12 



HYDROSTATICS. 



way down, their walls may taper from 

 the bottom upwards, provided the 

 thickness has been ascertained which 

 is sufficient to resist the pressure at the 

 greatest depth. So in a dam or flood- 

 gate, one side being perpendicular, the 

 other may slant towards the top. In 

 constructing a bank of a given quan- 

 tity of materials, against whose slop- 

 ing side the water presses, it is 

 found by mathematical reasoning, that 

 it will best resist the water, if the 

 square of its thickness at the base (that 

 is, the thickness multiplied by itself) is 

 to the square of its perpendicular height, 

 as the weight of a given bulk of water, 

 say a cubic foot, is to the weight of the 

 same bulk of the material the bank is 

 made of. Thus if the bank is of com- 

 mon stone, which is 2 times heavier 

 than water ; the thickness at the base 

 should be to the height nearly as 3 to. 

 43, (9, or 3 times 3, being to 22-^, or 

 4 multiplied by 43, nearly as 1 to'2i). 

 Therefore, a bank of 3 feet base and 

 4 feet 9 inches height, will answer the 

 purpose best. If the bank be of fir 

 timber, which is little more than half 

 the weight of water, the base being 

 a yard, the height should be about 

 2 feet 2 inches. 



If a fluid presses upon a surface, 

 there is a point of that surface at 

 which if a force be applied in the 

 same line with the pressure of the 

 fluid, and equal to the whole of that 

 pressure, but in a contrary direction, 

 it will exactly balance or counteract the 

 whole pressure of the fluid ; and this 

 point is called the centre of pressure. 

 If the water in the upper part of the 

 vessel A O C D Q H F (Jig. 14.) presses 

 against the surface B C D E, there is a 



fig. 14. 



point P, in that surface, against which, 

 if a force be applied in the opposite 

 direction P M, and equal to the whole 

 pressure of the water upon B C D E, it 

 will support B C D E, and prevent the 

 pressure from turning it or moving 

 jt in any way. It is here supposed 



that there is no water below the sur- 

 face B C D E, but that, if unsupported, 

 the water above would press it down. 

 If the opposite force were applied to any 

 other point than P, the centre of pres- 

 sure, the water would make the surface 

 turn round upon that point. 



To find this point often becomes of 

 great importance. It may be the best 

 means of propping a floodgate or other 

 surface from behind against the pressure 

 of water upon its face. The position of 

 the point varies according to the figure 

 of the surface and its depth under wa- 

 ter. If the surface is a rectangle of any 

 kind, as a square, standing upright, and 

 the water rises to its upper edge, the 

 centre of pressure is two-thirds down 

 the line, dropped perpendicularly from 

 the middle of that upper edge. If the 

 surface is a triangle, whose point is at 

 the surface of the water, the centre of 

 pressure is three-fourths down the per- 

 pendicular dropped from the point; 

 and if both sides of the triangle are 

 equal, the centre is in that perpendi- 

 cular. If the sides are unequal, but the 

 triangle is rightangled at the base, the 

 centre of pressure is three-fourths down 

 the perpendicular side, and three- 

 eighths of the base distant from that 

 side. 



Thus if A M (Jig. 15.) is the surface 

 of the water, A C D M the upright 



M_B 



and square surface against which the 

 water presses, N, the middle point of 

 A M, and N L a perpendicular from 

 N to the base of the surface, N P being 

 two-thirds of N L, P is the centre of 

 pressure upon A C D M. If A S D be 

 a triangle, and A S equal to A D, q 

 being taken at three-fourths down A C, 

 is the centre of pressure on the tri- 

 angle AS D. IfACD be a triangle, 

 rightangled at C, A q being three- 

 fourths of A C, and q u or C O three- 

 eighths of C D, u is the cenlre of pres- 

 sure upon A C D. Therefore, if a 

 force be applied on the opposite side of 

 A C D INI at P, of A S D at q, and of 

 A C D at u, equal to the pressure of the 

 water on these surfaces respectively, and 



