from the arm ought then, by the formula, to be found nearly equal to the difference of the elevations of the two centres of gravity multiplied by the weight of the body; and this was the test to which it was proposed that the formula should be subjected, with a view to its adoption by practical men as a principle of naval construction.

To give to the deflecting weight that instantaneous action on the ex tremity of the arm which was necessary to the accuracy of the experiment, a string was in the first place to be affixed to it and attached to a point vertically above, in the ceiling. When the deflecting weight was first applied this string would sustain its pressure, but this might be thrown at once upon the extremity of the arm by cutting it. A transverse sec tion of the vessel, with its mast and arm, was to be plotted on a large scale on a board, and the extreme position into which the vessel rolled being by some means observed, the water-line corresponding to this position was to be drawn. The position of the yard, in respect to the surface of the water in that position, would then be known, and the vertical descent of the deflecting weight could be measured, and also the vertical ascent of the centre of gravity of the immersed part or displacement.

To determine the position of the centre of gravity of the vessel, it was to be allowed to rest in an inclined position under the action of the deflecting weight; and the water-line corresponding to this position being drawn on the board, the corresponding position of the deflecting weight and of the centre of gravity of the immersion were thence to be determined. The determination of the position of the vertical passing through the centre of gravity of the body would thus become an elementary question of statics; and the intersection of this line, with that about which the section was symmetrical, would mark the position of the centre of gravity. This determination might be verified by a second similar experiment with a different deflecting weight.

These suggestions received a great development at the hands of Mr. RAWSON, and he adopted many new and ingenious expedients in carrying them out. Among these, that by which the position of the water-line was determined in the extreme position into which the vessel rolls, is specially worthy of observation. A strip of wood was fastened at right angles to that extremity of the yard to which the deflecting weight was attached, of sufficient length to dip into the water when the vessel rolled; on this slip of wood, and also on the side of the vessel nearest to it, a strip of glazed paper was fixed. The highest points at which these strips of paper were wetted in the rolling of the vessel, were obviously points in the water line in its extreme position, and being plotted upon the board, a line drawn through them determined that position with a degree of accuracy which left nothing to be desired

Two forms of vessels were used; one of them had a triangular and the other a semicircular section. The following table contains the general results of the experiments."

In the experiments with the smaller triangular model the differences between the results and those given by the formula are much greater than in the experiments with the heavier cylindrical vessel.

In explanation of this difference, it will be observed, first, that the conditions of the experiment with the cylindrical model more nearly approach to those which are assumed in the formula than those with the other; the disturbance of the water in the change of the position of the former being less, and therefore the work expended upon the inertia of the water, of which the formula takes no account, less in the one case than the other; and, secondly, that the weight of the model being greater, this inertia bears a less proportion to the amount of work required for inclining it than in the other case.

The effect of this inertia adding itself to the buoyancy of the fluid, cannot but be to lift the vessel out of the water and to cause the displacement to be less at the termination of each rolling oscillation than at its commencement.* This variation in volume of the displacement was apparent in all the experiments. Its amount was measured and is recorded in the last column of the Table; its tendency is to produce in the body vertical oscillations, which are so far independent of its rolling motion that they will not probably synchronise with it. The body, displacing, when rolling, less fluid than it would at rest, the effect of the weight ased in the experiments to incline it is thereby increased, and thus is explained the fact (apparent in the eighth and ninth columns of the Table) that the inclination by experiment is somewhat greater than the formula would make it.

The dynamical stability of a vessel whose athwart sections (where they

This result connects itself with the well-known fact of the rise of a vessel out of the water when propelled rapidly, which is so great in the case of fast track-boats, as considerably to reduce the resistance upon them.

are subject to immersion and emersion) are circular, having their centres in a common axis.

Let EDF, fig. 1. or 2., be an athwart section of such a vessel, the parts of whose periphery ES and FR, subject to immersion and emersion, are parts of the same circular arc ETF, whose centre is C. Let G, represent the projection of the centre of gravity of the vessel on this section, and G, that of the centre of gravity of the space whose section is SDRT, supposing it filled with water. The space lies wholly within the vessel in fig. 1. and without it in fig. 2. Let

W2 = weight of water occupying, or which would occupy, the space whose section is STRD.

0= the inclination from the vertical.

*

Since in the act of the inclination of the vessel the whole volume of the displaced fluid remains constant, and also that volume of which STRD is the section, it follows that the volume of that portion of which the circular area PSRQ is the section remains also constant, and that the water-line PQ, which is the chord of that area, remains at the same distance from C, so that the point C neither ascends nor descends. Now the forces which constituted the equilibrium of the vessel in its vertical position were its weight and that of the fluid it displaced. Since the point C is not vertically displaced, the work of the former force, as the body inclines through the angle 0, is represented by - W, h, vers. 6. The work of the latter is equal to that of the upward pressure of the water which would occupy the space of which the circular area PTQ is the section increased, in the case represented in fig. 1., by that of the water which would occupy STRD; and diminished by it in the case represented in fig. 2.

But since the space, of which the circular area PTQ is the section, remains similar and equal to itself, its centre of gravity remains always at the same distance from the centre C, and therefore neither ascends nor descends. Whence it follows that the work of the water which would occupy this space is zero; so that the work of the whole displaced fluid is equal to that of the part of it which occupies the space STRD,

* It will be observed that the space STRD is supposed always to be under

water.

taken in the case represented in fig. 1. with the positive, and in that represented in fig. 2. with the negative sign. It is represented, therefore, generally by the formula ±W, h, vers. 6. On the whole, therefore, the work Zu, of those forces, which in the vertical position of the body cons.ituted its equilibrium, is represented by the formula

Σκη = =― -W, h, vers. 0 W, h, vers. 9.

Representing, therefore, the dynamical stability Zu, by U (9), we have by equation (2. p. 122.)

in which expression the sign is to be taken according as the circular area ATB lies wholly within the area ADB, as in fig. 1, or partially without it, as in fig 2. Other things being the same, the latter is therefore a more stable form than the other.

13. The work of the upward pressure of the water upon the vessel represented in fig. 2. being a negative quantity, -W2h, vers. 9, it follows that the point of application of the pressure must be moved in a direction opposite to that in which the pressure acts; but the pressure acts upwards, therefore its point of application, i. e. the centre of gravity of the displaced fluid, descends. This property may be considered to distinguish mechanically the class of vessels whose type is fig. 1., from that class whose type is fig. 2.; as the property of including wholly or only partly, within the area of any of their athwart sections, the corresponding circular area ETF, distinguishes them geometrically.

The dynamical stability of a vessel of any given form subjected to a rolling or pitching motion.

Conceive the vessel, after having completed an oscillation in any given direction-being then about to return towards its vertical position-to be for an instant at rest, and let RS represent the intersection of its plane of flotation then, and PQ of its flotation when in its vertical position, with a section CAD of the vessel perpendicular to the mutual intersection O of these planes. The section CAD will then be a vertical section of the vessel.

Let G be the projection upon it of the vessel's centre of gravity when in its vertical position.

P

D

MN

ct ms

H, that of the centre of gravity of the fluid displaced by the vessel in the vertical position.

g, that of the fluid displaced by the portion of the vessel of which QOS. is a section.

h, that of the fluid which would be displaced by the portion, of which POR is a section, if it were immersed.

GM, HN, gm, hn, KL, perpendiculars upon the plane RS.

=

W weight of vessel or of displaced fluid.

wweight of water displaced by either of the equal portions of the vessel of which POR and QOS are sections.

H, depth of centre of gravity of vessel in vertical position.

H1 = depth of centre of gravity of displaced water in vertical position.

AHI,

AH,
P

=33

=

elevation of centre of gravity of vessel.

elevation of centre of gravity of displaced water.
area of plane PQ.

inclination of planes PQ and RS.

= inclination of line O in which planes PQ and RS intersect, to that line about which the plane PQ is symmetrical.

h = perpendicular distance of line O from centre of gravity of plane PQ.

= inclination to horizon of line about which the plane PQ is symmetrical.

x distance of section CAD, measured along the line whose projection is O, from the point where that line intersects the midship section.

y = 03.

y1 = PQ.

Y2 = RS.

2= hn + mg.

λ= = KL.

I = moment of inertia of plane PQ about axis O.

A and B = moments of inertia of PQ about its principal axes.

Suppose the water actually displaced by the vessel to be, on the contrary contained by it; and conceive that which occupies the space QOS to pass into the space POR, the whole becoming solid. Let All, represent the corresponding elevation of the centre of gravity of the whole contained fluid. Then will aH, + all, represent the total elevation of the centre of gravity of this fluid as it passes from the position it occupied when the vessel was vertical into the position PAQ. But this elevation is obviously the same as though the fluid had assumed the solid state in the vertical position of the body, and the latter had revolved with it, in that state, into its present position. It is therefore represented by KH —- NH ;

.. ΔΗ, + ΔΗ, =KH−NH and AH = KH – NH−nH. Since, moreover, by the elevation of the fluid in QOS, whose weight is w, into the space OPR, and of its centre of gravity through (gm + hu), the centre of gravity of mass of fluid of which it forms a part, and whose weight is W, is raised through the space AH,; it follows, by a well-known property of the centre of gravity of a system,* that

The line joining the centres of gravity of the vessel and its immersed part, in its vertical position, is parallel to the plane CAD, for it is perpendicular to the plane PQ, to whose intersection with the plane RS the plane CAD is per .. GK = H, and HK = H2