Part I1_—Magnetic Perturbations of the Spectral Lines. 15 
particle is attracted to a fixed centre with an acceleration 077, 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— 
&#=—- Oe + wt 2wy (3) 
y oP O?y Sr wy — Qwat 
So tliat, if (z, vy) = e’”’ be a solution, we have at once— 
p= Q+ W, 
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 the orbit, For if the orbit were fixed, the equations of motion 
would be (z, 7) =- A(z, y); hence the remaining terms on the right hand side 
of (3) must represent the perturbing forces. Of these the final terms 2ay and 
— 2w¢% are the w and y components of a force 2wv (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 2m be taken equal to the quantity / in equations (2). ‘The 
other pair of terms ww and wy represent a centrifugal force arising from the 
‘umposed rotation w. If we neglect ’, the equations (3) become identical in form 
with equations (1), and when 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 precessional 
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 V + x and WN — x 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 2wv sin 0; where v is the velocity of the particle and @ the angle its 
direction of motion makes with the axis round which the precession takes place. 
This force, which is represented by the terms in 2, y, 2 of equations (2), acts 
along the normal to the plane of 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 terms in ” are clearly the com- 
ponents of a centrifugal force arising from the rotation round the axis (/, m, n); 
and this is negligible only when o is relatively small. 
TRANS. ROY. DUB. SOC., N.S. VOL. VII., PART II. E 
