﻿548 Mr. W. Ellis Williams on the 



given above to Stokes's solution of the motion of a spherical 

 pendulum-bob given in his memoir of 1850. 



When the motion is not steady equation (8) takes the form 



D ( D *-^H sD * w 



Let us assume that the position and velocity of the sphere 

 are exponential functions of the time and write, following 

 Stokes, 



V = ce^ vt . 



Then if we write i|r = ce K2vt (f> (r) sin 2 #, 

 where 



^r is the solution for infinitely small velocity and satisfies 

 the left-hand side of (11) equated to zero. 



As before let ^=^0 + ^ be the solution of (11), then on 

 substituting and neglecting third-order terms we get 



D K-^H 3D ^ • • • (13) 



the right-hand side being a function of i/r only which may 

 be written 



X(r) ce 2K2vt sin 2 cos 0, 



Xi r ) being a known function obtained by operating on <£o( ? ')« 

 To solve (13) we assume 



^ = ( ? e W ^(r) sin 2 cos 0, 



&! and <f>i(r) being for the present undetermined. 

 Substituting in (13) we have 



V{ (*i"W- £-#iM-V<fr WK lS "'sin 2 cos 0\ 



= % (r> 2 ^sin 2 0cos0. 



Let us denote the function in the brackets by f (r), then the 

 above equation becomes 



D{%(r)e K i 2vt sin 2 cos 0} = x (r)e 2K%vi sin 2 cos 0, 

 ° r (t"r-^fr)*i*" ± xfr)*** 4 (U) 



