430 



HYDRODYNAMICS. 



Sptciiic 

 Gravities. 



PLATE 

 CCCXIIt. 

 Fig. 15. 



p;e. i*. 



] iart of the solid immersed is made up of these elementa- 

 , ry columns, it follows that the sum of all the pressures 

 exerted upon E n F is equal to a quantity of fluid of 

 the same size as the immersed part, which is the same 

 as the quantity of fluid displaced. 



When the body EF is wholly immersed, as in Fig. 

 15, it is obvious, that any part o is pressed down- 

 wards with a column of fluid whose height is mo, while 

 any part is pressed upwards with a column of fluid 

 whose height is mn; consequently the point n is pressed 

 upwards with a column noinn mo. But the sum 

 of all the elementary columns no, make up a quantity 

 of fluid equal to that which is displaced by the body. 



Cor. When a solid floats on a fluid, the quantity of 

 fluid which it displaces is equal to the xv eight of the bo- 

 dy. Since the whole weight of the solid pressing npon 

 the surface of water E n F is in equilibrio with the fluid 

 mass, it must be equal in weight to the quantity of fluid 

 E n F, which is also in equilibrium with the same fluid 

 mass, but this quantity of fluid is the quantity which 

 is displaced. 



PROP. II. 



When a body floats upon a fluid, the centre of gravi- 

 ty of the body and of the fluid displaced are in the same 

 vertical line. 



For since the upward pressure which supports the 

 floating body is the same as if it were applied to the 

 centre of gravity of the part immersed, or of the quan- 

 tity of fluid displaced, then since the whole floating bo- 

 dy is in equilibrio, its centre of gravity must be sup- 

 ported by this upward pressure ; that is, the centres of 

 gravity of the fluid displaced and of the floating body 

 must be in the same vertical line. 



PROP. III. 



The specific gravity of any floating body is to that of 

 the fluid, as the volume of the part immersed is to the 

 whole volume of the body. 



Calling S the specific gravity of the fluid, and s that 

 of the solid, we have by Cor. Prop. I. S X E n F = 

 s X E p F n, and therefore * : S=E n F : E p F n ; that 

 is, as the part immersed is to the whole volume of the 

 body. 



PROP. IV. 



If a solid is weighed in a fluid, it will lose as much 

 of its weight as is equal to the quantity of fluid dis- 

 placed. 



It appears from Prop. I. that the body is pressed up- 

 wards with a force equal to the weight of the fluid dis- 

 placed ; and as this force acts in opposition to the na- 

 tural gravity or absolute weight of the body, its abso- 

 lute weight must be diminished by a quantity equal to 

 the weight of the fluid displaced. The weight which 

 the body in this case loses is not destroyed, but is sus- 

 tained by an equal and opposite force. 



If we call * the specific gravity of the solid, S that of 

 the fluid, B the bulk of the solid, and m B the part of 

 it which is immersed ; then since B x * is the absolute 

 weight of the solid, and m B x S the absolute weight of 

 the quantity of fluid displaced, in order that an equili- 

 brium may take place, we must have B x S=; B X S, 

 and S : S=m B : B. Hence if s=S, we have m B=B ; 

 that is, tf the specific gravity of the solid is equal to that 



of the fluid, the part immersed is equal to lite tvhole body ; Specific 

 or, in other words, the solid will be completely immersed, Gravities. 

 and mill remain wherever it is placed. If s'^'S, then ""Y"* 1 

 m BT^B ; that is, when the spccifio gravity of the solid 

 is greater than that of the fluid, the body will sink to the 

 bottom : and if S~^-s, then mB^.E ; that is, n> ken the 

 specific gravity of the fluid is greater than that of the 

 solid, then t he part immersed is less' than that of the whole 

 solid, or the body will float. 



PROP. V. 



If a body is held beneath the surface of a fluid, the 

 force with which it will ascend, if it is lighter than the 

 fluid, or with which it will descend if it is heavier, is 

 equal to the difference between its own weight and the 

 weight of an equal quantity of the fluid. 



The body held beneath the water obviously descends 

 with its own weight =Bx, while it is pressed up- 

 wards with the weight of the quantity of fluid displaced 

 =B x S ; consequently the force with which it ascends 

 must be B x S B x*, and the force with which it de- 

 scends = B x * B X S, which are the differences between 

 the weight of the body and the weight of the fluid dis- 

 placed. 



SCHOLIUM. 



On the truth contained in this proposition is found- 

 ed the construction of the Camel for raising sunk ves- 

 sels, or for lifting ships over high sand banks. (See 

 our article CAMEL. ) A similar effect is exhibited in 

 some of the American rivers, where the ice is formed 

 upon the stones at their bottom. Ice is specifically 

 lighter than water, and therefore, when it accumulates 

 to a certain degree round the stones, the upward pres- 

 sure upon the stones exceeds their pressure downwards, 

 and they are brought to the surface, having been some- 

 times torn up with great force. Huge masses of stones 

 appear in many cases to have been floated by the ice 

 adhering to them, and carried to a great distance from 

 the place of their formation. 



PROP. VI. 



The specific gravity of a solid is to that of the fluid 

 in which it is weighed, as the absolute weight of the 

 solid is to the loss of weight which it sustains. 



In the equation B X s=m B X S, we have B=mB when 

 the body is weighed in a fluid, and of course wholly 

 immersed ; consequently if W be the weight of the bo- 

 dy in the fluid, or the weight necessary to keep it in 

 equilibrio with the fluid, then Bx* = BxS+W, 

 (and transposing and multiplying by s,) we have 

 s X B X * W=s X B X S, and (Euclid, Book VI. 16.) 

 s : S=B X s : B X s W; consequently since B x * W 

 is the loss of weight which it sustains, the specific gra- 

 vity of the solid is to that of the fluid, as the weight of 

 the solid is to its loss of weight. 



This Proposition may also be demonstrated, by consi- 

 dering that the weight lost, or B x * W, is the weight 

 of a bulk of fluid equal to the bulk of the solid, whose 

 weight is B x s ; and therefore as the specific gravities 

 are to one another, by the definition, as the weight of 

 equal bulks, we have * : S=B x * : B X * W. 



PROP. VII. 



If the same solid body is weighed in two fluids, the 

 specific gravities of the fluids are to one another as the 



