76 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 



Now, since by Problem XI, it has been proved that the pressure on 

 the entire parabola with its axis vertical, is equal to four fifths of that 

 upon its circumscribing parallelogram ; it follows, that the pressure 

 on the semi-parabola in the same position, is also equal to four fifths 

 of that upon its circumscribing parallelogram, it being manifestly 

 equal to half the pressure on the whole parabola. 



Divide the axis CD into two parts, such, that Dm and cm shall be 

 to one another in the ratio of 2 to 3 ; and in like manner, let the 

 ordinate or base BD be divided into two parts, such, that DW and vn 

 shall be to one another in the ratio of 3 to 5* ; then, through the 

 points m and n, and respectively parallel to DB and DC, draw the 

 straight lines WIG and no, meeting each other in G, the centre of 

 gravity of the semi-parabola DC B. 



Put &IZICF or DB, the horizontal breadth of the rectangle CFED, or 



the base of the semi-parabola c B D, 

 I =: CD or FB, the vertical depth of the rectangle, or axis of its 



inscribed semi-parabola, 



cZzz cm or EG, the perpendicular depth of the centre of gravity, 

 Pzzthe pressure on the rectangular parallelogram CFBD, 

 p zz the pressure on its inscribed semi-parabola, 

 A zz the area of the parallelogram, 

 a zz the area of the semi-parabola, and 



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

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

 rectangle is expressed by the product of its two dimensions ; that is, 

 by its length drawn into its breadth ; therefore, we have 



A = bl, 



and by Proposition (A), the pressure exerted by a fluid, perpendicu- 

 larly to any surface immersed in it, or otherwise exposed to its 

 influence, 



Is equal to the area of the surface pressed, drawn into the 



perpendicular depth of its centre of gravity, and again into 



the specific gravity of the fluid. 



Consequently, the pressure on the circumscribing rectangular paral- 

 lelogram CFBD, becomes 



* It is demonstrated by the writers on mechanics, that the centre of gravity of a 

 semi-parabola is situated in its plane, at the distance of three eighths of the ordinate 

 from the axis, and two fifths of the axis from the ordinate, or three fifths of the axis 

 from its vertex. 



