OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 67 



RULE. Multiply two fifths of the base of the parabola, by 

 the square of the length of its axis drawn into the specific 

 gravity of the fluid, and the product will express the pressure 

 sustained by the parabolic plane, in a direction perpendicular 

 to its surface. Or thus : 



Find the pressure on the circumscribing rectangular paral- 

 lelogram, according to the second case of the rule under the 

 third problem, and four fifths of the pressure so determined, 

 will express the pressure perpendicular to the parabolic 

 surface. 



79. EXAMPLE 14. A parabolic plane, whose base and vertical axis 

 are respectively equal to 28 and 42 feet, is perpendicularly immersed 

 in a reservoir of water, so that its vertex is just in contact with the 

 surface ; what weight is equivalent to the pressure on the plane, the 

 weight of a cubic foot of water being equal to 62 J Ibs. ? 



Here, according to the rule, we have 

 p = f x 28 X 42 2 X 62J = 1234800 Ibs., 



or by the second clause of the rule, it is 

 p \ X 28 x 42* X 62J X * = 1234800 Ibs. 

 Either of these methods is sufficiently simple for every practical 

 purpose ; but it will be found of essential advantage, to bear in mind 

 the relation between the pressure on the parabola and that on its 

 circumscribing rectangle ; for which reason, the latter method may 

 probably claim the preference. 



2. METHOD OF FINDING THE CENTRE OF GRAVITY OF THE SPACE 

 INCLUDED BETWEEN ANY RECTANGULAR PARALLELOGRAM AND 

 ITS INSCRIBED PARABOLA. 



80. It is a principle almost self-evident, that the centre of gravity, 

 and the centre of magnitude of a rectangular parallelogram, exist in 

 one and the same point ; consequently, admitting the position of the 

 centre of gravity of the rectangle to be known or determinable a priori, 

 the position of the centre of gravity of the inscribed parabola can 

 very readily be found. 



For by knowing the position of the centre of gravity of a rectangu- 

 lar surface, the pressure upon it can easily be ascertained, and we 

 have shown above, that the pressure upon a parabolic plane, and that 

 upon the surface of its circumscribing parallelogram, are to one 

 another in the ratio of 4 to 5 ; hence, when the pressure on the 



F2 



