(9)
Turning the axes to make them coincide with the principal axes of the area A, thus making ∫∫ xy dA = 0,
xh = −a2 cos α, yh = −b2 sin α,
(10)
where
∫ ∫ x2dA = Aa2, ∫ ∫ y2dA = Ab2,
(11)
a and b denoting the semi-axes of the momental ellipse of the area.
This shows that the C.P. is the antipole of the line of intersection of its plane with the free surface with respect to the momental ellipse at the C.G. of the area.
Thus the C.P. of a rectangle or parallelogram with a side in the surface is at 2⁄3 of the depth of the lower side; of a triangle with a vertex in the surface and base horizontal is ¾ of the depth of the base; but if the base is in the surface, the C·P. is at half the depth of the vertex; as on the faces of a tetrahedron, with one edge in the surface.