MOTION OF ELECTRIFIED SYSTEMS OF FINITE EXTENT, ETC. 179 



explain KAUFMANN'S results, and give the same value for the electric inertia without 

 limitation as to speed. 



10. Vibration of a Charged Conducting Sphere under a Periodic Force. While 

 the examination of rectilinear motion is important from the general standpoint of 

 electro dynamics, the problem of motion under periodic forces is no less important for 

 optical theory. 



Using the same notation as in Section 3, we shall assume the accelerating force to 

 be purely mechanical and equal to F cos nt. In this case the sphere may be supposed 

 never to move far from its original position, so that the approximation to the first 

 order remains valid for an unlimited time. 



The surface condition is, as before, 



and the equation of motion is 



m + 5 - - x" = F cos nt, 

 the integral of which is 



Hence, if 



M ' = -|fl 3 /aO a , 



my" mnC 



Rapidly damped harmonic vibrations are initially produced, and when these subside 

 we shall have only the residual effect of these with the particular integral. 



-L nils / ) o\ / f~\ i \ / ' f~\ \ T 



I / 1 4. ?H _ a ~ n \ ,n((jtr+a) an t - n((Jtr + a)\ 



eF eF |\ ~m~~O r / C C TT & C J * 



m' 



J- T " I 



m C 1 ' / C 2 



, m' a 2 n 2 \ an, ,1 



,-, ^ -, , H cos nt+ - -r sin nt > 



t_ F_ F cos /^ 7}i 7 F U m C 2 / C_ _J * 



/ /*V1 I 'I'l J * I 1'l~ /Wl 1 j * AV*JV1* / /111' ^WV1*\ * y~**li* 



mw m'n /, m_rr/r, ,v -. 



<-^2 ; i ^--(2 



It is possible to interpret this as a solution of the equation 



-&':= F cosnt* 



where 



22 



(m + m') (i 



' 



* In the corresponding expressions at ' Eoy. Soc. Proc.,' A, vol. 77, 1906, p. 272, note the error of sign. 

 Also in M the third term of numerator should be + . 



2 A 2 



