l45 'An Tnvesttgation of the Pressure sustained It/ 



D. In a flexible stave, forming part of the side of a cistern, 



and supported only at the ends, the inclination at the top is — 



as great as at the bottom. 



The centre of pressure being at one-third of the height, the 



upper support must withstand — , and the lower — of the whole 



force, which, if a be the height, may be called -^o*; and the 



strain at the distance x from the surface will be the difference 

 of the strains produced by the pressure of the fluid and the re- 

 sistance of the support, that is -^a^x rJ?', since the pressure 



of the fluid above the given point, that is — x% may be considered as 

 united in the centre of pressure, and therefore acting at the distance 

 —X. Hence, for the fluxion of the inclination of the stave, 



we have —a^xx -x^i, and the corrected fluent is — -a*x* — 



6 6 Iw 



—tX* 4- h : again, for the ordinate of the curve we find, by a 

 second integration, —a^x'— —a;' + ^x, which must vanish 

 wlien x = a. so that ttjt, a* + ^ = 0, and b= ——-- a-*: hence, 



^ S60 3hO 



^ 17 8 



when x = a, the inchnation becomes -Ta"*— t^^o^ = r?;;flS 



7 



while the initial inclination is represented by b= ^ a*. 



' - 360 



E. If a stave be supported by three fixed fulcrums or hoops, 

 one at each end, the other in the middle, the upper one will 



sustain r- of the whole pressure, the middle -g , and the lower- 



17 



most -r. 



4b 



If we call the distance from the surface x, the pressure at the 



top 1/, and at the middle z, the strain will be first yx r*^} 



and below the middle, calling half the height a)yx-\-z(x—a) — 

 — jc', whence the inclination will be first —yx^— — a;* + ij and 



secondly, ~yx^ — —^x''-\- w^^^ — azx •\- c, and the ordinate 

 first -^yx^— —x^ + bx, and secondly —yx^— 7:777^' + '7*^'"' 

 — azx* -{■ ex + d. Now the ordinate must vanish in the first 



expression 



