in Fluid Media. 59 



and if X' represents the co-ordinate of the centre of the ellipsoid referred 

 to the fixed origin, we shall have 



:<=-/ 



\dt (9.) 



Adding now to (f> the term — X oc due to the additional velocity, the ex- 

 pression (3.) will then become 



<P = ^fi 



df_ 



3 bc 



CC 



and the velocities of any point of the fluid will be given, by means of the 

 differentials of this last function. But and its differentials evidently va- 

 nish at an infinite distance from the solid, where f= go ; and consequently, 

 the case now under consideration is that of an indefinitely extended fluid, 

 of which the exterior limits are at rest, whilst the parts in the vicinity of 

 the moving body are agitated by its motions. 



It will now be requisite to determine the pressure p at any point of the 

 fluid mass. But, by supposing this mass free from all extraneous action 

 V = 0, and if the excursions of the solid are always exceedingly small, com- 

 pared with its dimensions, the last term of the second member of the equa- 

 tion (1.) may evidently be neglected, and thus we shall have, without sen- 

 sible error, 



p dd> dd> 



p dt r r dt 



or, by substitution from the last value of <p, 



dtx Pdf 



* dt r J a 3 be 



co 



Having thus ascertained all the circumstances of the fluid's motion, let 

 us now calculate its total action upon the moving solid. Then, the pressure 

 upon any point on its surface will be had by making f= in the last ex- 

 pression, and is 



& dt r J -^r; 



a*bc 



H 2 



