703 
MECHANICS. 
pendicular to that furface is as-»2X«**5 confequently the 
whole preffure exerted on ACB, in a direction perpen¬ 
dicular to the furface at every point, will be as mXmx 
•\-nXny+ «X« Z + & C - But, if we confider m, n, o, Sc c. as 
reprefenting weights in proportion to their refpeftive mag¬ 
nitudes-, and the furface AC B to reprefent a proportional 
weight, then »2X mx + nXny + oXoz + Scc.= ACBxGQj 
hence, the whole preffure perpendicular to the iurface 
varies as A C BxGQ. 
Cor. i. The fame preffure is equal to the weight of a 
cylinder of the fame fluid, the area of whofe bottom is 
equal to the furfqce ACB, and altitude G Q. For the 
areas preffed, and the depths of the centres of gravity, 
are equal in the two cafes ; therefore the preffures are 
equal. But the preffure on the bottom of a cylinder is 
equal to the weight of the fluid contained in it; confe¬ 
quently the preffure on the furface ACB is equal to the 
fame weight. 
Cor. z. The preffures of different fluids againft different 
furfaces, will be as the areas X depths of their centres of 
gravity X fpecific gravities of the fluids. For it is ma- 
nifelt that, cater is paribus, the preffures mull be as the 
weights, or as the fpecific gravities. Thus, if the fame 
vefi'el be filled with mercury and water, the fpecific gra¬ 
vity, or weight, of the former, will be 14. times that of 
the latter, and confequently the preffure mult be 14 times 
greater. Hence, for different veffels, combining this ratio 
with that in the Prop, we have the whole preffure as above. 
Cor. 3. The preffure againft the fide of a cubical velfel 
■filled with a fluid 2=the preffure againft the bottom ; for 
the area preffed is the fame, and the depth of the centre of 
gravity in the former cafe — \ that in the latter. Hence, 
the preffure againft the fide =2 £ the weight of the fluid. 
Cor. 4. Let a cylinder, the altitude of which is a, and 
diameter of its bafe d, be filled with a fluid ; then, if 
/= 3-14159, &c. the area of the bafe — \p d 2 , and the 
area of the fides ; hence, The preffute on the bot¬ 
tom : the preffure on the fide :: %pd 2 a : \pda 2 :: d : za. 
If two equal cylinders be filled, one with mercury and 
the other with water, then, The prefi'ure on the bafe of 
the former : the preffure on the lides of the latter :: 
14^ : za :: yd : a. 
Cor. 5. Let xy, fig. 4. be the furface of a fluid, A B C D 
a □ perpendicular to it ; draw E F parallel to A B, and 
bileft AE in v, and E D in w ; then At, Aw, will be the 
depths of the centres of gravities of A B F E and E F C D. 
Hence, the preffures on thefe C3 s are as A B F E x A v : 
EFCDxAit :: AExAt : EDxAa :: AExfAE : 
A D — A EX -- D ^ : A E 2 : AD 2 -*-AE 2 . 
2 , 
Prop. VII. If a vejfel be filled with a fluid, The preffure 
on any part : the whole weight of the fluid :: the area of that 
part X the depth of its centre of gravity : the folid content of 
the fluid .—By Prop. II. the preffure on the bafe of a cy- 
lindei filled with fluid ~ its weight; alfo, the weight of 
the fame fluid is in proportion to its folid content. Let 
a veffel of any form be filled with a fluid, and let A = 
any pait of its lurtace, D 22: the depth of the centre of 
gravity, P= the preffure upon it, W 2= the whole weight 
of the fluid, and S = its folid content; and let a cy¬ 
linder, whole bafe =a and altitude d, be filled with the 
fame fluid, and let p be the prefi'ure on its bottom, w the 
—P\ 
p 
: p : 
AXD 
axd 
w 
W : 
S 
p 
W : 
A X D 
S 
i alfo s=zaxd ; hence 
. - on ns uaic, ue nuea with t 
fluid, and A = the bafe, we have S = fAxD; confe- 
quently the preffure on the bafe 2= three times the weight 
of the fluid. 0 
C°f‘ Z ’r.^ a ! lollow fphere be filled with a fluid, the 
prellure P againft the whole internal furface (A) 22 
tnree times the weight of the fluid ; for the centre oJ 
gravity of the furface is in the centre of the fphere, 
whofe depth D below the upper point of the fluid mult 
therefore be equal to the radius R of the fphere; and 
S = iA X R = iAxD. 
Hitherto we have confidered the whole preffure of a 
fluid againft any furface in a direction perpendicular to 
every point of it; but in this cafe, the effeft on one part 
may partly deftroy the effect on another, by their not aft- 
ing in the fame direction. Since we do not therefore 
thus get the whole joint effeft in any direction, let us 
next confider what is the whole preffure againft any plane 
in the direction of gravity. 
Prop. VIII. The preffure of a fluid downwards againft 
the Jides and bottom of any vejfel , is equal to the weight of the 
whole fluid, provided there be, over every part of the fides and 
bottom, a perpendicular column of the fluid reaching to the fur. 
face.—Cut A w zon B, fig. 5. be fuch a veffel filled with a 
fluid whofe Iurface is AB, perpendicular to which, con¬ 
ceive vmno, q x wz. See. to be indefinitely-fmall prifmatic 
columnsqnto which the whole is divided ; then (Prop. V.) 
the preffure downwards of every column is equal to the 
weight of the column of fluid ; hence, the whole preffure 
downwards is equal to the whole weight of the fluid. 
Cor. 1. As the preffure of every column downwards is 
equal to its gravity, the joint eft-eft of all the columns, 
or of the_ whole fluid, is the fame as the gravity of the 
whole if it had been folid, and confequently the effeft is 
the fame as if all the power was concentrated in the centre 
of gravity. 
Cor. z. IfABCD, fig. 6. be a veffel of fluid, and a cb 
be any part G of the fluid, its aftion downwards muft be 
equal to the re-aftion of the fluid under it upwards ; but 
the effeft of G downwards is juft the fame as if the whole 
effeft took place at its centre of gravity ; therefore the 
effeft of the re-aftion of the fluid under G muft be the 
fame as if it took place at the fame point. 
Prop. IX. If two fluids communicate in a bent tube, their 
perpendicular altitudes above the plane where they meet are in - 
verfely as their fpecific gravities .—Let ABC, fig. 7. be the 
tube, ux the plane where the two fluids meet. Handing 
at v and w, draw muxn parallel to the horizon, and vm, 
wn, perpendicular to it. Let S, s, reprefent the fpecific 
gravities of the two fluids in vu, uw ; then, the area of 
the feftion ux being common to both fluids, the preffure 
of each fluid at that feftion will (Prop. VL Cor. 2.) be as 
its perpendicular depth X its fpecific gravity; but, as the 
fluids are at reft, their preffures muft be equal; hence, 
SXvm — sXwn, therefore S : s :: wn : vm. 
. Cor. 1. Hence, the fame fluid will Hand at the fame al¬ 
titude on each fide; for, if S = s, then wn—vm. If there¬ 
fore a pipe convey a fluid from a refervoir, it can never 
carry it to a place higher than the furface of the fluid in 
the refervoir. 
Cor. z. If ABCD, fig. 8. be a veffel of fluid, mxwn 
a hollow cylinder, to whofe bottom a cylindrical body 
wxyz, of greater fpecific gravity than the fluid, may be 
fo clofely fitted that the fluid cannot enter; then, if this 
body be kept in that pofition by the firing s, and the 
whole be immerfed perpendicularly in the veffel, until y? 
be to jrx as the fpecific gravity of the body to that of the 
fluid, the body will remain iufpended without the aflift- 
ance of the firing. _ For, we may confider the body wxyz 
juft the fame as if it were a fluid of the lame fpecific gra¬ 
vity, and confequently it will reft when the altitudes of 
the body and fluid above y z are in verfely as their fpecific 
gravities. 
Prop. X. The afeent of a body in a fluid of greater fpecific 
gravity than itfelf, arifesfrom the preffure of the fluid upwards- 
againft the under furface of the body.— For, it the body be 
placed upon the bottom of the veffel in which the fluid 
is, and be fo clofely fitted to it that no part of the fluid 
can get under it, it will remain at reft; but, if it be lifted 
up, fo that tlie fluid can get under it, it will immedi¬ 
ately rile. 
Ths. 
