124 
MR. IVORY ON THE EQUILIBRIUM OF FLUIDS, 
unless the pressures propagated inward, which increase as any point sinks 
deeper below the surface, mutually compensate and destroy one another. 
Some further discussion is therefore necessary in order to prove that the equi- 
librium is completely established. 
The function <p, in which we may suppose there is no constant quantity, can 
contain no term having the coordinates for divisors ; for, were this the case, 
the pressure would be infinite at all those points where such coordinates are 
equal to zero. Let the terms of <p be arranged in homogeneous expressions of 
one, two, three, &c. dimensions ; then 
p=(A l x + A 2 y + A 3 z) 
+ (B 3 x 2 + B 2 y 2 -f B 3 * 2 + B 4 xy + B 5 x z + B & y z) 
-f (D 4 x 3 -{- D 2 y 3 -f- D 3 z 3 + D 4 x 2 y -f- &c.) 
+ &c. 
Differentiate this expression, and after the operations put x = 0, y — 0, 
% — 0 : then 
— a ^ = A 
dx D dy 2 ’ dz 3 ’ 
But the differentials of <p are no other than the expressions of the accelerating 
forces acting on a particle ; consequently A ls A 2 , A 3 are the forces in action at 
the origin of the coordinates, that is, at the centre of gravity of the mass. 
Wherefore, according to what was observed, we shall have 
Aj — 0, A 2 — 0, A 3 = 0, 
<t> = (Bj x 2 + B 2 y 2 + B 3 z 2 + V>±xy + B 5 xz + B 6 yz) 
+ (Dj x? + D 2 y 3 + D 3 2 3 + D 4 x 2 y + &c.) 
4- &c. 
That the expression of <p must be of this form is required by the nature of the 
problem : for <p must be always positive, and it must increase continually from 
the centre of gravity to the surface of the fluid. 
Let us now put 
x = r cos 0 = r 
y = r sin 0 cos \|/ = r ■/), 
z — r sin 0 sin 4 = r Z., 
