of the Motion of Fluids. 9 



r Acad. des Scien. Ann. 1818), also conducts to a particular 

 form of the arbitrary functions, when treated according to the 

 same principles : and from the discussion it appears that <p has 



the same value as would be obtained by integrating -~- + 

 + \ = 0, on the supposition that <p is a function of 



V x z 4- y* -f- z 2 or r, and t ; in which case the velocity is di- 

 rected to or from a centre, and varies inversely as the square 

 of the distance from the centre. The instances in which 

 udx + vdy + wdz, or d<j> 9 is known not to be a complete dif- 

 ferential of a function of x,y,z 9 confirm the view here taken. 

 When a mass of incompressible fluid revolves uniformly, with- 

 out changing form, about a fixed axis, d<p is not an exact dif- 

 ferential ; and it is plain that the motions of elementary por- 

 tions of the fluid are not directed to fixed or moveable cen- 

 tres, because the whole mass moves in such a manner that it 

 may be considered solid. Again, Euler has shown that d$ is 

 not an exact differential, when a mass of incompressible fluid 

 revolves round a fixed axis, and the velocity is any function 

 of the distance from the axis, and yet that the general equa- 

 tions are satisfied by this case of motion. We may in this 

 instance conceive the fluid to be divided into portions, in- 

 cluded between cylindrical surfaces indefinitely near each 

 other, having in common the axis of revolution: the motion 

 of each portion will be the same as if it were solid. No part 

 of the motion will be such as is directed to a fixed or move- 

 able centre, that is, the motion will not be that which pecu- 

 liarly belongs to fluids. 



The principle I have endeavoured to establish, will only in 

 particular cases facilitate the solution of hydrodynamical pro- 

 blems, on account of the difficulty of ascertaining the centres 

 to or from which the motion is directed. The following ex- 

 ample, selected for illustration, may perhaps be interesting; 

 because no principles, that I am aware of, have hitherto been 

 advanced by which such a problem could be solved. 



Water contained in a conical vessel, the axis of which is 

 vertical, descends with its upper surface always horizontal : 

 It is required to find the velocity and direction of the velocity 

 at any given point in the interior of the fluid mass at a given 

 time, when the velocity of the descending surface is given. 



In this example, all the particles must be moving in vertical 

 planes passing through the axis of the cone : hence the centres 

 to which the motion is directed, must be situated in the axis. 



The integral of -^- + ^~ + ~ - = 0, supposing <J> to be 

 N. S. Vol. 9. No. 49. Jan. 1831. C a function 



