SPECIFIC OKA ITT.] MECHANICAL PHILOSOPHY. HYDROSTATICS. 



757 



Qt*\ "\7' /CI" ^^ Q'^ 



Hence, adding 1 to each side, 



V + V S S' 



V S" S' , 



T57= e eT 1 t " e ratl 



V o O 



V S ! 



If the solid float on one fluid, then in these results 

 8 = 0, that is, the upper portion V of the body is in 



V S' S" V + V S' 



vacuo ; therefore = gj 



Also- 



V 



S" 



Consequently, if a body float on a fluid in vacuo, anc 

 then the air or any other lighter fluid be admitted, the 

 body will rise, and leave a less portion of it immersed in 

 the original fluid. Also, if a body float on a fluid, the 

 part immersed V is to the whole body V + V as the 

 specific gravity S" of the body to the specific gravity S' ol 

 the fluid. 



Consequently if the body float upon a second fluid oi 

 specific gravity S,, and if V t be the part of the body 

 immersed, then we have the two proportions 



and V. 



: V + V 

 : V, 



'-VtS, 



That is, the parts of the body immersed, or the 

 volumes of the fluids displaced, are inversely as the 

 specific gravities of those fluids. Upon this prin- 

 ciple the instrument called the hydrometer, for measur- 

 ing the specific gravities of different liquids, is con- 

 structed. 



THE HYDROMETER There are several modifi- 

 cations of this useful instrument ; in 

 its simplest form, the hydrometer con- 

 sists of two hollow spheres, usually of 

 glass a larger, and then smaller one 

 below it, communicating with each 

 other. Into the lower sphere mercury 

 or small shot is introduced, in order 

 that the spheres may sink in the liquid. 

 From the upper sphere rises a cylin- 

 drical stem, the axis of which is in the 

 straight line through the centres of 

 gravity of the spheres ; so that when 

 the instrument rests in the fluid, the 

 stem may be vertical. Let the instru- 

 ment be placed in water, the stem will 

 sink to some point A ; let it then be 

 placed in a liquid L that will cause the 

 stem to sink to B ; then, by the above- 

 mentioned proposition, the specific gravity of L is to 

 that of water (1), as the volume or magnitude of the 

 body A C , to the volume or magnitude of B 0. 



.'. specific gravity of L, TolumeofAO 

 volume of BC 



volume of (D D A) 

 "volume of (D C^D B) 



The whole volume of the instrument that is, the 

 volume of water it would displace if wholly immersed in 

 is regarded as consisting of 4,000 parts, and the stem 

 is divided so that each division is one of those parts. 

 Suppose the stem to contain 50 of the parts, numbered 

 from D downwards, and that it sinks to 30 in one liquid 

 L,, and to 20 in another liquid L s : then 



Specific gravity ofJL ; _ 4000 20 _ 3980 

 Specific gravity of L 3 ~4000 30 = 3970 



And in this way are the specific gravities of different 

 liquids compared. 



Me | IOI.SON 'S HYDROMETER. This differs from 

 the common hydrometer chiefly in having a dish at each 

 end, and in serving for measuring the specific gravities 

 of solids as well as fluids. The stem in this instru- 

 ment is a slender wire of hardened steel ; and the 



adjustment is such that when a given weight 1000 

 grains for instance is placed in the upper dish A (Fig. 

 171), the instrument will sink, in distilled water at 

 Fig. 171. the temperature of 60 Fah., to the point P in 

 the middle of the stem. It is used as fol- 

 lows : 



To Compare the Specific Gravities of Two 

 Lirjuids L, L'. Let W be the weight of the 

 hydrometer, w the weight which must be placed 

 in the dish A to sink the instrument to the point 

 P in the liquid L, to,- the weight which must be 

 placed in A to siuk the instrument to P in the 

 liquid L': then 



weight of the volume of L displaced = W -f- w 

 weight of same volume of L' displaced = W-f- w t 



specific gravity of L W -f- to 

 specific gravity of L' W + Wj 



To find the Specific Gravity of a Solid. Let w be the 

 weight which, placed in the dish A, causes the instrument 

 to sink to P in the water. Place the solid to be examined, 

 or a fragment of it, in A, and let w' be the weight to be 

 added to sink the instrument to P. Then remove the 

 solid to the lower dish B, and let w" be the additional 

 weight to be added to the upper dish to sink the instru- 

 ment to P. 



It is plain that the weight of the solid in air, 



in A=i0 to' 

 and that its weight in water, in B = to w" 



Hence the weight of water displaced by the solid 



. specific gravity of the solid^to - vf ^ ^ 

 specific gravity of water w" w' 



vity of the solid, the specific gravity of water being 1. 



Or, without placing any additional weight in the 

 upper dish, observe to what depth the solid alone sinks 

 the instrument in water ; then remove the solid to the 

 lower dish, and some additional weight to will be neces- 

 sary to sink the instrument to the same depth ; so that, 

 calling the weight of the solid W, we shall have 



specific gravity of the solid = . 



10 



The weight W of the solid under examination is readily 

 ascertained by finding what weight supplying its place 



Ji the upper dish will sink the instrument to the same 

 depth. 



It is evident that the instrument becomes more sen- 

 sitive the slenderer the wire stem be made. If an 

 additional small weight a grain for instance be placed 

 in the dish A, an additional grain weight of the liquid 

 must be displaced by the sinking of a suitable additional 



ength of the stem ; so that the finer this stem is, the 

 greater must the additional length be. The diameter of 

 stem is in general j\yth of an inch ; and the instru- 

 ment will rise or fall nearly one inch by the abstraction 

 or addition of -^th of a grain. For the weight of a cubic 

 inch of water is 253' 1808 grains troy ; and therefore the 



olume of 



of 



a grain is - of 



cubic inch. 



<bw the area of a transverse section of the wire stem 



X '7854= 



7854 



-, and if the volume due to j'^th 



40 2 " 16000000' 



of a grain, namely _, - , be divided by this area, 



- , 

 *808 



he quotient will be about 4ths of an inch, for the length 

 )f wire that will rise or fall in water when the weight 

 liffers by y'^th of a grain. In a liquid lighter than water 

 he length will of course be greater. 



The hydrometer is the instrument used by officers of 

 he Excise to ascertain the strength of spirituous liquors, 

 >r in what degree they are above or below proof. Vroof- 

 pirits consist of half alcohol, or pure spirit, and half 



