264.

PI. XV,Fig. 2.

Hydrostatics.

X. The Weight, Pressure , or EffeB ofa Fluid upon the Bottom DE of any Vessel

ACDEF,

I d, &c. equal to those Altitudes respectively, and sup -pose the same Thins; done for every other Point in theLine AK, it is evident the Triangle A « K will be asthe whole Pressure on the Line A K. But the saidTriangle is equal to AK X j e K ; and supposing AG=rGH=HI=IK, and c f drawn parallel to AK ; weshall have | e K=/'K=rH=CH ; therefore AKXCH,will be as the total Pressure on the Line AK.

13. If AK were the Section of a Plane, then theSurface of that Plane multiplied by CH, will be the Ex-pression for all the Pressure on that Plane, provided CHbe the Depth of the Centre of Gravity H, from theSurface of Water. But to give a more general Theo-rem of the above Rule, lets, b, be two Weights hang-ing from an horizontal Plane, at the Distances ac, bd\and join their Centres by the Line s b, and let x betheir common Centre of Gravity, and x 0 its Distancefrom the Plane c d, perpendicular to which draw ay andbz ; then since a ; b :: b x : a x, by the Property of thecommon Centre of Gravity ; and by similar Triangleswe have b x : a x :: x z ; xy. Therefore a X x y — b Xx z ; but xy — x 0—y 0 — xo — ac ; and x z — z 0 —-

x 0 —bd — xo-, therefore aX xo — ac — b X b d — xo-,that is, a X a c -f- b X b d — a -j- b X x 0. That is,in Words, The Product of the Weights multiplied by theirDijlanccs from the Plane is equal to their Sum multipliedby the Distance of their common Centre of Gravity fromthe Plane.

14. Now that this holds true in Lines and Planes isevident, because the indefinitely small Particles of Linesand Planes may be considcr’d as very small Weights,and as what has been demonstrated of two, holdsequally for all, therefore the above Rule is applicableto all Sorts of Surfaces, or the Pressure upon the Bot-toms of Vessels, however posited or figured, may beexactly computed thereby.

15. Thus suppose A B C D E F represent a Vessel ofa Prismatic Form, whose Bottom is an oblique trian-gular