HYDROSTATICS. 



depth ; except for the violence of the waves, the 

 same strength of dike would resist the weight in 

 both cases. 



At the depth of a foot, the pressure on a square 

 inch of surface is a column of 12 cubic inches, 

 weighing, in the case of fresh water, very nearly 

 7 oz. ; and in the case of salt water, slightly more 

 than 7 oz. This gives, at the depth of 7 feet, a pres- 

 sure of 49 oz. or about 3 Ibs. As water is so very 

 slightly compressible, we may assume, without 

 any great error, that its density is uniform at all 

 depths, and, therefore, that the pressure will go 

 on increasing regularly at the rate of 3 Ibs. on the 

 square inch for every 7 feet in depth. It is thus 

 easy to calculate the pressure at any given depth. 



At a thousand feet, for instance, it will be 



7 

 X 3 = 428'6 Ibs. or about 29 atmospheres. At 



the depth of a mile, or 5280 feet, it will be 



X 3 = 2263 Ibs. or 150 atmospheres. 



Centre of Pressure. We have seen that a verti- 

 cal line or section AC (fig. 8) of a vessel contain- 

 ing a liquid, is pressed outwards by a set of forces 

 increasing in intensity at every point from A to C. 

 Now, there is evidently a point somewhere in AC 

 at which, if a single pressure of sufficient strength 

 is applied on the outside in the opposite direction, 

 it will counteract the effect of all the outward 

 pressures at the several points on the inside, and 

 hold the line in its place. This point is called the 

 Centre of Pressure; it is the point of application 

 of the resultant of all the elementary pressures 

 exerted at the different points. 



As the pressures on the opposite sides of the 

 centre of pressure must balance as on a fulcrum, 

 it is clear that it cannot be on the middle of AC, 

 but nearer the bottom, where the pressures are 

 the greatest. And since the distribution of the 

 pressure is represented by the triangular area 

 AC<? (fig. 8), the centre of pressure will be deter- 

 mined by that of the centre of gravity of the 

 triangle ; in other words, it will be at the point 

 4, which is \ of the height of the triangle (see 

 NATURAL PHILOSOPHY). 



For surfaces of other forms than rectangular, 

 the investigation of the centre of pressure is too 

 difficult for introduction here. On a triangular 

 side in the position of ABC (fig. 10), it is deter- 

 mined to be at m, % the depth ; in the position of 

 fig. 11, it is at #, t from the bottom. 



Each stave of a vat or cask may be considered 

 as a rectangular side ; the position, therefore, 

 where a single hoop has the greatest effect in 

 resisting the pressure, is at i from the bottom. 



THE SURFACE OF LIQUIDS. 



The surface of a mass of liquid at rest is a 

 perfect level. Two or more points are on the 

 same level when they are equally distant from 

 the earth's centre. Let EAPBS represent a 

 portion of the earth's circumference, C its centre, 

 and L, T, two points above the surface equally 

 <L slant from C ; then L and T are on the same 

 level ; and if an arc of a circle is described from 

 C through these two points, any other points, as 

 Q, R, in that arc are on the same level with L 

 and T. In like manner, the points A, P, B, in 

 the earth's circumference, are on the same level ; 

 and so are m and ;/, below it. A level line, then, 



is, strictly speaking, not a straight line, but a 

 portion of a circle ; and similarly, a level surface 

 is not a plane, but a portion of a spherical surface. 

 The direction of gravity that is, the direction 

 of a plumb-line is always towards the earth's 

 centre. Thus, at T or at B, the direction of 



JPH 



gravity is the line TBC ; at Q, it is the line QC ; 

 and so of the other points in the figure. These 

 lines marking the direction of gravity are perpen- 

 dicular to the earth's surface, or to any other level 

 surface, as LHT. A straight line, as TB, is per- 

 pendicular to a curve when it is perpendicular to 

 the line B drawn touching the curve at the point 

 B. The lines TB, HP, LA, that mark the direc- 

 tion of gravity, are called vertical lines ; and lines 

 at right angles to them, such as o~Qu, LPT, vQw, 

 are horizontal lines. 



When we confine our attention to a small arc 

 of a very large circle, it does not differ sensibly 

 from a straight line. Owing to this, ' horizontal ' 

 and ' level ' are used, in common language, to 

 signify the same thing. It is only when we take 

 in a considerable space that the difference becomes 

 appreciable. When considering a mass of water 

 in a vessel or pond, then, the surface may be 

 assumed to be straight or plane, and all vertical 

 lines, or lines shewing the direction of gravity, as 

 parallel to one another. 



The familiar fact, that the surface of a liquid at 

 rest is always level or horizontal in the popular 

 sense admits of explanation ; it can be shewn to 

 be a consequence of the primary property of 

 ' pressing equally in all directions.' For let da and 

 cb be vertical lines, or lines in the direction of 

 gravity ; and ab a plane at right angles to that 

 direction, or horizontal. A particle of the liquid 

 at a is pressed by the column of particles above 

 it from a to d; and the like is the case at b. 

 Now, since the liquid is at 

 rest, these pressures must be 

 equal ; for if the pressure at 

 b, for instance, were greater 

 than at a, there would be a 

 flow of the water from a to- 

 wards b. It follows that the 

 line ad is equal to be, and 

 hence that dc is parallel to 

 ab, and therefore horizontal. 

 The same might be proved 

 of any two points in the 

 surface ; therefore the whole 

 is in the same horizontal 



Tig. 13- 



plane. Or we might prove the same thing by the 



229 



