GO 



OF PKKUMATIC EQUILIBRIUM. — OF HYDRAULICS. 



may be the weight of the body, and will support an equal 

 weight in both cases. 



Scholium. Hence the specific gravity of any fluid may 

 be determined by finding how much weight it deducts from 

 a body of known dimensions immersed in it. And the di- 

 mensions of a solid may be found by weighing it in a fluid 

 of known specific gravity, and thence its specific gravity 

 may also be ascertained. 



SECTION IV. OF PNEUMATIC EQUILIBRIUM. 



378. Definition. Elastic fluids are such 

 as have a tendency to expand when at liberty, 

 with a force which is proportional to their 

 density. 



Atmospheric air, gases, and vapours, are examples of such 

 fluids. 



379. Theorem. Supposing the force of 

 gravitation constant, the logarithm of the 

 rarity of the atmosphere must be in a con- 

 stant ratio to the height. 



Let the length of a column of air increase equably in de- 

 scending, then the densities at each point being as the pres- 

 sures which counteract the expansive force (378), and the 

 increments of the pressures being as these densities, the 

 pressures vary proportionally while the heights vary equa- 

 bly ; therefore the heights are in a constant ratio to the lo- 

 garithms of the numbers representing the pressures (50) : 

 and in ascending, the same logarithms, taken negatively, 

 are the logarithms of the reciprocals of the densities (38), 

 or of the rarities. So that if a, I; h, r; be corresponding 

 heights and rarities, a : l.t :: A : l.r. Suppose a 7 miles, 



and /-zlt.thcn 7 -.lA-.-.h: l.r, and l.rlz-^Ar: , and 



' 7 1.1026 



h 

 h 

 *=!. 16-26 (l.r), also since l.r=:(1.4). — , r=(4)7. Practi- 

 cally 7i:i;ioooo (l.r), in fathoms. 



380. Theorem. The height of a column 

 of air, of equal density with the atmosphere 

 at any part, capable of producing a pressure 

 equal to the atmospheric pressure at that 

 part, is the same at all distances above the 

 earth's surface. 



The height of such a homogeoeous atmosphere must b« 

 directly as the pressure to be produced, and inversely as the 

 density, and since the density varies as the pressure, the 

 result will be constant. This height is found to be some- 

 what more than 5 miles : in very great elevations it proba- 

 bly varies. 



381. Theorem. Ifa fluid be contained 

 in a tube closed at the top, it will be support- 

 ed by the pressure of the atmosphere at such 

 a height, that its weight will be equal to that 

 of a column of air on the same base, and of 

 the height of the atmosphere. 



For if an upright tube be partly immersed in a fliiid, a 

 heavier fluid will be sustained in it at a proportionate height, 

 provided that the access of the fluid to its upper part be 

 prevented ; and in this case, the pressure of the atmosphere 

 is as effectually removed from the upper surface of the fluid 

 in the tube, as if the tube were continued throughout thi 

 height of the atmosphere. 



SECTION V. OF HYDRAULICS* 



382. Theorem. The velocity of a small 

 jet of water issuing in any direction from a 

 reservoir, is nearly equal, in favourable cir- 

 cumstances, to the velocity acquired by a 

 body in falling through the height of the 

 surface of the reservoir above the orifice. 



Supposing a very small plate of water immediately with- 

 in the orifice, to be put in motion at each instant by means 

 of the whole pressure of the fluid; which is equal to the 

 weight of a column on the same bascj of the height of the 

 reservoir, and supposing the whole pressure to be employed 

 in generating the velocity of the thin stratum, neglecting 

 the motion of the surrounding fluid, this stratum would be 

 urged by a force as much greater than its own weight as the 

 column is higher than its thickness, through a space which 

 is shorter than the heighl of the column in the same ratio. 

 But the spaces being inversely as the forces, the final velo- 

 cities are equal ; and the velocity thus generated would be 

 -equal to that of a body falling through the height of the 

 column. And although a part of the pressure of the co- 

 lumn is expended in producing motion in its own particlesi 

 this part is not wholly lost, because tlie velocity of these 



