114 MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
pressure. By differentiating the equation of the interior surface, we obtain 
Y±dx-\-Ydy-\-Zdz — 0; 
from which we deduce, by the like reasoning as before, that such surfaces are 
perpendicular to the resultant of the accelerating forces urging the particles 
contained in them. The interior surfaces in question were named level sur- 
faces by Clairaut ; and they are distinguished by the two properties of being 
equally pressed at all their points, and of cutting the resultant of the forces at 
right angles. They spread through the mass, and ultimately coincide with 
those parts of the outer surface which are at liberty. It may be observed, that 
what essentially constitutes a level surface is its equation, which must differ 
from the equation of the outer surface at liberty in no respect, except that the 
constant C — p takes the place of the constant C ; for we shall afterwards find 
that, in some cases of the equilibrium of a fluid, the two properties of being 
equally pressed, and of cutting the resultant of the forces at right angles, belong 
to more sets of interior surfaces than one. 
4. In what goes before, we have supposed that the density is constant, but 
it is easy to extend the investigation to heterogeneous fluids. Let g be put for 
the function of the co-ordinates which expresses the variable density ; then 
admitting that g has the same value at every point of the small elementary 
cylinder or prism, we shall have 
dm — gals; 
but, /being the whole accelerating force, urging every particle of dm in the 
direction of l s, we have 
if , yLv i 7 if. 
J ~ A 8s + Y 8s + Z 8s’ 
wherefore, 
f dm = g a (Kl x Y ly + Z l z). 
The equation expressing that the action of the accelerating forces is equal 
and opposite to the variation of pressure, is the same as before, viz. 
a X Ip f dm — 0 ; 
and by substituting the value of / dm, we deduce 
&/>-|-f(XS.r-l-Y&3/-|-Z&2) = 0. 
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