vessel on stout paper, which on being cut out with a knife and swung in two positions, the points being intersected afterwards, will give the centre of gravity (buoyancy) very accurately. Various approximate methods are in vogue for finding this centre, some of which are fairly accurate. (1) Approx. C. B. above base = d 1 (5a-28). 1/d 3 (2) Approx. C.B. below L.W.L. = + where A is the area of load-water plane. A This centre, as will be explained, has an important bearing on the stability of the ship. = Common Interval = 10 ft. x .06 0.6 ft. C.B. abaft No. 5. The locus of the centre of buoyancy in a fore-and-aft direction is of course the centre of gravity of the displacement, and is the pivotal point or fulcrum for the moments of all weights placed forward or aft of this position. It will be obvious, therefore, that its location is of great value in determining the trim of the vessel, and the various alterations thereof due to rearrangements of weights on board. Its position is calculated by taking the areas of the sections and putting them through the multipliers; these functions of areas are in turn multiplied by the number of intervals, (each one is forward or aft of the mid-ordinate,) and the difference between these forward and after moments divided by the sum of the area functions. The quotient resulting is the number (or fraction) of intervals that the centre of buoyancy is forward or aft of the length according as the moment preponderates forward or aft respectively. This centre should be calculated for various draughts, as of course it changes with different draughts and alterations of trim, owing to the changing relationship between the fineness of fore and after bodies at different immersions and trims. Transverse Metacentre (M.C.) The position of this element is, in conjunction with the centre of gravity, the most vital in the design of the ship. As its name implies, it is the centre or point beyond which the centre of gravity of the ship may not be raised without producing unstable equilibrium in the upright position, or, otherwise stated, if the ship be inclined transversely to a small angle of heel, the centre of buoyancy which originally was on the centre line will move outboard to a new position; but, as it acts vertically upward, it must somewhere intersect the centre line. This point of intersection is known as the metacentre. One of the factors in the determination of its location above the centre of buoyancy has already been calculated, viz: the volume of displacement V; the other, the moment of inertia of the water plane about the centre line of ship, we shall proceed to compute. The height M above the C.B. or B.M. is found by : Moment of Inertia of Water Plane Volume of Displacement I The moment of inertia of the water plane is a geometrical measure of the resistance of that plane to "upsetting," or when taken about the centre line, as in the case of calculating for transverse metacentre, to "careening." So that the greater the waterline breadth the higher will be its value; for we must imagine the water plane as being divided into a great number of small areas, and each of these multiplied by the square of its distance from the centre line of ship, when the sum of these products will equal the moment of inertia of half the water plane, about the middle line of vessel as an axis. As both sides of the water plane are symmetrical, the total I will be this result multiplied by 2. Applying this principle to W.L. 4 in the example with which we are concerned, we get the following tabular arrangement:— ORDINATES. Moment of Inertia of Water Plane (I). CUBES OF SIMPSON'S PRODUCTS. HALF- The calculation for Moment of Inertia and Transverse Metacentre above C. B. may be more easily remembered if we treat the cubes of water line half-breadths as the ordinates of a curve twothirds the area of which will equal I, and this, in turn, divided by V will give B.M. However, when we know a, or the coefficient of water line, we may arrive very accurately at the moment of inertia of the water plane, and consequently at the B. M. without the labor of the foregoing calculation by multiplying the Length by the Breadth3 by a coefficient, which coefficient will be determined by a and selected from the table given on page 48. By referring to this table, we find for a (value .694) that the coefficient "i" (inertia coefficient) is equal to .0414, whence we get I=LX B3 × i = 100 × 128 × .04147154 moment of inertia, which is sufficiently close for all purposes, and :— By transposing and taking the calculated I, we find Longitudinal Metacentre (L.M.C.) From the definition given for the transverse metacentre it will be seen that if the ship be inclined longitudinally, instead of, as in the former case, transversely, through a small angle that the point in which the vertical through the altered C. B. intersects the original one will also give a metacentre known as the longitudinal, or L.M.C. Its principal use and value are in the determination of the moment to alter trim and the pitching qualities of the vessel, or longitudinal stability. It will be obvious that the moment of inertia of the water plane must be taken through an axis at right angles to the previous case, viz., at right angles to the centre line through the centre of gravity of water plane, which will be where the original and new water planes cross one another in a longitudinal view. L.M.C. above C.B. = I1 of Water Plane about its C.G.. Therefore, to calculate the M I1, we must figure the moment of inertia with, say, ordinate 5 (or any other one) as an axis when the moment about a parallel axis through the centre of gravity plus the product of the area of water plane multiplied by the square of the distance between the two axes will equal the moment about ordinate 5. The moment of inertia about the midship ordinate we shall call I, and the distance of the centre of gravity from this station = x. The moment of inertia about the centre of gravity of plane = 11. We then have I=I1+ Ax2, or I1=I-Ax2. A clearer conception of this will be obtained from the tabulated arrangement. Area of water plane = 62.41 × ( × 10) × 2. =832.14 square feet. Distance of centre of flotation abaft ordinate 5 = (67.5752.41) 10 =2.42 feet. Moment of inertia of water plane about ordinate 5 = 324.13 × (4 × 10) × 102 × 2 = 432,172 = I. Moment of inertia of water plane about axis through its centre of flotation. = 432,172 - (832.14 × 2.422) = 427,304 = I1. Longitudinal metacentre above C.B. I 427,304 = T 2583.7 165 feet Longitudinal B. M. = |