16 Preston — Radiating Phenomena in a Strong Magnetic Field. 



Explanation of the Quahtet and other Forms. 



If the direction I, m, n be taken to be that of the lines of magnetic force, and 

 if the axis of s be taken to coincide with this direction, then the equations (2' 

 simplify into 



Jr = - [Q,- - W-) X + 2wf/ \ 



// = -(S2= - to')// - 2c../' I ... (4), 



~ = - Q-: ] 



and these are the equations of motion of a particle describing an elliptic orbit 

 which processes with angular velocity w round the axis of s. The two first of 

 these equations contain x and g, 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 &>, and gives 

 rise to the two side lines of the normal triplet of frequencies (H ± w)/27r. On the 

 other hand, the vibration parallel to the axis of z is unaffected by the processional 

 motion, and gives rise to the central line of the triplet of frequency Ii/7r. 



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 C are left 

 undisturbed. That is, we want to introduce a double period into the last of 

 equations (4) while the fir.st and second remain unchanged. This is easily done 

 if we write the equation for z in the form 



z = A^mQ.t . . . (5), 



and remark that this will represent two superposed vibrations of different periods 

 if we regard -4 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 * 



z = a siu nt sin Q,t = - [cos (Q - n) t - cos (Q + n) <], 



which represents two vibrations of equal amplitude and of frequencies (fl - n) J^tt 

 and (fl + n) I2tt as required to produce the quartet. The magnitude of n 

 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 tlie form (« + J sin nt) tlie central line will be a triplet of frequencies iV+ «, N, JV- n, 

 and by a similar supposition regarding tbe perturbating forces, the side Lines may become replaced by 

 doublets or triplets. Such doublets and triplets exist, and the amplitudes of their constituents arc 

 detei-mined by the quantities a and b. 



