440 Dr. J. W. Nicholson on tlie Damping of the 



Lamb, the vibrations would be very persistent when started. 

 Walker quotes the result without proof, and perhaps only a 

 misprint has occurred, for the true formula., as will appear 

 below, is of the form 



\= + 4-493 uc~i+ (4-493) 4 *-«, . . . . (3) 



and the resulting decrement is thus increased in the ratio 10 12 , 

 so that the free vibrations would not have such a pronounced 

 degree of permanence. 



When the sphere is capable of free motion while the 

 vibrations on it are taking place, and has a surface charge 

 distributed over it, the field will set it into motion, and 

 the electromagnetic inertia, given for small motions by 

 m' = 2e 2 /3<2c 2 , where e is the charge, will enter into the 

 question. The more general formula valid in this case is 

 found by Walker to be 



(tanh if\)/Ai.=: I + kX 2 (l + ™ ~x\ I | (/c- 1) (\ + ^ -x\ 



-jbX-+*\ 8 ); . . . (4) 

 m J 



and if m! is negligible in comparison with m, this is Lamb's 

 formula (1). We shall calculate the decrement corresponding 

 to any root of this equation when /c is large. In this case, 

 the main term of X, expressed as a series of inverse powers 

 of k% is of order k~K The roots are therefore, to a first 

 approximation, those of 



tanh k*X = k^X, (5) 



whose first root is well known to be given by k%X = + 4=*4932, 

 being purely imaginary. Let k?\= ±ip denote any pair of 

 roots, and let k^X — ±ip 4 er be the corresponding roots of the 

 equation (4), a denoting merely the leading term of the 

 real portion. That cr must be positive is known from the 

 physical consideration that the vibration must die out ulti- 

 mately, and the equation being real, the roots must occur in 

 pairs in this way. It is evident that a will be of lower 

 magnitude than p. Then 



/• , x ,/ t , ™' ip+<r\ 

 tanh(y + < Q =1| fr + *>\ 1+ » ~ Vj 



