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therefore, the filament EI alone keeps in equilibrium all the other filaments of the mass AGHB; consequently, the mass GCDH being still supposed without weight, there will not result any other pressure on the bottom CD than that of a single filament EI, which, being transmitted equally to all the points of CD, will make the pressure upon CD to that upon the base I of the fila ment El as the area of CD to the area I. If therefore, we imagine (fig. 11. pl. 87.) a heavy fluid contained in ACB to be divided into horizontal lamina, the upper lamina will communicate to the bottom CD no other action than would be communicated by the single filament a b: and the same thing obtaining with respect to each lamina, the bottom, therefore, is pressed in the same degree as it would be by the combined ope. ration of the filaments a b, bc, ed, &c. Whence, as this pressure is transmitted equally to all the points of CD, it will be equal to the product of CD into the sum of the pressures which the filaments a b, bc, ed, are capable of exercising on the same point, or it will be proportional to CDX (a bxb cxc dx&c.)
Cor. 1. Hence, if the fluid contained in the vessel ABDC is homogeneous, the pressure on the bottom CD will be expressed by CDXEC; and will be measured by the weight of the prism or cylinder whose base is CD and height EC.
Cor. 2. Hence also, when the heights are equal, the pressures (of the same fluid) are as the bases: when the bases are equal, the pressures are as the heights: when both heights and bases are equal, the pressures on the horizontal bottoms are equal in all, however irregular the shape and different the capacities of the vessels may be.
Cor. 3. In different vessels containing different fluids, the pressures are as the areas of the bottomx depths specific gravities.
Cor. 4. If the laminæ AH, GK, &c. be of different densities, or specific gravities, D, d, 3, &c. then will the pressure on the hot tom CD be equal to CD × (a b. Dxbc.dx cd .:X&c.).
Scholium. Upon the two principles that fluids press equally in all directions, and in proportion to their perpendicular depths, depends the explanation of the circumstance know by the title of the hydrostatic paradox, which is this: any quantity of water or other fluid, how small soever, may be made to balance and support any quantity or any weight, however great: a circumstance which has been converted to a useful purpose in the construction of some chines (See BRAMAH'S MACHINE). A wellknown contrivance to illustrate this principle is the hydrostatic bellows. It consists of two thick boards EF, CD (fig. 10, pl. 87.), about sixteen or eighteen inches diameter, covered or connected firmly with pliable leather round the edges, to open and shut
common bellows, but without valves;
but there is a pipe AB about three feet high fixed into the bellows at B. Now let water be poured into the pipe at A, and it will run into the bellows, gradually separating the boards by raising the upper one. Then, if several weights (three hundred weights, for instance) be laid upon the upper board, the water being poured in at the pipe till it be full, will sustain all the weights, though the water in the pipe should not weigh a quarter of a pound. For the narrrower the pipe the better (beyond certain limits), provided we make it long enough, the proportion being always this:
As the area of the orifice or section of the pipe, To the area of the bellows boards, FE: So is the weight of water in the pipe, AG, To the weight it will sustain on the board. For the fluid at B, the bottom of the tube, is pressed with a force varying as its altitude AB: and this pressure is communicated horizontally to all the particles in the space FE, and then distributed equally throughout the fluid in the bellows: consequently, the pressure upwards at FE is equal to the weight of a cylinder of fluid whose base is FE and altitude AB; while the actual weight of water borne up is only that of the cylinder, whose base is FE and height BG; and hence no weights laid upon CD that do not exceed the weight of a cylinder, of the fluid whose base is EF and altitude AG, will disturb the equilibrium.
Prop. If two immisceable fluids are included in a bent tube, and balance cach other, their perpendicular altitudes, estimat ed from a horizontal plane drawn through the common surface where they are in contact, will be reciprocally as their specific gravities.
Let ABCD (fig. 1. pl. 88.) be such a bent tube, its form and dimensions being arbitrary; and let the common surface of the two immisceable fluids be GH; one fluid occupying the space EFHG, the other the space GHBCKI. Let the specific gravity of the fluid in EFHG be s, that of the other S. Through the surface GH draw the horizontal plane GHLOM, then it is manifest that the part GHBCML is naturally in equilibrio: in order, therefore, that the equilibrium may exist in the whole, the pressures exerted upon GH by the fluids contained in EFHG, IKML, must be equal. Now (prop. 2. cor. 3.) the former of these pressures is denoted by GHXFHX8, and the latter by GHNÓ×S. Consequently GHXFHX8=GH×NO×S, or FHXs=NÖ ×S; whence flows the proposition, i. e. NO::S: 8.
Scholium. Before we commence the investigation of the pressure of fluids on oblique and curvilinear surfaces, we may just remark, with respect to pressures upon the horizontal bottoms of vessels, that it is necessary to distinguish between the pressure which the plane CD (fig. 8. pl. 87.) would sustain as arising from the fluid, and that
which it would have to sustain if it carried the vessel. If the bottom CD were detached from the vessel, in order to prevent the escape of the water there, the bottom CD must be pressed upwards with a force equal to the weight of the cylinder CDEF of the fluid: but if we would support the vessel, it will require a force equal to the weight both of the vessel and the fluid it contains. Thus, when the vessel is narrowest at bottom, it will require more force to support the vessel than to keep its bottom from falling: while, if the vessel is widest at bottom, it may be supported with a less effort than would be necessary to prevent the bottom from separating from the sides of the vessel. But the pressure of the fluid on the bottom of an upright prismatic vessel is equal to its weight.
Prop. Any plane surface immersed in a heavy fluid, of which the upper surface is horizontal, is perpendicularly pressed with a force equal to the weight of a column of that fluid, having the surface pressed for its base, and the depth of its centre of gravity under the surface of the fluid for its altitude.
Let ABCD (fig. 12. 13. pl. 87.) be a vertical section of a vessel terminated by surfaces either plane or curved, and any way inclined to the horizon; and let the vessel be filled with a fluid whose upper surface intersects the section ABCD in the horizontal line AB. If GHhg be an indefinitely thin lamina of the fluid, we may consider it, abstractly from its weight, and then conceive this lamina as pressed by the superior Now this pressure is distributed equally through all the particles of the lamina, and acts perpendicularly and equally upon all the points of the faces G g, Hh: bence, because this force is the same as would be occasioned by the filament El alone, the pressure which is exerted per pendicularly upon Gg will be expressed by GgxEI: and the same will manifestly obtain, if, instead of Gn as an evanescent right line, we consider it as an evanescent surface. Therefore, in general, the pressure which is exerted perpendicularly upon any evanescent surface, by a heavy homogene. ous fluid, is estimated by the continual product of that surface, its distance from the horizontal surface, and the specific gravity of the fluid.
by the continual product of the surface pressed, the distance of its centre of gravity from the upper surface, and the specific gravity of the fluid; which is the proposition in other words.
Cor. 1. The entire lateral pressure of a vessel whose sides are perpendicular to the base is equal to the weight of the fluid contained in a rectangular prism, whose altitude is that of the fluid, and base is a parallelogram, one side of which is equal to the altitude of the fluid, and the other to the semiperimeter of the vessel.
Cor. 2. The pressure against one side of a cubical vessel filled with a fluid is equal to half the pressure against the bottom. And the whole pressure against the sides and bottom is equal to three times the weight of the fluid in the vessel.
Cor. 3. If ABCD, CDEF (fig. 2. pl. 88.) are two rectangles whose common breadth is CD, standing vertically in a fluid, whose upper surface is SS', then will the pressures upon the rectangles ABCD and CDEF be as AC2 and AE2-AÇ2.
For if G and g be the respective centres of gravity of the two rectangles, we shall have pressure upon ABCD: pressure upon CDEF:: ABCD×IG: CDEF×Ig:: AC×}} AC: CE× (AC×¿CE) :: AC×¦ÃC : (AEAC) ×}(AE×AC) :: AC2 : AÊ2—AC2.
Cor. 4. Hence, if AE be to AC as✅ 2 to 1, the pressures upon ABCD and CDEF will be equal.
Def. The centre of pressure is that point of a surface against which any fluid presses, through which the resultant of all the individual pressures passes, or to which, if a force equal to the whole pressure were applied in a contrary direction, it would keep the surface at rest.
Prop. If a plane surface which is pressed by a fluid be produced to the horizontal surface of it, and their common intersection be made the axis of suspension, the centre of percussion will be the centre of pressure.
Let ABCD (fig. 3. pl. 88.) be the horizontal surface of the fluid which presses upon the plane EIF: produce this plane till it meets the surface of the fluid in the line MN; and let O be the centre of pressure. From any point p of the surface pressed draw the vertical pm, meeting the hori zontal surface in m; and in the plane CB draw from m the line m M perpendicular to Hence it will follow that the total pres- MN. The pressure upon p is as p.pm, and sure exerted upon any plane surface what its effect to turn the plane about MN is as ever, whether vertical or oblique, is equal p.pm.p M, by the nature of the lever: to the product of the specific gravity into also, its effect to turn the plane about NI is the sum of the products of the evanescent as p.pm. MN. In like manner, if the plane parts of this surface into their respective EIF be supposed to revolve about the axis distances from the upper surface of the MN, and to strike an obstacle at O, the flaid: but, by the nature of the centre of percussive force of the particle p, by which gravity, the sum of these latter products is it endeavours to move the plane about MN, equal to the product of the whole surface will be as p. pm, or as p.pm.p M; and its into the distance of its centre of gravity force to turn the plane about NI will be as from the horizontal surface of the fluid: so p.p M. MN. or asp.pm. MN. And the like that the whole pressure will be denoted correspondence between the percussive and
the pressive forces, of any other particles in 2 (d3—a3)
the plane EF, may be shown in the same man- 3 (d2—a2); or, when the variable d be
ner. Consequently, the percussive forces of the whole of the particles, whereby they endeavour to move the plane in the two directions, have the same relation as the forces of pressure, and therefore the centres of pressure and percussion are coincident.
Cor. 1. Hence, the theorems given for the centre of percussion may be applied to the determination of the centre of pres
Cor. 2. Hence also appears the mistake of those who assert that the centres of percussion and of pressure do not coincide. They are the centres of oscillation and of pressure which do not coincide universally.
Scholium. To adopt the general formulæ for the centre of percussion to the instance of the centre of pressure, it will be proper to make a slight change in the notation. Let d be the distance from MN of any particle, or of any horizontal lamina of the fluid in contact with the plane EIF; let / be the length of such lamina, and d its depth (being considered as evanescent), then will ld be its area; also, let & be the distance of the centre of gravity of the plane EIF from the line MN, where that plane intersects the surface; and let the horizontal distance of Id from the line NI be denoted by h: then, with respect to the line MN the
formula will become l d2 à fl d2 å Sia a EIFX83
comes equal to e we have
2 (e3-a3) 3 (e2—a2)' distance of the centre of pressure from the horizontal surface of the fluid. When one of the extremities of the line a coincides with this surface we have a=0, and e=x, and the distance of the centre of pressure becomes x.
II. If upon the vertical line a we construct a rectangle, of which the horizontal base is h, the whole pressure upon it will be shλ, and the distance of the centre of pressure from the surface of the fluid will be 2 (e3—a3) 3(e2-a)
the same as we have just found.
And this centre must evidently be found upon the vertical line which divides the parallelogram into two equal parts.
If the upper horizontal side of the parallelogram coincides with the surface of the ABCD, fig. 2. pl. 88. its tendency to turn fluid (as the side AB of the parallelogram about its base will be sh× 3x= ± 8 hṛ3, and its tendency to turn about one of its vertical sides will be 8 h x2 × h = ± 8 h2x2 ; thus the first of these efforts will be to the second as 8h 23:48 h 2x2, or as 2x: 3h; which reduces to 2: 3, when the rectangle
becomes a squere.
III. To determine the centre of pressure in the triangle CAB (fig. 4.) whose side AB is horizontal, and which is placed vertically in a fluid whose horizontal surface is SS. and with respect to the line NI, the draw the vertical line SCP, also the line Through C, the summit of the triangle,
See CENTRE of PERCUSSION.
A few examples are here added to illustrate the use of these theorems.
I. Let a reservoir which contains water, or any other fluid (its specific gravity being s), have one of its sides plane and vertical: if we imagine a right line drawn vertically
CQ bisecting the base AB, and any line TR parallel to AB. Make CP=λ, AB=h, CS=a, SP=a+λ=e, the distance of the horizontal line in which lies the centre of gravity of the triangle from S= (that is, if Cg CQ, the distance between SS' and g=3), the angle _PCQ=k. SM=d, TR=l, CM=c=d-a.
The whole pressure upon this triangle And to find the depth of the centre of pres will be represented by 8× × 8—1 s ô x h. sure below SS' we must find the fluent of the
upon this plane, its length being, and the expression de d, orf_ld
to this we have CP: AB:: CM: TR, or a:
distance of its centre of gravity from the
'1 d2 d, gives ƒ (d3—a d3)
· (‡d1—ž a d 3) + C. The constant
(1 being constant) which is ing that the fluent must vanish at the point
quantity C may be determined by consider
C, that is, when =0, or when d-a-0,
Hence, making the the expression for the centre of pressure's di
a and the correct fluent is
‡ do — ƒ a d3 + 1⁄2 a4). Hence, then, for the
stance from SP
is at the surface of SS' of the fluid, a 0, 3. Finally, when the vertex of the triangle e=,=, and the expression becomes tang. k; which, for the right-angled triangle, reduces to h.
4. When the triangle has its vertex at the horizontal surface of the fluid, the tendency to turn about the base is to that to turn about the perpendicular let fall from the vertex upon the base as 1 to 3 tang. k; and in the case of the right-angled triangle, as 2 to 3 h: which, when the legs of the triangle are equal, reduces to the ratio of 2 to 3, as in the case of the rectangle and
If C and AB lie in different sides of SS', that is, if part of the triangle is out of the fluid, no other change will be necessary in the preceding expressions than a change of signs in those terms which contain uneven powers of a. So that this simple transformation will accommodate the preceding general theorems to the case of trapezoids.
5. If the radius of a circle be r, and 8 the distance of its centre below the surface of
the fluid in the plane of the circle, then is the distance of the centre of pressure from the upper surface, in the same plane, ex
Prop. To inquire generally into the results of all the pressures, upon any surface, plane or curved, regular or irregular, both in the vertical and the horizontal direc
This is usually performed by foreign authors by means of the calculus of partial differences. But another mode of investigation is pursued here, as we think it likely to carry more conviction to the mind of a learner. Had the vertical pressure alone been the object of investigation, it might be determined far more concisely.
1. Whatever the figure of a body may be, we may always imagine it to consist of an assemblage of an infinite number of indefinitely small laminæ respectively parallel, and the surface of each lamina as an assemblage of many trapezoids, their number indeed being infinite likewise, when the surface in contact with the fluid is curved. Hence, to estimate the result of the pressures, of a fluid, whether upon the interior
surface of a vessel which contains it, or upon the exterior surface of a body immersed in it, we must first estimate the result of the pressure upon the surface of a trapezoid whose height is evanescent.
Conceive, therefore, a b c d (fig. 5. pl. 88.) a trapezoid whose two parallel sides are a b, ed, and whose height HK is infinitely small with regard to those sides. To resist the pressure upon this surface we must apply to the centre of gravity g of the trapezoid a force P perpendicularly to its plane, the value of which is expressed by the product of the surface of the trapezoid into the distance Gg of its centre of gravity from the horizontal surface ABCD of the fluid.
To determine the effect of this force P both in the vertical and the horizontal direction, conceive a vertical plane c d FE to pass through the line ed, and a horizontal plane a b EF through the line ab; the common intersection of these planes being EF: then, having drawn the vertical lines c E, d F, meeting the horizontal plane in E and F, join E and a F: again, through the direc tion P g of the force P, conceive a plane KIH perpendicular to ed, and of which Hg Kand HI are the intersections with the two planes a bed and FE ed: this plane will be perpendicular to the planes a bed and FE c d, because ed is their common intersection: finally, from the point K, where a b and Hg meet, draw KI perpendicular to the plane FE ed, this line must necessarily be perpendicular to HI.
The construction effected, resolve the force P (represented by g N) into two others which are also in the plane KIH, of which the horizontal one is g L, and the vertical one g M. Calling these component forces L and M, we have, by the nature of the parallelogram of forces P: L: M::g N: g Lig M::gNg L:LN:: HK: HI: IK, the triangles g LN, HIK, being evidently similar. Multiplying the three latter terms a b+c d
by a by
× G g, which will not change the
ratio, we shall have P : L ab+c d
a b c d
a b+c d
× Gg: HIx
xG g. 1st. That HK ×·
×G g: IK ×
Now it may be observed,
is the surface of the
have c d=FE, therefore IK+" ab+ EF
=IK which of consequence, is the
surface of the trapezoid a b EF. 3dly. That, because the height of the trapezoid a b c d is evanescent with respect to the sides a b and ed EF, which is equal to c d, may be taken
surface of the rectangle E c d F. We have, therefore, P: L:M: :abed+Gg: E c d F Gg: a FEbGg. But we have supposed that the force P is expressed by a b c d×6 g; consequently, the horizontal force L is denoted by Ec dFxG g, and the vertical force M by a FE b×G g.
As the triangle may be considered as a trapezoid, of which one of the parallel sides vanishes, the same thing, therefore, obtains for any evanescent triangle.
Conceiving, now, that from the angles a, d, e, b, lines are drawn to fall perpendicularly upon the plane ABCD, these perpendiculars will be the edges of a prismatic frustrum, of which the horizontal base is equal to a FE b, and the inclined base a b cd; or, as a banded are supposed indefinitely near, the solidity of the prismatic frustrum will not differ sensibly from that of the prism which has the same horizontal base, and whose height is Gg: but this latter is equal to a FEb×Gg which is precisely the expression above found for the vertical force M. Hence it appears, that this force is equal to the weight of a prismatic frustrum of the fluid whose inclined base is abcd, and horizontal base the projection of a bed upon the horizontal surface ABCD.
H. Let us next consider any solid whatever cut into an indefinite number of hori zontal laminæ, such as ABDE a b d e (fig. 6. pl. 88.), and that perpendicularly to the centre of gravity of the surface of each trapezoid into which the contour of the la mine is divided, forces are applied, each represented by the product of the surface of the corresponding trapezoid into the distance of its centre of gravity from the horizontal surface A'D'. These forces are the pressures of a heavy fluid, sustained by the interior surface of the lamina ABDE a b d e of a vessel which contains it: they are also the pressures of such a fluid which would be sustained by the exterior surface of a solid whose contour is the same, and which is immersed to the same depth. But it is manifest that if each into two others, the one vertical, the other of those forces P, P' P", &c. were decomposed horizontal, each vertical force would be represented by the weight of a prismatic frus of the trapezoids in the contour of the lamina, trum of the fluid whose inclined base is one and its horizontal base the projection of that trapezoid upon the upper surface of the fluid. Therefore the sum of these vertical forces, or the single vertical resulting force, will be represented by the sum of the weights of all those prismatic frustrums: and the same property may obviously be extended to every other horizontal lamina, we may conclude,
1. That, if a vessel, of any figure, be