simple addition for a vessel built up in different lengths for the different forms. Thus, the values in lines 1 to 8 of column (2) are simply the means of the corresponding values in columns (1) and (3). We cannot apply this process to the longitudinal stability because here the length element enters as a cubic factor. If we were so to compound the moments of length, what we should really do would be equivalent to screwing together two longitudinal halves of different vessels; in the case before mentioned, screwing half a box to half a tub; not introducing a flat midship length between two semicircular ends.

Explanation of Table Giving Effect of Form of Water Line on Position of Metacenter.

Explanation of Table. By the preceding table we can at once make an approximate estimate of the value of any proposed form of water line, by selecting that form in the table to which it comes nearest. From this table we gather that the more nearly the water line approaches to a right parallelogram, the more it will contribute to the stability of a ship. No. 9, on the contrary, the straight line wedge form, is the least stable of these water lines, and from the comparison of the successive groups of lines on the table we shall see exactly how this comes about.

Areas on Water Lines. The first and second lines in the table give the measures simply of the areas of those water lines. From lines 3 and 4 we see that, Fig. 1 being taken as the standard of comparison, Fig. 2 only contains 89 per cent of the rectangular area, and this diminution is effected merely by rounding off the rectangular corners, the length and breadth remaining the same in both. In Fig. 3, when the curvature of the ends extends quite to the middle of the water line, its area is reduced to 69 per cent. In Fig. 6, by forming the water line of parabolic arcs, a favorite form of some builders, the area is reduced to twothirds of the rectangle. Figs. 7 and 8 are the lines used for a wave stern and a wave bow; from which it appears at once how much more powerful the stern contributed to the stability of a ship than the bow; the stern line being 62 per cent, and the bow line only 50 per cent.

Metacentric Moments. Lines 5 and 6 are the actual measure of the stability (by its moments) for small inclinations. For example: in the rectangle, the moment is one-twelfth part of the product of the length by the cube of the breadth, or .08 of that product; and as we pass along line 6 we find it gradually diminish, until, in the wedge form, it is only .02, showing that a sharp wedge form has only one-fourth part of the power to carry top weight that the rectangular form has, although its power of buoyancy, or power to carry absolute load, is one-half. This is set out more fully in lines 7 and 8; so that by carefully comparing together line 4 and line 8, the relative values of all those figures for carrying absolute weight and for carrying top weight may be clearly seen.

Metacentric Intervals. Lines 9 and 10 measure the powers of ships, formed on these water lines only to carry top weight without upsetting.

*The length L appears simply as a factor. The numerical factor in the table, therefore, remains unchanged if the proportion of L to B be altered, as in passing from the square to the rectangle, or from the circle to the ellipse.

That is to say, a trochoid twice the length of a cycloid of the same width,

The entries in these lines assume that the vessel is flat-bottomed, with vertical sides. The other entries hold good whatever may be the shape of the vessel under water. In general, the height of the metacenter may be found by dividing the entry in lines 5 or 6 by the displacement.

*The length L appears simply as a factor. The numerical factor in the table, therefore, remains unchanged, if the proportion of L to B be altered, as in passing from the square to the rectangle, or from the circle to the ellipse.

That is to say, a trochoid twice the length of the cycloid of the same width.