LIMIT TO THE REQUISITE THICKNESS OF FLOOD-GATES. 59 



therefore, the breadths of the respective sections, estimated in order 

 from the surface of the fluid, are 



11.18034, 4.63155, 3.55352, 2.99576, 2.63932 feet. 

 Let A BCD be the rectangular parallelogram, whose length AD or 

 B c is equal to 25 feet, and the breadth A B or DC 

 of any convenient magnitude at pleasure. Upon 

 the length A D, set off the distances AE, A a, A b 

 and AC, respectively equal to the preceding num- 

 bers, taken in the order of their arrangement, 

 and through the points E, a, b and c, draw the 

 straight lines EF, af, be and cd, respectively 

 parallel to AB or DC the horizontal sides of the parallelogram ; then 

 are the rectangular spaces AF, ae, bd and cc, the respective portions 

 into which the given parallelogram ABCD is divided, and on which, 

 according to the conditions of the problem, the perpendicular pres- 

 sures are equal among themselves. 



OF THE REQUISITE THICKNESS OF FLOOD-GATES, &C. 



73. The problem which we have just resolved is a veiy important 

 one; by it we can determine a limit to the requisite thickness of 

 flood-gates and other constructions of a similar nature, and also the 

 form which the section ought to assume, in order that the strength in 

 every part may be proportional to the pressure sustained. 



For according to the preceding notation, and by equation (7) under 

 the third problem, the pressure on the rectangle ABFE, is 



p J6v 2 ssin.^, 

 and the pressure on the whole rectangle ABCD, is 



Pzz: ^bFssin.ty; 

 consequently, by analogy and comparison, we get 



P:p:i?i v\ 



And in like manner it may be shown, that the same relation 

 obtains in respect of any other rectangle AB/a, when compared 

 with the entire figure ABCD; consequently, the pressures are uni- 

 versally as the squares of the depths, the breadth being constant ; 

 therefore, the thickness should be as the square of the depth, being 

 greatest at the bottom and decreasing upwards to the surface of the 

 fluid. 



Thus for example, let the flood-gate be of the same depth as the 

 rectangle in the foregoing question, and let the thickness at the 

 bottom be equal to one foot or twelve inches ; then, the corresponding 

 thicknesses for the several feet of ascent estimated upwards, will be as 

 follows. 



