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 all 

 points. Any portion of the liquid volume may then be denoted by the 



expression / dx dy dz, the integral being taken between the proper 

 limits. It will then be necessary to find an expression Avhich will re- 

 present I dx dy dz as, suffering no diminution during any displace- 

 ment that the liquid undergoes. For this purpose, let J x, Sy, S z re- 

 present the projection of the space which the point (x, 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 jjoint would after the displacement (whether positive 

 or not) correspond to the coordinates x + S x, y + S y, z -\- o z, which, 

 for the sake of brevity, we shall represent by X, Y, Z respectively. 



Every other point of the volume / dx dy d z being similarly dis- 

 placed, the whole volume would assume another position in space, and 

 its different points would be determined by the coordinates X, Y, Z, 

 which maybe regarded as functions (of a;, y, 2;) altogether arbitrary. 



The volume / d x dy d z would, in its new position, become 

 / d'S. dY dZ, and consequently undergo the variation / d\ dY dZ 

 — j dx dy dz, which we are now about to develop. 



In order to effect a better comparison of the integrals / d^dY dZ 

 and / d X 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 X, y, z by means of the known formula. For this pur- 

 pose we have 



dXdYdZ= fdXdYdZ _d_XdYdZ dXdYd^ 



\^dx dy dz dx dz dy dy dz dx 



_dX dYdZ dXdYdZ 



dy dx dz dz dx dy 



dXdYdZ} , , - 



— _- r- y dx dy dz, 



dz dy dxj '' 



2p2 



