must be conceived of as acting at P in the direction of the tangent to its path. P is only a force in the generalised sense; when y is an angular magnitude, as in the present case, is the moment of the impressed forces about the axis of rotation. This being understood, the point P need not be at unit distance from the axis.] For an elementary mass m, of the first disc, at the distance r, from the axis, the velocity is v=rq, and the kinetic energy mr.2q2; for an element of mass m of the second disc at the distance r from the second axis of rotation these quantities become respectively R1/r.ra, m (R1/r2)2r/2q2. The speed-ratio Rr is a constant characteristic of the particular piece of mechanism.

In the cycle now under consideration the velocity at any point i takes the form v=aq, where a is a factor depending on the manner in which the points i and P are connected with one another. For all points on the first disc, a, is numerically equal to the distance from the axis of rotation; for points in the connecting band, whose motion is in part rectilinear, in part circular, the value of a is simply R1, the radius of the first disc; for all points in the second disc a, is equal to the distance from the axis of rotation, multiplied by the speed-ratio R1/r2.

All arrangements of homogeneous discs connected by bands, and quite generally all combinations of homogeneous bodies turning about axes of symmetry connected by bands. or strings (without slipping), give rise to pure cyclic motions.

Thus a cyclic motion is not necessarily a simple rotation about an axis; cyclic motions about axes any distance apart may be so combined as to constitute a single cycle. It is not even necessary that all the motions concerned in the cycle should take place along circular paths, some of them may be in part rectilinear, or (when the connecting bands are slack) may take place along catenaries. The fundamental character of the motion is still maintained, each element of the band, as it moves along, being always replaced by an identically similar element.

Again, substances of widely different kinds may take part in the cyclic motion: the substances of the discs and of the connecting band. Finally the axes about which the different motions take place need not be parallel to one another, and the directions of some of the rotations may be reversed by putting the bands

on cross-wise, but still the fundamental character of the cyclic motion is maintained.

We have thus reached a considerable extension of the conception of cyclic motions. We may conceive of a region (field), filled with an immense number of cyclic motions, gyrostatic or vortical, all of which taken together constitute a single cyclic motion. The condition for this is that all the separate motions are determinable from a single cyclic velocity.

In the second section we shall become acquainted with such a system of coupled cyclic motions. It is not to be supposed that in nature the cycles concerned are coupled together in the simple manner of our discs with a connecting band, or that they have the same simple character. Our model only serves as an illustration of the principles, and helps to explain the manner in which the energy of the field is distributed.

When the velocities of the separate particles which take part in such a cyclic motion are proportional to the cyclic velocity, the formulæ 19 to 22 hold good which we obtained in a very simple case.

For example, the total kinetic energy of the coupled cycles in fig. 45 is

2

L=} { 2mr?+ (黒)
(B1) * 22m{r(2)q2

where the first summation is extended to all particles of the first disc, and the second summation similarly refers to the double disc. The bracketed expression may be regarded as the moment of inertia T corresponding to the whole system. From 2= Tq2, however, all other formulæ follow, including equation (26).

If we compare the formula 20 and 22 for the energy of a cyclic motion with the formulæ 17 and 16, § 119, for the energy of a field, we see that the analogy is complete, a result which adds greatly to the plausibility of our attempt to explain magnetic phenomena by means of cyclic motions. If we do not wish to conceive of the energy of the magnetic

field as being of some definite mechanical form (such as the kinetic energy in the present illustration) we must look upon the cyclic motions as constituting only instructive energy models, showing how we may imagine a realisable distribution of energy in the field.

142. Imperfect cycles.-The following example will serve to illustrate how the conception of a cycle may be extended in yet another direction. A wheel with spokes is made to rotate about its axle with constant angular velocity. Can the system of moving mass-elements be considered to constitute a cycle, as defined in § 136? In the rim of the wheel and in the hub, similar particles having similar motion succeed one another without intermission, so that to this extent the system is purely cyclic. But at any given place in the track of the spokes it is only during a limited time that particles of like nature (such as those of the spokes) continuously succeed one another; after the lapse of this time other particles appear in their stead, in the present case the irregularly moving particles of air. At each such place of the region in question this process is repeated periodically, so long as the character of the motion is maintained.

The configuration of the whole system, in regard to the point considered, is identically repeated after the lapse of each period. Here again the velocity of any particle i of mass m; is v=r.q, where r; is the distance of i from the axis of rotation and q denotes the angular velocity of the wheel. Thus the kinetic energies of the separate particles, and their moments of inertia, as well as the kinetic energy and moment of inertia of the entire system, depend only on the cyclic velocity, just as in the case of a pure cycle. The wheel with spokes is an example of an imperfect cycle. The same holds good for a toothed wheel, or for a system of toothed wheels geared together. In all such cases v=aq, where a, is a factor depending on the nature of the mechanism.

Even a non-homogeneous body, which turns with constant velocity about any given fixed axis, constitutes an

imperfect cycle, provided that no change takes place within the body during the rotation. For after the lapse of the period of rotation the same state of things is always exactly reproduced; and through a large number of such periods. the character of the motion is maintained unaltered. In the case of imperfect as of perfect cycles, we may make the assumption that the cyclic velocity is accurately constant; or if any change takes place in this magnitude, it must be a very slow change.

143. Monocycles, dicycles and polycycles. So far we have confined our attention to cyclic systems in which the velocities and energies of all mass-elements are determined by the rate of variation q of a single co-ordinate p. Such systems are called by Helmholtz monocycles.' They are characterised by the relation vag.

Bodies of any shape rotating about fixed axes, and any combinations of such bodies, coupled or geared together, may be regarded as monocyclic systems, provided the progressive changes taking place in their configuration are slow compared with the cyclic motion proper.

In Section II., when we come to deal with the electric current, we shall find fields whose properties present the closest analogy to the mechanism of a monocyclic motion. In dealing with induction' we shall have to recur to equations (20)-(26), especially the last.

But in addition to monocycles there are other cyclic systems in which the velocity of each mass-element is determined by the values of two independent cyclic velocities q, and q2, so that in the simplest case

91

Such a system is called by Helmholtz a dicycle.'

If the velocities of the separate masses of a system involve the simultaneous values of more than two independent cyclic velocities,

the arrangement is called a 'polycyclic system' or 'polycycle.'

144. Slowly varying parameters.-The cyclic motions. occurring in nature are commonly of such a kind that they are repeated many times over before any considerable change takes place in the character of the motion. But a gradual change may be going on all the time, so that the later cycles are accomplished under conditions different from those of the earlier cycles. These gradual progressive changes we may suppose to be slow in comparison with the more rapid cyclic motion itself; the value of the cyclic velocity determines whether the change in question is to be considered slow or rapid.

Thus the rotation of the earth about its axis is a cyclic motion completed in a day. After the lapse of many days and years, precession and nutation will have produced a small change in the direction of the earth's axis; the change being slow' in comparison with the diurnal cyclic motion.

Quantities which determine the condition of a body or of a system of bodies are called 'parameters.' The parameter whose variations correspond to the slow progressive changes in the condition of the system we may refer to as 'slowly varying parameters.' If we denote them by the italic letter p, in distinction to the cyclic co-ordinate, these variables will be characterised by the following properties (compare § 140):

Parameter p
q=rate of change of p

appears in the dynamical equations.
is small in comparison with q.

dq

=rate of change of q

is very small.

dt