SPECIFIC GRAVITY.] MECHANICAL PHILOSOPHY HYDROSTATICS. 



755 



sequently the centre of pressure of the parallelogram 

 A J is the centre of gravity of the trapezium A m n B. 



As, in the first case, when the upper side of the paral- 

 lelogram is at the surface of the fluid, the centre of the 

 pressure is always on the same line E F, and at the 

 same depth, however close the middle point E is to the 

 extremities D, C that is, however slender the parallelo- 

 gram may be it follows, that the centre of pressure of a 

 straight line E F, having one extremity at the surface of 

 the fluid, is at frds the length of K F below E. The 

 centre of pressure being the point about which all the 

 pressing forces balance, it is, evidently, the point most in 

 need of support, or where the opposing force should be 

 more especially applied, to resist the pressure of the fluid 

 on the surface, and thus to prevent rupture. The staves 

 of vats and casks, which may be regarded as so many 

 rectangles, should be each more especially strengthened 

 at one-third of their lengths from the bottom. 



SPECIFIC GRAVITIES OF BODIES. When 

 bodies are compared together, having the same specific 

 gravity, any volume of one must of course have the 

 same weight as an equal volume of each of the others ; 

 so that the weights of such bodies are to one another as 

 their volumes. The volume of any irregular body may 

 be ascertained by the bulk of water it displaces by being 

 immersed (though more conveniently from its weight and 

 specific gravity) ; or, for the purpose of comparing dif- 

 ferent volumes, we may merely observe the height to 

 which the water rises in the two cases in a cylindrical 

 TOMel, upon the immersion of the bodies. The following 

 is an example. 



Example. A mass of gold immersed in a cylinder 

 containing water, caused the surface to rise a inches ; 

 a mass of silver of the same weight caused it to rise ft 

 inches ; and a mass still of the same weight, but com- 

 posed of gold and silver, caused it to rise c inches. 

 What was the proportion of gold and silver in the com- 

 pound mass ? 



Let x be the volume of the gold, and y that of the 

 silver : then as the volumes immersed are as the eleva- 

 tions of the surface caused by the immersions, we have 



(J 



c : a : : x -\- y : - (x + y) the volume of the mass of gold. 



c:b::x 



c 



y : c (z 



silver. 



The weights of these masses being equal, let W be the 

 weight of each ; then the weights being as the volumes 

 when the specific gravities are the same, 



y) : * : : 





Vfy 

 



the weight of silver 



Wy 



W 



the compound, 



*(*-r-y):y::W:^. 

 in the compound. 



c _!^.j. 

 'a ' x + V 6 



.'. ftcz + acy 06 (at + y) 



." . (ftc aft) x = (aft ac)y 



/. Z : y : : a (ft c) : 6 (c a) 



It is probable that in some such way as this Archi- 

 medes solved the problem proposed to him by Hiero, 

 King of Syracuse, who having ordered a crown of gold 

 to be made, suspected that the crown furnished to him 

 was a mixture of gold and silver, and wished the truth 

 to be ascertained without injuring the workmanship. 



As the weight of one volume is to that of another of 

 different specific gravity as the product of volume and 

 specific gravity of the former is to the product of volume 

 and specific gravity of the latter ; therefore, if S, S' 

 represent the specific gravity of gold and silver respec- 

 tively, the weights of x and y will be the one to the 

 otlT as Sj; to S'y ; therefore, by the foregoing pro- 

 portion, 



wt. of the gold (x) : wt. of the silver (y) 

 ::a(6 c) S : 6 (o a) S' 

 6-c) S 



ft(c a)' S' - 

 Consequently the weight of the compound is 



W 



.'. the weight of the silver in the compound ia 



PKOBLKM. Having given the volumes and specific 

 gravities, or the weights and specific gravities of several 

 bodies, to find the specific gravity of the compound. 



1. Let the volumes V T , Vz, V 3 , &c., be given with 

 the specific gravities S,, S 2 , S 3 , <fcc. ; and let S be the 

 specific gravity of the compound ; then since W=SVg, 

 we have for the weight of Vj + V 2 + V 3 + <fec. 



(S V 1 + S 2 V 2 + S., 



2. Let the weights W n W 2 , W 3 , <bc., with the specific 

 gravities S^ S 2 , S 3 , &c. be given, S being the specific 



gravity of the compound as before : then since V = , 



off 

 we have for the volume of W, + W 2 + \V 3 + &c. 



(\V i -f-W 2 +W 3 -|->bc.) S t S 2 S 3 .fee. 



s = w^g s7&c7-fw7s7s 3 &o. + w^s 3 s t & c . 



If m equal volumes are mixed, the specific gravity of 

 the compound is 



o s 1 +s,+s a +&& 



m 



If m equal weights are mixed, the specific gravity of 

 the compound is 



WIO' Qo O-> . . . . >O| 

 Q L & n - 



When there are only two bodies to be compounded, 



S I 

 ' 



; and for equal 



then for equal volumes, S 



2S S 

 weights, n- ^-, the former value being an arithmetic, 



"iT-Sa 



and the latter an harmonic mean, between the specific 

 gravities of the two substances. 



PBOBLEM. To determine the volume of any substance, 

 however irregular, of known specific gravity. 



Let S be its specific gravity, and W its weight in 

 ounces ; then, since 



cubic feet. 



In a similar way may the capacity of an irregular 

 vessel be ascertained. Let the weight of the water that 

 will fill th* vessel be to ounces, then the capacity or 



Iff 



volume of the vessel will be cubic feet. If the 



result is to be in cubic inches instead of in cubic feet, 

 the divisor 1000 must be replaced by the multiplier 1 '728, 



1728 



because =- '.: = T728 : so that w being the number of 

 1000 



the avoirdupois ounces in the weight, the volume or 

 capacity will be V=V728 ^ cubic inches. 



Should to be the number of troy, instead of the num- 

 ber of avoirdupois ounces, then sinco 



1 oz. troy : 1 oz. avoirdupois : : 480 : 437 '5 



