OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 79 



Again, the area of the semi-parabola BCD, is equal to two thirds of 

 its circumscribing rectangular parallelogram BFCD ; therefore we have 



a = l X bl=$bl, 

 and the pressure perpendicular to its surface, is 



p = &Vl8. 



This is manifest, for according to the construction and the nature 

 of the figure of the parabola, BW or EG is equal to five eighths of BD ; 

 therefore, we have 



p ^blXibX s = -frb*ls', (40). 



consequently, by analogy, we obtain 



p : P : : &&ls : b*ls. 



Therefore, by suppressing the common factors, and rendering the 

 fractions T \ and | similar, we shall get 



p:P::5: 6; 



hence it appears, that when the ordinate of the semi-parabola is 

 vertical, and its upper extremity in contact with the surface of the 

 fluid : 



The pressure upon the semi-parabola, is to that upon its 

 circumscribing rectangular parallelogram, as 5 is to 6, or as 

 1 is to 1.2. 



92. Consequently, the practical rule for determining the pressure 

 in the present instance, as deduced from the equation marked (40), 

 or from the above analogy, may be expressed as fdllows. 



RULE. Multiply the square of the given ordinate by the 

 axis of the semi-parabola, and again by the specific gravity 

 of the fluid ; then, Jive twelfths of the result will give the 

 pressure sought. Or thus, 



Find the pressure on the circumscribing parallelogram, and 

 take five sixths of the pressure thus found, for the pressure on 

 the semi-parabola. 



93. EXAMPLE 18. Let the numerical values of the axis and ordi- 

 nate remain as in the preceding example ; what will be the pressure 

 on the surface of the semi-parabola, supposing the axis to be hori- 

 zontal, the ordinate vertical, and its remote extremity in contact with 

 the surface of the fluid ? 



If the operation be performed according to the rule deduced from 

 equation (40), we shall obtain 



p= 16 2 X 40 X 62 X T V = 2666661- Ibs. 



