HYDRODYNAMICS. 429 



PROP. I. of the strata are perpendicular to the direction of gra- 



vity. 

 If two fluids of different densities are included in the 



Specific 



Gravities. 



PtAT* 



CCCXIIL 



rtf.it 



'ui 11 



separate branches of a syphon, they will be in equili- 

 brio, when the altitudes aliove their common junc- 

 tion are reciprocally proportional to their specific gra- 

 vities. 



Let ABCD Fig. 12, be the syphon, and AB, CD the 

 heights of the two fluids of different densities, which 

 may be supposed separated from each other by the 

 common junction m H. Then, if G is the centre of gra- 

 vity of the surface m , the pressure exerted by the 

 fluid in AB is. by Prop. IV. m x < ', *nd that of the 

 n CD, m n x GB ; but since their specific gravi- 

 ties S, S' are, by the hypothesis, reciprocally as their alti- 

 tudes, that is, S:S'=Gr:Gi, we have SxG'=S' 

 X G r but the pressure of the one column is m M x 

 G * x S= n x G r x S, the pressure of the other, con- 

 sequently they will be in equilibrio. 



If o p be the junction of the two fluids, then, by 

 drawing the horizontal line opytv, we may regard the 

 columns op m m, {pumas balancing one another, since 

 they are composed of the same fluid, and then consider 

 the columns ABpo, CD r I of different densities, as 

 resting upon the surfaces op, I e. In this case, y t x S' 

 =y r x S , and consequently the pressure* op x y * X S 

 sropxyxS'. 



In the case of water and mercury, the values of 

 arc I and 13.58 at a temperature of OCf, consequently 

 G/ : G r=13-5 : 1 ; ami therefore to balance S3 feet 

 of water, a column of mercury 29 16 inches will be 

 required. 



PROP. II. 



The pressure on the bottom of a vessel containing 

 horizontal strata of fluids of different densities, is equal 

 to the area of the bottom multiplied by the sum of the 

 products of the thickness of every stratum and their 

 specific gravities. 



Let ABCD, Fig. IS, be the vesael, and AB/e, efk t, f* * 

 kit, and * / e D strata of different densities, S, S', S ', A e w 



If the lower stratum of IclcD (Fig. 13.) were placed cccxill. 

 alone in the vessel, its surface k I would be horizontal. Kig. 13, 

 Let us now suppose that every point of the surface i I 

 is pressed downwards by equal forces, which it will be 

 when it is pressed down by the weight of the superior 

 strata with horizontal surfaces, then since there can be 

 no reason why one point should yield more than ano- 

 ther to these forces, it follows that the stratum /. C will 

 still be in equilibrio. In like manner, it may be 

 shewn, that the stratum g k I k will be in equilibrio, 

 and so on with all those above it, so that we may con- 

 clude that the whole fluid in the vessel is in equili- 

 brio. 



POP. IV. 



To find the pressure exerted by a fluid composed of 

 an infinite number of strata of variable density against 

 any part of the vesael which contains it. 



Let S, S', S", Ac. be the specific gravities of the dif- F* 1* 

 ferent strata qp, p o, on, &c. then since the point q 

 sustains the weight of all the columns q p, po. Sic. above 

 it, it will be pressed down by a force equal t* S x qp 



= Sx(^- X -^ + 

 \ 



S'x 



;x o, &c. 



* 



&c. 



f* 



Then since, by the last Prop, any column of fluid 

 is in equilibrio with another, or hastae same pressure 

 when their heights are reciprocally M their densities, 

 calling I i , H , H", &c the heights of the strata, we have 



S:S'=II : H, and H = ^J-. We may therefore J* * ~J- 

 kubstitute a column of fluid of the same kind as the lower 



Hence it fol- 

 a" r 3" 



Iowa, that we may substitute in place of the fluid of 

 variable density a fluid whose density is uniform through 

 the whole height q m. 



Let us take an infinitely small elementary stratum 

 e nff f i contained between the horizontal lines e nf, 

 i' 9, then the pressure upon e t is the absolute weight of 

 the column m r ; but making m n =i, and X = the spe- 

 cific gravity of the fluid in n, the weight of mn will be 

 id therefore the sum of the pressures against 

 ill bey A e x /I x. Thus to find the pressure 

 upon the bottom DC, let 9 m = a, and S = the speci- 

 fic gravity of the fluid in DC, then x =-'"*-, and 



S* 



and since, in the case of a 



S' x H' 

 itratum 1C, and having an altitude instead 



of the column g I, and in like manner, instead of the 

 other columns eh, A /"we may substitute columns whose 



S" v II" "^"^ v H** 

 heights are - , , and therefore calling 



** *" 



DC the area of the bottom, the whole pressure on the 

 bottom wfll be=SxDCxH+(x H + 



S^_x H" 9-xI! 

 8~~ ~&~ 



II + S'" x H'"), 

 in the proposition. 



PHOP. III. 



If a fluid contained in a vessel consists of an infinite 

 number of strata whose densities vary according to any 

 law, the fluid will be in equilibrio, when the surface* 



= DC x (S ; 



which is the truth announced 



pressure upon the bottom, * = a, we have -^-= 5 



Sa 

 and the whole pressure upon the bottom = X DC. 



CHAP II. 

 ON SPECIFIC GRAVITIES. 



PROP. I. 



If any object floats upon a fluid, or is wholly im. < 

 mersed in it without sinking, it is pressed upward* gnriuct. 

 with a force equal to the weight of the fluid displaced. 



Let EF (Fig] U.) be a body floating i 

 ABCD. Then by Prop III. Chap. I. any pom 

 ticle n is pressed upwards with a force equal to a co- 



in the vessel Fig. 

 point or par- 



lumn of particles whose height it m n, and as this is 

 true of every part of the surface EnF, then since the 



