disturbed by the sun and moon. But the same equations will also determine the rotation of a solid, when not disturbed in its rotation.

Rotation of a Solid not subject to the action of Disturbing Forces, and at liberty to revolve freely about a Fixed Point, being its Centre of Gravity, or not.

202. Values of p, q, r in terms of the time must be obtained, in order to ascertain all the circumstances of rotation at every instant. If we suppose that there are no disturbing forces, the areas are constant: hence the equations (40) become

If the first be multiplied by p, the second by q, and the third by r,

their sum is

Apdp + Bqdq + Crdr = 0,

and its integral is

Ap2 + Bq2 + Cr2 = k2,

(46)

being a constant quantity. Again, if the three equations be multiplied respectively by Ap, Bq, Cr, and integrated, they give

Ap+ Bq + C2r2 = h2,

(47)

a constant quantity. This equation contains the principle of the preservation of impetus or living force which is constant in conFrom these two integrals are obtained:

formity with article 148.

By the substitution of these values of p and q, the last of equations (45) when resolved according to dt, gives

dt=

Cdr. AB

(49)

√{ (h2 — Bk+(B− C). Cr2). ( − h2+Ak+(C − A)). Cr2} This equation will give by quadratures the value of t in r, and reciprocally the value of r in t; and thus by the substitution of this value of r in equations (48) the three quantities p, q and r become known in functions of the time.

This equation can only be integrated when any two of the moments of inertia are equal, either when

A = B, B = C, or A = C;

in each of these cases the solid is a spheroid of revolution.

203. Thus p, q, r, being known functions of the time, the angular velocity of the solid, and its rotation with regard to the principal axes, are known at every instant.

204. This however is not sufficient. To become acquainted with all the circumstances of rotation, it is requisite to know the position of the principal axes themselves with regard to quiescent space, that is, their position relatively to the fixed axes x, y, z. But for that purpose the angles 4, 4, and 0, must be determined in functions of the time, or, which is the same thing, in functions of p, q, r, which may now be regarded as known quantities.

If the first of equations (45) be multiplied by a, the second by b, and the third by c, their sum when integrated, in consequence of the relations between the angles in article 194, is

aAp+bBq+cCr = 1, by a similar process a'Apb'Bq + c'Crl',

a"Ap+b"Bq + c"Cr=1",

(50)

l, l', '', being arbitrary constant quantities. These equations coincide with those in article 195, and contain the principle of areas. They are not however three distinct integrals, for the sum of their squares is

in

A2p2 + B2q2 + C2 r2 = l2 + 112 + 1'1o,

consequence of the equations in article 194. But this is the same with (47); hence 12 + 1/2 + 1/22 = h2

being an equation of condition, equations (50) will only give values of two of the angles p, 4, and 0.

The constant quantities 1, l', ', correspond with c, c', c", in article 164, therefore t√√2+1/2 + 1/12

is the sum of the areas described in the time t by the projection of each particle of the body on the plane on which that sum is a maximum. If xoy be that plane, and are zero: therefore, in every solid body in rotation about an axis, there exists a plane, on which the sum of the areas described by the projections of the particles of the solid during a finite time is a maximum. It is called the Invariable Plane, because it remains parallel to itself during

the motion of the body: it is also named the plane of the Greatest Rotatory Pressure.

if the first of equations (50) be multiplied by a, the second by a', and the third by a", in consequence of the equations in article 194, their

The accented angles ', ', ', relate to the invariable plane, and

0,,, to the fixed plane.

Because p, q, r, are known functions of the time, ' and ' are determined, and if do be eliminated between the two first of equation (41), the result will be

sin e'.dy' sin e'. sin p'.pdt + sin e'. cos p'.qdt. But in consequence of equations (51), and because

and as r is given in functions of the time by equation (49), y'is determined.

Thus, p, q, r, y', O', and p', are given in terms of the time: so that the position of the three principal axes with regard to the fixed axes, or, oy, oz; and the angular velocity of the body, are known at every instant.

205. As there are six integrations, there must be six arbitrary constant quantities for the complete solution of the problem. Besides h and k, two more will be introduced by the integration of dt and dy'. Hence two are still required, because by the assumption of for the invariable plane, 7 and l' become zero.

xoy

Now the three angles, ', p', ', are given in terms of p, q, r, and these last are known in terms of the time; hence y', ', 0', (fig. 49,) are known with regard to the invariable plane roy: and

by trigonometry it will be easy to determine values of 4, 4, 0, with regard to any fixed plane whatever, which will introduce two new arbitrary quantities, making in all six, which are requisite for the complete solution of the problem.

206. These two new arbitrary quantities are the inclination of the invariable plane on the fixed plane in question, and the angular distance of the line of intersection of these two planes from a line arbitrarily assumed on the fixed plane; and as the initial position of the fixed plane is supposed to be given, the two arbitrary quantities are known.

If the position of the three principal axes with regard to the invariable plane be known at the origin of the motion, Ø', e', will be given, and therefore p, q, r, will be known at that time; and then equation (46) will give the value of k.

The constant quantity arising from the integration of dt depends on the arbitrary origin or instant whence the time is estimated, and that introduced by the integration of dy' depends on the origin of the angle ', which may be assumed at pleasure on the invariable plane.

207. The determination of the sixth constant quantity h is very interesting, as it affords the means of ascertaining the point in which the sun and planets may be supposed to have received a primitive impulse, capable of communicating to them at once their rectilinear and rotatory motions.

The sum of the areas described round the centre of gravity of the solid by the radius of each particle projected on a fixed plane, and respectively multiplied by the particles, is proportional to the moment of the primitive force projected on the same plane; but this moment is a maximum relatively to the plane which passes through the point of primitive impulse and centre of gravity, hence it is the invariable plane.

B

208. Let G, fig. 52, be the centre of gravity of a body of which ABC is a section, and suppose that it has received an impulse in the plane ABC fig. 52. at the distance GF, from its centre of gravity; it will move forward in space at the same time that it will rotate about an axis perpendicular to the plane ABC. Let v be the velocity generated in the centre of gravity by the primitive impulse; then if m be the mass of the body, m.v. GF will be the moment of this

F

A

impulse, and multiplying it by t, the product will be equal to the sum of the areas described during the time t; but this sum was shown to be t√+2+1112 ;

hence

√12 + 112 + 1112=m.v.GFh; which determines the sixth arbitrary constant quantity h. Were the angular velocity of rotation, the mass of the body and the velocity of its centre of gravity known, the distance GF, the point of primitive impulse, might be determined.

209. It is not probable that the primitive impulse of the planets and other bodies of the system passed exactly through their centres of gravity; most of them are observed to have a rotatory motion, though in others it has not been ascertained, on account of their immense distances, and the smallness of their volumes. As the sun rotates about an axis, he must have received a primitive impulse not passing through his centre of gravity, and therefore it would cause him to move forward in space accompanied by the planetary system, unless an impulse in the contrary direction had destroyed that motion, which is by no means likely. Thus the sun's rotation leads us to presume that the solar system may be in motion.

210. Suppose a planet of uniform density, whose radius is R, to be a sphere revolving round the sun in S, at the distance SG or 7, with an angular velocity represented by u, then the velocity of the centre of gravity will be vur.

Imagine the planet to be put in motion by a primitive impulse, passing through the point F, fig. 53, then the sphere will rotate about an axis perpendicular to the invariable plane, with an angular velocity equal to r, for the components q and p at right angles to that plane will be zero; hence, the equation

√√12+112+1112 = m.v.GF.

becomes = mur. FG; and " = rC.

fig. 53.

G

The centre of gyration is that point of a body in rotation, into which, if all the particles were condensed, it would retain the same degree of rotatory power. It is found that the square of the radius of gyration in a sphere, is equal to 3 of the square of its semi-diameter;