434 APPLIED MECHANICS 
Let MN (Fig. 707) be a plane surface immersed in a liquid, and k 
the straight line ‘in which the plane of the surface intersects the fre 
surface of the liquid be taken as 
the axis about which moments are 
to be taken. In what follows this 
axis will be referred to as the 
axis. ~ Consider a narrow _hori- 
zontal strip EF which is at a 
distance x from the axis, and let 
a be the true area of this strip. 
The total pressure on the strip EF is 
wax sin 0, where w is the weight of a unit of volume of the liquid, ai 
is the inclination of the surface MN to the horizontal, The Soa 
the total pressure on EF about the axis is wax? sin 6, and the sum of a 
such quantities for the whole area of the given surface is Dwax? sin 4, ¢ 
w sin 02ax?, and this must be equal to the moment of the esultant t 
pressure about the axis. i 
The magnitude of the resultant pressure on- the whole area 
wAx, sin 0, where A is the true area of the given surface, aad a ty t 
distance of its centre of gravity from the axis. If % is the distance of € the 
centre of pressure from the axis, then a 
FACE VIEW. 
Fig. 707. 
Yaa? 
ZwAx, sin 0=w sin Zaz", therefore = oie 
- AX 
But Sax? is the moment of inertia of the surface about the axis; call ing z 
: I = 
this'I,, #=—2. 
Te dn, Ait a 
If the moment of inertia of the surface about an axis parallel to the 
above-mentioned axis, and passing through the centre of gravity of the 
surface, be denoted by I, then (Art. 68) since I) =1+Az2, 
T+ Avg A+ Ach W+eh q 
Ax, Ax Ly a 
where é& is the radius of gyration of the surface voferred to the « 
through its centre of gravity. 
The foregoing demonstration shows that the position of the centre of 
pressure is independent of the inclination of the immersed surface if 1 
distances of the various points of the surface from the free surface 0! a 
liquid measured in the plane of the surface remain unaltered. aa 
The depth of the centre of pressure below the free surface of th 
liquid is obviously @ sin @. p. 
As to the lateral position of the centre of pressure; if the locus 0 
the middle points of the horizontal lines which can be drawn on fl 
surface is a straight line, this line will obviously contain the cent o ¢ 
pressure, and in practically all cases where the centre of pressure 
required in practice, the surface satisfies this condition. : _ 
369. Examples of Centre of Pressure.—The following are the SE 
of most frequent occurrence in practice. In the illustrations, ¢ is 
centre of pressure in each case. The distances d, h, and # are me 
in the plarie of the figure or surface. = 
Fig. 708. A rectangle or parallelogram, with its highest side belo 
z= 
