340 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE 



4Tr-'/«V iirat , . , ... j , , • e ., , „ m~- r* 



- „ — cos , which, with regard to «- is of the order of — - x — , 



XX a X 



that is, of the fourth order; whilst the first term in the expression for 



-r- is with regard to a of the order of — x — , that is, of the second 

 dt fo a X 



order. Hence we may conclude as before that the first term introduced 



into the expression for the pressure by the term of equation (7) just 



considered, is of the fourth order. 



4. Lastly, let us retain the term ; . -, ™ of equation (7), and 



reject the other small terms. We shall then have, 



#£ 1 cP$ 2 d±_ Q . Qr d\r<f> d\r<p 



The known integral of this equation is 



r<p =f(r - at) + F(r + at). 



The second arbitrary function applies to a disturbance which causes 

 propagation towards the centre; and as such a motion is excluded by 

 the nature of the question to the solution of which the present reasoning 

 is directed, I shall suppose this function to vanish. Then, 



f(r - at) 

 <*> = - r • 



d< t> - „ f'( r - at ) 

 dt r 



d£ _ fir -at) f(r - at) 

 dr ' r r 2 



As an application of this solution, let us suppose the velocity impressed 

 at any time t in the direction of r, and at the distance r from the centre, 

 to be m(p(t). Then, putting for shortness' sake u for f{r — at), we 

 shall have 



1 du u 

 r ar dt r 



du a .,-.■« 



or —. — h - . u 4- marcb (t) = 0, 

 at r 



