190 MR. GEORGE W. WALKER ON THE INITIAL ACCELERATED 



much from unity when K is large. It corresponds to a purely exponential 

 disturbance. 



Again, if K is very great and X very small, but so that K 1 2 X is finite, the second 

 form is approximately 



i | fr i f\ ~rr \ ->\ C //(' / / K ilv "IT" \ *) \ 



tanh K X = K. "Xf / -f KX . 



m I \ m I 



Hence 



X 

 where 



tan K l2 z = K 1J 2f - 



m \ m 



the roots of which in general make K l2 z finite. These correspond to a vibratory 

 disturbance, and a nearer approximation gives the damping coefficient. 



The problem of rotation of a vibratory character may also be treated by aid of the 

 equations. In dealing with the problem of linear vibration, approximate treatment 

 for large values of K was possible since i/i and i/ 2 were small compared with ^. 



With rotary vibration i//,, i// 2 , and ^ are of the same order, and approximate treat- 

 ment is no longer possible. 1 do not propose to give the results of this problem, as 

 they are somewhat cumbrous, and do not appear to present any important optical 

 features. Such a conclusion may be expected from the investigations of HAYLEIGH 

 (' Phil. Mag.,' p. ,379, 1899), LAMB, and LOVE (/oc. cit.), which show that the radiation 

 for disturbance of the second type is insignificant compared with radiation for 

 disturbance of the first type. In this connexion it is interesting to observe the 

 nature of the mode of linking of the sphere to the tether in the tw r o types. In both 

 we have the slowly damped vibrations which may be of wave-length large compared 

 with the diameter of the sphere if K is great. The main link is, however, through 

 the rapidly damped vibration of wave-length comparable with the diameter, in the 

 case of a linear vibration, and the purely exponential disturbance in the case of 

 rotaiy vibration. 



The considerations in Section <J prevent us from attempting to extend the results 

 to the case of a high speed of rotation. 



The independence of small linear motion and small rotary motion will be apparent 

 from the method of examining the two cases, and we are thus able to present in a 

 purely dynamical form the equations of motion of a sphere in general, provided the 

 velocities are small compared with that of radiation. 



The disturbance of exponential type, which occurs in the problem just treated, 

 arises with all disturbances of magnetic type associated with zonal harmonics of odd 

 order. It also occurs with all disturbances of electric type associated with zonal 

 harmonics of even order. Pure damped harmonic vibrations, on the other hand, occur 

 with disturbances of magnetic type associated with even order zonal harmonics, and 

 disturbances of electric type associated with odd order zonal harmonics. 



Changes of the linear motion of a charged sphere give rise to disturbances of 



