176 MR. GEORGE W. WALKER ON THE INITIAL ACCELERATED 



Inspection shows that there can be no purely imaginary root, and further that 

 there must be an even number of positive roots, if any. 



Again, for even moderate values of K the roots will be nearly those of 



provided the root makes e very small. 



Now this equation suggests a single positive root, but examination of the initial 

 form of the function shows that this root cannot differ from 0. As regards the 

 remaining two roots of the cubic, it is readily shown that real positive values of X are 

 impossible, while real negative values cannot make c small. Hence we have a pair of 

 complex roots, the real part being positive. 



For still greater values of K, the equation approximates to the form for a conductor, 

 namely, 



] + -X + X- = 0. 



There is, of course, also the possibility of other complex roots, just as in the 

 dynamical case of an elastic sphere vibrating in air. Without entering on the 

 determination of these, it seems reasonable to expect that the vibrations are of a 

 damped harmonic type and rapidly subside. They will be considered in Section 11. 

 When this condition has been secured, we get the solution in the form 



,, v , 2ma(Ctr+a) , ., 

 t r + ti)- '- 2a~- 



and hence 



oil 



- (K-l)(w+ra' 



The result is very similar to that for a conducting sphere, differing only in the 

 contribution to apparent initial displacement. We therefore conclude that an 

 insulating sphere and a conducting sphere of equal radius and with equal surface 

 charge possess equal electric inertia for slow speeds. 



9. Fundamental Equation.* for a Moving Dielectric. It would clearly be of 

 considerable value if we could determine the accelerated motion of an insulating body 

 at any speed, in the way that we have been able to determine it for a conductor. 



As has been already mentioned in the preceding section, there is as yet no great 

 degree of certainty as to the fundamental equations. 



In addition to conforming to the ascertained fundamental laws and the laws for the 

 fether as a special case, the equations must explain FRESNEL'S assumption as to the 

 velocity of radiation in a moving body. 



Two systems of equations may be proposed : 



