28 OF THE PRESSURE OF INCOMPRESSIBLE FLUIDS 



Draw the diagonals d/and/a meeting the straight linear A in the 

 points m and n ; then are m and n the centres of gravity of the respec- 

 tive portions into which the parallelogram is divided by the line ef. 



Through the point g and in the plane of the fluid surface ABEF, 

 draw gsr at right angles to a b, and the angle mgr will be the incli- 

 nation or obliquity of the plane ; then, through the points m and n, 

 draw the straight lines mr and ns respectively perpendicular to gsr, 

 and rm and sn will be the vertical depths of the centres of gravity 

 below the upper surface of the fluid. 



Put b=iab, the horizontal breadth of the given parallelogram, 

 / zr ad or be, the immersed length tending downwards, 

 d~rm, the vertical depth of the centre of gravity of the lower 



portion efcd, 

 $ ~ sn, the vertical depth of the centre of gravity of the upper 



portion abfe, 



(j>=:mgr, the inclination of the plane to the surface of the fluid, 

 Pn= the pressure on the whole parallelogram abed, 

 p the pressure on the lower portion efcd, 

 and p the pressure on the upper portion abfe. 



Then, because the straight line gh is bisected in 0, and each of the 

 portions g and h respectively bisected in the points n and m ; 

 it follows that g n zz |, and gm % of gh; that is 



g n nz J I, and g m j / ; 

 consequently, by the principles of Plane Trigonometry, we have 



swrzrSzz \l sin. <j>, and rm d=%l sin. 0; 



therefore, since the area of each portion of the parallelogram is 

 expressed by \ bl, the pressure on each portion is as below, viz. 

 The pressure perpendicular to the surface a bfe, is 



p'zz: bl z sin. 0, 

 and the pressure perpendicular to the surface efcd, is 



p m %bl* sin.0 ; 



consequently, by comparison, the pressures on the upper and lower 

 portions of the parallelogram, are to each other as the numbers 1 and 

 3 ; that is 



pT:p::l:3. 



But according to the third problem, the aggregate pressure sustained 

 by the plane, in a direction perpendicular to its surface, is 



consequently, the pressures on the two portions and on the whole 

 plane, are to one another as the numbers 1, 3 and 4. 



