To investigate this new condition, let G denote the centre of gravity of the entire solid, and 0 that of its immersed volume; and let the line connecting those points be produced, so as to divide the entire solid into the two portions APQB, and DPQC. Now although, from the laws of hydrostatic equilibrium, the points G, 0 are situated in the same vertical plane, it by no means follows that the portion of the entire solid, APQB, and its immersed volume RSQB, in the present condition of the whole body, can have their centres of gravity in the same vertical plane. Hence we may imagine the centre of gravity of the former to be situated as at a, and of the latter as at the centre a, being, from the form of the body, ne cessarily farther from the vertical line PQ, than the centre N. In like manner, may we suppose the point to be more distant from the same vertical line PQ, than the point Now, the gravitating force of the solid APQB ope rating at a, and the upward pressure of the fluid at ,2, and that these forces act in opposite and not coinci dent directions; from those centres, let lines be drawn perpendicular to the fluid surface; and let \V repre sent the gravitating effort the body APQB, and to that of the fluid acting through R. Let also a similar construction be made with respect to the centres 1, a, the weights AN", to' representing the mechanical efforts operating at those points.
If we now contemplate the effect of the forces act ing at the points a, 7, it is apparent that their tend ency is to depress the extremities of the body; whereas the action of the forces operating at the points g,,r, have a tendency to elevate its middle parts; and we know, from the theory of the neutral axis, that the ultimate effect of these forces must be, to make the entire body turn about a line of that nature, situated somewhere within the area of fracture, if the forces applied are capable of breaking the body; and which line, for the sake of illustration, we may suppose to be situated in the plane of the fluid surface at S; there by causing all the fibres above that surface to be in a state of extension, and all below in a state of compres sion; while those fibres which pass through the neu tral point S, undergo neither extension nor compres sion. hut are in a state perfectly neutral with regard to both.
But if, in addition to the want of support afforded by the fluid to the inclined surfaces All, CD, we im agine the vertical sections between II and C so con stituted, as not in every case to be exactly in equi librium with the columns of fluid displaced respec tively by them; it is obvious that an additional ten dency to arching will be created, and which we shall now more particularly investigate, by contemplating the actual conditions of a ship.
To discover the law which influences a ship, whether laden or unladen, when floating quiescently in water, we shall, in the first place, consider it with respect to its length, and afterwards in relation to its breadth.
To accomplish this, let us suppose a vessel to be di% ided into vertical sections of an indefinitely small constant thickness, perpendicular to the vertical lon gitudinal plane. If we commence our consideration at the stern, and advance gradually forward, it is evi dent that the sections comprising the counter and its connecting parts, being free from the water, will be subject to no reaction from it; and when at last any reaction of the fluid does take place, it must at first, from the peculiar form of the body, be infinitely less than the weight of the section whose displacement occasions it. As we approach, however, nearer the midship section of the vessel, the upward section of the fluid will approach more and more to an equality with the weight of its corresponding section, and ul timately become equal to it; and if we pass beyond this section, and which may be denominated the Sec tion of Hydrostatic Equilibrium, we shall find the weight of the water displaced, become greater than the weight of the section above it. In like manner, if we commence at the bow of the vessel, we shall find a similar section of' hydrostatic equilibrium, and afterwards a like increase of the weight of the water displaced above the weight of the section reposing on it.
Let us, therefore. In this very interesting inquiry adopt the investigat,ion of Dnpin, contained in his paper Sur la Structure des raisseaux .thiglais," in the Philosophical Transactions for 1817.
For this purpose, let x represent the distance of any part of a vessel, from a vertical plane assumed as a standard of reference for the different moments, and let d x be the thickness of the infinitely small sections parallel thereto. Let also 5' (X) d X denote the weights of those sections, and 4 (x) d x that of the water which they displace. Then will the total moment of these forces be x . . d X 4 (x). d x, and, consequently, the integral of the same system of moments X . (x) . x — x (x) . d x Now, in order that this function may be either a maximum or a minimum, it is necessary that its va riation should be zero, and which, therefore, produces, the equation x (x) . d — x 4 (x). d x = o.
But in this latter expression, neither of the original sections alters its weight; and. the functions (.r) and 4 (x) remain constant, as well as the thickness d x of the sections, only by removing the plane, with respect to which the moments are taken, to the distance ,r the section of which x) represents the weight, and (ct x) its displacement is added.