M E C II A N I C S. 
705 
deduced. What is done above is intended chiefly as a 
fpecimen of the method ; but it ferves at the fame time 
to (how the importance of an acquaintance with the fpe- 
cific gravities of different fubftances. We now proceed 
to exhibit the molt ufeful propofitions in this branch of 
our fubje£h 
Prop. XII. A body immerfed in a Jluid will, when left to 
it [elf. Jink, if its fpecific gravity be greater than that of the 
jluid : it will rife to the furface and float there, if its fpecific 
gravity be lefs than that of the fluids but, if the fpecific gra¬ 
vities of the folid and fluid be equal, the body will refl in any 
part wherever it happens to be placed. 
r. For the body endeavours to defeend by its own 
weight, and is fupported by a force equivalent to the 
weight of an equal bulk of fluid, or of as much fluid as 
will fill the fpace occupied by the body. If, therefore, 
the body be heavier than the fluid, bulk for bulk, its 
weight will be greater than the upward preffure of the 
fluid which is to counteract it 5 and, consequently, this 
latter prelfure is not lufficient to prevent the body from 
finking. 
a. If the body be fpecifically lighter than the fluid, its 
preffure downwards will be lefs than the upward preffure 
of the water at the fame depth ; confequently, in this 
cafe alfo the greater force will overcome the lefs, and the 
furplus caufe the body to rife. 
3. When the body and the fluid are of the fame fpecific 
gravity, equal maffes of each are of the fame weight, and 
confequently the force with which the body endeavours 
to defeend, and the force which oppofes the defeent, are 
equal to each other; and, as they act in contrary direc¬ 
tions, the body will reft between them, fo as neither to 
fink by its own weight, nor to afeend by the upward 
prelfure of the fluid. 
Cor. 1. If by any contrivance the fpecific gravity of a 
folid can be fo varied, as to be at one time greater, at 
another lefs, and then equal, to the fpecific gravity of the 
fluid wherein it is immerfed, the body will fink, or rife, 
or remain fufpended, according to the variations of its 
fpecific gravity. This is the cafe in the experiment udth 
thofe little glafs images which fome philofophers exhibit 
in their ledtures, which are made to afeend or defeend, or 
remain fufpended, at pleafure. 
Cor. 2. If a folid fpecifically heavier than a fluid be im¬ 
merfed to a depth which is to its thicknefs as the’fpecific 
gravity of the folid to that of the fluid, and the preffure 
of the fluid from above be removed, the body will be fuf- 
tained by the fluid ; for, the preffure from above being re¬ 
moved, the body is in the fame date, with refpedt to the 
contrary preflure,as though with the fame weight it filled 
the whole fpace to the furface of the fluid 3 that is, as 
though its fpecific gravity and that of the fluid were equal. 
This ferves for the explication of the common experiment 
of making lead fwim, in confequence of being fitted to the 
bottom of a glafs tube. 
Cor. 3. Hence alfo we fee the meaning of the propofition. 
that “ All bodies when immerfed in a fluid lofe the weight 
of an equal bulk of that fluid.” The weight is not other- 
wife lolt than.as it is Curtained by the attion of a contrary 
force. And it therefore becomes obvious, why the weight 
of a bucket of water is not perceived while it is in the 
water; it is not becaufe that weight is defrayed, but be- 
caufe it is Jupported-, not becaufe fluids do not gravitate 
when they are in fluids of the fame fort, but becaufe there 
is a preffure in a contrary direction which is precifely 
equal to their gravity. 
Cor. 4. The weights thus lofl, by immerfing the fame 
body in different fluids, are as the fpecific gravities of the 
fluids. 
Cor. 5. Bodies of equal weight, but different bulk, lofe 
in the (ame fluid weights which are reciprocally as the 
fpecific gravities of the bodies, or directly as their bulks. 
Cor. 6. i he whole weight of a body which will float in 
a. fluid, is equal to the weight of as much of the fluid as 
the immerfed part of the body difplaces. 
Cor. 7. Hence the magnitude of the whole body is to 
Vox,. XIV. No. 1006. 
that of the part immerfed, as the fpecific gravity of the 
fluid to that of the body. And, if the body be any prifrn 
with its bafe kept horizontal, the altitude of the prifin 
will be to the depth immerfed, as the fpecific gravity of the 
fluid to that of the body. 
Cor. 8. And becaufe, when the weight of a body taken 
in a fluid is fubtradled from its weight out of the fluid, 
the difference is the weight of an equal volume of the 
fluid ; this, therefore, is to its weight in the air, as the 
fpecific gravity of the fluid is to that of the body. Con¬ 
fequently, if W be the weight of a body in air, W' its 
weight in water, or any other fluid ; S the fpecific gravity 
of the body, s the fpecific gravity of the fluid, we (hall 
have W—W' : W :: s : S; whence 
W 
S r=—— r7T. s > the fpecific gravity of the body. 
and s= 
W—W' 
W—W' 
W 
S, the fpecific gravity of the fluid. 
1 |/=its fpec. gravity ; 
So that the fpecific gravities of bodies areas their weights in the 
air directly, and their lofs in one and the fame fluid inverfely. 
Cor. 9. Hence, for two bodies connected together, or 
mixed together into one compound of different fpecific 
gravities, we may, fuppofing there is no penetration of di- 
menfions, eafily deduce the neceflary equations. 
Let the refpeftive weights and fpecific gravities be de¬ 
noted thus : 
H=weight of the heavier body in air, 7 s _. f ravit 
H = weight of the fame m water, i f fa / » 
weight of the lighter body in air, 7 S '=its fpec. gravity; 
L'= weight of the lame in water, 5 r & /# 
C= weight of the compound in air, 
C'= weight of the fame in water, 
s — the fpecific gravity of water. Then, 
1 ft. (H—H')S=Hs. 
ed. (L—L') S'=Ls. 
3d. (C— C)f=Cs. 
4th. H + L=C 
s th. H'+L'=C'. 
H L C 
6tl, -+T—7’ 
From which equations any of the above quantities may be 
found in terms of the refl. 
If the body L be of lefs fpecific gravity than water, 
then L' muft be confidered as negative; and to find its 
fpecific gravity we nuift have recourfe to a compound 
mafs, as C ; thus, becaufe from equa. 4 and 5, L—L'= 
Ls 
(C—C') —(H—H') ; and from equa. 2. S'=--— ; con- 
L— L> 
value, w< 
have 
fequently, fubftituting for L — L' its 
S'=———-——rr-r. Or, if we deduce the value 
S/L 
( C—C')—(H—H') 
of S' from the laft equa. we (hall thence find S'=r- „ 
H CS—Hfl 
Prop. XIII. fa vejfel contain two immifcible fluids (fuch as 
water and mercury), and a Jolid of fome intermediate fpecific 
gravity be immerfed under the furface of the lighter jluid and, 
float on the heavier ; the part of the folid immerfed in the heavier 
fluid is to the whole folid as the difference between the fpecific 
gravities of the folid and the lighter jluid to the difference be¬ 
tween the fpecific gravities of the two fluids. —Let the fpecific 
gravity ot the heavier fluid be s, the part of the body im¬ 
merfed in it ; the fpecific gravity of the lighter fluid 
— s', the part of the body immerfed in it=:B'j and let 
the fpecific gravity of the folid body be S. Let.alfo the 
area of the horizontal feftion of the folid coinciding with 
the contiguous furfacesof the two fluids be — A, and its 
perpendicular difiance from the upper furface of the lighter 
fluid be=cf. Then the preffure againft the feftion £%. 
from the lighter fluid will be Ad —B 7 s'; which, added to 
the weight of the folid (B-}-B') S, will give the whole 
force by which this lection of the folid is urged down, 
wards. And the preffure upward againfi the fame febtioa 
is Afl'-J-Bs. But, as the folid is fultained in equilibrio 
by thefe contrary forces, they mult be equal; that is, 
8 I< A& 
