Let ACDB, Fig. 1, Plate CCCCXCI. be a plane in clined at any angle to the horizon, and let the fluid im pinge on it with a force represented by EF, in a hori zontal direction from E to F.
From F, let Fl be drawn parallel to the horizon, and from E, the line EG perpendicular to the assumed plane. Then will EG represent the measure of the force which acts at right angles to the plane. From G draw GH perpendicular to EF. Then will EH de note the relative action of the direct force. From E also, draw El perpendicular to Fl, and join the points G and 1; then will the plane EGI be perpendicular to the horizon. Moreover, from G let a perpendicular be drawn to El, or, which is the same thing, to the horizon, and then will GK represent the relative ver tical force. Join the points K and H, and which be ing perpendicular to EF, will represent the relative lateral force. Hence the relative direct, vertical, and lateral forces, acting at the point F, are represented in value by the lines EH, GK, and KU.
To apply these principles to the fore-part a egof a ship, Fig. 2, let a a, b b, c c, Cc. represent some of its water lines, at equal distances from each other; and AK, BL, MN, Ste. vertical sections, also at equal dis tances from each other. The intersections of these water lines and vertical sections, will form a series of trapeziums, each of which must be divided into tri angles, by having its diagonal drawn. Draw, for ex ample, the diagonal All of the trapezium ABCD; and from D and A the extremities of the same diagonal, draw the lines DF and AE perpendicular to AC and DB; and from E and F, in like manner, let fall the perpendiculars Eli and PG to the water lines a a and b b.
In the next place, in Fig. 5, draw the parallel lines RS and PQ at the same distance from each other, as the vertical sections, and to these lines draw the per pendicular RP. Transfer the distance DF in Fig. 2, from P to T in Fig. 3. Draw TR, and from T draw TU perpendicular to TR, to meet RP produced in U.
If, now, UR. be supposed to express the absolute force of the fluid, UP will represent its relative direct action; FG, Fig. 2, its vertical action, and GD, in the same figure, its lateral action, those lines representing the fluid's action on the triangle ACD. Now, as the absolute force is constant, we may adopt for its repre sentative the interval between the sections; and, for this purpose, draw PTV at right angles to Ri', and AVX perpendicular to RP. Then will PX denote the relative direct force.
From Fig. 2, transfer the distance FG, which is the vertical force, when the absolute force is denoted by RU, from Z to Y in Fig. 3. And as the force RU has been reduced to RP, so the force, whose measure is ZY, should be reduced in the same ratio. Draw, therefore, the line TX; and since YZ : Q a : : 111) : RX : RI', it follows that Y 0 will represent the relative vertical force on the triangle ACD.
Again, transfer DG from P toy in Fig. 3, and draw Then, in the same matinee, will c)`X be the mea sure of the lateral force on the same triangle; and it will moreover appear, that RP being the measure of the absolute force, l'x, 0Y, and J'X will be the meastn es of the relative direct, the relative vertical, and the rela tive lateral forces; and we shall obtain the entire effect of these forces by multiplying each of them respectively by the triangular area ACD. In a similar manner may the effect of the fluid be ascertained on the triangle A DB, and on every other triangular area which may be found on the surface of the vessel.
If, therefore, the effect of the resistance on the fore part of the vessel be represented by and that on the afterpart by N, it follows that by applying the co-effi cients of resistance, the whole resistance will be pro portional to 6:41 + 7N. But as we have taken for the representative of the absolute force, the distance be tween the sections, and which may be greater in one place than in another, the relative force on one ship cannot be compared with that of another. It hence be comes necessary to find a plane figure, which, being moved in the water with the same velocity us the ship, shall meet with the same resistance. Such surface is called the plane of resistance. o determine the value of this plane of resistance, let x represent it, and let m be the absolute force of a particle. Then will nix be the resistance on the forepart of the vessel, and the sante quantity nix be also the measure of the negative resist ance, or the diminution of the pressure rot ward, in con sequence of the motion. Hence, multiplying by the co efficients of resistance, we shall have 6nix + 7 nix = t3nis, for the measure of the whole resistance on the plane represented by x; that is 13mx = 6:41 7N, and x = 6M + 7N . If we suppose 14I = 18, N = 16, 13m and m = 5, we shall have x = 3.38, that is, the ship will meet with the same resistance as a plane whose area is 3.38 feet, when moved with the same velocity as the ship.