HYDRODYNAMICS. 



431 



Pscdfe losses of weight which the solids respectively sustain 

 *>" in each. 

 Y - 



Making B the bulk of the body as before, S, S' the 

 snecitic gravities of the two fluids, and W, nr the weights 

 of the solids in each fluid, then the weight of the quan- 

 tities of fluid displaced will be BxS, BxS', and since 

 these weights are the weights lost by the body, if we 

 add the weights of the body in the fluid, viz. W, te, to 

 these weights, we shall have an expression which will 

 be equal to B X i, the real weight of the body. Thus : 



BxS+W = BX*andBx-S'+w=BX'- 

 Hence we have the two equations, 



B X S=B x * W and B x S'=B X * ; hence 

 ;S'xB=Bx W:BxS' w, conse- 



quently, (End. B. V. l6.)S:S'=Bx* W:Bx* * 

 that is, the specific gravities, or as the losses of weight 

 attained by the solid, for these losses must always be 

 equal to the difference between the real weights and 

 the weights W, *> in the fluid. 



Cor. Hence, if two solid bodies lose equal parts of 

 their weights in the same fluid, they have equal vo- 

 lumes. 



PROP. VIII. 



If a solid body is immersed in two fluids of different 

 specific gravities, so as to be partly in the one and part- 

 ly in the other, it will be in equilibrio, if the part in the 

 lighter fluid is to the part in the heavier fluid, as the 

 difference between the specific gravities of the solid and 

 the heavier fluid is to the difference of the 

 vities of the solid and the lighter fluid. 



Let EF, Fig. 16, be the solid immersed in two fluids, 

 and having the part M in the lighter fluid, whose pe- 

 cine gravity is S, and the part Nin the heavier fluid, 

 whose specific gravity is S', and let * be the specific 

 gravity of the solid. Now the weight of the solid is 

 X M + N. the weight of the heavier fluid displaced by 

 N is S' x N, and the weight of the lighter fluid displaced 

 by M is S X M. But as the solid is, by the hypothesis, 

 suspended in the fluids, the whole of its weight is lost ; 

 and consequently the part lost in the lighter fluid, ad- 

 ded to the part lost in the heavier fluid, most be equal to 

 its whole weight, that is, SxTf-f ff~xTT= i X "STpJ ; 

 then, by transposition, and Euclid, B. VI. 16, we have 

 M x S^i=N X S'i and M : S'=S' * : S . 



l. Since M N ' = j : S , then, by inver- 



sion and composition, EucL B. V. Prop. B and 18, 



M : M + N ' - N _ , . > _s. that it, tkcpart in tkeligklcr 



Jbud it to M kdt ttU a, Ilit difference bttmtn Ike 



mtfifc gnrtXut of tke tolid and ike keavitr jfimU it If 



Ut difference fcftswm the tpectfc eravilie* of Uu Iteo 



i ,. ' 



Cor.*. If the specifc gravity S of the lighter fluid 

 is very small when compared to S', as in the case of 



air and water, then we may, for ordinary purpose?, take 

 the analogy M -,:*. 



Paor. IX. PHOB. 

 To detect the adulteration of the precious metals. 



Let us suppose, as in the caw of Hiero's crown, that 

 a mass of pure gold is adulterated by the admixture of 

 silver. If we take a "quantity of pure gold of the same 

 weight as the adulterated mass, it will obviously have less 

 bulk, as its specific gravity is greater than that of the 

 mixture, (silver having a las* specific gravity than gold) 

 and therefore the Quantity of pure gold, when weighed 

 in water, will displace lew of the fluid than the adulte- 



rated mass. Hence it follows, that we hare only to 

 weigh the susjiected mass, and a mass of pure gold of 

 the same weight ; nnd if there is any difference in their 

 weight, we must conclude that the mass is adulterated. 

 If the gold is heavier in water tlian the suspected 

 mass, it lias obviously lost less weight, and has there- 

 fore lew bulk than the mass ; consequently the adul- 

 terating mixture has a less specific gravity than gold. 

 If, on the contrary, the gold loses more weight than 

 the mass, it will have a greater bulk, and therefore the 

 adulterating metal must nave a higher specific gravity 

 than gold, such as platinum. 



PROF. X. 



If two substances of any kind be compounded toge- 

 ther, the bulk of the heavier of the two ingredients is 

 to the bulk of the lighter ingredient as the difference 

 between the specific gravities of the compound and the 

 lighter ingredient is to the difference between the spe- 

 cific gravities of the compound and the heavier ingre- 

 dient. 



Calling S, S' the specific gravities of the ingredients, 

 B, B' their bulks, and the specific gravity or the com- 

 pound, then the weight of the compound is x B-f-B', 

 and as the weight of the compound must be equal to 

 the sum of the weights of its ingredients, we have 

 zU+lB'=BX-f>H'xV. and by transposition, &x. 

 we have B X S S = B' X S' Z and (Eud. VI. 16.) 

 B:B=.>- 



ScilOUVH. 



The supposition in the preceding reasoning, that 

 the bulk of the compound is equal to the sum of 

 the bulks of its ingredients, is not physically true. 

 A pint of alcohol or of sulphuric acid, mixed with a 

 pint of water, will not make so much as two pints of 

 the compound fluid ; and, on the other hand, a cubical 

 inch of tin, mixed with a cubical inch of lead, will 

 make a compound containing more than two cubical 

 inches of metal. A certain bulk of water is diminished 

 by the addition of ^ of sal ammoniac ; and 40 parts 

 of platinum and 5 of iron will make but 39 parts by 

 measure. 



PROP. XI. PROS. 



To determine accurately the specific gravity of ga- 

 or aeriform bodies. 



As the specific gravities of gaseous bodies are mea- 

 sured in relation te that of air, we must first determine 

 the weight of a given volume of this gas. In order to 

 do this, take a large glass vessel, containing at least 

 five or six litres, and having exhausted it of its air by 

 good air-pump, weigh it in a delicate balance, and call 

 its weight W. Let the air be now admitted to the gJajM 

 vessel, and let its weight, a* ascertained by the lislsssan, 

 be now called W'. The difference between these weights, 

 or \V W', will obviously be the weight of the atmo- 

 spherical air contained in the vessel. Let it now be 

 required to measure the* specific gravity of another gas. 

 Weigh the glass vessel when empty as formerly, ana al- 

 so when full of the gas, and let these weights be ie, t', 



then tc w' will be the weight of the gas, and \ v _ u . 



will be its specific gravity compared with that of the 

 air, which is taken at 1.000. This specific gravity is 

 that which corresponds with the state of the atmo- 

 sphere at the time when the experiment was made. 



