OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 81 



It is further manifest, that while the semi-parabola revolves about 

 the point D, from the position BCD to that of A ED, the points B and c, 

 the extremities of the ordinate and axis, describe respectively, the 

 circular quadrants BA and CE, while the point F describes another 

 quadrant, whose containing radii are the diagonals DF and DH. 



Put 6:zr BD or AD, the ordinate of the semi-parabola in either posi- 



tion, 



I n= CD or ED, the corresponding axis, 

 d nG, the depth of the centre of gravity of the space BFC, 



when the axis is vertical, 

 3 rz m G, or A ft, the depth of the centre of gravity of the space 



AHE or BFC, when the axis is horizontal, 

 Pzz: the pressure on the circumscribing rectangular parallelogram 



BDCF, Or AH ED, 



p the pressure on the inscribed semi-parabola, and 



p the pressure on the space comprehended between the semi- 



parabola and its circumscribing rectangular parallelo- 



gram. 



Then, according to equation (8) under the third problem, the 

 pressure on the circumscribing rectangular parallelogram when the 

 length is vertical, is 



and agreeably to equation (38) under the eleventh problem, the 

 pressure on the inscribed semi-parabola with the axis vertical, is 



consequently, by subtraction, the pressure upon the space BFC, com- 

 prehended between the sides of the parallelogram BF, FC and the 

 curve of the parabola BC, is 



hence, by suppressing the symbol for the specific gravity, we get 

 p' bl^(l f)rr T V^ 2 . (41). 



Now, according to the writers on mensuration, the area of the 

 semi-parabola is equal to two thirds of the area of the circumscribing 

 parallelogram ; it therefore follows, that the area of the space BFC, is 

 equal to one third of the rectangle BDCF; that is, 

 bi $bl=;$bl. 



But it has been demonstrated, that the pressure upon any surface, 

 is equal to the area of that surface, drawn into the perpendicular 

 depth of the centre of gravity ; consequently, we have 



VOL. i. G 



