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102 OF FLUID PRESSURE UPON THE ANNULT OF A CYLINDER. 



Put D = AB or DC, the diameter of the proposed cylindrical vessel, 

 A =. the area of its bottom, 

 d =nm, the whole perpendicular depth, or altitude of the 



cylinder, 



x ~Aa, the breadth of the first annulus, 

 x' z=ab, the breadth of the second, 

 x" c, the breadth of the third, 

 x'" CD, the breadth of the fourth, and so on, to any number 



of annuli n, 

 P the pressure on the concave surface of the cylinder, or the 



sum of the pressures on the several annuli into which 



it is divided, 

 p the pressure on the bottom of the vessel, and each of the 



several annuli, 

 TT =z 3.1416 the circumference of a circle whose diameter is 



unity, and 

 s zr the specific gravity of the fluid. 



Then we have, nk \x ; ni in x -\- \x ; n h x -f- x' -|- \x", 

 and ng x -\- x 1 -|- x" + k x> " an( * by the principles of mensuration, 

 the area of the bottom of the vessel, is 



AZrjTTD*, 



and by Proposition (1), the pressure upon it, is 



p l7n>*ds= .7854 i>M$, (62). 



Again, by the principles of mensuration, the concave surfaces of the 

 respective annuli are as follows, viz. 



For the first annulus, we have TT D x 3.1416 D x, the surface, 



second TTDO;' ~ 3.1416 D x', 



third TTDX" 3.1416 DX", 



fourth TT DB"' = 3.1416 Da? 1 ", 



&c. &c. &c. 



And by Proposition (1), the pressures perpendicular to these sur- 

 faces, are respectively as below, viz. 

 For the first annulus, the pressure isp=zl.5708Da: a s, 



second j0=3.1416D*' s(x+ J#'), 



third p 3.1416Dx"s(aj+x / +Ja: 1 '), 



fourth - - P =i3.U\6x'"s(x+x'+x"+%x f "), 



&c. &c. &c. 



Now each of these pressures, according to the conditions of the 

 problem, is equal to the pressure upon the bottom, exhibited in the 

 equation (62) ; consequently, by comparison, we have 



