water; and the work of these, U(9,) — U(9), done between the inclinations and, when the vessel was in the act of receding from the vertical, was shown to be represented by (W,h,‡W2h2) (vers. 0 vers. 0); therefore the work, between the same inclinations, when the motion is in the opposite direction, is represented by the same expression with the sign changed;

and since the axis about which the vessel is revolving is perpendicular to the plane EDF, and passes through the point O, if W,2 represents its moment of inertia about an axis perpendicular to the plane EDF, and passing through its centre of gravity G1,

Substituting in equation 38. and writing for OG, its value h, sin. e, we have

or assuming to be so small that the fourth and all higher powers of

sin. may be neglected, and observing that, this being the case,

1

2

The sign

πλ

4h+k2

1+

sin.

....

(40).

gh, (1+

Wh

being taken according as the centre of gravity of the displaced fluid ascends or descends.

The time of a vessel's rolling or pitching through a small angle, its jorm and dimensions being any whatever.

Let EDF (figs. 1. or 2.) represent the midship section of such a vessel, supposed to be rolling about an axis whose projection is O; and let C represent the centre of the circle of curvature of the surface of its planes of flotation at the point M where that surface is touched by the plane PQ, being above the load water-line AB in fig. 1, and beneath it in fig. 2. Let the radius of curvature CM be represented by p; then adopting the same notation as in the last article, and observing that the axis O about which the vessel is turning is perpendicular to EDF, we shall find its moment of inertia to be represented by

where H, represents the depth of the centre of gravity in the vertical position of the vessel.

.

Also, by equation 35.

1

£u1=U(9)—U(0)=W1(H, — H2) (cos. —cos.,) +μA (cos.16—cos.?e,),

.. by equation 38.

1

W

W1(H‚— HI,)(cos. § — cos. §,) + ¦ μA (cos.' 0—cos.' 0,) = { '+ (H‚—p)* sin.* 0

2g

Assuming and , to be so small that cos. + cos. 9, 2, and observing

that

cos. 0.

cos. vers. 01· -vers. 9,

=

√20{1

1

Supposing, moreover, p to remain constant between the limits-0, and +91, and integrating as in equation 39.

πλ

k3 + (H,—p)2 sin.*0

do.

vers. -vers. 0

1+

sin.2

(41).

4/2

W

1
2

Since the value of sin., is exceedingly small, the oscillations are

nearly tautochronous, and the period of each is nearly represented by the

The following method is given by M. DUPIN for determining the value of p*:

"If the periphery of the plane of flotation be imagined to be loaded at every point with a weight represented by the tangent of the inclination of the sides of the vessel at that point to the vertical, then will the moments of inertia of that curve, so loaded, about its two principal axes, when divided by the area of the plane of flotation, represent the radii of greatest and least curvature of the envelope of the planes of flotation."

If p be taken to represent the radius of greatest curvature, the formula 41. will represent the time of the vessel's rolling; if the radius of least curvature (B being also substituted for A), it will represent the time of pitching.

NOTE D.

On the conditions of the equilibrium of any number of pressures in the same plane, applied to a body moveable about a cylindrical axis in the state bordering upon motion. (From a memoir on the Theory of Mechanics, printed in the second part of the Transactions of the Royal Society for 1841.) LET P1, P2, P3, &c. represent these pressures, and R their resultant. Also let a1, a, a, represent the perpendiculars let fall upon them severally from the centre of the axis, those perpendiculars being taken with the positive signs whose corresponding pressures tend to turn the system in the same direction as the pressure P., and those negatively which tend to turn it in the opposite direction. Also let a represent the perpendicular distance of the direction of the resultant R from the centre of the axis, then, since R is equal and opposite to the resistance of the axis, and that this resistance and the pressures P1, P2, P3, &c. are pressures in equilibrium, we have by the principle of the equality of moments,

་་

Pa+Pa+Pa ̧ + &c. =¿R.

Representing, therefore, the inclinations of the directions of the pressures P1, P2, P3, &c. to one another by 412, 13, 123, t, &c., &c., and substituting for the value of R.

* Applications de Géométrie, p. 47.

The inclination ↳ 2 of the directions of any two pressures in the above expression is taken on the supposition that both the pressures act from, or both towards the point in which they intersect, and not one towards, and the other from, that point; so that in the case represented in the figure in the note at p. 175., the inclination,, of the pressures P, and P,, represented by the arrows, is not the angle P, IP, but the angle PIQ, since IQ and IP, are directions of these pressures, both tending from this point of intersection, whilst the direc tions of P,I and IP, are one of them towards that point, and the other from it. POISSON, Mécanique, Art. 33.

If the value of P, involved in this equation be expanded by Lagrange's theorem *, in a series ascending by powers of a, and terms involving powers above the first be omitted, we shall obtain the following value of that quantity:

:

Now a-2a,a, cos. 12+a represents the square of the line joining the feet of the perpendiculars a, and a, let fall from the centre of the axis upon P, and P2; similarly a-2a,a, cos. 43+ a represents the square of the line joining the feet of the perpendiculars let fall upon P, and P„, and

This expansion may be effected by squaring both sides of the equation, solving the quadratic in respect to P,, neglecting powers of above the first and reducing; this method, however, is exceedingly laborious.