174 Dr. T. Preston on Radiation Phenomena 



moving with angular velocity co round a line whose direction- 

 cosines are /, m, n, the equations of motion 



x = — fl 2 x + Zco(ny — mz) + co 2 x — co 2 l(lx-\- my + nz) , 



" = — Q?y + 2co(lz — nx) + co 2 y — co 2 m(lx -J- my + nz), J> (4) 



z = — O 2 ? + 2co(mx — ly) + co 2 z — co' 2 n(lx + my -f nz) 



show that the perturbing forces consist firstly of a force 

 2cov sin 6 ; where v is the velocity of the particle and 6 the 

 angle which its direction makes with the axis round which the 

 precession co takes place. This force acts along the normal to 

 the plane of v and co (the direction of the velocity and the axis 

 of rotation), and is precisely the force experienced by an 

 ion moving in a magnetic field in Larmor's theory. The 

 remaining terms containing co 2 are the components of a 

 centrifugal force arising from the rotation round the axis 

 (I, m, n), and this is negligible only when co is relatively 

 small. 



If the direction I, 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 (4) simplify into 



x=-(Q?-co' ;L )x + 2coy~\ 

 y=-(W-co*)ij-2cox\ (5) 



s = -n% j 



and these are the equations of motion of a particle describing 

 an elliptic orbit which precesses with angular velocity co round 

 the axis of z. The two first of these equations contain x 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 pro- 

 jection is an ellipse revolving in its own plane with an apsidal 

 angular velocity co, and gives rise to the two side lines of the 

 normal triplet of frequencies (£1 ± co)I'Itt. 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 H/2ir. 



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 must introduce a double period into the last of equations 

 (5) while the first and second remain unchanged. This is 

 easily done if we write the equation for z in the form 



2 = AsinX2£ . . . . . ... (6) 



