336 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE 



2. From the equations (1), (2), (3), others more immediately appli- 

 cable to the question proposed to be discussed will now be deduced. 



The equation (1) is equivalent to 

 dp dv vdp 2v 



W; 



pdt dr pdr r 



and by substituting for p in (3) from (2) there results, 



a' dp dv vdv 

 Jt 



;d7 + ( rt + -dT = ° < 5) - 



If now we assume <p' to be such a function of r and t that the 

 partial differential coefficient -T~ is equal to v, and substitute this ex- 

 pression for v in (5), the equation is integrable with respect to r. The 

 result is, 



* Nap. log. P + *£ + ^ =f(t). 



To get rid of the arbitrary function of the time, suppose 



$'=<!> + ff(t)dt. 



Then ^ = ^4-/(0, and^' = ^t Hence 

 dt at ° v ' dr dr 



" n "p- •* * + ft + ^ - ° < 6 > 



Obtaining from this equation — j and — p- , substituting their values in 



(4), and putting ~ for v, the result will be, 



d*<p l d<p \ 1 d'(p 2 d<p d 2 <j> 2^ = , 



dr 2 \ (fdr*) a* dt 1 a? dr drdt r dr 



3. Before making use of this equation it will be necessary to 

 consider the comparative values of its terms under the circumstances 

 in which we propose to apply it. The circumstances are, that v is 

 very small compared to a, and r always exceedingly small compared 

 to the breadths of the waves whose dynamical action is to be inves- 

 tigated. 



