OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 77 



Now, since the pressure on the semi-parabola is equal to four fifths 

 of the pressure on its circumscribing parallelogram, we shall obtain 



f P = $ X Ibl^s ^b^sp. (38). 



The expression which we have here determined for the pressure on 

 the surface of the semi-parabola D c B, is precisely the same as that which 

 we have given in equation (35) for the entire figure ; only in the pre* 

 sent instance, the value of 6, the horizontal breadth of the parallelogram, 

 is but one half the value as applied to the parabola, when placed 

 under the conditions specified in the eleventh problem foregoing. 



89. If the symbol b retain its former value, that is, if it be referred 

 to the base of the entire parabola, or to the breadth of the parallelo- 

 gram circumscribing the entire parabola, then, the pressure on the 

 semi-parabola, becomes ' 



p \bl^s. (39). 



Consequently, the practical rule for determining the pressure on the 

 semi-parabola as deduced from this equation, may be expressed as 

 follows. 



RULE. Multiply one fifth of the base, or double ordinate of 

 the whole parabola, by the square of the length of its axis, 

 and again by the specific gravity of the fluid, and the product 

 will express the pressure on the semi-parabola in a direction 

 perpendicular to its surface. 



But if the symbol b refer to the ordinate, or base of the semi- 

 parabola, then, the rule as deduced from the equation (38), will be 

 precisely the same as that which we have given under the equation 

 numbered (35) in Problem XI, to which place the reader is referred 

 for the purpose of avoiding a direct repetition. 



90. EXAMPLE 17. A plane in the form of a semi-parabola whose 

 base or ordinate is 16 feet, and its axis 40 feet, is perpendicularly 

 immersed in a cistern of water, in such a manner, that its axis is 

 vertical, and its vertex in contact with the surface of the fluid ; what 

 pressure does it sustain, the weight of a cubic foot of water being 

 equal to 62 Jibs.? 



The equation in its present state, supposes the ordinate, or base of 

 the semi-parabola to be given, and therefore, the pressure is deter- 

 mined by the rule to the equation (35) or (38), in the following 

 manner : 



pl x 16 X40 2 X 621= 640000 Ibs. 



But in order to determine the pressure by the rule immediately 

 preceding, we must suppose the breadth or base of the figure to be 



