x cos [alpha] + y sin [alpha] - h = 0, (3) _ _ / / R = | | [rho](h - x cos [alpha] - y sin [alpha]) dx dy, (4) _/_/ _ _ / / [=x]R = | | [rho]x(h - x cos [alpha] - y sin [alpha]) dx dy, (5) _/_/ _ _ / / [=y]R = | | [rho]y(h - x cos [alpha] - y sin [alpha]) dx dy. _/_/ Placing the new origin at the C.G. of the area A, _ _ _ _ / / / / | | xd x dy = 0, | | y dx dy = 0, (6) _/_/ _/_/ R = [rho]hA, (7) _ _ _ _ / / / / [=x]hA = -cos [alpha] | | x² dA - sin [alpha] | | xy dA, (8) _/_/ _/_/ _ _ _ _ / / / / [=y]hA = -cos [alpha] | | xy dA - sin [alpha] | | y² dA. (9) _/_/ _/_/ Turning the axes to make them coincide with the principal axes of the area A, thus making [int][int] xy dA = 0, [=x]h = -a² cos [alpha], [=y]h = -b² sin [alpha], (10) where _ _ _ _ / / / / | | x² dA = Aa², | | y² dA = 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. The _core_ of an area is the name given to the limited area round its C.G. within which the C·P. must lie when the area is immersed completely; the boundary of the core is therefore the locus of the antipodes with respect to the momental ellipse of water lines which touch the boundary of the area. Thus the core of a circle or an ellipse is a concentric circle or ellipse of one quarter the size. The C.P. of water lines passing through a fixed point lies on a straight line, the antipolar of the point; and thus the core of a triangle is a similar triangle of one quarter the size, and the core of a parallelogram is another parallelogram, the diagonals of which are the middle third of the median lines. In the design of a structure such as a tall reservoir dam it is important that the line of thrust in the material should pass inside the core of a section, so that the material should not be in a state of tension anywhere and so liable to open and admit the water.