16 Preston—Radiating Phenomena in a Strong Magnetic Field. 
EXPLANATION OF THE QUARTET AND OTHER FORMS. 
If the direction /, m, n be taken to be that of the lines of magnetic force, and 
if the axis of z be taken to coincide with this direction, then the equations (2° 
simplify ito 
io (P= a)e & WJ 
y = —(Q? - w’) y -— 200% ee 
g = — QO’ 
and these are the equations of motion of a particle describing an elliptic orbit 
which precesses with angular velocity » round the axis of zg. The two first of 
these equations contain w and y, and give the projection of the orbit on the plane 
x, y at right angles to the axis of the magnetic field. This projection is an 
ellipse revolving on its own plane, with an apsidal angular velocity w, and gives 
rise to the two side lines of the normal triplet of frequencies (Q + w)/27. On the 
other hand, the vibration parallel to the axis of z is unaffected by the precessional 
motion, and gives rise to the central line of the triplet of frequency 0/7. 
Now, in order to account for the quartet (fig. 2), we must introduce some 
action which will double the central line A while the side lines B and Care left 
undisturbed. That is, we want to introduce a double period into the last of 
equations (4) while the first and second remain unchanged. ‘This is easily done 
if we write the equation for z in the form 
Bs ALAMO s o o (W, 
and remark that this will represent two superposed vibrations of different periods 
if we regard A as a peroidic function of the time instead of a constant. That is, 
if we take A to be of the form a@ sin nt, we will have * 
S=asin nt sin Ot = 5 [cos (Q — n) ¢ — cos (Q + 2) ¢] 
a? 
which represents two vibrations of equal amplitude and of frequencies (Q — 2) /27 
and (© + n)/2m as required to produce the quartet. The magnitude of x 
determines whether the separation of the constituents of the central line A (fig. 2) 
shall be less than, or greater than, the separation of the side lines B and C; and 
if the former is sensible while the latter is insensible, we are presented with the 
* If A be taken of the form (a + 6 sin nt) the central line will be a triplet of frequencies V+», V, V-n, 
and by a similar supposition regarding the perturbating forces, the side lines may become replaced by 
doublets or triplets. Such doublets and triplets exist, and the amplitudes of their constituents are 
determined by the quantities a and 6. 
