MOTION OF ELECTRIFIED SYSTEMS OF FINITE EXTENT, ETC. 195 



Equating the coefficients of the various powers of x to zero, we get 



^^-S^) = 0, 

 - A, (0^-2^) = 0, 



Si/g = 0. 



Eliminating A,,, Aj, and A 2 , we get the period equation 



X 2 (fly- 50* 

 = 2" (0V-8 



Now, when X is entirely neglected, we get 



(9Y,,-/'i = 0, or a 2 0- + a#+l = 0, 

 which is our former period equation for a fixed sphere. The roots are 



We can now proceed to an approximation to ad in X. 2 by putting 



on the left-hand side of the period equation, and substituting the value 



in the term on the right-hand side, which is of order X 2 . In this term we may use 

 the approximate results 



4 



and 



Hence we find 



, _X^ _ ; .X 2 59\/3 



260 4 780 

 Thus 



- 

 ISO/' 2 \ 390 / 



We could, in a similar manner, proceed to the period equation for higher order 

 vibrations, and to calculate higher approximations to the roots in powers of X 2 . The 

 process would-be tedious, and I have not as yet discovered any artifice for effecting 

 the summation, 



2 c 2 



