Part II. — Magnetic Perturbations of the Spectral Lines. 15 



particle is attracted to a fixed centre with an acceleration nV, while its orbit 

 revolves in its own plane with angular velocity w, then, by taking the moving 

 axes of the orbit as axes of reference, the equations of motion become — 



.'■■ = - H'x + w-x + 2(1)1/ ) 



• • • (3). 

 // + - O'// + ii)-!/ - 2w.r ' 



So that, if (x, ?/) = e'''' be a solution, we have at once — 



!> = H + (1), 



which shows at once the doubly periodic character of the motion, and also 

 exhibits the character of the perturbing forces necessary to produce the given 

 apsidal motion of tlie orljit. For if the orbit were fixed, the equations of motion 

 would be (iv, ij) — - A'^ (.r, y) ; hence the remaining terms on the right hand side 

 of (3) must represent the perturbing forces. Of these the final terms 2wy and 

 - 2(ox are the x and y components of a force 2wy (where v is the velocity of the 

 particle) acting in the plane of the orbit and along the normal to the path of the 

 particle. This represents the force which a charged particle or an ion experiences 

 in traversing a magnetic field when the lines of force are perpendicular to the 

 plane of the orbit, if 2(u be taken equal to the quantity k in equations (2). The 

 other pair of terms orx and w^y represent a centrifugal force arising from the 

 imposed rotation w. If we neglect w^, the equations (3) become identical in form 

 with equations (1), and when o? is not negligible they are identical in form with 

 equations (2), for the values (/, m, n) = (0, 0, 1), as they obviously should, for an 

 apsidal motion of an orbit in its own plane is the same thing as a processional 

 motion of the orbit round an axis perpendicular to its plane. In this case the 

 motion has no component in the direction of the axis round which precession 

 takes place, and consequently the frequencies N + n and N - n alone exist, so 

 that the central line of the precessional triplet is absent, and a doublet alone is 

 produced. 



In the same way, if we work backwards from the general equations (2) of 

 uniform precessional motion, we see that the perturbing forces consist, firstly, of 

 a force 2a)y sin 6 ; where v is the velocity of the j^article and 6 the angle its 

 direction of motion makes with the axis round which the precession w takes place. 

 This force, which is represented by the terms in i, i/, s of equations (2), acts 

 along the normal to the plane of w and v (the axis of rotation and the direction 

 of the velocity), and is precisely the force experienced by an unconstrained ion 

 moving in a magnetic field. The remaining tei'nis in oy' are clearly the com- 

 ponents of a centrifugal force arising from the rotation round the axis (/, m, n) ; 

 and this is negligible only when w is relatively small. 



TRANS. ROY. DUB. SOC, N.S. VOL. VII., PART IF. E 



