AND THE FIGURE OF A HOMOGENEOUS PLANET. 
117 
pressure at the point (x, y, z), the like pressure at the distance of S s will be 
p -f Ip ; therefore the opposing pressures which act upon the two ends of the 
part of the canal in the length h s, will be p X a and (p + hp) X a ; and 
hp x a will be the effective pressure which pushes d rn towards the point 
(x, y, z). Because every part of the canal is supposed at rest, the tendencies 
of d m to move in opposite directions must be equal, and we shall have this 
equation, 
}>p X u -f- f dm = 0 ; 
consequently, 
* , fdm 
S?+ =°; 
and by taking the sum of the similar quantities in all the parts of the canal, we 
obtain 
/ Sp+ //^=°. 
Butp being a function of three independent variables, the sum of its variations, 
supposing the flowing quantities to follow any arbitrary law of increase or de- 
crease, is equal to the difference of p' and p°, the final and initial values of the 
function ; wherefore we have 
p' -p° +f-~r = <>■ 
Now /dm, that is the quantity of matter multiplied by the accelerating force, 
is the impulse or pressure in the direction of the canal caused by all the forces 
urging d m ; and as this pressure is exerted on the surface u, is the same 
pressure reduced to the unit of surface. Therefore, whatever be the figure of 
the canal, it follows from the foregoing investigation, that the difference of the 
pressures at its two extremities is equal to the sum of the impulses of all the 
contained molecules of fluid, every impulse being reduced to the direction of 
the canal and to the unit of surface. 
If the extremities of the canal be both in the parts of the outer surface which 
are at liberty, the pressures p' and p° will be both evanescent, and there will 
be no effort of the fluid either way, and no tendency to run out at one end. 
Further, if a canal be continued through the fluid till it return into itself, the 
