82 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 



and we have shown above in equation (41), that when the axis of the 

 semi-parabola is vertical, the pressure on the space BFC is 



consequently, by comparison, we obtain 

 #*!=&*+ 



and finally, dividing by ^bl, we shall have 



Again, when the axis of the semi-parabola is horizontal, as indicated 

 by A ED, the pressure on the circumscribing rectangle, according to 

 equation (10) under the third problem, is 



and the pressure upon the inscribed parabola, according to equation 

 (40) under the eleventh problem, is 



therefore, by subtraction, the pressure upon the space comprehended 

 between the rectangular parallelogram and its inscribed semi-parabola, 



p' P p = J6* /* T M> 2 ' , 

 and by suppressing the symbol for the specific gravity, we have 



Now, the area of the inscribed semi-parabola is, as we have seen 

 above, equal to two thirds of its bounding rectangle, and the area of 

 the space comprehended between the rectangle and the semi-parabola, 

 is therefore, equal to one third of the same quantity ; that is, 



bllbl = *bl; 

 consequently, the pressure on the irregular space A HE, is 



p'=ibtl; 

 hence, by comparison, we shall have 



^= T V^; 



therefore, by division, we obtain 



Having thus determined the values of the rectangular co-ordinates, 

 as represented by the equations (42) and (43), the position of the 

 centre of gravity can easily be found ; for, from the point F, set off 

 rm equal to three tenths of the axis CD, and jfn equal to one fourth 

 of the ordinate BD, or its equal FC ; then, through the points m and n, 

 and parallel respectively to the ordinate BD and axis CD, draw the 



