Proof of reality of adynamic and of precessional periods when

system's irrotational periods are either all

real or all imaginary.

345 It remains to prove that the roots λ of (71), and of (72) also when,, .,..., . are all of one sign, are essentially real and negative. (71) is the determinantal equation of § 345 (42) with any even number of equations instead of only four. The treatment of SS 345 and 345 is all directly applicable without change to this extension; and it proves that the roots λ are real and negative by bringing the problem to that of the orthogonal reduction of the essentially positive quadratic function

G(aa)=} {(12a,+13a ̧+etc.)2+(21α ̧+23a ̧+etc.)2+(31a,+32a,+etc.)2+etc.} (76):

Algebraic theorem.

it proves also the equalities of energies of (56), § 345, and the orthogonalities of (55), (58) § 345: also the curious algebraic theorem that the determinantal roots of the quadratic function consist of i pairs of equals.

Inasmuch as (72) is the same as (71) with A' put for λ and 12', 13', 23', etc. for 12, 13, 23, etc., all the formulas and propositions which we have proved for (71) hold correspondingly for (72) when 12', 13', 23', etc. are all real, as they are when ,,,..., are all of one sign.

345. Going back now to § 345, and taking advantage of what we have learned in § 345 and the consequent treatment of the problem, particularly that in § 345, we see now how to simplify equations (14) of § 345 otherwise than was done in § 345, by a new method which has the advantage of being applicable also to materially simplify the general equations (13) of $ 345". Apply orthogonal transformation of the co-ordinates to reduce to a sum of squares of simple co-ordinates, the quadratic function (76). Thus denoting by G (44) what G (aa) becomes when 4,,,, etc. are substituted for a,, a,, etc.; and denoting by n,, n,...,n the values of the pairs of roots of the determinantal equation of degree i, which are simply the negative of the roots λ of equation (71) of degree li in λ2; and denoting by έ1, n1, É。, N.,..., the fresh co-ordinates, we have

2

2

2

G (44) = 1 {n ̧2 (§ ̧2 + n ̧3) + n ̧2 (§ ̧2 + n ̧2) + ... +Nzï3 (§şi2 + N ̧ï3)}... (77).

2

It is easy to see that the general equations of cycloidal motion (13) of § 345 transformed to the έ-co-ordinates come out in i pairs as follows:

345xxviil

+

dt dig

dn

Considerations of space and time prevent us from detailed treatment at present of gyrostatic systems with odd numbers of degrees of freedom, but it is obvious from § 345 and 345 that the general equations (13) of § 345" may, when i the number of freedoms is odd, by proper transformation from coordinates 4,,,, etc. to a set of co-ordinates, É1, „,... (i-1), (-i) be reduced to the following form:

Kinetic stability.

tive dis

turbance.

346. There is scarcely any question in dynamics more important for Natural Philosophy than the stability or instability of motion. We therefore, before concluding this chapter, propose to give some general explanations and leading principles regarding it.

A "conservative disturbance of motion" is a disturbance in the motion or configuration of a conservative system, not Conserva altering the sum of the potential and kinetic energies. A conservative disturbance of the motion through any particular configuration is a change in velocities, or component velocities, not altering the whole kinetic energy. Thus, for example, a conservative disturbance of the motion of a particle through any point, is a change in the direction of its motion, unaccompanied by change of speed.

Kinetic stability and

discrimi

nated.

347. The actual motion of a system, from any particular instability configuration, is said to be stable if every possible infinitely small conservative disturbance of its motion through that configuration may be compounded of conservative disturbances, any one of which would give rise to an alteration of motion which would bring the system again to some configuration belonging to the undisturbed path, in a finite time, and without more than an infinitely small digression. If this condition is not fulfilled, the motion is said to be unstable.

Examples.

348. For example, if a body, A, be supported on a fixed vertical axis; if a second, B, be supported on a parallel axis belonging to the first; a third, C, similarly supported on B, and so on; and if B, C, etc., be so placed as to have each its centre of inertia as far as possible from the fixed axis, and the whole set in motion with a common angular velocity about this axis, the motion will be stable, from every configuration, as is evident from the principles regarding the resultant centrifugal force on a rigid body, to be proved later. If, for instance, each of the bodies is a flat rectangular board hinged on one edge, it is obvious that the whole system will be kept stable by centrifugal force, when all are in one plane and as far out from the axis as possible. But if A consist partly of a shaft and crank, as a common spinning-wheel, or the fly-wheel and crank of a

steam-engine, and if B be supported on the crank-pin as axis, Examples. and turned inwards (towards the fixed axis, or across the fixed axis), then, even although the centres of inertia of C, D, etc., are placed as far from the fixed axis as possible, consistent with this position of B, the motion of the system will be unstable.

example.

349. The rectilinear motion of an elongated body lengthwise, or of a flat disc edgewise, through a fluid is unstable. But the motion of either body, with its length or its broadside perpendicular to the direction of motion, is stable. This is demonstrated for the ideal case of a perfect liquid (§ 320), in § 321, Example (2); and the results explained in § 322 show, for a Kinetic stability. Hysolid of revolution, the precise character of the motion con- drodynamic sequent upon an infinitely small disturbance in the direction of the motion from being exactly along or exactly perpendicular to the axis of figure; whether the infinitely small oscillation, in a definite period of time, when the rectilineal motion is stable, or the swing round to an infinitely nearly inverted position when the rectilineal motion is unstable. Observation proves the assertion we have just made, for real fluids, air and water, and for a great variety of circumstances affecting the motion. Several illustrations have been referred to in § 325; and it is probable we shall return to the subject later, as being not only of great practical importance, but profoundly interesting although very difficult in theory.

simple

350. The motion of a single particle affords simpler and not less instructive illustrations of stability and instability. Thus if a weight, hung from a fixed point by a light inexten- Circular sible cord, be set in motion so as to describe a circle about a pendulum. vertical line through its position of equilibrium, its motion is stable. For, as we shall see later, if disturbed infinitely little in direction without gain or loss of energy, it will describe a sinuous path, cutting the undisturbed circle at points successively distant from one another by definite fractions of the circumference, depending upon the angle of inclination of the string to the vertical. When this angle is very small, the motion is sensibly the same as that of a particle confined to one plane and moving under the influence of an attractive 27

VOL. I.

Circular orbit.

Kinetic stability in cir

cular orbit.

force towards a fixed point, simply proportional to the distance; and the disturbed path cuts the undisturbed circle four times in a revolution. Or if a particle confined to one plane, move under the influence of a centre in this plane, attracting with a force inversely as the square of the distance, a path infinitely little disturbed from a circle will cut the circle twice in a revolution. Or if the law of central force be the nth power of the distance, and if n +3 be positive, the disturbed path will cut the undisturbed circular orbit at successive angular intervals, each equal to π/+3. But the motion will be unstable if n be negative, and - n > 3.

The criterion of stability is easily investigated for circular motion round a centre of force from the differential equation of the general orbit (§ 36),

-1

Let the value of h be such that motion in a circle of radius a satisfies this equation. That is to say, let P/h3u2 = u, when u = a. Let now u = a + p, p being infinitely small. We shall have

Hence we see that the circular motion is stable in the former case, and unstable in the latter.