 metrical with respect to that axis,, the lines of magnetic force 

 will be circles round the axis, and the force will be specified 

 by a single variable, its intensity H. The current, whether 

 in dielectric or in conducting media, will circulate in wedge- 

 shaped sheets with their edges on the axis, and may be 

 specified by a stream function, as in fact will appear below. 

 If we employ cylindrical coordinates p, 0, z, and apply the 

 Amperean circuital relation (Viz : circulation of magnetic 

 force equals Air times current) to the faces of the element 

 of volume 8p.pS8.Sz, we obtain for the components P, R of the 

 electric force 



d? 2 (TKp dR = 2 (Mp 



dt pdz * dt pdp ' 



so that Hp plays the part of a stream function ; while by fhe 

 circuital relation of Faraday 



dz dp dt 



Thus the characteristic equation for H is 



