fix the position of the instantaneous axis oz"; three will determine its place with regard to the principal axes or', oy', oz', and three more are necessary to determine the position of the principal axes themselves in space, that is, with regard to the fixed co-ordinates ox, oy, oz. The permanency of rotation is not the same for all the three axes, as will now be shown.

189. The principal axes possess this property-that the moment of inertia of the solid is a maximum for one of these, and a minimum for another. Let a', y', z', be the co-ordinates of dm, relative to the three principal axes, and let x, y, z, be the co-ordinates of the same element referred to any axes whatever having the same origin. Now if C'S (x2 + y2) dm

be the moment of inertia relatively to one of these new axes, as z, then substituting for x and y their values from article 163, and making A = S (y'2+z2).dm; B = S (x2+z12)dm; C = $ (x22+y'2)dm ; the value of C' will become

C' A sin20 sin2 + B sin2 e cos2 + C cos2 0,

sin sin, sin2 e cos p, cos2 0,

in which are the squares of the cosines of the angles made by or', oy', oz', with oz; and A, B, C, are the moments of inertia of the solid with regard to the axes x', y', and z', respectively. The quantity C' is less than the greatest of the three quantities A, B, C, and exceeds the least of them; the greatest and the least moments of inertia belong therefore, to the principal axes. In fact, C' must be less than the greatest of the three quantities A, B, C, because their joint coefficients are always equal to unity; and for a similar reason it is always greater than the least.

190. When A B = C, then all the axes of the solid are principal axes, and it will rotate permanently about any one of them. The sphere of uniform density is a solid of this kind, but there are many others.

191. When two of the moments of inertia are equal, as A=B, then C' A sin+C cos2 0 ;

=

and all the moments of inertia in the same plane with these are equal: hence all the axes situate in that plane are principal axes. The ellipsoid of revolution of uniform density is of this kind; all the axes in the plane of its equator being principal axes.

192. An ellipsoid of revolution is formed by the rotation of an ellipse ABCD about its minor axis BD. Then AC is its equator.

B

fig. 49.

C

When the moments of inertia are unequal, the rotation round the axes which have their moment of inertia a A maximum or minimum is stable, that is, round the least or greatest axis; but the rotation is unstable round the third, and may be destroyed by the slightest

cause. If stable rotation be slightly deranged, the body will never deviate far from its equilibrium; whereas in unstable rotation, if it be disturbed, it will deviate more and more, and will never return to its former state.

193. This theorem is chiefly of importance with regard to the rotation of the earth. If roy (fig. 46) be the plane of the ecliptic, and z its pole; r'oy' the plane of the equator, and z' its pole: then oz' is the axis of the earth's rotation, zoz'e is the obliquity of the ecliptic, N the line of the equinoxes, and y the first point of Aries: hence xoy is the longitude of or, and x'oy is the longitude of the principal revolving axis or', or the measure of the earth's rotation: oz' is therefore one of the permanent axes of rotation.

The earth is flattened at the poles, therefore oz' is the least of the permanent axes of rotation, and the moment of inertia with regard to it, is a maximum. Were there no disturbing forces, the earth would rotate permanently about it; but the sun and moon, acting unequally on the different particles, disturb its rotation. These disturbing forces do not sensibly alter the velocity of rotation, in which neither theory nor observation have detected any appreciable variation; nor do they sensibly displace the poles of rotation on the surface of the earth; that is to say, the axis of rotation, and the plane of the equator which is perpen

dicular to it, always meet the surface in the same points; but these forces alter the direction of the polar axis in space, and produce the phenomena of precession and nutation; for the earth rotates about oz", fig. 50, while oz" revolves about its mean place

oz', and at the same time oz'

fig. 50.

describes a cone about oz; so that the motion of the axis of rotation

*

is very complicated. That axis of rotation, of which all the points. remain at rest during the time dt, is called an instantaneous axis of rotation, for the solid revolves about it during that short interval, as it would do about a fixed axis.

The equations (32) must now be so transformed as to give all the circumstances of rotatory motion.

194. The equations in article 163, for changing the co-ordinates, will become x = ax' + by' + cz'

where a, b, c are the cosines of the angles made by x with x', y', z'; a', b', c' are the cosines of the angles made by y with x', y', z'; and a", b", c" are the cosines of the angles made by z with the same axes Whatever the co-ordinates of dm may be, since they have the same origin, x2 + y2 + z2 = x12 + y12 + z'2. By means of these it may be found that

In the same manner, to obtain x', y', z', in functions of x, y, z,

six of the quantities a, b, c, a', b', c', a'', b'', c'', are determined by the preceding equations, and three remain arbitrary.

If values of x', y', z', found from equations (36) be compared with their values in equations (37), there will result

195. The axes x',y', z' retain the same position in the interior of the body during its rotation, and are therefore independent of the time; but the angles a, b, c, a', b', c', a", b", c", vary with the time; hence, if dy dz dt dt'

values of y, z,

from equations (36,) be substituted in the first

If a', a", b', &c. be eliminated from this equation by their values in

aAp+bBq + cCr = Mdt;

by the same process it may be found that

a'Ap + b'Bq + c'Cr = SM'dt,

a"Ap + b′′Bq + c"Cr = SM"dt.

196. If the differentials of these three equations be taken, making all the quantities vary except A, B, and C, then the sum of the first differential multiplied by a, plus the second multiplied by a', plus the third multiplied by a", will be

A

dp

dt

+ (CB).qr = aM + a'M' + a'"M",

in consequence of the preceding relations between a a' a", b b' b", c c' d'', and their differentials. By a similar process the coefficients b b'b'', &c., may be made to vanish, and then if

aM + a'M' + a" M" = N

bM + b'M' + b′′ M" = N'
cMc'M' + c" M" = N"

the equations in question are transformed to

And if a, a', a'', b, b', &c., and their differentials, be replaced by their functions in 4, 4, and 0, given in article 194, the equations (39) become

197. These six equations contain the whole theory of the rotation of the planets and their satellites, and as they have been determined in the hypothesis of the rotatory pressures being zero, they will give their rotation nearly about their principal axes.

198. The quantities p, q, r determine oz", the position of the real and instantaneous axis of rotation, with regard to its principal axis oz'; when a body has no motion but that of rotation, all the points in a permanent axis of rotation remain at rest; but in an instantaneous axis of rotation the axis can only be regarded as at rest from one instant to another.

If the equations (36) for changing the co-ordinates, be resumed, then with regard to the axis of rotation,

dx = 0, dy = 0, dz = 0,

since all its points are at rest; therefore the indefinitely small spaces moved over by that axis in the direction of these co-ordinates being zero, the equations in question become,