HOMOGENEOUS FLUID AT LIBERTY. 493 



continue to be equal ? Thus it may be possible that the assumed principle may be 

 fulfilled in the same body of fluid under different forms. 



The difficulties which must be overcome before this subject can be freed from in- 

 accurate and insufficient reasoning, have occurred in problems relating to fluids of 

 uniform density ; and for this reason homogeneous fluids are alone treated of in what 

 follows. 



]. Suppose that ABC represents a mass of homogeneous fluid entirely at liberty, 

 the particles of which are urged by accelerating forces ; let 

 all the forces which act upon any element of the mass, as 

 dm, be reduced to the directions of three rectangular co- 

 ordinates x,y, %', and put X, Y, Z for the sums of the par- 

 tial forces respectively parallel to x, y, %. Now, if A a be 

 an infinitely slender prism of the fluid parallel to .r, passing 

 completely through the mass, and divided in its whole length 

 into elementary portions, it is obviously a condition necessary to the equilibrium of 

 the body of fluid, that the forces X, acting upon all the elements of A a, mutually 

 destroy one another. 



What has been enunciated of a prism parallel to x, must hold equally of prisms 

 parallel to y and z. 



Any element dm may be conceived as formed by the intersection of three slender 

 prisms parallel to x,y,%\ and, as the pressures in the whole extent of each prism 

 balance another, the element will be at rest, having no tendency to move parallel to 

 X, or to y, or to z. But no proof is required to s'how that an elementary portion of 

 a fluid in equilibrium must be pressed equally on all sides. 



The forces which act upon the elements at the ends of any prism, A a, passing com- 

 pletely through the mass parallel to x, are necessarily directed inward, and have op- 

 posite directions ; wherefore the force X, in varying through the whole length of A a, 

 must first decrease, then become equal to zero, and afterwards changing its sign, 

 increase in approaching the other surface of the fluid. Thus, in every slender prism 

 parallel to x, there is a point at which the force X is equal to zero ; and if the whole 

 body of fluid be divided into such prisms, all the zero points will form a continuous 

 surface stretching completely through the mass. In like manner there will be two 

 other internal surfaces containing all the points at which the forces Y and Z are 

 evanescent. The intersection of the three surfaces will determine a point G within 

 the body of fluid at which all the three forces X, Y, Z, vanish, and which may be 

 called the centre of the mass in equilibrium. 



In considering the equilibrium of a mass of fluid entirely at liberty, it is obvious 

 that we may abstract from any motion common to all the particles, and from any 

 forces acting upon them all with equal intensity in the same direction. The forces 

 that must be balanced and rendered ineffective to produce motion, are such only as 

 tend to change the relative position of the particles with respect to one another ; 



MDCCCXXXIV. 3 s 



