132 
MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
to be destroyed, which will produce its full effect according to the figure and 
quantity of the attracting matter and the situation of the attracted point. The 
equilibrium will therefore be absolutely impossible, unless such a figure can be 
induced on the mass of fluid as will set free every particle in the surface ab c 
from the attraction of the stratum. If such a figure can be found, every mole- 
cule of the mass will be urged by the principal forces only; because a surface 
such as ab c, at every point of which these forces alone will be in action, may 
be described through any interior molecule a m arbitrarily assumed. We must 
therefore turn our attention to investigate such figures, if there be any, as will 
make the irregular attraction in the interior parts disappear, so as to leave the 
principal forces alone in action ; for, unless this can be effected, the fluid can- 
not maintain a permanent form. 
According to the notation we have used, if r denote the distance of am from 
G, V (r) will represent the sum of the quotients of all the molecules of the 
whole mass divided by their respective distances from a m ; let V' (r) denote 
the same thing, relatively to the interior mass ab c, that V ( r ) does, relatively 
to the whole mass ABC; then V (/•) — V' (r) will denote the sum of the quo- 
tients of all the molecules of the stratum divided by their respective distances 
from am. Take a point (x + dx, y -|- dy, z + dz) in the surface ab c infi- 
nitely near a m ; and, differentiating in the surface, the expressions, 
d.(Y{r) - V'(r)) d. (V (r) — V' (r)) d. (V (r) - V' (r)) 
dx ’ dy ’ d z 
will be equal to the attractive forces of the stratum upon the particles of am, 
in the respective directions of x, y, z: but, as we have shown, the equilibrium 
indispensably requires that these attractions be evanescent, so that we have 
these equations, 
’.(V(r)-V'(r)) d.(\[r) - (r)) d . (V (r) - V' (r)Y 
— 7 = 0, — j = 0, — - = 0, 
dx ’ dy ’ " ^ 
d z 
* The perpendicularity to the surface ab c, of the attraction of the stratum upon am, is expressed by 
this equation, 
<f.(V(r)- V'(r)) d.(V(r)-V'(r)) d . (V (r) - V' (r)) 
dx dy v dz ’ 
and it is a consequence of the differential equations in the text. The neglect of this consideration, and 
the assumption that the level surfaces depend solely upon the outer surface in every case, is the great 
blemish of Clairaut’s theory. 
