we have o (.r) — 4 (x) xd x=f 4 (d x)] kx)— 4 (x)] d x .
But since the functions 4 (x) and 4 (x) become zero, when we cause x to vanish,—these expressions repre sent the weight and displacement of a vanishing sec tion; and hence we may perceive that (d x) — (dx) becomes infinitely small when compared with 4) (x) 4 (x).
If, therefore, the expression cp (cf x) — x) may he neglected, much more may the functions E, — 4 (s •)] d x .
From this last equation of condition, we learn that the sum of the moments which tend to produce arch ing is either a maximum or a minimum, when the weight of the part of the vessel, either before or be hind the plane of the moments, is equal to the weight of the water displaced by the same part of the ship.
It may be necessary, however, to distinguish the condition of the maximum from that of the minimum, and which may be discovered by the circumstance, that according as the term of the formula neglected has the same or a contrary sign from the expression for the total moment the sum of the moments, with relation to the plane determined, will be a minimum or a maximum.
But since cp (cr x) Sx is the weight of the section having s x for its thickness, and 4 (cr x) d x the weight of the water displaced by the same section, it follows that the quantity (ci — (ci 3..)1 .ex•dx will be positive or negative, according as the weight of the infinitely small section, which commences at the plane of the moments, is greater or less than the weight of the water displaced by the section itself. From these principles we deduce the following theo rems: I. Thal when a vertical plane divides a vessel into two parts, so that the weight of each part is equal to the weight of water displaced by it, the moments of those parts estimated in relation to the same plane, to produce what we have denominated arching, will be either a maximum or a minimum.
II. That this (Ifeet will be a maximum, when the in finitely small section contiguous to the plane of the mo ments,hus its own moment in a contrary direction to that of the total moment.
III. That the effect will be a minimum when this see• lion has its own moment acting in the same direction as the total moment.
We shall proceed to the farther illustration of these theorems by their application to a seventy-four guy ship, the elements of which are derived from Dr. Young's paper on the employment of Oblique Riders, contained in the Philosophical Transactions for 1814 In order to distribute with some uniformity, the positive and negative differences recorded in the last column of the preceding table, M. Dupin has had re course to a geometrical figure, and by an hypothesis, originally adopted by Dr. Young, in the paper before quoted, has assumed in the line AO, Fig. 3, Plate CCCCXCV. supposed coincident with the plane of the water's surface, certain segments AC, CE, EG, GH, HK, KM, and MO, having values equivalent to the quantities recorded in the first column of the suc ceeding table; and on those segments has supposed certain triangular areas to be formed, equivalent to the positive and negative quantities recorded in the last column of the former table. For example, on the segment AC is formed the right angled triangle ABC =+72; on CE, the isosceles triangle CDE =-108; on EG, the isosceles triangle EFG'=+11 8; on HK the right angled H1K =-1 19; and on KM, MO the right angled triangles IKM, 'ION, the area of the former being-155, and of the latter+192; the dif ference of these areas being equivalent to 37, the last member of the column last refered to. The po sitive areas, it will be perceived, are formed below the plane of flotations, and the negative areas above it. These particulars are recorded in the next table.
In the next place, let the centres of gravity of the several triangular areas referred to, be determined, and from them let perpendiculars be demitted on the primitive line AO, meeting it in the points b, d, f, r, s, and n. The determination of these centres will furnish the elements of the following table; the com parison of the first and last columns of which, gives the respective distances of the common origin of the horizontal ordinates from the centres of gravity of the triangles.
This construction, therefore, furnishes for the equi librium of the forces operating on the vessel, the equation