OF FT.OATING BODIES. — OF SPECIFIC, CRAVITIES. 



60 



their heights above the common surface are 

 inversely as their specific gravities. 



For if the tube be equable, and its arms similar, the actual 

 weights above the common surface will be equal ; and if 

 otherwise, the efficient weights will be equal, since, in 

 either fluid, the pressures on the common surEa(^ are simply 

 as the heights. 



SECTION II. or FLOATING BODIES. 



374. Theorem. If an}' body floats on 

 a fluid, it displaces a quantity of the fluid 

 equal to itself in weight. 



- For since the body is supposed to remain at rest, and to 

 retain the pressure of the fluid below it in equilibrium, it 

 must exert by its weight a pressure downwards, equal to 

 that of the quantity of fluid which would retain the same 

 pressure in equilibrium, or to the quantity displaced. 



375. Theorem. When the centre of 

 gravity of the floating body is in the same 

 vertical hne with the centre of gravity of the 

 fluid displaced, the body remains in equili- 

 brium. .:• - • 



If a uniform fluid, of the same specific gravity as the fluid, 

 occupied the place of the portion removed, it would remain 

 at rest, in consequence of the contrary actions of the fluid 

 and of gravity. Now the efTect of any forces on the motion 

 of the centre of gravity of a compound body, is the same as 

 if they were applied to the same mass placed in the centre 

 ^f gravity ; therefore since the direction of gravitation is 

 .•vertical, the. result of the combined pressures of the fluid 

 which counteract it, would, if united at the centre of gra- 

 vity, be also vertical: and if the actual centre of gravity of 

 the body of equal wdight be placed in this line, there will 

 be an equilibrium; but if otherwise, the centre of gravity 

 will descend towards this line, and a part of the immersed 

 portion will in the mean time be somewhat raised by the 

 • pressure of the fluid. 



376. Theorem. If a floating body have 

 its section, made by the surface of the fluid, a 

 |iarallelogram, its equilibrium will be stable 

 or tottering, accordingly as the height of its 

 centre of gravity, above that of the portion of 

 the fluid displaced, is smaller or greater than 



one twelfth of the cube of the breadth, divid- 

 ed by the area of the transverse vertical sec- 

 tion of the immersed part. 



Let the body be inclin- 

 ed in a small degree from 

 the position of equilibrium 

 ABC, into the position 

 DEF; then the triangles 

 GHI and KHL will be 

 equal, since the area of the 

 section immersed must 

 remain constant, and GK 

 and IL will ultimately bi- 

 sect each other in H. Now the centre of gravity of thu 

 section ILF, is the common centre of gravity of its parts 

 IHMF and LHM, making K,M=GI ; but N the centre of 

 gravity of IHMF is in the line HF bisecting it, and the com- 

 mon centre of gravity may be found by making NO parallel 

 to HKortoHL,in the same ratio to the distance of the centre 

 of gravity of LHM from H that LHM bears to IFL. Now 

 the distance of the centre of gravity of any triangle from the 

 vertex is two thirds of the line which bisects the base (277) ; 

 that is in this case |HK, and the area of the triangle LHM 

 is *1fK.KP, therefore NO : SHK :: HK.KP : LFI, NO=: 



^HKo KP 



•2 — ; but drawing Oa vertically through O, NO : 



NO :: KP :. HK and ^^^^2^^mS^=^.'}f^. 

 KP IFL '' 11-L 



If therefore the centre of gravity be in Q, the body will 



remain in its position in any small inclination; since the 



result of the pressure of the fluid acts in the direction OQ, 



if the centre of gravity be below Q, it will descend towards 



the line €lO, and the body will recover its situation; if 



above Q, the body will overset. . Hence the point Q is 



sometimes called the centre of pressure, or the mctaccntrc. 



The theorem may be easily accommodated to bodies of 



other forms. 



SECTION III. OF SIT.CIFIC GRAVITIE'S. 



377. Theorem. If a body is immersed 

 in a fluid, it loses as much qf its weight as is 

 equivalent to an equal bulk of the fluid. 



For if the body were of the same specific gravity with 

 the fluid, it would remain at rest, without any tendency to 

 ascend or to descend,' the pressure of the fluid counteracting 

 its whole weight: butthat pressure will belhesamewhatever 



