74 OF THE PARALLELOGRAM AND ITS INSCRIBED PARABOLA. 



Consequently, by analogy, the comparative pressures on the para- 

 bola and its circumscribing rectangle, are as follows : 



p: P ::ibl*s:%bl*s; 

 and from this, by casting out the common quantities and assimilating 



the fractions, we get 



p : P : : 2 : 3. 



COROL. Hence it appears, that when the axis of the parabola is 

 horizontal, and its base perpendicular to the horizon ; the pressure 

 perpendicular to its surface, when compared with that on its circum- 

 scribing parallelogram, bears precisely the same relation, that its area 

 bears to the area of the rectangle by which it is circumscribed. 



85. The practical rule for determining the pressure on the parabolic 

 plane, when placed in the position specified in the problem, may be 

 expressed in words at length in the following manner. 



RULE. Multiply the horizontal axis, by the square of the 

 vertical base or double ordinate, and again by the specific 

 gravity of the fluid ; then, take one third of the product for 

 the pressure perpendicular to the surface of the parabolic 

 plane. 



Or thus, Calculate the pressure on the circumscribing 

 rectangle, and take two thirds of the result for the pressure 

 on the parabola. 



86. EXAMPLE 16. The data remaining as in the example to the 

 foregoing problem, it is required to determine the pressure on the 

 parabolic plane, supposing its axis to be horizontal, its base or double 

 ordinate vertical, and the upper extremity of the base in contact with 

 the surface of the fluid, which, according to the conditions of the 

 previous question, is water, whose specific gravity is expressed by 

 unity, and the weight of one cubic foot of which is equal to 62 Ibs. 

 avoirdupois ? 



Referring the numerical data to the same parts of the figure, as in 



the preceding cases, we have given 6 = 42 feet; /iz:28 feet, and 



s zz: 62 j Ibs. ; therefore, by proceeding according to the rule, we have 



42 X 28 X 28 X 62 J = 2058000, 



which being divided by 3, gives 



p = 2058000 -r 3 = 686000 Ibs. 



Hence it appears, that the total pressures perpendicular to the 

 parabolic surface, according to the several positions in which we have 

 placed it, are to one another respectively as the numbers 

 3087, 2058 and 1715; 



