01' THE- CENTRE OF PRESSURE. 



407 



Let ABC be a cistern filled with an incompressible and non-elastic 

 fluid, and let abed be a rectangular 

 plane immersed in it at a given 

 angle of inclination to its surface ; 

 produce the sides da and cb di- 

 rectly forward to meet the surface 

 of the fluid in the points e and /; 

 join ef, and through the points e 

 and /, draw es and/r respectively 

 perpendicular to the plane produced, 

 and coinciding with the surface of 

 the fluid in ef; draw also ds and cr, 



meeting es andyY at right angles in the points s and r\ then is des 

 or c/r, the angle of the plane's inclination, and ds, cr are the per- 

 pendicular depths of the points d and c. 



Let P be the position of the centre of pressure, and through p draw 

 pm and PW, respectively perpendicular to cb and cd the sides of the 

 rectangular plane ; then are cb and cd the axes of rectangular co-ordi- 

 nates originating at c, and pm, PW are the corresponding co-ordinates, 

 passing through p the centre of pressure, supposed to be situated in 

 that point. 



Now, it is manifest from the nature of fluid pressure demonstrated 

 in the first chapter, that the force of the fluid against d : 



Is equal to the weight of a column of the fluid, whose base 

 is the point d, and altitude the perpendicular depth of that 

 point below the upper surface of the fluid. ^ 



Consequently, the force against the point d, varies as dx ds ; but 

 by the principles of Plane Trigonometry, we have 

 rad. : ed : : sin. des : ds; 

 hence by reduction, we get 



ds=:ed$m.des; 

 therefore the pressure on the point d varies as 



d XedXsin.des, 



and the effort or momentum of this force, to turn the plane about the 

 ordinate pm, manifestly varies as 



d Xee?Xsin.c?esXPw, 



where p n is the length of the lever on Avhich the force acts. 

 But by subtraction, p n -=.ed /m, for edfc; therefore by sub- 

 stitution, the force to turn the plane about the ordinate pm, varies as 



