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which, regarded as a force tending to turn the system round y, gives rotatory pressure = wirzdm, because it acts at the distance z from the axis y. Therefore when S.rzdm = 0, the whole effect is zero. Similarly, when S. yzdm 0, the whole effect of the revolving system to turn round a vanishes. Therefore, in order that should be

permanent axis of rotation,

S.xzdm0, Syzdm = 0.

In like manner, in order that y should be so,

S.rydm0, Szydm = 0

must exist; and in order that x should be so

Syxdm 0, S.zxdm = 0


must exist, all of which are in fact only three different equations, namely, S. rydm= 0, Sxzdm0, S. yzdm = 0; and if these hold at once, x, y, z, will all be permanent axes of rotation. Thus the impetus is as the square of the distance from the axis of rotation, and the rotatory pressures are simply as the distance from the same axis.


182. In order to ascertain whether a solid possesses permanent axes of rotation, let the origin be a fixed point, and let x', y', z', be the co-ordinates of a particle dm, fixed in the solid, but revolving with it about its centre of gravity. The whole theory of rotation is derived from the equations (32) containing the principle of areas. These are the areas projected on the fixed co-ordinate planes roy, xoz, yoz, fig. 48; but if ox', oy', oz' be the new axes that revolve with the solid, and if the values of x, y, z, given in article 163, be substituted, they will be the same sums, when projected on the new co-ordinate planes x'oy', x'oz', y'oz'. The angles 0, 4, and 4, introduced by this change are arbitrary, so that the position of the new axes ox', oy', oz', in the solid, remains indeterminate; and these three angles may be made to fulfil any conditions of the problem.

183. The equations of rotation will take the most simple form if we suppose x'y' z' to be the principal axes of rotation, which they will become if the values of 0, 4, and can be so assumed as to make the rotatory pressures S. x'z'dm, Sx'y'dm, Sy'z'dm, zero at once, then the equations (32) of the areas, when transformed to functions of r', y', z', and deprived of these terms, will determine the rotation of the body about its principal, or permanent axes of rotation, x', y', z'.

184. If the body has no principal axes of rotation, it will be impossible to obtain such values of 0, ø and y, as will make the rotatory pressures zero; it must therefore be demonstrated whether or not it be possible to determine the angles in question, so as to fulfil the requisite condition.

185. To determine the existence and position of the principal axes of the body, or the angles o, p, and 4, so that

S.x'y'dm0; Sx'z'dm = 0; Sy'z'dm = 0.

Let values of x', y', z', in functions of x, y, z, determined from the equations in article 163 be substituted in the preceding expressions, then if to abridge,

S.xdm=1a Sy'dm = n2 Sz3dm = s2
S.xydmf Sxzdm=g

there will result


cos p.S.x'z'dm-sin p.S.y'z'dm=

(2n2) sin o sin cos y+fsin e (cos y

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sin cos 0 {siny+n cosys+2f sin y cos 4}

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If the second members of these be made equal to zero, there will be

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by the arithmetic of sines; hence, equating these two values of tan 20, and substituting for tan 6 its value in ; then if to abridge, utany, after some reduction it will be found that

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{ (l2—n2) u +ƒ·(1 — u2)} . { (hs2 — hl2 +ƒg) u + gn2 – gs2 – hf} ; where u is of the third degree. This equation having at least one real root, it is always possible to render the first members of the two equations (35) zero at the same time, and consequently

(S. x'z'dm)2 + (S . y'z'dm)2 = 0.

But that can only be the case when Sa'z'dm = 0, Sy'z'dm = 0. The value of u = tan 4, gives y, consequently tan ✪ and ✪ become known.

It yet remains to determine the condition S. x'y'dm = 0, and the angle. If substitution be made in S. a'y'dm = 0, for x' and y' from article 163, it will take the form H sin 20+ L cos 2p, H and L being functions of the known quantities 0 and 4; as it must be zero,

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and thus the three axes ox', oy', oz', determined by the preceding values of 0, 4, and p, satisfy the equations

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186. The equation of the third degree in u seems to give three systems of principal axes, one for each value of u; but u is the tangent of the angle formed by the axis a with the line of intersection of the plane xy with that of x'y'; and as any one of the three axes, x', y', z', may be changed into any other of them, since the preceding equations will still be satisfied, therefore the equation in u will determine the tangent of the angle formed by the axis a with the line of intersection of xy and x'y', with that of xy and a'z', or with that of xy and y'z'. Consequently the three roots of the equation in u are real, and belong to the same system of axes.

187. Whence every body has at least one system of principal and rectangular axes, round any one of which if the body rotates, the opposite centrifugal forces balance each other. This theorem was first proposed by Segner in the year 1755, and was demonstrated by Albert Euler in 1760.

188. The position of the principal axes ox', oy', oz', in the interior of the solid, is now completely fixed; and if there be no disturbing

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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 x', 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 AS (y + z). dm; BS (x2+ z)dm; CS (x2+y2)dm; the value of C' will become

in which

C' A sin20 sin+B sino 0 cos2 + C cos2 0,


sin sin, sin2 e cos p, cos2 0, 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 ABC, 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 sin20+ 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.


fig. 49.


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; x'oy' the plane of the equator, and z' its pole: then oz' is the axis of the earth's rotation, zoz' is the obliquity of the ecliptic, N the line of the equinoxes, and y the first point of Aries : hence roy 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


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