OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 75 



the first two of which, by reason of the parts of the figure being the 

 same in each, are obviously dependent upon one another; but the 

 third, in which the parts of the figure are reversed, is wholly inde- 

 pendent and distinct from the other two. 



87. COROL. 1. Admitting therefore, that the pressure upon the 

 parabolic surface, under the three circumstances of position in which 

 we have considered it, is represented by the equations (35), 36) and 

 (37) ; it follows, that the situation of the centre of gravity can easily 

 be ascertained ; for the pressure in each case, as we have elsewhere 

 shown, is represented by the area of the figure, drawn into the per- 

 pendicular depth of its centre of gravity ; consequently by reversing 

 the process, the depth of the centre of gravity will become known, if 

 the pressure be divided by the area of the surface on which the fluid 

 presses. 



COROL. 2. Since the parabola is a figure symmetrically situated with 

 respect to its axis, it is obvious, that the centre of gravity of its surface 

 must occur at the same point, in whatsoever position it may be placed; 

 but when the place of its centre is referred to the surface of the fluid 

 in which it is immersed, the distance varies for each particular case: 

 thus, 



In the first instance, the perpendicular distance, is in f ths of the axis, 



second, izrfths 



third, m ^ the base. 



But as we have just stated, the centre of gravity of the parabolic sur- 

 face as referred to its vertex, or any other fixed point, in all these cases, 

 remains unaltered, in whatever position the figure itself may be placed. 



5. THE METHOD OF DETERMINING THE PRESSURE OF THE FLUID UPON 

 A SEMI-PARABOLIC PLANE AS COMPARED WITH THAT ON THE 

 CIRCUMSCRIBING RECTANGULAR PARALLELOGRAM. 



88. When the semi-parabola only is considered, the determina- 

 tion of its centre of gravity, and consequently, of the pressure on 

 its surface becomes more difficult ; for, since the figure is not sym- 

 metrical with respect to its axis, we are under the necessity of com- 

 puting the two rectangular co-ordinates, whose in- 

 tersection determines the place of the required centre. 



Let CBD be a semi-parabola, perpendicularly 

 immersed in a fluid, so that its axis CD is vertical, 

 and the vertex in contact with CF the surface of the 

 fluid, and let CFBD be the circumscribing rectan- 

 gular parallelogram. 



