OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 73 



Divide the axis DC into two parts Dm and cm such, that Dm is to 

 cm, in the ratio of 2 to 3; and the point m thus determined, is the 

 place of the centre of gravity of the parabola ACF. Through the 

 points m and n, draw the straight lines mr and ns respectively per- 

 pendicular to the axis DC; then are mr and ns 7 the perpendicular 

 depths of the points m and n below AB, the surface of the fluid, and 

 which in the present case are equal to one another. 



Put &ZZIAB, the horizontal breadth of the rectangular parallelogram 

 ABEF, or the axis of its inscribed parabola ACF, 



I AF, the length of the circumscribing rectangle, or the base 

 of the inscribed parabola, 



d rm or sn, the vertical depths of the centres of gravity, below 

 AB the surface of the fluid, 



Pmthe pressure on the rectangle ABEF, 



/> = that on the inscribed parabola ACF, 



A = the area of the circumscribing rectangular parallelogram, 



a the area of the parabola, and 



s the specific gravity of the fluid in which they are immersed. 



Then, according to the principles of mensuration, the area of the 

 rectangular parallelogram ABEF, is equal to the product of the 

 breadth AB drawn into the depth AF ; that is, 



A = bl; 

 and by the property of the parabola, its area is 



a = \b I. 



But the pressure perpendicular to the surface of the rectangular 

 parallelogram, is, as we have already frequently stated, expressed by 

 the area drawn into the perpendicular depth of the centre of gravity ; 

 and this being the case, whatever may be the form of the surface 

 pressed, it follows, that the pressure on the rectangle ABEF, is 



P bl X d X s bdls' y 

 and that on the parabola ACF is 



Now, it is manifest from the nature of the figure, and from the 

 principles upon which it is constructed, that rm and sn are each of 

 them equal to JAF; that is, d=L\l\ therefore, let \l be substituted 

 for d in each of the above equations, and we shall obtain 

 For the rectangle ABEF, 



and for the parabola ACF, it is 



(37). 



