50 MR. A. P. BASSET ON THE MOTION OF 



then (16) becomes 



(18) 



This is the equation of motion of the sphere, from which F (t) or v must be 

 determined. 



7. Up to the present time we have supposed the motion to have commenced from 

 rest, so that F (0) = 0. Let us now suppose that the sphere was initially projected 

 with velocity V. In order to obtain the equation of motion in this case we may 

 divide the time, t, into two intervals, h and t h, where h is a very small quantity, 

 which ultimately vanishes. During the first interval* let the sphere move from rest 

 under the action of gravity and a very large constant force, which is equal to 

 (M + ^M')X, and then let the large force cease to act. This force must be such as to 

 produce a velocity, V, at the end of the interval, h, whence we must have V = XA, 

 v = Xt ; and, therefore, v = "Vt/h. Changing f into f-\- X in (18), multiplying by 

 c w , and integrating between the limits t and 0, we obtain 



= - ha \/ - f du PV" F'(tt - T) ~ + fxe^M + / f f^du. (19) 



V TJ-JO JQ vT JQ Jy 



Now F'(t) is composed of two parts : a large part which depends upon X, and which 

 is equal to V/A ; and another part which depends uponf, and which we shall continue 

 to denote by F'(l). Hence (19) may be written 





= * (e" - 1) -H {(e - 1) - ka A/ - f du [>( - T) 



X X V V 7TJ JQ 



u}du, . . (20) 



where 



Vdr 



Now x ( u ) depends on X, and therefore vanishes when u > h. When u < h, 



x (u) = 2V tt ,Yi ; 



therefore 



ft f* 2V 



^"X (") d u ~T u^'du = 0, when h = 0. 

 Jo Jo' 1 



Hence, in the limit when h vanishes, (20) becomes 



v = Vc-" + {(1 - -") - ka A/- f du re-* ( '>F (u - T) -J- , . (21) 



" V 7T JQ Jo V T 



* The following procedure, suggested in a Report upon this paper, has been substituted for the 

 remainder of this uectiou as originally written. 



