320 ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA. 



will still remain satisfied. Moreover, if a/, ?/, z are now referred 

 to three axes fixed in space, we shall have 



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

 referred to the fixed origin, we shall have 



j\dt (9). 



= - \\dt 



Adding now to <f> the term \x due to the additional ve- 

 locity, the expression (3) will then become 



X^' 



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

 means of the differentials of this last function. But <j> and its 

 differentials evidently vanish at an infinite distance from the 

 solid, where /= co; and consequently, the case now under con- 

 sideration 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^ at any 

 point of the fluid mass. But, by supposing this mass free from 

 all extraneous action, F= 0, and if the excursions of the solid 

 are always exceedingly small, compared with its dimensions, 

 the last term of the second member of the equation (1) may 

 evidently be neglected, and thus we shall have, without sensible 

 error, 



j) d(p . u(p 



or, by substitution from the last value of </>, 



Having thus ascertained all the circumstances of the fluid's 

 motion, let us now calculate its total action upon the moving 



