THE NAVAL CONSTRUCTOR CHAPTER I. DISPLACEMENT (D). The displacement of any floating body whether it be a ship, a barrel, a log of lumber or, as in the case of the great Philosopher who first discovered its law, the human person, is simply the amount of water forced or squeezed aside by the body immersed. The Archimedian law on which it is based may be stated as :Al floating bodies on being immersed in a liquid push aside a volume of the liquid equal in weight to the weight of the body immersed. From which it will be evident that the depth to which the body will be immersed in the fluid will depend entirely on the density of the same, as for example in mercury the immersion would be very little indeed compared with salt water, and slightly less in salt water than in fresh. It is from this principle that we are enabled to arrive at the exact weight of a ship, because it is obvious that if we can determine the number of cubic feet, or volume as it is called, in the immersed body of a ship, then, knowing as we do that there are 35 cubic feet of salt water in one ton, this volume divided by 35 will equal the weight or displacement in tons of the vessel. If the vessel were of box form, this would be a simple enough matter, being merely the length by breadth by draught divided by 35, but as the immersed body is of curvilinear form, the problem resolves itself into one requiring the application of one of a number of ingenious methods of calculation, the principal ones in use being (1) The Trapezoidal Rule, (2) Simpson's Rules, and (3) Tchibyscheff's method. Simpson's First Rule. The calculation of a curvilinear area by this rule is usually defined as dividing the base into a suitable even number of ecial parts, erecting perpendicular ordinates from the base to the curve, and after measuring off the lengths of these ordinates, to the sum 1 of the end ones; a ld four times the odd and twice the even ordinates. The total sum multiplied by one third the common interval between these ordinates, will produce the area. It should, however, be stated that the number of equal parts need not necessarily be even, and as it is sometimes desirable to calculate the area to an odd ordinate by taking the sum of the first ordinate and adding to it four times the odd ones, and twice the last as well as the even ordinates into one third the common interval, the area may be calculated accurately. In the foregoing definition it should be noted that the first ordinate is numbered “0," and that the number of intervals multiplied by 3 should equal the sum of the multipliers. Area of ABCD= (90 + 441 + 2 y2 + 4 y3 + y4). 3 And if half ordinates be inserted between yo and Y1 and between Y: and Y4 we should then have : Area = (1 4+2 4++11 4+4 y+11 33+2 434 + } 44). Should, however, we desire to calculate the area embraced within the limits of y; only, omitting the half ordinate ył, then :Area . 3 So that it is immaterial what subdivision of parts we may use as long as the multiplier is given the relative value to the space it represents as exemplified in the subjoined table. It will be obvious that we may also give multiplier only half its value, as 140 +241 +1 42 +243 + } 74, and multiply the sum by of x, which will be found the more convenient way to use the rule, involving as it does figuring with smaller values. As proof of the rule let us deal with an example : 3 (yo + 4y1 + y2). Assume curve DFC is part of a common parabola ; area DKCFD is į area of parallelogram. Join DC, and draw parallel = Yi FIG. 2. to GH touching curve. If DFC be part of parabola area, DFC is of parallelogram DCHG. yo+Y2 EK = } (Yo + yı). FK = 2 Parallelograms on same base and between same parallels are equal. Draw through G and H two lines parallel to base as GM and DL, then area DCHG = area DLMG = 2 X X DG Yo + y2 (Yo + 4 yı + y2). 3 Simpson's second rule for determining areas bounded by a parabola of the third order and the “five eight” rule applicable to the calculation of one of the subdivided areas are given in most text-books, but are omitted here as superfluous, Simpson's first rule being adaptable to either of these cases, so that for all ship calculations where areas, volumes, or moments are required, the first rule, or as hereafter explained Tchibyscheff's rule, are recommended. We have seen, then, how the area or surface may be calculated by this rule, and as the volume is the area by the thickness, it will be evident that if the areas be calculated at various levels or water lines, as shown in the figure, and these areas in turn treated as a curve and integrated by means of the rule, that the result will be the volume of the body. Let the Figs. 3 and 4 represent the immersed half longitudinal body of a vessel 100 feet long by 12 feet broad submerged to 5 feet draught as represented by L.W.L. It is required to calculate the volume of water displaced by Simpson's first rule. The base line length between perpendiculars should be divided into an equal number of intervals, and as advocated in the chapter on Design, it will be well to have a definite number and retain same for all designs, as by so doing it will facilitate comparisons and working from one design to another. Ten such intervals with half-end ordinates is a very convenient division, and in this case |