(9) Turning the axes to make them coincide with the principal axes of the area A, thus making f f xydA = o, xh = - a 2 cos a, y h = - b 2 sin a, (io) where ffx2dA=Aa2, ffy 2 dA= Ab 2, (II) 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.
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.
An inclining couple due to moving a weight about in a ship will heel the ship about an axis perpendicular to the plane of the couple, only when this axis is a principal axis at F of the momental ellipse of the water-line area A.
Of the wedge of immersion and emersion, will be the C.P. with respect to FF' of the two parts of the water-line area, so that b 1 b 2 will be conjugate to FF' with respect to the momental ellipse at F.
Of the fluid, equal to the weight vertically upward through the movement of a weight P through a distance c will cause the ship to heel through an angle 0 about an axis FF' through F, which is conjugate to the direction of the movement of P with respect to an ellipse, not the momental ellipse of the water-line area A, but a confocal to it, of squared semi-axes a 2 -hV/A, b 2 - hV/A, (I) h denoting the vertical height BG between C.G.
Since they are essentially positive the quadric is an ellipsoid; it is called the momental ellipsoid at 0.
A limitation is thus imposed on the possible forms of the momental ellipsoid; e.g.
If A = B = C, the momental ellipsoid becomes a sphere; all axes through 0 are then principal axes, and the moment of inertia is the same for each.
If all the masses lie in a plane (1=0) we have, in the notation of (25), c2 = o, and therefore A = Mb, B = Ma, C = M (a +b), so that the equation of the momental ellipsoid takes the form b2x2+a y2+(a2+b2) z1=s4.
Which may be called the momental ellipse at 0.
It possesses thi property that the radius of gyration about any diameter is half thi distance between the two tangents which are parallel to that diameter, In the case of a uniform triangular plate it may be shown that thi momental ellipse at G is concentric, similar and similarly situatec to the ellipse which touches the sides of the triangle at their middle points.
The relation between these axes may be expressed by means of the momental ellipsoid at 0.
If p be the radius-vector 0J of the momental ellipsoid Ax+By+Czf=Me4 (I)
We have seen (~ 18) that this vector coincides in direction with the perpendicular OH to the tangent plane of the momental ellipsoid at J; also that ~ (2)
The motion of the body relative to 0 is therefore completely represented if we imagine the momental ellipsoid at 0 to roll without sliding on a plane fixed in space, with an angular velocity proportional at each instant to the radius-vector of the point of contact.
It has been shown by Dc Sparre that, owing to the limitation imposed on the possible forms of the momental ellipsoid by the relation B+C>A, the curve has no points of inflexion.