110 
MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
and other liquids, a very great external force must be applied to produce an 
almost imperceptible variation of bulk ; while in others, such as air and the 
gases, very notable changes of volume are caused by moderate compression. 
In the investigation of the properties of the first sort of fluids, to which our 
attention is here exclusively directed, we shall throw out of view the very small 
degree of compressibility they possess, and shall suppose them to retain the 
same bulk whatever changes of figure or pressure they may undergo. 
In a fluid in equilibrium, the action of the accelerating forces that urge the 
particles must be counterbalanced by the pressure propagated through the 
mass : to find the relation between these opposite forces must therefore be the 
first object of research. 
2 . Assuming three planes intersecting at right angles which, by the co-ordi- 
nates drawn to them, ascertain the position of the particles of the fluid, we 
shall suppose two points or particles {x, y, 2) and (x $x, y + ly, z + S 2) 
at the infinitely small distance ci 5 from one another ; and we shall put &> for 
the small base of an upright cylinder or prism of the fluid placed between 
the two points, and having ci s for its length : then the density of the fluid being 
invariable and represented by unit, and the quantity of matter of the cylinder 
or prism being denoted by d m, we shall have 
dm = & x 
Let all the accelerating forces which act upon the particle (x,y, z) be 
reduced to the directions of the coordinates ; and put X, Y, Z for the sums of 
the reduced forces respectively parallel to x,y,z ; then because || are 
the cosines of the angles which the line S s makes with x, y , z, the partial forces 
urging the particle in the direction of Ss, will be X , Y gp, Z g -, and, if 
we put 
/=x^ + y!? + z 8 -* 
J Ac 1 OS 1 * c 
is' 
the whole accelerating force urging the particle (x, y , %) in the direction of l s, 
will be equal to f. Multiply now by the equal quantities dm and uls, and 
the result will be 
f dm u x -\-Yly -\- Zlz). 
