[ 274 ] 



XXXVII. A Remark on an Article of M. Poisson's Traite 

 de Mecaniqiie {Edition 2nd. No. 593.). Bj/ James Ivory, 

 K.H.,RR.S.,^c.* 



TN speculations of difficulty it is of great importance to note 

 -*■ such points as are susceptible of clear demonstration. 

 What is thus established by undoubted evidence, is not liable 

 to be misapprehended or inadvertently misapplied. In this 

 view it may be useful to demonstrate the following theorem, 

 relating to the equilibrium of incompressible fluids, the par- 

 ticles of which are urged by accelerating forces : If an in- 

 terior level surface be extended through tJie mass, the hody of 

 fluid isoithin the level surface 'will be in equilibrium independently 

 of the rest of the mass, and supposing the incumbent fuid ivere 

 removed. 



In order to demonstrate this theorem, suppose a canal 

 to be conducted from an orifice in the upper surface of the 

 fluid to the central point within all the level surfaces: the 

 pressure of this canal at the centre, caused by all the forces 

 which urge its elementary portions in the direction of the canal, 

 and estimated on the unit of surface, will be the same, what- 

 ever be the position of the initial point in the upper surface : 

 the symbol B may be used to denote the intensity of the 

 pressure at the centre resulting from this canal, which is no 

 other than the central column of Newton. In like manner if 

 a similar canal be drawn to the centre from any orifice in an 

 interior level surface, the intensity of pressure at the centre, 

 represented by b, will be a constant quantity. The intensity 

 of the exterior pressure at all the points of the level surface, 

 caused by all the forces that urge the particles of the incum- 

 bent fluid, will be equal to B— ^. Using the same symbols 

 as M. Poisson (Edition 2nd, No. 583), a;y, z will represent 

 the three rectangular coordinates of a particle of the fluid ; 

 X, Y, Z, the accelerating forces acting on the particle in the 

 directions of a:,y,z; and, as unit may stand for the density 

 of an incompressible fluid, we shall have 



B-b =f{Kdx + Ydy + Zdz) (a.) 



the integral representing the intensity with which any canal, 

 having one orifice in the upper surface of the fluid and the 

 other in the level surface, presses upon the level surface. 

 By differentiating the equation, supposing the coordinates to 

 vary in the level surface, we obtain 



Xdx + Ydy + Zdz = {b.) 



* Communicated by the Author. 



