in the Magnetic Field. 173 



plane while the particle is attracted to a fixed centre with a 

 force OV 3 then, by taking the moving axes of the orbit as 

 axes of reference, the equations of motion are 



x — — £l 2 x + co 2 x + 2coy. \ , „. 



y = — 12^+ coy— 2 cox, J 



so that if (#, y)=e ipt be a solution, we have at once 



p = £l ±co, 



which shows the doubly periodic character of the motion, and 

 also exhibits the character of the perturbing forces necessary 

 to produce the given apsidal motion of the orbit. For if the 

 orbit were fixed, the equations of motion would be (x } y) 

 = —£l 2 (x,y): hence the remaining terms on the right-hand 

 side of the above equations must represent the perturbing 

 forces. Of these the final terms 2 coy and —2 cox are the x 

 and y components of a force 2cov, where v is the velocity of 

 the particle, acting in a direction perpendicular to v 9 that is 

 along the normal to the path of the particle, and represent 

 the forces which a charged ion would experience in moving 

 through a magnetic field with the lines of force at right 

 angles to the plane of the orbit, if 2co be taken equal to k in 

 Dr. Larmor's equations (2). The other pair of terms, co 2 x and 

 co 2 y, represent a centrifugal force arising from the imposed 

 rotation co. If we neglect g> 2 , the above equations become 

 identical with those which hold in Larmor's theory for the 

 moving ion, as they obviously should, for an apsidal motion in 

 the plane of the orbit is the same thing as a precession about a 

 line perpendicular to the plane of the orbit, and in this case 

 there will be no component in the direction of the axis round 

 which precession takes place ; accordingly the middle line of 

 the precession triplet will be absent, and we are furnished 

 merely with a doublet. 



Now in the magnetic field the perturbing force, being the 

 magnetic force, is fixed in direction, and on this account the 

 doublets and triplets arising from perturbations caused by it 

 are polarized. On the other hand, if the perturbing forces be 

 not constant in direction, this polarization should cease to 

 exist, and polarization should not be expected in the case of 

 any lines of the normal spectrum, even though these happen 

 to be derived from other lines by perturbations in the manner 

 conceived by Dr. Stoney. 



In the same way the general case of precessional motion 

 may be worked backwards in order to discover the types of 

 force which produce the perturbation. Thus, taking the axes 



