84 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 



Put 26 = AB or ab, the base of the parabola ACB, or the horizontal 



breadth of the circumscribing parallelogram OAB&, 

 I zz CD or a A, the vertical axis of the parabola ACB, or the 



depth of the rectangle by which it is encompassed, 

 2y zz EF or cd, the base of the parabola ECF, or the breadth of 



the rectangle c E F d, 

 x zz cm or CE, the axis of the parabola ECF, or the depth of its 



circumscribing rectangle CEFC?, 

 P zzthe pressure on the rectangular parallelogram CEAB&, cir- 



cumscribing the parabola ACB, 

 p zz the pressure on the inscribed parabola, 

 P' zz the pressure on the rectangular parallelogram CEF^, cir- 



cumscribing the parabola ECF, 

 p' zz the pressure on the inscribed parabola, and 

 * zz the specific gravity of the fluid. 



Then since ABzz26 and EFZz2y, it follows, that AD 6 and 

 EWI zz y ; therefore, by the property of the parabola, we have 



and consequently, by equating the products of the extreme and mean 

 terms, we shall obtain 



y=p*, 



and by division, it is 



b*x 



y =T> 

 and this, by extracting the square root, becomes 



and finally, multiplying by 2, we obtain 

 EFZZ ^ ^ 



Now, the pressure perpendicular to the surface of the rectangular 

 parallelogram aAB#, according to equation (8) under the third 

 problem, is 



but we have seen elsewhere, that the pressure on the surface of a 

 parabola, is equal to four fifths of that upon its circumscribing paral- 

 lelogram ; consequently, the pressure on the parabola ACB, is 



(44). 



