heel over; and the vessel does heel over to leeward until the moment of the righting couple, or moment of stability, is sufficient to balance the heeling moment. Thus the power of a vessel to carry sail depends on her stability, and must be kept in view in designing her. It is evident that, for the purpose of propulsion by sails, it is not sufficient that the hull of the vessel should be of such a shape as to meet with little resistance to forward motion through the water; its shape should also be such as to meet with great resistance to transverse motion or leeway. CHAPTER VII. Working or Manoeuvring Qualities of a Ship-Design. THE working or manoeuvring qualities of a ship depend on the combined action of the rudder, the propelling apparatus, the sails, and the figure of the ship's hull. In vessels under steam alone, the working qualities chiefly required are those of going readily astern as well as ahead, and of turning quickly and accurately, and in as small a circle as possible, under the action of the rudder. Sometimes, as in vessels driven by two screws with independent engines, the propelling apparatus may be used so as to turn the ship when required. In vessels under sail, a much more complex combination of qualities is required. One of the most important is that of working well to windward, for which purpose it is essential that the ship should make little leeway, and should carry a weather helm; that is to say, that the action of the sails, unaided by that of the helm, should tend of itself to bring the ship's head towards the point from which the wind blows; this property, when in excess, is called ardency. It depends in a great measure on the position of the centre of effort of the sails, relatively to the resultant of the resistance to leeway. The manoeuvring of a ship under sail, too, depends not merely on the action of the rudder and of the ship's hull on the water, but on the positions of the masts, and the figure and dimensions of the sails. The working qualities of a ship are materially affected by her trim, or the position in which she floats. This depends on the position of her centre of gravity relatively to her centre of buoyancy; and the position of the centre of gravity depends partly on the stowage of the lading, as has been already stated in Chap. III. Design.-From the brief summary which has been given of the qualities sought in a ship, and the means of obtaining them, it is evident that every one of those qualities is more or less affected by every circumstance in the figure and dimensions of the ship, the distribution of her weight, and the nature and arrangement of her means of propulsion; and, consequently, that the naval architect must keep the whole of those qualities, and the whole of those circumstances, before his mind at once, in designing a ship. CHAPTER VIII. Areas of Arbitrary Plane Figures-Trapezoidal Rule--Simpson's First and Second Rules. THE area of a plane figure bounded by a curved line of arbitrary form, such as a frame or a water-line of a ship, is found by measuring a sufficient number of parallel and equidistant ordinates, conceiving the figure to be divided by certain of those ordinates into figures of the parabolic kind, computing the areas of those figures, and adding them together; or else computing the sum of those areas at one operation. In Fig. 7, let ABCD be the plane figure whose area is to be measured, bounded by the straight base-line or axis of abscissæ, AD, by two ordinates, AB and DC, at right angles to the base and by the curved line, BC. Divide the base into a sufficient number of equal intervals, and draw and measure ordinates at the points of division. The total number of ordinates, including the two endmost ordinates, will of course be one more than the number of intervals. If the area is to be measured by conceiving it to be divided into trapezoids (that is, by conceiving BC to be made up of straight lines), the number of intervals into which the base is divided may be either odd or even. If the area is to be measured by conceiving it to be divided into parabolic areas of the second order, the number of intervals must be even; if into parabolic areas of the third order, the number of intervals must be a multiple of three. In the example represented in the figure, the base is divided into twelve equal intervals, which will suit any one of those three methods. For the sake of uniformity in stating the rules for calculation, the ordinates which separate the parabolic areas into which the figure is conceived to be divided from each other will be called dividing ordinates, and all the other ordinates, except the endmost ordinates, intermediate ordinates. I. Trapezoidal Rule.-Here every ordinate, except the endmost ordinates, is a dividing ordinate. Add together all the dividing ordinates, and one-half of the endmost ordinates; multiply the sum by the common interval : the product will be the required area, nearly. This is the simplest rule; but for figures whose boundaries present long sweeps of convexity or concavity, it is only a rough approximation. II. Parabolic Rule of the Second Order, or Simpson's First Rule. Here the number of intervals must be even; and the dividing ordinates are at the distance of a double interval from each other, being those at the points 2, 4, 6, &c., in Fig. 7. The intermediate ordinates are those at 1, 3, 5, &c. Add together the endmost ordinates, double the dividing ordinates, and four times the intermediate ordinates; multiply the sum by one-third of the common interval: the product will be the required area, nearly. This is the most generally useful of all rules for measuring areas. It is capable of any required degree of accuracy, if the ordinates are made numerous and close enough. III. Parabolic Rule of the Third Order, or Simpson's Second Rule. Here the number of intervals must be a multiple of three; and the dividing ordinates are at the distance of a treble interval from each other, being at the points marked 3, 6, 9, in Fig. 7. The intermediate ordinates are at 1, 2, 4, 5, &c. Add together the endmost ordinates, double the dividing ordinates, and three times the intermediate ordinates; multiply the sum by three-eighths of the common interval: the product will be the required area, nearly. This rule is more complex than Simpson's first rule, and not more accurate. [In algebraical symbols, those three rules for mensuration are expressed as follows:-Let L denote the whole length of the base, and n the number of intervals into which it is to be divided; then the common interval is given by the formula n being a multiple of 2 or 3, according to the order of the parabolic curves of which the boundary of the figure is held tc consist. Let the ordinates corresponding to the following abscissæ, 0, ▲ x, 2 ▲ x, 3▲x, &c. n^x=L, be denoted as follows Y1, 92, 93, Y49 &c. Yn+1. This mode of numbering the ordinates is that practised by naval architects. Amongst pure mathematicians it is more common to number them as follows- because of the convenience of having each ordinate marked by a number proportional to the corresponding abscissa; but the former method of numbering is adopted in the following equations :— ƒ ydx= 3 Ax (Y1 + 3 Y2 +3 Y3 +2у1+3y+3у+2y7+&c. Mathematical principles might here be explained, for determining how close together the ordinates ought to be, in order to give an approximate area of any required degree of accuracy; but it is unnecessary to do so; because the constructor, after a little experience in the use of the rules, learns to judge by the eye whether the ordinates are close enough. Where the curved boundary of the figure is nearly at right angles to the ordinates, and where it is nearly straight, the ordinates may be far apart. Where the curved boundary is very A |