116 
MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
must be an exact differential, for p must be a function of the co-ordinates ; 
which condition will not be fulfilled unless § be a function of <p. Thus both 
the pressure p and the density § are functions of the same quantity <p, and they 
are both invariable where (p is constant. The density is therefore the same at 
all the points of any level surface. If we conceive a heterogeneous fluid in 
equilibrium to be divided into thin strata by level surfaces infinitely near one 
another, the density will be the same throughout every stratum, but it will 
vary from one stratum to another. 
5. We have now placed before the reader the general points of the theory of 
the equilibrium of fluids. What has been said comprehends all that can be 
determined when a fluid is conceived to extend indefinitely; but in applying 
•the theory to limited masses, it is necessary besides, that the pressures propa- 
gated through the interior parts either be supported or mutually balance one 
another. 
In treating of the equilibrium of fluids, another mode of investigation is 
sometimes employed, which it would be improper to pass by without notice, 
as it is useful on many occasions to fix the imagination, although it leads to no 
new results. We allude to the narrow canals supposed to traverse the mass 
in various ways, of which so much use has been made by Clairaut and other 
authors. 
Let two points (x°, y ° , *°) and {x', y l , z') be assumed in the interior of a 
mass of fluid in equilibrium, and conceive an infinitely narrow canal of any 
figure to pass between them ; we may suppose that the whole fluid, except the 
portion within the canal, becomes solid without any change taking place in 
the position of the particles, or in their mutual action upon one another ; for, 
as this supposition makes no alteration of the forces urging the particles con- 
tained within the canal, these particles will remain at rest after the solidifica- 
tion as they were at first. Suppose that the canal is divided into infinitely 
small parts by sections perpendicular to its sides ; at any point (x, y, z ) let u 
be the section ; Is the infinitely small part of the length of the canal ; dm the 
quantity of matter in the length h s, that is, the product of the volume and the 
density, or g X v X %s; and /"the sum of all the partial forces that urge the 
particles of dm in the direction of the canal ; then, the motive force of dm, or 
its effort to move, will be equal to f dm. Further, p being the hydrostatic 
