53 



OF HYDROSTATIC EQUILIBRIUM. 



constant force of gravity is represented by the subnormal, 

 or the portion of the axis intercepted between the ordinate 

 and the perpendicular to the curve, the perpendicular or 

 normal being the result of the two forces. But the curvp , 

 in which the subnormal is constant is a parabola ; for the 

 triangle composed by the increments of the curve, the ordi- 

 "nate and the absciss, is similar to that which is formed by 

 the normal, the subnormal, and the ordinate, consequently 



"yii 



y':3f::i:y, i) : i :: s : y, yil=si, and s—~ ; but m 



. . 2?/y 



the parabola, since «x=yi/ (204}, ai—-2yy, ^—~' ^"° 



a 



J— — , which IS constant. 



370. Theorem. The pressure of a fluid 

 on every particle of the vessel containing it, 

 or of any other surface, real or imaginjiry, in 

 contact with it, is equal to the weight of a co- 

 . lunin of the fluid of which the base is equal 

 to that particle, and the height to its depth 

 below the surface of the fluid. 



Imagine an equable tube to be so 

 bent that one of its arms may be ver- 

 tical, and the other perpendicular to 

 the given surface : then drawing a 

 horizontal line AB, the fluid in the 

 portion of the tube AB will remain in 

 cquinbrium, and will only transmit 

 the pressure of BC to the surface at A, and this will be 

 true whatever be the position of the imaginary tube ; and 

 since some particles of the fluid may be so arranged as to 

 be no more disturbed in their initial tendency to motion 

 than the fluid in such a tube would be, the equilibrium 

 can never be permanent unless the pressures be such as 

 are here assigned. 



Scholium. If therefore any portion of the superior 

 part of a fluid be replaced by a part of the vessel, the pres- 

 sure against this from below will be the same which before 

 supported the weight of the fluid removed, and, every part 

 remaining in equilibrium, the pressure on the bottom will 

 be the sarhe as if the horizontal section of the vessel were 

 every where of equal dimensions. In this manner the 

 smallest given quantity of a fluid may be made to produce 

 a pressure capable of sustaining a weight of any magnitude, 

 either by diifiinishing the diameter of the column and in- 

 creasing its height, or by increasing the surface which sup- 

 ports the weight. 



371. Theorem, The pressure on any 

 vertical or oblique, plane surface is equal to 



the weight of a column of a fluid of which 

 the base is equal to the surface, and the 

 height to the distance of its centre of gravity 

 below the level surface 'bf the fluid. 



Suppose the surface to be divided into a number of equal 

 evanescent portions, then the number of particles in each 

 column standing^ on the same base being as its length, the 

 weight will be is Ihe: length and the base conjointly, or as 

 the numerical product of the base and length : but from the 

 property of the centre of gravity or of inertia, the sum of 

 tlie products of each particle of the surface into its depth, 

 is equal to the product of the whole into the distance of the 

 centre of gravity (276), which represents a column of the 

 same height, and on the same base. 



372. Theorem. A hemisphere or semi- 

 cylinder of uniform densitj', having its axis 

 fi.xed in the surface of a fluid, and remaining 

 in equilibrium in any one position, will re- 

 main in equilibrium when its position is 

 changed by the iiijcrease or diminution of the 

 quantity of the fluid. 



The pressure of the fluid on 



the convex surffi.ce.of the solid 



will have no effect .in turning 



it round its axis, consequently 



we have only to consider the 



pressure exerted on its plane 



surface. Thecentre of gravity 



of this^urface AB or CD being 



at A or C, the pressure of the 



fluid will be always as the depth AB, CD : but the eflfect of 

 the weight of the solid will be always as EB, FB, the distance 

 of the centre of gravity E, G from the vertical line AB : but 

 the triangle BDC is always similar to the triangle GFB, con- 

 setjuently CD varies always as FB, and if the forces are 

 once equal, they will remain always equal in any position 

 of the solid. 



Scholium. If the surface of the fluid be below the 



axis, and there be an equilibrium, for instance when the 



surface is at A, there will be an equilibrium, for a similar 



reason, when the fluid rises to C in the oblique position of 



he solid. 



373. Theorem. If fluids are of different 

 specific gravities, that is, if equal bulks of 

 them have different weights, they will coun- 

 terbalance each other in a bent tube, whea 



