Let a b, Fig. 5. Plate CCCCLXXXVIII, represent an element of the ship's body, and c b the direction of the motion and the height due to the velocity of a par ticle of water, which meets this element. By resolv ing c b into c d and d b, c d, which is perpendicular to a b, is supposed to be destroyed, and the particle of water glides along the surface of the ship's body with a velocity d b. Let e b be equal to z, and the depth of the particle below the surface of the water be equal to k, and the angle a b c equal to i; then d b = z. cos. i. The pressure of the particle of water on the part of the fore body before the neutral line will then be pro portional to k z. cos. i; the pressure of the particle of water on the part of the fore body between the neutral line and the midship section, will be propor tional to k—z. cos. i; and the pressure of the par ticle of water on the after body will be proportional, according to the experiment on the bent tube, to Suppose the ship to be placed with its fare part op posed to a current, the direction of which is that of the ship's keel, and the velocity that which is due to the height z. Suppose the surface of the ship's body below the surface of the water to be divided into an infinite number of small surfaces; let x be the hori zontal distance of one of these small surfaces from the midship section, and y its transverse distance from the longitudinal vertical plane, dividing the ship into two equal and similar parts, and k its distance front the plane of floatation. The projection of this small surface on the plane of floatation is d x. d 'y; and re presenting the specific gravity of the water by I', the pressure of the water on this small surface, in a ver tical direction, in the after body, P el x . d y .
and the vertical pressure of the water on a small sur face, in the part contained between the midship sec tion and the neutral section in the fore body, is P d x . d y . cos . i); and the vertical pressure of the water on a small surface in the part of the fore body be fore the neutral section, by taking x for the distance of this small surface from the neutral section, is P it x . d y . (k+z . cos . i). The vertical pressure of the wa ter on the whole of the fore part of the ship's body (taking both sides of the ship), is therefore 2 Pfdxfdy. (k + z. cos. i) + 2 Pfdx COS. i); and the vertical pressure of the water on the whole of the after part of the ship's body, is P f d xf dy.
The sum of these terms, 2 Pf dx.dy. (k+ z . cos. i)-1- f d xf d y.(k—z. cos. I, +2P fdx.dy.
is the total vertical pressure of the water on the ship's body, lying opposed to a current, whose velocity is that due to the height z.
If the water were at rest, the total vertical pressure of the fluid on the ship's body, supposed to be sunk to the same depth below the surface of the water, tak ing z for the whole length of the ship, would be 2 P fdxfdy k. In this expression x is equal to the sum of the three lengths expressed by x in the former expression. This quantity representing the vertical pressure of the water on the ship's body when at rest, is evidently greater than the quantity representing the vertical pressure of the water on the ship's body when in motion, in consequence of the first term of the ex pression 2P fdxf dy cos. i)+2 Pf d xf
d y . . cos. 2 Pf dxf dy. which is Less than the sum of the two other terms, ing increased in a less proportion by the addition of . cos. i to k, than the sum of the last two terms is diminished by z . cos.•i being subtracted from k in the first, which is the smaller of the last two terms, and by z being subtracted from k in the last term. The vertical pressure of the water, therefore, being less on the ship's body when placed in a current than in still water, estimated to the same draught of water in both cases, the ship must sink deeper in the cur rent than in still water. The distance it sinks de pends on the value of z, the height due to the veloci ty of the current, and i, representing the different an gles at which the particles of water strike the ship's body, which depend on the form of the body. Sup posing the values of all the terms known, and subtract ing the expression for the vertical pressure of the water in motion on the ship's body, from the expres sion for the vertical pressure of the water at rest, es timated at the same draught of water in both cases, the remainder will be the quantity to be taken from the expression for the vertical pressure of the water at rest, measuring from the line of floatation downwards, which determines the distance which the ship will sink deeper in the water in motion than at rest.
This expression represents the vertical pressure of the water, under the consideration that each particle of the fluid in motion impinges on the surface of the ship's body; neglecting the circumstance of those par ticles which meet the body at the middle of the fore part, escaping along the surface of the body, and pre venting many of the particles farther removed from the middle froM impinging on the surface, and com municating their action to the body only, through the medium of those particles in contact with it; the particles still farther removed from the middle, com municating in the same manner their action to the body through the medium of a greater number of in tervening particles. The investigation is conducted with immediate reference to the results of the expe riments with the tubes, instead of forming an inde pendent theory on the hypothetical action of fluids on floating bodies. The division of the fore part of the ship's body into two parts, by the section at the lines on the surface of the body, at which the effect of the pressure of the water in motion is the same as that of the pressure of the water at rest, is introduced by the writer of these remarks, as being directed by the re sults of the experiments with the bent tube; consider ing that the whole pressure of the water on the fore part is not increased by the motion of the fluid, but only the part before the neutral section, the pressure on the remainder being diminished.