of a homogeneous fluid mass that revolves upon an axis. 109 
worth while to set before the reader very briefly the steps 
of Euler’s investigation, for the purpose of pointing out the 
omission with which it is chargeable. 
Suppose that the fluid mass in equilibrio is divided into in- 
definitely small rectangular parallelopipeds by planes parallel 
to those of the co-ordinates ; and let x, y, z, be the co-ordi- 
nates of one angle of a parallelopiped which has dx, dy, dz, 
for its sides, and which we may conceive to be so placed, 
that x -|- dx, y -f- dy, z -J- dz, are the co-ordinates of the 
opposite angle. The forces that act upon the parallelopiped 
are; the pressure of the adjacent fluid upon its six faces; 
and the accelerating forces X, Y, Z, urging every particle in 
directions parallel to x, y, z, and tending to increase these 
lines. The pressure at any point of the fluid must depend 
upon the situation of that point, or it must be some function 
q> of the co-ordinates x,y, Z: and, according to the principles 
of the differential calculus, <p will retain the same value over 
all the three faces of the parallelopiped that comprehend any 
one of the solid angles. Now, y and z remaining constant, 
if we substitute x + d x in place of x, <p will be changed into 
dx; and the two quantities <p and <p ~ x dx, will 
represent the intensities of pressure upon the opposite rec- 
tangles comprehended by dy and d z : the forces compres- 
sing the parallelopiped are therefore < py.dy dz, and (q> -{- dx) 
dy dz; and the difference, or ^ dx dy dz is the force causing 
the parallelopiped to move in a direction tending to diminish 
x. In like manner, the pressures on the other sides produce 
the forces j- dxdydz and ~ dxdydz, causing the parallelo- 
