180 Mr. G. A. Schott on the 
the equations for an electron slightly disturbed from uniform 
circular motion. Next we calculate the changes in the forces 
exerted by the controlling field, which are due to the 
displacement of the given electron from its position in steady 
motion, and, when necessary, those due to displacements of 
neighbouring rings. Lastly, we calculate the changes in the 
forces exerted by the remaining electrons of the ring. The 
last problem is the most troublesome when the velocity of 
the ring is not small. 
§ 13. With the notation of § 7 the equations for the 
disturbance (f, y, f), whether periodic or not, are 
i 
^ ., r 2^/S 5 »nf 
+w,g+ l 3 P '(i-/y)' +/ " B ^;« = 
2e 2 /3 3 t ■-, / 2^' 
iy(l-/3*) S* +m?+ 13A1-/8")' 4 
m is the transverse mass as before, in which a small secular 
change is admitted as possible for the sake of generality ; 
fi is the longitudinal mass and is given by fj,= \a ; ST, 
8¥, SS are the components of the disturbing force due to all 
causes. The electron is supposed to be slightly disturbed from 
steady motion in a circle of radius p with angular velocity &>. 
§ 14. if the controlling field be due to fixed charges, the 
changes in the forces P= -^— , S= — ^— (this being zero 
in the steady motion) are due to the displacement (f, rj, f) 
of the electron, which is also its displacement relative to the 
controlling field. Since the field is symmetrical about the 
axis of the ring, the component displacement along the 
ring, £, produces no change, and for the same reason no 
force is produced along the ring itself. A force of this 
latter type would be produced by induction if the charges 
producing the field themselves moved. 
For shortness write 
Vt-^ ^-^± T- 3 -!* 
dp 2 ' ""C^:' V 
