262 Prof, Challis on Hijdro dynamics. 



it was proposed to obtain, viz. 



1 d 2 qr _ d* 2 . qr 1 fd 2 . qr d . qr - #r \ ■ 



Now this equation admits of being exactly integrated by sup- 

 posing that qr— cf> l sin (9 + </> 2 sm (? cos ^ an d that <f> l and <£ 2 are 

 functions of r and / only. For on substituting this value of qr 

 in the equation, the result is 



which shows that the equation (rj) is satisfied by the assumed 

 value of qr, if the quantities cf> l and </> 2 be determined by inte- 

 grating the equations 



1 d 2 <£, _d 9 ^ , ?0i =o 4 > /^ 



dp dr< 



1 d 2 <£ 2 ^ 2 $ 2 6<£ 2 



^ 2 rfr« 



+ ^=o W 



Although the integral thus obtained may not be the most 

 general that the equation {rf) admits of, yet if it can be shown 

 to satisfy the conditions of the proposed problem, it must be the 

 particular integral which is alone appropriate to the question. 

 I propose, therefore, to inquire next what is indicated by this 

 integral. 



In the first place it may be remarked that whether qr be sup- 

 posed equal to (/^ sin or to </> 2 sin 6 cos 6, or to the sum of these 

 two quantities, the equation (rj) is satisfied. This analytical 

 circumstance shows that the motions indicated by the two func- 

 tions are independent of each other, and may exist either singly 

 or conjointly. Let us first suppose that qr = cf) 1 smd. The 

 solution thence derived, which satisfies also the equation (J ), has 

 been discussed in the communication entitled <( Researches in 

 Hydrodynamics," contained in the Philosophical Magazine for 

 June 1864. The case there considered is that of a sphere of 

 radius c vibrating in the fluid with the velocity T, and the 

 results obtained are 



The investigation also showed that if the velocity of the sphere 

 be supposed to be impressed on the sphere and the whole of the 

 fluid, in the direction contrary -to that of the sphere's actual 



