## Confocal Sentence Examples

- 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 twocan be drawn through a specified point, one being an ellipse, the other a hyperbola, and they intersect orthogonally.**confocals**