one particle, will be immediately occupied by another, having the same properties and the same motion. If we leave out of account the minute heterogeneity of the material, we may say that the configuration of the whole system remains unchanged during the motion. If the rotation is made sufficiently rapid, it ceases to be evident to our observation, and we have in a certain sense an example of 'hidden motion,' of whose existence we may satisfy ourselves through the medium of the sense of touch.

137. Cyclic co-ordinates and cyclic velocities.-The positions occupied by the constituent parts of any movable system may be defined by fixing the magnitudes of certain quantities which vary with the time, and are called coordinates.'

This term, then, has here a more general meaning than in Cartesian co-ordinate geometry, where the rectangular co-ordinates of a point, for example, are its distances from three co-ordinate planes. A co-ordinate is any quantity (variable with the time) which helps to define the position of a body or of one of its parts, or in any way to determine the configuration of the system, whether the quantity by itself is sufficient to fix the configuration completely, or whether other co-ordinates must be known in addition.

In order to follow as closely as possible the notation used by Helmholtz and by Hertz, we shall use the symbol p to denote co-ordinates of a system in cyclic motion, so that p then represents a 'cyclic co-ordinate.'

In the example of the model in § 136 the cyclic co-ordinate p will be most conveniently chosen as the ever-increasing angular displacement of the peg in the rotating disc from its original position.

If dy is the change occurring in the value of p during the infinitesimal time dt, then

is the rate of change of the cyclic co-ordinate, that is the cyclic velocity.

In our example, q is the angular velocity with which the disc

rotates.

The velocity of any point in the rotating disc distant r, from the axis is v1=riq. The velocities of all elements of mass taking part in the cyclic motion are thus expressible in terms of the single cyclic velocity q, to which they are directly proportional. The factor depends only on invariable geometrical circumstances, namely, the distance of the point in question from the axis of rotation.

An essential property of the Helmholtzian cyclic system is this: that its condition does not depend on the instantaneous value of the cyclic co-ordinate p itself, but only on its rate of variation with respect to time, that is, on the cyclic velocity q.

As a matter of fact, if every part of the system is exactly similar to every other part, and even the point of application of the applied force is not directly evident to our senses, there is nothing to tell us at any given moment what value p has attained, since like particles continually follow like. Thus the absolute value of p does not concern us; when the cyclic velocity a is known, everything relating to the system is determinate.

In our example of a uniform circular disc, rotating with constant velocity q, this is immediately evident.

138. Energy of cyclic systems; the cyclic moment.- Since in a cyclic system the velocity of each constituent part is determined by a single cyclic velocity, the same will hold good for the energy of each element of mass, and consequently for the energy of the whole system.

This may be immediately seen in the case of a body rotating about an axis, for example, our circular disc, § 136. A particle of mass m whose distance from the axis is r, will be moving with the velocity v=rq, and its kinetic energy will be mr2q2; the kinetic energy of the whole system being thus

where T is the moment of inertia of the disc

Thus in the case of rotation about a fixed axis, the expression for the kinetic energy involves the moment of inertia in place of the mass.

The differential coefficient of the kinetic energy with respect to the velocity q is called the cyclic moment

dL

f=

dq

also, £= 9.

(21)

In our example, s=Tq, so that £=qf. Again_ds=Tdq (T constant), and the energy differential has the form

This expression like the differential expressions in § 118 and in § 119 (16) is the product of two factors.

139. Forces corresponding to cyclic co-ordinates.--Let the motion in the cycle be frictionless, and determined by the cyclic velocity q; the system will possess a certain amount of kinetic energy. If q is increased to q+dq by allowing a force P to act in a definite manner, a certain amount of work d will be performed, and since nothing in the system is changed except c, the work dll done on the system must be equal to the increase de in the kinetic energy.

We define the (generalised) force (Kraftmomente,' HELMHOLTZ) corresponding to any one of the variables by supposing this variable alone to change by an infinitesimal amount and dividing the infinitesimal work done by the infinitesimal change in the variable. In our case the work du is performed on the point of application (the peg inserted in the disc) whose position is determined by the co-ordinate p. We have thus for the generalised force (in this case the moment of the applied force about the axis of rotation)

This definition of the force is more general than that in ordinary mechanics; it includes not only forces, and moments of forces, but also applies to cases in which the condition of a substance alters, and where we do not ordinarily speak of forces at all; for example there is the pressure of a gas, which may be deduced from the work required to produce an infinitesimal decrease of volume. A comparison of (25) and (23) gives for our case

The time-rate of change of the cyclic moment is equal to the (generalised) force acting upon the corresponding cyclic co-ordinate.

140. Extension of the conception of a cycle.-The type of the cyclic motion remains unaltered, even when in course of time a change takes place in the cyclic velocity q. We may consider a system to have at each instant a purely cyclic motion even when the cyclic velocity q is changing, provided that the rate of change da/dt=q is sufficiently small. In this case the representation of the system (configuration of all elements of mass, their velocity and kinetic energy, and the energy of the entire system) involves the following quantities:

Co-ordinate p does not enter into the dynamical expressions.

Velocity q (rate of change of p): determines the condition of the system.

141. Coupled cycles. So far we have always had in mind a very simple case of cyclic motion. But there are also very complex systems whose motion is cyclic in character, so that the conception, as well as the formulæ given above, have a very wide range of application. If we

have two discs, of different diameters and of different materials, capable of turning about two parallel axes, and if we connect them by an endless band, the definition of § 136 is strictly applicable to this system, whose motion constitutes a pure, simple cycle; for each particle as it moves forward is replaced by another, similar in properties and moving in the same manner, so that in spite of the motion no change takes place in the configuration of the system. Even in the connecting band one part as it moves along is replaced by another, and if the band is quite uniform throughout its length, there is no change in its entire configuration, externally considered.

As cyclic co-ordinate it will be simplest to take the angular displacement p of a definite point P of one disc from the position which it initially occupied. Since there is nothing to distinguish one particle from another, the absolute value of the co-ordinate p has no dynamical significance. But its differential coefficient q with respect to the time determines the velocities of all the particles, and hence their kinetic energy, as well as the energy of the whole system.

The gearing together of two or more circular motions in this way is called 'coupling.'

Model of coupled cycles. To a baseboard two axles are fastened. About one of these turns a simple wooden disc (a) of radius R1 with a groove cut in its rim; about the other turns a double disc, consisting of a smaller disc of radius r1⁄2 with a grooved rim, and a large disc of radius R2 rigidly connected with it. The circumferences of R1 and r2 are connected by means of an endless rubber band vv. The dotted line a denotes the arbitrarily chosen initial position of a point P (not shown) which is fixed with respect to the disc, and is at unit distance from its axis of rotation. Its angular displacement p from the position a is the cyclic co-ordinate of the whole system, and the force P

mi

FIG. 45

R2