28 



HYDROSTATICS. 



a 2 d x 1.57079, and that on the bottom 

 half as much, or ad 2 x. 78539. 



If ABC D (fig. 28.) be a vertical 

 section of the vessel, and A MM" a 

 parabola, the ordinates P M, P M ', 

 P"M", will measure the relations of 

 the pressure upon the whole vertical 

 surface above the points P, P', P", 

 respectively. 



jig. 28. 



5. As the pressure on any particle 

 of fluid at rest is equal to the weight 

 of a column of that fluid whose base is 

 the particle, and whose altitude is the 

 depth of the particle below the surface ; 

 if we call those particles p,p',p". &c., 

 and the depths d, d', d", &c. respectively, 

 the pressure upon the whole of any 

 vertical or oblique lamina of the fluid 

 on each side of it, will be a weight of the 

 fluid equal to p d-rp' d'+p" d"+ , 

 &c., and this is also the pressure with 

 which the lamina next to the vertical 

 side weighs upon that side. But if D 

 be the distance of the centre of .gravity 

 of all the particles from the horizontal 

 surface of the liquid, it is a property of 

 this point that D x (p+p'+p"+ , &c.) is 

 equal to p d + p' d' + p" d" +, 

 &c. ; and as p + p' + p" + , &c. is the 

 area of the plane or surface formed by 

 all the p, p', p", &c. therefore the 

 pressure upon the whole area, whether 

 vertical, oblique, or horizontal of any 

 surface under water, is equal to D 

 times that area, or the area multiplied 

 by the distance of its centre of gravity, 

 below the plane of the fluids' surface. 



6. It follows from hence that the 

 pressure upon the sides, whose breadth 

 is b, of a rectangular vessel, as a cube 

 or paralellopiped, at different depths 



D, D 1 , D" is as b x D x -^ , b x D' 



x | , b x D" x ~ ; or as D*, D'*, 



D" 2 ; that is, as the squares of the 

 depths. Hence, if a conic parabola be 

 described, whose axis is the upper edge 

 of the vessel A B, (fig. 29.) the pressure 

 upon the whole of the side A T, above 



P, will be as the ordinate P M ; upon 

 fig. 29. 



the portion above P', P'M' ; upon the 

 portion above P", P"M". 



7. If a body, whose specific gravity 

 is , be plunged in two fluids of diffe- 

 rent specific gravities, G and G, the 

 heavier being G', it will float in equi- 

 librio, if the part P in the lighter be to 

 the part P' in the heavier, as G'S' is 

 to G S. For the weight of the 

 whole body is 5 x (P + P'), the weight 

 of the fluid displaced by P is GxP', 

 and that displaced by P' is G'P' ; and 

 because the body floats, and its weight 

 is suspended, G x P + G' x P'=S x 

 (P+P'); therefore Px (G S) = P' 

 x (G'-S) and P : P':: G'S '. G-S. 

 It follows also that P:PxP'::GS: 

 G' G, or the part in the lighter is 

 to the whole body as the difference 

 between the specific gravities of the 

 solid and the heavier fluid to the diffe- 

 rence between the specific gravities of 

 the two fluids. When that difference 

 is very great, G may be neglected as 

 evanescent in the last term of the pro- 

 portion ; or the part of the body out 

 of water is to the whole body, as the 

 difference between the specific gravities 

 of the body and water to the specific 

 gravity of water ; or (because P' '. P+ 

 P' ; : S : G) the part plunged is to the 

 whole as the specific gravity of the 

 solid to that of the fluid ; and, in like 

 manner, neglecting G in the former 

 proportion, and multiplying the two 

 last terms by - 1, P : P' : : S- G' : S ; 

 or the part out of water is to the part 

 plunged, as the difference of the spe- 

 cific gravities of the solid and fluid 

 to the specific gravity of the solid. 



8. The centre of pressure of any tri- 

 angle ABC (fig. 30.) placed vertically at 

 the depth C W, below the surface W R 

 of the water is found thus. Let A S = 



SB, andjGS- \ C S ; G will be the 



centre of gravity of A B C ; and G M 

 being perpendicular to W R, the pres- 

 sure on A B C will be equal to a co- 



