68 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 



rectangular parallelogram is known, the pressure on the inscribed 

 parabola is also known, being equal to four fifths of that upon the 

 parallelogram. 



Now, it has already been demonstrated, that the pressure upon any 

 surface, whatever may be its form, is always equal to its area, drawn 

 into the perpendicular depth of the centre of gravity, below the upper 

 surface of the fluid ; therefore, conversely, the perpendicular depth 

 of the centre of gravity of any surface, is equal to the pressure which 

 it sustains, divided by the area. 



But from what, has been demonstrated above, it is manifest that 

 the area of the parabola and the pressure upon it, are respectively 

 expressed by 



f&Z, and f&/ 2 s; 



consequently, by division, we obtain 



and when s is expressed by unity, as in the case of water, we get 



a=# 



Now, because the parabola ACB is symmetrically divided by the 

 axis CD, it follows, that the centre of gravity occurs in that line, and 

 we have just shown, that it occurs at the distance of three fifths of its 

 length from the vertex ; hence, the position is determined, and that 

 independently of computing the corresponding horizontal rectangular 

 co-ordinate, whose intersection with the axis fixes the place of the 

 centre sought. 



The aggregate pressure upon the two equal and similar spaces A E c 

 and BFC, is obviously equal to the difference between the pressures 

 on the rectangular parallelogram ABFE, and that on the inscribed 

 parabola A c D ; that is, 



where p' denotes the pressure on the spaces A EC and BFC. 



Again, the area of the spaces A EC and BFC, is equal to the dif- 

 ference between the area of the parallelogram ABFE, and that of the 

 inscribed parabola ACB; therefore, if a 1 denote the area of the trian- 

 gular spaces, we have 



a' A a=zbl %bl = %bl. 



But the depth of the centre of gravity of any surface, is equal to 

 the pressure upon that surface divided by its area ; consequently, the 

 depth of the centre of gravity of the figure A EC FB, which is composed 

 of the two triangular spaces AEC and BFC, is 



