EQUILIBRIUM OF INCOMPRESSIBLE FLUIDS. 549 
directly expressed that any portion (whether finite or infinitely small) 
of the liquid mass cannot be diminished. It is undoubtedly true that, 
as any volume may be supposed to consist of differential elements, the 
incompressibility of these elements involves that of the volume as a 
necessary consequence. But it would still be desirable to see how the 
calculus would directly express the incompressibility of any portion of 
the liquid volume. 
In order to show this, let x, y, z represent the coordinates of a point 
of the liquid, which, because of their variability, will belong to alk 
points. Any portion of the liquid volume may then be denoted by the 
expression Ef dx dy dz, the integral being taken between the proper 
limits. It will then be necessary to find an expression which will re- 
present Sf dxdydz as suffering no diminution during any displace- 
ment that the liquid undergoes. For this purpose, let da, dy, dz re- 
present the projection of the space which the point (a, y, z) should 
have traversed in consequence of a displacement either actually made 
or only supposed in the liquid, on the coordinate axes x, y, z respect- 
ively. The point would after the displacement (whether positive 
or not) correspond to the coordinates 2 + dx, y +4 y, z+ 22, which, 
for the sake of brevity, we shall represent by X, Y, Z respectively. 
Every other point of the volume { dx dy dz being similarly dis- 
e 
placed, the whole volume would assume another position in space, and 
its different points would be determined by the coordinates X, Y, Z, 
which may be regarded as functions (of x, y, 2) altogether arbitrary. 
The volume dx dy dz would, in its new position, become 
f dX dY dZ, and consequently undergo the variation aXdYdZ 
_ f dx dy dz, which we are now about to develop. 
In order to effect a better comparison of the integrals dXdYdZ 
and hf dx dy dz with one another, we must reduce them to the same 
variables and the same limits. This will be done by transforming 
X, Y, Z into 2, y, z by means of the known formule. For this pur- 
pose we have 
Red Wiad: 7ie— os dYdZ dXdYdZ dXdYdZ 
dady. de dz dzdy) dydede 
_dXdYdZ , dXdYdZ 
dy dx dz dzdxdy 
_ AXdYdZ 
dz dy my i ucclnb 
bo 
a=] 
bo 
