We shall now proceed to apply the formula just de termined to the ease of a csscl whose sides are pa rallel to the plane of the masts, both above and below the plane of flotation.
Let QBOAH, (Fig. 14.) represent a vertical sec tion of the vessel, when it floats in an upright or qui escent position, BA denoting the plane of flotation. Let also G be the centre of gravity of the entire body, and E that of the portion immersed in the fluid. Let V as before represent the magnitude of the volume immersed.
Bisect the line BA passing through the plane of flotation in S; and through S draw CSH, forming with BA an angle equal to the vessel's inclination. Then since BSC is the triangular area raised above the fluid surface, in consequence of the inclination, and ASH the similar and equal surface depressed be low it from the same cause; bisect BC and AH in the points F and N. Join FS and NS, and take SI to SF, and likewise SM to SN in the ratio of 2 to 3; then will 1 and be the centres of gravity of the triangu lar spaces. From these centres, let fall 1K and IML perpendicular to CH. Through E, draw E V parallel to CH, and take ET to KL, in the ratio of the volume ASH to the whole volume displaced. Through G draw GU parallel to CH, and through T the line TZ perpendicular to GU, and GR parallel to TZ. Then will RT or GZ be the measure of the vessel's stability.
To determine the value of GZ analytically, and thence numerically, let BA, the breadth of the water section be denoted by 4 b, and GE the interval be tween the centres of gravity of the entire body, and of the volume immersed by a; also the angle of incli nation ASH by c. Then 1 : 2 b :: tan. c : 2 b tan. a= AFT, whence AN= b tan. 9, and + Also, as SN : HN : sin. NHS : sin. NSH; or as sin, c b + tan. 2 : b tan. 4 :: cos. 9 (4 + 9) cos. a + sec. a Hence cos. NSH = Now SM + 2 2 b = -3 (4 + ; and therefore SL 2 b (cos. 4+ sec. 4). And since the triangles SLM, SKI are equal and similar, ICI.. = 2 SL = (cos. c + sec. 4). Also the area of the triangle ASH =2 tan. c. Therefore by the mechanical theorem referred to in the demonstration, Also because GE : ER :: 1 : sin. therefore ER = a sin. p; 8 whence RT = GZ3 V tan. 5, (cos. p sec. — a sin. 5,, is the analytical value of the proposed vessel's stability.
To determine the value of G Z numerically, let the breadth of the vessel at the water's surface, or All be 100, and the interval GE between the centres of gra vity of the entire body, and of the volume immersed be 13; that is, let b = 25, and c = 13. Suppose also
the angle of inclination p = and let V the area of the section of the volume displaced be represented by 3600. Then we shall have cos. 4 + sec.cp = cos. + sec. =2.0012020 8 tan. q, 125000 tan. 15° 3 V = 3 X 3600 Hence ET == 2.0012 x 3.1012 =6.2062555 and a sin.p = 13 sin. 15° =3.3646470 which gives the measure of stability GZ =2.8416085 From this result therefore it appears, that when the proposed vessel has been inclined from its position of equilibrium through an angle of 15°, the action of the fluid to restore it to its quiescent posi tion, will pass at the distance of estimated hori zontally, when the breadth of the water section is de noted by 100. And this result will be the same whatever be the length of the axis.
The absolute pressure of the fluid, is in reality the total volume displaced by the body. Suppose this quantity to be 1000 tons. Then since by this hypo thesis, the stability of the vessel, when inclined at an angle of 15°, is equivalent to the force of 1000 tons, 2.84 acting at the distance of parts of the breadth of 10000 the water section from the axis, to restore the vessel to its primitive state of equilibrium; the effect will be 1000 X 2'84 the same as if a force --- = 56•S tons were 50 applied to turn the vessel at the distance of 50 from the axis.* If therefore the wind should act on the sails of the vessel with a force of 56•8 tons, at the mean distance of 50 from the axis, the force of sta bility would just balance it, so as to preserve an equi librium, the vessel still preserving its inclination of Such is nearly the method pursued by Mr. Atwood, to illustrate the general question of stability; and we have introduced the example to the attention of our readers, to enable them to form some idea of the mode pursued by that celebrated man in this very interest ing inquiry. It would very far exceed the limits of the Encyclonxdia, to follow him through all the cases and forms of bodies he has chosen to illustrate his subject; but we will endeavour, by tabulating some of his leading results, to afford our readers every assist ance we are able on so important a question. We recommend, however, most earnestly to every one interested in the inquiry the two papers of Mr. At wood contained in the Philosophical Transactions for 1796 and 1798.