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Cycloidal motion. Conditions of pyrostatic do mination.

Suppose for simplicity i to be even. All the roots l' of (71) are (S 345xvi below) essentially real and negative. So are those of (72) provided w,, W.,..., 0, are all of one sign as we now suppose them to be. Hence the smallest root - d' of (71) is less than li (12.34 · 56,..., 1-1, )'

.(74), (12. 34. 56,..., i - 3, 1-2) and the greatest root - 18 of (72) is greater than ( 12'. 34' 56',..., i - 3, i – 2')*

....(75). di (12'. 34' 56',..., 1 – 1, 8)

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Hence the conditions for gyrostatic domination are that (74) must be much greater than the greatest of the positive quantities w,

and that (75) must be very much less than the least of these positive quantities. When these conditions are fulfilled the i roots of (18) $ 345's equated to zero are separable into two groups of și roots which are infinitely nearly equal to the roots of equations (71) and (72) respectively, conditions of reality of which are investigated in § 345"vi below. The interpretation leads to the following interesting conclusions:

Gyrostatic 345tir. Consider a cycloidal system provided with nonlinks explained. rotating flywheels mounted on frames so connected with the

moving parts as to give infinitesimal angular motions to the axes of the flywheels proportional to the motions of the system. Let the number of freedoms of the system exclusive of the ignored co-ordinates [$ 319, Ex. (G)] of the flywheels relatively to their frames be even. Let the forces of the system be such that when the flywheels are given at rest, when the system is at rest, the equilibrium is either stable for all the freedoms, or unstable for all the freedoms. Let the number and connexions of the gyrostatic links be such as to permit gyrostatic domina

tion (8 345**) when each of the flywheels is set into sufficiently Gyrostati- rapid rotation. Now let the flywheels be set each into sufcally do

ficiently rapid rotation to fulfil the conditions of gyrostatic system:

domination ($ 345"): the equilibrium of the system becomes stable: with balf the whole number i of its modes of vibration exceedingly rapid, with frequencies equal to the roots of a certain algebraic equation of the degree fi; and the other half of

minated

system:

its modes of vibration very slow, with frequencies given by the Gyrostatiroots of another algebraic equation of degree fi. The first class minated of fundamental modes may be called adynamic because they are the same as if no forces were applied to the system, or acted between its moving parts, except actions and reactions in its adynathe normals between mutually pressing parts (depending on the lations(very inertias of the moving parts). The second class of fundamental modes may be called precessional because the precession of the and proequinoxes, and the slow precession of a rapidly spinning top oscillaticns supported on a very fine point, are familiar instances of it. Remark however that the obliquity of the ecliptic should be infinitely small to bring the precession of the equinoxes precisely within the scope of the equations of our "cycloidal system."

(very slow).

345". If the angular velocities of all the flywheels be altered in the same proportion the frequencies of the adynamic oscillations will be altered in the same proportion directly, and those of the precessional modes in the same proportion inversely. Now suppose there to be either no inertia in the system except that of the flywheels round their pivoted axes and round their equatorial diameters, or suppose the effective inertia of the connecting parts to be comparable with that of the flywheels when given without rotation. The period of each Comparison of the adynamic modes is comparable with the periods of the adynamiq flywheels. And the periods of the precessional modes are com- Totational parable with a third proportional to a mean of the periods of these the flywheels and a mean of the irrotational periods of the sys- Frequencies tem, if the system be stable when the flywheels are deprived of rotation. For the last mentioned term of the proportion we queres may, in the case of irrotational instability, substitute the time of ities of the increasing a displacement a thousandfold, supposing the system with fly, to be falling away from its configuration of equilibrium prived of according to one of its fundamental modes of motion (Al). The reciprocal of this time we shall call, for brevity, the rapidity of the system, for convenience of comparison with the frequency of a vibrator or of a rotator, which is the name commonly given to the reciprocal of its period.

wheels,

of the system,

rotation,

Proof of reality of adynamic and of preCissional periods when system's irrotational periods are either all real or all imaginary.

345v! It remains to prove that the roots lof (71), and of (72) also when , gg..., , are all of one sign, are essentially real and negative. (71) is the determinantal equation of $ 345xiv (42) with any even number of equations instead of only four. The treatment of 345xiv and 315is all directly applicable without change to this extension; and it proves that the roots l’ are real and negative by bringing the problem to that of the orthogonal reduction of the essentially positive quadratic function

Algebraic theorem,

Gaa)=){(120, +134, +etc.)'+(210,+2 39, +etc.)'+(310, +32a,+etc.)'+etc.} (76):

it proves also the equalities of energies of (56), $ 345", and the orthogonalities of (55), (58) S 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 1- put for å 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 Wys To 2..., are all of one sign.

345vvi Going back now to $ 345 h, and taking advantage of what we have learned in $ 345ix and the consequent treatment of the problem, particularly that in § 345*), we see now how to simplify equations (14) of $ 345 til otherwise than was done in $ 345i", by a new method which has the advantage of being applicable also to materially simplify the general equations (13) of 8 345"! 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 y,, 4s, etc. are substituted for a,, Q,, 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 l' of equation (71) of degree li in do; and denoting by &,,,,$, ner...

Si, the fresh co-ordinates, we have
G (44) = {n, ($,° +7,?) + n,' (?, ? + m2°) + ... + nji' ($1° + nai?)}... (77).

It is easy to see that the general equations of cycloidal motion (13) of 8 345" transformed to the &-co-ordinates come out in i pairs as follows:

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345xvil 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 $ 345ull and 3154 that the general equations (13) of $ 345" may, when i the number of freedoms is odd, by proper transformation from coordinates 41,4,, etc. to a set of co-ordinates $, &, 13... És (i-1), M(i-1) be reduced to the following form:

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tive dis

nated.

Kinetic 346. There is scarcely any question in dynamics more imstability.

portant 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 turbance. 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, unaccom

panied by change of speed. Kinetic sta- 347. The actual motion of a system, from any particular bility and 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

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