66 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 



parabolic space ACB ; and if the axis CD be bisected in the point n ; 

 n is the centre of gravity of the circumscribing rectangular parallelo- 

 gram ABEF. 



Put b zz A B, the base of the parabola, or the horizontal breadth of its 

 circumscribing rectangular parallelogram, 



/CD, the vertical axis of the parabola, or the depth of its sur- 

 rounding rectangle, 



dm en, the depth of the centre of gravity of the parallelogram 

 ABFE below the upper surface of the fluid, 



S zz cm, the, depth of the centre of gravity of the parabola ACB, 



Pzz the pressure on the circumscribing rectangle, 



p zz the pressure on the parabolic plane, 



A zz the area of the parallelogram, 



a zz the area of the parabola, and 



s the specific gravity of the fluid. 



Then, according to the writers on mensuration, the area of the cir- 

 cumscribing rectangular parallelogram, is 



A = bl', 



but the area of a parabola, is equal to two thirds of the area of its 

 circumscribing rectangle ; therefore, we have 



a zz %bl. 



Now, 5 zz %l by the construction, and we have shown in Proposition 

 (A), that the pressure upon any surface, 



Is expressed by the area of that surface, drawn into the 

 perpendicular depth of its centre of gravity, and also into the 

 specific gravity of the fluid. 



Consequently, the pressure perpendicular to the surface of the 

 parabolic plane, is 



/>==#/ X#X * = lbP*. (34). 



But in order to compare the pressure on the parabola, as repre- 

 sented by, or implied in the above equation, with that upon its cir- 

 cumscribing parallelogram, we have only to recollect, thatc?zz|/, 

 and consequently, the pressure on the rectangle, is 



consequently, by omitting the common factors, and rendering the 

 fractions similar, we have 



p : P : : 4 : 5. 



78. Now, the practical rule for determining the pressure on the 

 parabolic plane, as deduced from the equation (34), or from the pre- 

 ceding analogy, may be expressed in words, as follows. 



