550 M. OSTROGRADSKY ON A SINGULAR CASE OF 
and the variation of the volume will consequently become 
daXdYdZ dXdYdZ ,dXdYdZ adXdYdZ 
dzdxdy dzdydzx 
by substituting for X, Y, Z their values a + da, y + dy,z +z, and 
rejecting (on the same principle as they are rejected in the differential 
calculus) all the infinitely small quantities, except those of the lowest 
order, we have as the variation of the volume 
dix doy , doz 
‘dat dy | dz 
or, if we consider the possible displacement alone, the volume can only 
increase or continue unchanged. The foregoing variation must then be 
either zero or positive for all the possible displacements, and must be so 
whatever be the volume under consideration, that is to say, whatever 
be the limits of the integral 
ddx  ddy , dbz ; : 
s(Getaet p=) ae dy dz: 
which cannot be the case unless we have 
déx .ddydibz 
ai dy dz 
for all possible displacements. We might have employed the polar or 
any other coordinates whatever. We might likewise, if it were neces- 
sary, express the invariability of a portion of a mass, &c. 
The geometers who have treated of the equilibrium of fluids in Euler’s 
manner have considered the equilibrium of the differential parallelopi- 
peds also, but the equilibrium might be determined for any portion of 
the volume, whether finite or infinitely small. Let us imagine, in the 
interior of the liquid, any volume at pleasure. The condition of equi- 
librium of this volume must be established in virtue of the moving forces 
applied to it and of the pressures on its surface. If we employ dm to 
represent an element of its mass answering to the coordinates x, y, 2, 
and X, Y, Z to represent the accelerative forces parallel to the coordi- 
nate axes, the moving forces will be Xdm, Ydm, Zdm, and even 
other elements will be acted upon by similar forces. 
This being supposed, let p be the pressure at the surface of the vo- 
lume in question. If ds represents an element of this surface, and 
A, w, v the angles formed with the coordinate axes by the normal to 
ds produced beyond the volume, p ds will be the pressure sustained 
by the element ds, and — pcos.Ads, —pcospds, — p cosy ds 
the components of that pressure. Now, each element of the volume 
S ) de dy de: 
7 =0 
