In another case he has also proved, that if the sides of one vessel coincide with the curve of a conic para bola, and the sides of another vessel with a conic parabola of any other form, but having a different parameter, the breadths of the water sections, the weights of the vessels, and the other conditions being the same, the stabilities of the two vessels at all equal angles of inclination will be equal. This he infers from the principle, that in proportion as the dimen sicms of the parabolic curve are augmented, the 114re more closely approximates to a sectangular parallelo gram; and that when they are increased sine the form ultimately coincides with a body of that kind. And that as we have before seen that the sta bility of the conic parabola is the same with that of the rectangular parallelogram, so must the stability of a body whose form is that of a parabolic curve of the highest possible dimensions, approach to the same identity. As a matter of useful reference, illustrat ing the remarkable property alluded to, and because the forms of vessels approximate in many cases to the parabolic figure, we insert Fig. 17, in which the curve cBCo is a conical or appollonian parabola.
elfEo a biquadratie parabola.
f BF° a parabola of 8 dimensions.
and gliGo a parabola of 50 dimensions.
Having made these general observations on the subject of stability, we shall in the next place proceed to the application of the principles that ha-se been developed, to the computation of the stability of the ship whose displacement we before computed.
For this purpose let BDQA, Fig. 18, represent that portion of the principal vertical section of the vessel proposed, which is situated below the water line BA, when the plane of the masts is at right angles to the fluid surface; and let DC represent the line which coincides with the surface of the water, when the ves sel is inclined at an angle of 10°, at which angle we propose to compute the stability. This plane DC, from the conditions of must be so situated as to cause the volumes immersed and emerged in con sequence of the inclination, to be equal in solidity; and it will follow, from the varieties of form which the different transverse sections of a vessel present, that the areas of the figures SAbC, SlIeD can in no case he equal; although, in the previous investigations, from the perfect equality and similarity supposed to exist among the transverse sections, the areas of im mersion and emersion were properly regarded as equal. And it is farther evident, that at whatever
distance the point S is situated from the middle point X of the water's surface, in any one section, the same distance XS will be preserved in every other section; for by the supposition the vessel is inclined round the longer axis, and therefore the intersection of the planes which pass through the lines BA and DC will he parallel to the longer axis, and consequently par allel to a line drawn through all the points X, from one extremity of the vessel to the other.
To show, in the next place, by what means the magnitude of XS is to be determined, through X draw the line NXW inclined to the water's surface at the given angle; and let a plane be supposed to pass through it, so as to cut all the sections in like man ner. Then by means of the ordinary rules of mensu ration, let the area of the figure AX\Vb be computed fur each section; and from these equidistant areas, let the solidity of the volume comprised between the two planes XW and XA be calculated by means of the formula 4 S and let the same operation 3 be performed for the solid contained between the planes BX and XN. Let the former of these solidi ties be denoted by A'. Then will the difference of these solidities be equal to the solid comprised be tween the two planes NW and DC. And if we re present the area of the section NW by W, this dif ference will be equivalent to W x SO; that is A—A' W x SO.
But by trigonometry XS : SO : : I : sin. SXO.
and therefore SO=XS x sin. SXO.
This value of SO being substituted in the preceding equation, gives A—A'=\VXXSXsin. SXO, and from which we obtain A