ON HYDROSTATICS. 203 



motion, the force of gravity being partly employed in generating momentum 

 either in the fluid or in the solid, the whole pressure on the bottom of the 

 vessel is necessarily somewhat lessened. Hence the apparent weight of a 

 jar of water will suffer a slight diminution, while a bullet is descending in 

 it, or while bubbles of air are rising in it, but the difference can seldom be 

 great enough to be rendered easily discoverable to the senses. 



It sometimes happens that a solid body is partly supported by a fluid, 

 and partly by another solid ; of this we have an example in one of 

 Dr. Hooke's* ingenious inventions for keeping a vessel always full. A 

 half cylinder, or a hemisphere, being partly supported on an axis, which is 

 in the plane of the surface of the fluid, its weight is so adjusted as to be 

 equal to that of a portion of the fluid of half its magnitude : when the 

 vessel is full it is half immersed, and exerts no pressure on the axis : it 

 descends as the fluid is exhausted, and its tendency to turn round its axis 

 can only be counteracted by the pressure of the fluid on its flat side, as long 

 as the surface of the remaining portion of the fluid retains its original level. 

 (Plate XIX. Fig. 249.) 



When a fluid is contained in a vessel of a flexible nature, the sides of the 

 vessel will always become curved, in consequence of the pressure, and the 

 more in proportion as the pressure is greater ; the form of the curved sur- 

 face will also be such that the common centre of gravity of the fluid and 

 the vessel may descend to the lowest point that the circumstances of the 

 case allow ; this form is generally of too intricate a nature to be determined 

 by calculation : no mathematician has hitherto been able to investigate, for 

 example, the curvature which a square or rectangular bag of leather will 

 assume when filled with water or with corn. "When, indeed, one dimension 

 only of a vessel is considered, for instance, when the bottom of a cistern is 

 supposed to be flexible, and to be fixed at two opposite sides, while the ends 

 are simply in contact with upright walls, without allowing the water to 

 run out, the nature of the curve may be determined with tolerable facility, 

 whether the weight of the bottom itself be considered or not. If the weight 

 be exactly equal to that of the water, the form of a semicircle will agree 

 with the conditions of equilibrium, as Bernoulli has demonstrated, sup- 

 posing the fixed points at the distance of its diameter ; but if the weight of 

 the bottom be neglected, the curvature will be everywhere proportional to 

 the distance below the surface, the form being the same as that of an elastic 

 rod, bent by two forces in the direction of the surface. The same principles, 

 with a slight difference in the calculations, will serve to determine the forms 

 adapted to the equilibrium of arches intended for supporting the weight of 

 superincumbent fluids, or of such soft materials as approach nearly in their 

 operation to more perfect fluids. (Plate XIX. Fig. 250.) 



LECT. XXI. ADDITIONAL AUTHORITIES. 



Works on Hydrodynamics, 8fc. not referred to in the Lectures. Switzer's 

 Hydrostatics, 2 vols. 4to, Loud. 1729. Wolfius, Elementa Matheseos, 5 vols. 4to, 

 Geneva, 1732-41. D'Alembert, TraitS de 1'Equilibre et du Mouvement des Fluides, 



* Birch's History of the Royal Society, ii. 155. 



