24 OF THE AGGREGATE PRESSURE OF FLUIDS 



sn. 0, 

 and in like manner, the pressure on the triangle bdc is 



p' ~\bl* sin. <f>. 



These equations, however, express the pressures simply by the 

 magnitude of a fluid column, whose base is the area pressed, and 

 whose altitude is equal to the depth of the centre of gravity below the 

 upper surface of the fluid. In order, therefore, to have the pressures 

 expressed in general terms, the specific gravity of the fluid must be 

 taken into the account; in which case, the pressure on the triangle 

 abd becomes 



p=bl*ssm.<t>, (11). 



and the pressure on the triangle bdc is 



p'i=i&Z 2 ssin.;>. (12). 



COROL. Hence it appears, that the pressure perpendicular to the 

 plane of a triangle, when its vertex is upwards and coincident with 

 the surface of the fluid, is double the pressure on the same triangle, 

 when its base is upwards, and placed under the same circumstances. 



39. If the immersed plane be perpendicular to the surface of the 

 fluid, then <b nz 90, and sin. ty ml; therefore, by substitution, the 

 preceding equations become 



p = bl*s, andp'=ibr~s; 



here again, the pressure in the one case is double the pressure in the 

 other, and the same thing will obtain, whatever may be the inclination 

 of the plane, provided only that a b coincides with the surface of the 

 fluid ; for then, the perpendicular depths of the centres of gravity will 

 vary in a given ratio. 



When the immersed plane is a square, that is, when b and I are 

 equal to one another, the equations for the pressures in the oblique 

 position become 



p :z: %b s s sin. 0, and p' b 3 s sin. 0, 



and when the plane is perpendicular to the surface of the fluid, we 

 have 



p = b s s, and p' | b 3 s* 



Since the aggregate pressure upon the plane is equal to the sum 

 of the pressures on the constituent triangles, the expression for the 

 aggregate pressure in the oblique position, becomes in the case of a 

 rectangle 



P =p +p f ; that is, P z= bl*s sin. -f bl*s sin. = bl*s sin. 0. 

 COROL. Hence it appears, that the pressures on the constituent 

 triangles and that on the entire plane, are to one another as the 

 numbers 1 , 2 and 3 ; and the same thing obtains in the case of a 

 square, whatever may be the inclination of the plane. 



