the first shows the angular velocity to be uniform, and the two last

217. If oz" the real axis of rotation coincides with oz', the principal axis in the beginning of the motion, then q and p are zero; hence also, M = 0, and M' = 0. It follows therefore, that in this case q and P will always be zero, and the axis oz" will always coincide with oz'; whence, if the body begins to turn round one of its principal axes, it will continue to rotate uniformly about that axis for ever. On account of this remarkable property these are called the natural axes of rotation; it belongs to them exclusively, for if the position of the real axis of rotation oz" be invariable on the surface of the body, the angular velocity will be constant ;

hence

dp0, dq= 0, dr = 0,

and (C — B) qrdt = 0, (A — C) rpdt = 0, (B — A) pqdt = 0. 218. If A, B, C, be unequal, these equations will only be zero in every case when two of the quantities p, q, r, are zero; but then, the real axis coincides with one of the principal axes.

If two of the moments of inertia be equal, as A = B, the three equations are reduced to rp = 0, qr = 0; both of which will be satisfied, that is, they will both be zero for every value of q and p, if r0. The axis of rotation is, therefore, in a plane at right angles to the third principal axis; but as the body is then a solid of revolution, every axis in that plane is a principal axis.

219. When ABC, the three preceding equations are zero, whatever may be the values of p, q, r, then all the axes of the body will be principal axes. Thus the principal axes alone have the property of permanent rotation, though they do not possess that property in the same degree.

220. Suppose the real axis of rotation oz', fig. 50, to deviate by an indefinitely small quantity from oz', the third principal axis, the coefficients M and M' will then be indefinitely small, since q = M'× cos (nt+g), and p= M sin (nt + g) are indefinitely small. Now if n be a real quantity, sin (nt + g), cos (nt + g), will never exceed very narrow limits, therefore q and p will remain indefinitely small; so that the real axis oz" will make indefinitely small oscillations about the third principal axis. But if n be imaginary, by article 215, sin (nt+g), cos (nt + g),

will be changed into quantities which increase with the time, and the real axis of rotation will deviate more and more from the third principal axis, so that the motion will have no stability. The value of n will decide that important point.

Since

n=r

(A − B) (B—C),
-

AB

it will be a real quantity when C the moment of inertia with regard to oz', is either the greatest or the least of the three moments of inertia A, B, C, for then the product (AC) (BC) will be positive; but if C have a value that is between those of A and B, that product will be negative, and n imaginary. Hence the rotation will be stable about the greatest and least of the principal axes, but unstable about the third.

221. Having determined the rotation of the solid, it only remains to ascertain the position of the principal axis with regard to quiescent space, that is, with regard to the fixed axes ox, oy, oz. That evidently depends on the angles p, 4, and 0.

If the third principal axis oz', fig. 50, be assumed to be nearly at right angles to the plane roy, the angle zoz', or 0, will be so very small that its square may be omitted, and its cosine assumed equal to unity; then the equations (41)

be a constant quantity, at + €.

sin sinu, the two first of equations

The integrals of these two quantities are obtained by the method

in article 214, and are

6 and

being new arbitrary quantities introduced by integration.

The problem is completely solved, since s and u give 0 and in values of the time, and y is given in values of and the time.

Compound Pendulums.

222. Hitherto the rotation of a has been considered either when the fig. 54.

y

-y

solid about its centre of gravity body is free, or when the centre of gravity is fixed; but imagine a solid OP, fig. 54, to revolve about a fixed axis in o which does not pass through its centre of gravity. If the body be drawn aside from the vertical oz, and then left to itself, it will oscillate about that axis by the action of gravitation

alone. This solid body of any form whatever is the compound pendulum, and its motion is perfectly similar to that of the simple pendulum already described, depending on the property of areas.

The motion being in the plane zoy, the sums of the areas in the other two planes are zero; so that the motion of the pendulum is de

rived from the equation S (ydz-zdy) dm = S (yZ− zY) dm.

dt

In order to adapt that equation to the motion of the pendulum, let

y

fig. 55.

be represented in the diagram.

oyy, oP z, Ao=z', Ay=y', hence PA-y', fig. 55; and let the angle PoA be represented by 0. P is the centre of gravity of the pendulum, which is supposed to rotate about the axis ox, passing through o at right angles to the plane zoy, and therefore it cannot

If the first of these four equations be multiplied by sin 0, and the second by cos 0, their sum is

If these values be substituted in the equation of areas it becomes

If the value of y be substituted in this it becomes,

Y = 0

dt2

force of gravitation alone in the z = g.

Because

passes through the centre of gravity of the pendulum,

the rotatory pressure S.y'dm is zero; hence

If L be the distance of the centre of gravity of the pendulum from the axis of rotation or, the rotatory pressure S. z'dm becomes Lm, in which m is the whole mass of the pendulum; hence

C being an arbitrary constant quantity.

223. If a simple pendulum be considered, of which all the atoms are united in a point at the distance of / from the axis of rotation or, its rotatory inertia will be A = ml, m being the mass of the body,

andzy. In this case L. Substituting this value for A, we find

224. Thus it appears, that if the angular velocities of the compound and simple pendulums be equal when their centres of gravity are in the vertical, their oscillations will be exactly the same, provided also that the length of the simple pendulum be equal to the rotatory inertia of the solid body with regard to the axis of motion, divided by the product of the mass by the distance of its centre of gravity from A the axis, or = mL

Thus such a relation is established between the lengths of the two pendulums, that the length of a simple pendulum may be found, whose oscillations are performed in the same time with those of a compound pendulum.

In this manner the length of the simple pendulum beating seconds has been determined from observations on the oscillations of the compound pendulum.