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:

Assume curve DFC is part of a common parabola; area DKCFD is area of parallelogram. Join DC, and draw parallel

to GH touching curve. If DFC be part of parabola area, DFC is of parallelogram DCHG.

EK = (yo+Y1).

Yo + y2

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

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

will give a common interval of 10 feet. The draught of 5 feet must likewise be subdivided into a certain number of equal intervals, which in this case we will fix at 4, so that

interval between water lines. These divisions of water lines must be drawn across the body plan of ten sections, and the half breadths read off with a scale and tabulated as in table on following page.

It should be stated in connection with the subdivision of the base line that the length taken for displacement is measured by some designers from the after side of body post i.e., ignoring the propeller aperture; and by others from the fore side of body post to the after side of stem omitting the moulded size of these forgings. Both of these methods are inaccurate besides leading to confusion, as, in the first case, the displacement of the propeller with its boss will equal the displacement cut out for aperture not to mention the volume of the rudder, which is rarely, if ever, taken into account. And in the second case the tiny amount of displacement added at the knuckle formed by the bearding line of plating when the length is taken to forward and after sides of stem and stern post respectively, is compensated for by the gudgeons on stern post. Therefore the most correct and also the most convenient length is from after side of rudder post to forward side of stem at load water line.

Where vessels have a very flat floor line a half water line should be taken be

Required the half-breadth x at the keel for the displacement sheet, where 10 feet is the actual scaled length L, 6′′ the rise of floor, 7" the distance from the rise line to first water line at moulded half-breadth of ship and, of course, 13 inches the water line interval, then :

( W.L. interval) × (3 ordinate interval) × 2 (both sides) 35 (cub. ft. of S. W. in a ton)

= coeff.

The displacement to the load water line being 73.82 tons it is useful to know what relation that weight bears to the vessel if she were of box section, in other words, the amount that has been cut off the rectangular block formed by the length, breadth, and draught, to fine it to the required form, or the block coefficient or coefficient of displacement represented by the symbol "". It will be evident that this coefficient may readily be computed by multiplying the length × breadth × draught, and dividing the product, which is the volume of the box in cubic feet, by 35 to get the tons displaced by the rectangular block. The displacement as calculated, divided by this result, will give the block coefficient "8", or,

66

The range of this coefficient for various types is given elsewhere in the Table of Element Coefficients.

The area of any of the water planes in the specimen displacement table will simply be the sum of the products of the particular