:J38 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE 



an exact equation, which gives the pressure at the distance r at any 

 time. The two terms will be of the same order if m be not very 

 small compared to 2br. 



Next, let us try the effect of retaining one of the omitted terms 

 of the equation (7) and neglecting the others. Retaining, first, the 



term -£ x „ , „ , and putting v for -^ , we shall have, 

 dr" adr 2 l ° dr 



dv t v~\ 2v _ 



dr \ a 2 ) r 



dv vdv 2 



or — —j- + - = 0. 



vdr a'dr r 



Integrating with respect to r, 



v* 

 Nap. log. vr ! - -3 = Nap. log./(0 ; 



.-. e*» =f(t) A = /'(/) (1 + £ + &c.) 



Hence neglecting terms removed in order by two degrees from those 



fit) 

 retained, v = ^—^ • This is the same result as in the case of the in- 

 r 



compressible fluid, and by reasoning in the same manner as for that 



case, it will be found from equation (6) that 



o«Nap. log. P =-^-£- **(/). 



If p = 1 + o-, and we assume that the value of a depends only on 

 a disturbance in the direction of r, it will follow that F' (t) = 0, v and 

 ,f'(t) being supposed to vanish when p — 1. 



Hence P = e* r ™ = 1 +«-^ ; , nearly, 



J 2 fit) V* , 



and (i"cr =' / —^-L nearly. 



r 2 J 



It appears, therefore, by the foregoing reasoning, that whenever 



— J, is of the same order as t> 2 , the first term introduced into the 

 r 



expression for the pressure by the term of equation (7) which has now 

 been considered, is of the order of v\ 



