Suppose, in the next place, a force to be externally applied to the solid, su as to cause it to move through any finite angle KGS, round the axis of motion before referred to; and let the line KC, which in the state of equilibrium was vertical, now assume the position of SGL. Let also I XN denote the new situation assum ed by AXB, and WRMNP, the new position of the immersed volume, in consequence of the In the line SL take GE equal to GO; then it is evident that 0, the centre of gravity of the immersed volume All1111, will be transferred to E, the centre of gravity of the equal space IRMN; and the action of the fluid on the immersed volume, would be in the direction of a vertical line passing through F., if IRMN" repre sented the volume immersed in the fluid. But from the inclined position of the solid, the volume NXP, which in the original position of the solid was above the fluid surface, is now immersed in it; and, on the contrary, the volume IWX, which in the position of equilibrium was surrounded by the water, in the new position, is elevated above it. Hence it follows, that the new condition of the solid will cause the centre of gravity of the immersed volume to approach towards that part of it which is most immersed in the fluid.
Suppose, therefore, the centre of gravity of the im mersed volume WRMP to be situated at the point Q, and through Q draw PS parallel to GO, or which is the same thing, perpendicular to the plane of flota tion. Through E and G, draw EY and GZ parallel to the last mentioned plane. Then since Q is the centre of gravity of the volume immersed, the pres sure of the fluid will act in the direction of the verti cal line QS, passing through that centre, with a force equivalent to the body's weight; and by the principles of mechanics, will have precisely the same effect to turn the solid round its axis, as if the same force was applied immediately at the point Z, and acting in the same direction QS. Since, therefore, the effect of the fluid's pressure acting in the direction of a vertical line passing through the centre of gravity Q. no way depends un the absolute position of that point, but on the horizontal distance between the vertical lines GO and SF, which pass through the primitive centre of displacement and the 11C NV position of that centre cre ated by the circumstances of inclination, it follows, that in any attempt to determine the stability of a float ing body, our object must be to determine the magni tude of the line GZ.
The volume immersed under the conditions of in clination being is manifestly equal to the volume immersed under the original circumstances of hydrostatic equilibrium, diminished by the trilateral space IWX, and augmented by the trilateral figure NXP. But since the volume immersed must always preserve the same constant magnitude as long as the whole weight of the body subjected to examination remains unaltered, and which, in every inquiry of the kind, is a necessary and essential condition, it follows, that whatever may be the position of the point of in tersection X, the trilateral areas before alluded to most be equal. Having made these fewgenert.I
servations, we proceed to the following construction for the purpose of determining the magnitude of GZ.
Find a and d the centres of gravity of the spaces IWX and NXP; and from those centres, let perpen diculars a b, c d be drawn to the fluid surface; and in the line EV, take ET a fourth proportional to the whole volume immersed WRMP, the trilateral area IWX or NXP, and the distance b c between the per pendiculars demitted from the centres of gravity a and d. Through the point T thus found, draw FTS parallel to GO, intersecting GZ in Z. Then will GZ represent the measure of stability.
Let the total volume immersed be represented by V, and the volume NXP immersed in consequence of the inclination by v. Let also the distance GO = GE, between the centres of gravity of the entire body, and of the volume immersed, be denoted by d, and the sine of the angle of inclination KGS to radius unity, by c. Let also the distance b c between the perpendicular a b and d c be represented Then by the con struction which is a general formula for the stability of a float ing body of any magnitude and form, at any finite angle of inclination.
It is demonstrated by the writers on mechanics, (Wood's Mechanics, art. 178, third edition,) that in any system of bodies given in position, if the situation of one of them be changed, the corresponding motion of the common centre of gravity estimated in any given direction, will be to the motion of the centre of gra vity of the part of the system moved, estimated in the same direction, as the weight of the body moved is to the weight of the whole system. In the present instance, the volume TRAIN may be regarded as a sys tem of bodies, whose common centre of gravity is E. The centre of gravity a of one of the bodies IWX composing this system, is transferred, in consequence of the inclination of the entire body, to the point d, the centre of gravity of the equal volume NXP. Then since the translation of the volume IWX has occa sioned a motion in its centre of gravity from a to d, and which estimated horizontally on the plane of flo tation, is b c; by the mechanical theorem quoted, the entire volume \VR MP, is to the volume IWX or NXP, as b c to ET; which is the measure of the space the centre of gravity of the entire volume has passed through, when estimated in the same horizontal di rection. If therefore a vertical line FTS be drawn through the point T, it must also pass through the centre of gravity of the immersed volume; and since the line ER is known in terms of the radius GE or GO, and the sine of the angle of inclination EGO, it follows that by subtracting its value from ET, there will remain RT, or its equal GZ, the measure of the stability required.