54 MR. A. B. BASSET ON THE MOTION OF 



v =/-' - VAe-*' - fka ( - \t)<t>(t) + v / +yW^- A ' (1 - |Xi), (39) 



Mr M . . (40) 



These equations determine to a third approximation the values of the acceleration 

 and velocity of the sphere, when it is projected vertically downwards with velocity, 

 V, and allowed to descend under the action of gravity. If the sphere is ascending 

 the sign of g must be reversed. 



If no forces are in action we must put/= 0, and the preceding equations give the 

 values of v and v to a first approximation only; but, on referring to (21) and (22), it 

 will be seen that the values of these quantities to a third approximation may be 

 obtained in this case from (39) and (40) by changing / into V\ and expunging the 

 terms/~ xt and/X" 1 (1 c~ At ). We thus obtain, since X = lev, 



(4 ! 



9. It appears from the preceding equations that the successive terms are multiplied 

 by some power of A; as well as of v. If k is not a very large quantity, and the velocity 

 of the sphere is not very great, the foregoing equations may be expected to give fairly 

 correct results ; but if k is a very large quant : ty, it may happen that, notwithstanding 

 the smallness of v, kv may be so large that some of the terms neglected may be of 

 equal or greater importance than those retained. Now, from (17), k= 9/a(2o-+ p)~ l a~*; 

 if, therefore, the sphere is considerably denser than the liquid, k will be small provided 

 a be not very small ; but if the sphere be considerably less dense than the liquid, k 

 will approximate towards the limit 9a~ z , and this will be very large if a be small, and kv 

 may therefore be large. On the other hand, it should be noticed that when kv or X is 

 large the quantities c~ xl and <f>(t) diminish witli great rapidity, and it is therefore by 

 no means impossible that the formulae may give a fairly accurate representation of the 

 motion even in this case. 



All that we can therefore safely infer is this, that in the case of a sphere ascending 

 or descending in a liquid whose kinematic coefficient of viscosity is small compared 

 with the radius of the sphere (all quantities being of course referred to the same units), 

 the formulae would give approximately correct results, provided the velocity of the 

 sphere were not too great. But, in the case of small bodies descending in a highly 

 viscous liquid, it is possible that the motion represented by the formulae may be very 



