Suppose a ship is found to answer well at some given water line AC, Fig. 4, Plate CCCCXCV. Let the areas of the transverse vertical sections be divided by some constant quantity, as, for instance, the breadth, and suppose the distances a b, c (1, &c. equal to the quotients, to be set off on the respective sec tions, from the water line; then a curve drawn through the points b, (I, Re. will be the curve of sections. It will he found to be convex to the water line at the extremities.
The order of the parabola which coincides for the greatest distance with this line may easily be found.
Let the general equation to the parabola be ex pressed by y" =- e x: then it is always possible to de termine n and a, so that the parabola shall pass th•ough two points besides the vertex; any two points between and C may be taken, but it is evident that the farther apart the three points are taken, the longer will the parabola coincide with the line of sections; of course neither point may be in the convex part of the line of sections. It will be found that the point g, at the foremost frame, and h in the middle between g and b, are the p nets which should be taken.
Draw a tangent to the curve at the point b, which will be of course para.11el to the water line; then in Is and n g are abscissas; b in and b n ordinates to a para bola passing through b, h, and put in n g = x", b 922 = y', and b is = y"; then substituting these values in the equation to the parabola, we have y'" = a x', and y"„ =-- a x" or n . log . y' = log . a log . x' and n log y" = log . a + log . x" x' — Io7 tc" hence n = log. y — log. y" ?in .= x' log. x' . log,. y" — log. x" . y' and log. = log. y' — log. y" We have now the value of n and a, and by calculating several other abscissas we can trace the parabolic curve. The same operations applied to the after body will give the exponent and parameter of the parabola, which is the most similar to the curve of sections in that body.
It generally happens that the exponents are nearly the same in both bodies, if the place of the midship section be determined in the manner shown in the sequel.
It will be found that the parabola and the line of sections very nearly coincide; the former being some times a little within the latter between g and lt, and without at the foreside of It; and sometimes, but much more seldom, the contrary. The parabola always cuts the water line at a short distance from the rabbets, this distance being rather greater forward than abaft.
Several American ships of war have been submit ted to this method of investigation. which was found to answer very well h their bodies: indeed there can be no great deviation, as the parabola varies ac cording t its exponent and parameter; if the ship is full, a exponent adapts it to that shape; and if the ship is lean, a small one. If the body has a long
straight of breadth, and sharpens quickly at the ex tremities, by deducting. a part in midships from the comparison, the system may still be applied; or if, as is the case generally with English merchant ships, there is a very great draught of water in proportion to the breadth, by deducting a part from the water line downwards, this method may be applied to the remainder.
From this reasoning it appears that ships may be constructed to coincide exactly with the parabolic line, without deviating from the forms which experi ence has proved to be the most conducive to giving ships good qualities. Chapman stated that this sys tem would most probably be superior to the old one, and the result has confirmed his statement; for ships of the line, frigates, and merchantmen, have been constructed after it, all of which have been very fine vessels.
From the manner in which the curve of secticus is formed, it follows that its area, multiplied by the breadth, is equal to the displacement, and that the centre of gravity of the area is in the same transverse section as the centre of gravity of the body: but the area of this curve, supposing it to he a parabola of a certain power, is a known part of the rectangle formed by the greatest ordinate and the abscissa; hence. by making the areas of the sections decrease in the ratio of the abscissas in the parabola, we obtain certain equations between the quantities. To find these equa tions, suppose the parabolic line, now also represent ing the line of sections, to be ACB; Fig. 5, Plate CCCCXCV., cutting the water line at some distance from both rabbets; let C be the place of the midship section, and DC the greatest abscissa. Put AB = 1 and DC = d, let the exponent of the parabola before and abaft = n, and the displacement =-- D; then the area of the parabolic line BDACB = n . 1. (1, and the displacement = n+ I 1 . d . B (B representing the breadth;) but t/ B = area of the midship section; hence . 1 . (area of niidship section) = D . . . ( 1).
Let E be the middle point of the water line AB, which we may call the construction water line, F the place of the centre of gravity in point of length; let ED, the distance the midship section is before the middle of the water line, = k, and EF, the distance the centre of gravity is before the middle, = we will now determine the place of the midship section in reference to the situation of the centre of gravity F.
As BCD represents the displacement of the fore body. and CDA that of the after body, the moments of these two parts will give the common moment.
The centre of gravity of the parabolic area is at a distance from the abscissa DC 22• ± 1 n X the ordinate DB; and for the parabolic area DCA it is n} 1 DA.
2 n + The moment of DCB from the point E n + I