140 
MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
d . V (r) 
d x 
, ( d . V (r) „ \ , /<2 . V (r) ,, \ 1 
dx -\r~dj- +Jy) d y~ +•/*) dz = °> 
( 8 ) 
in which the co-efficients of the differentials express the whole forces urging 
the molecule in the directions of x,y, z. 
It is obvious that the equations (7) and (8) must be identical ; for they are 
both true at every point of the same surface ab c. But if the co-efficients of 
the differentials of these two equations be identical, the like co-efficients in the 
equation (5) must be separately equal to zero ; and this proves that the co- 
ordinates of the surface ab c do not enter into the equation (6), which there- 
fore contains such quantities only as remain invariably the same at all the 
points of that surface. 
The equations (7) and (8) being identical, the latter will belong indifferently 
to all the similar surfaces in the interior parts, and to the outer surface which is 
their limit. Wherefore, if for the sake of distinction we suppose that r becomes 
R at the upper surface, we shall obtain the equation of that surface by inte- 
grating, viz. 
C = V(R ) + £(/+ z2 )- (9) 
The integral of (8) will likewise give the equation of any of the interior sur- 
faces, as a b c, viz. 
p = V (»•) + 4 (</ 2 + * 2 ) - c, (10) 
the quantity C being absolutely constant in all circumstances, and the same as 
in the equation of the upper surface, and/? being a new quantity which is con- 
stant when the co-ordinates are taken in the surface ab c, but varies when 
the co-ordinates belong to any point of the mass not contained in that surface. 
At the upper surface /? vanishes ; it changes its value in passing from one of 
the interior surfaces to another ; and it is evidently the hydrostatic pressure at 
every point of the mass, because — are equal to the co-effi- 
cients of the differentials in equation (8) and to the accelerating forces which 
oppose and destroy the variation of pressure. The equations (6) and (9) and 
(10) at which we have arrived by this new train of reasoning are the very same 
with the equations (4) and (1) and (3) of the first investigation; and as the 
