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.
The varying direction of the inclining couple Pc may be realized by swinging the weight P from a crane on the ship, in a circle of radius c. But if the weight P was lowered on the ship from a crane on shore, the vessel would sink bodily a distance P/wA if P was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of Q with respect to the confocal ellipse, parallel to FF', at a distance FK from F FK= (k2-hV/A)/FQ sin QFF' (2) through an angle 0 or a slope of one in m, given by P sin B= m wA FK - W'Ak 2V hV FQ sin QFF', (3) where k denotes the radius of gyration about FF' of the water-line area.
- Confocal Elliptic Cylinders.
- Employ the elliptic coordinates n,, and -=n+Vi, such that z=cch?, cchncos,y=cshnsin-; (1) then the curves for which n and are constant are confocal ellipses and hyperbolas, and -d(n,) =c 2 (ch 2 n - cost) = 2c 2 (ch2n-cos2) = r i r 2 = OD 2, (2) if OD is the semi-diameter conjugate to OP, and ri, r 2 the focal distances, rl,r2 = c (ch n cos 0; r 2 = x2 +y2 = c 2 (ch 2 n - sin20 = 1c 2 (ch 2 7 7 +cos 2?).
In a similar way the more general state of motion may be analysed, given by w =r ch2('-y), y =a+, i, (26) as giving a homogeneous strain velocity to the confocal system; to which may be added a circulation, represented by an additional term in w.
A system of confocal ellipsoids is taken y2 (3) a 2 +X b 2 +X c2 + A= I, and a velocity function of the form = x1 P, (4) where 4' is a function of X only, so that 4) is constant over an ellipsoid; and we seek to determine the motion set up, and the form of >G which will satisfy the equation of continuity.
When the liquid is bounded externally by the fixed ellipsoid A = A I, a slight extension will give the velocity function 4 of the liquid in the interspace as the ellipsoid A=o is passing with velocity U through the confocal position; 4 must now take the formx(1'+N), and will satisfy the conditions in the shape CM abcdX ¢ = Ux - Ux a b x 2+X)P Bo+CoB I - C 1 (A 1 abcdX, I a1b1cl - J o (a2+ A)P and any'confocal ellipsoid defined by A, internal or external to A=A 1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox BA+CA-B 1 -C1 W'.
The extension to the case where the liquid is bounded externally by a fixed ellipsoid X= X is made in a similar manner, by putting 4 = x y (x+ 11), (io) and the ratio of the effective angular inertia in (9) is changed to 2 (B0-A0) (B 1A1) +.a12 - a 2 +b 2 a b1c1 a -b -b12 abc (Bo-Ao)+(B1-A1) a 2 + b 2 a1 2 + b1 2 alblcl Make c= CO for confocal elliptic cylinders; and then _, 2 A? ?
The potential of such a shell at any internal point is constant, and the equi-potential surfaces for external space are ellipsoids confocal with the ellipsoidal shell.
Legendre shows that Maclaurin's theorem with respect to confocal ellipsoids is true for any position of the external point when the ellipsoids are solids of revolution.
The third memoir relates to Laplace's theorem respecting confocal ellipsoids.
For different values of 0 this represents a system of quadrics confocal with the ellipsoid ~f+~1+~I, (~4)
Now consider the tangent plane w at any point P of a confocal, the tangent plane fii at an adjacent point N, and a plane of through P parallel to of.
Confocal conics are conics having the same foci.
If one of the foci be at infinity, the conics are confocal parabolas, which may also be regarded as parabolas having a common focus and axis.
An important property of confocal systems is that only two confocals can be drawn through a specified point, one being an ellipse, the other a hyperbola, and they intersect orthogonally.