(2)
The C·P. is thus the C·G. of a plane lamina bounded by the area, in which the surface density is p.
If p is uniform, the C·P. and C·G. of the area coincide.
For a homogeneous liquid at rest under gravity, p is proportional to the depth below the surface, i.e. to the perpendicular distance from the line of intersection of the plane of the area with the free surface of the liquid.
If the equation of this line, referred to new coordinate axes in the plane area, is written
x cos α + y sin α − h = 0,
(3)
R = ∫ ∫ ρ (h − x cos α − y sin α) dx dy,
(4)
xR = ∫ ∫ ρx (h − x cos α − y sin α) dx dy,