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= Sum of Ordinates on Displacement Table. 2,=Sum of Moments (differences x lever) for Centers of Flotation.
Transverse Metacenters, by Tchibyscheff's Rule.
* See Table of Center of Buoyancy and Displacement, pp. 24 to 27.
EXPLANATION OF TABLE, GIVING EFFECT OF FORM OF WATER LINE ON POSITION
OF LONGITUDINAL METACENTER.
Longitudinal and Lateral Stability Compared. The first four lines are exactly the same as those in the other table; and the last eight lines differ only in having length and breadth interchanged, so as to give pitching instead of rolling.
On comparing them with the following table, it will be noticed that, in the algebraic factor, the length and breadth always interchange; and that the numerical factor remains unchanged for forms (1), (3), and (A), namely, the square or rectangle, the circle or ellipse, and the wedge. Of the nine forms selected, these are obviously the only ones in which breadth and length are absolutely interchangeable.
With respect to the comparison of the different forms, one with another, if we disregard the wave-bow No. (8), the variation of stability follows much the same sequence for longitudinal as for lateral stability, but with a somewhat less absolute value. This result might be expected à priori, because the extreme breadth ordinate cuts the outline at right angles in all but the wedge form (9); while the extreme length ordinate meets the outline more sharply. In forms (2) and (4) this difference is only of the second order ; but, as the figures show, it is quite sufficient to be of practical importance even in these.
Differ Chiefly in Wave-Bow.. The wave-bow form (8) falls altogether out of its sequence, and its stability is less than the wedge form (9) as regards pitching. This is due to the sudden falling off of the extreme ordinate length, which meets the curve tangentially, instead of normally, as the extreme breadth ordinate.
Fine Bow Affects Pitch More than Rolling. - If we consider rolling on any given axis, it is easily seen from geometrical considerations, and also from the algebraic form of the integral, that the instantaneous stability depends, firstly, on the length of the transverse axis, and, secondly, on the slowness of the rate of diminution of that axis, as we pass along that axis of motion. Hence sharp bows have less stability for pitching than bluff bows, while their lateral stability for rolling is not so very different.
Caution in Use of Table. In the table of lateral stability, the element of length only appears as a simple factor ; therefore, as regards lateral stability, we may compound the moments by