Under the given conditions, then, any vibratory motion of the pendulum masses to and from the central axis would be accompanied by alternate retardation and acceleration of the moving system. The storage of energy in the springs (central and radial) produces retardation, the restoration of this energy gives rise to a corresponding acceleration. The angular velocity of the system would rise and fall accordingly. These are the natural conditions of working of the system. As already pointed out, the motion of the pendulum masses may be regarded as executed over the surface of an imaginary sphere. Their motion against the radial springs would therefore correspond to a displacement outwards or upwards from the spherical surface. A definite part of the effect of retardation is, of course, due to this outward or radial displacement of the masses.
Assuming still the property of constancy of energy of rotation, let it now be supposed that in such a vibratory movement of the pendulum masses as described above, the energy required merely for the displacement of the masses against the radial springs is not withdrawn from and obtained at the expense of the original rotatory energy of the system, but is obtained from some energy agency, completely external to the system, and to which energy cannot be returned. The retardation, normally due to the outward displacement of the masses against the radial springs, would not then take place. But the energy is, nevertheless, stored in the springs. It now, therefore, forms part of the energy of the system, and consequently, on the returning or inward movement of vibration of the masses towards the central axis, this energy, received from the external source, would pass directly from the springs to the rotational energy of the system. It is clear, then, that while the introduction of energy in this fashion from an external source has in part eliminated the effect of retardation, the accelerating effect must still operate as before. Each vibratory movement of the pendulum system, under the given conditions, will lead to a definite increase in its energy of rotation by the amount stored in the radial springs. If the vibratory movement is continuous, the rotatory velocity of the system will steadily increase in value. Energy once stored in the radial springs can only be released by the return movement of the masses and in the form of energy of rotation; the nature of the mechanical machine is, in fact, such that if any incremental energy is applied to the displacement of the masses against the radial springs, it can only be returned in this form of energy of motion.
These features of this experimental system are of vital importance to the author's scheme. They may be illustrated more completely, however, and in a form more suitable for their most general application, by the hypothetical system now to be described. This system is, of course, devised for purely illustrative purposes, but the general principles of working of pendulum systems and of energy return, as demonstrated above, will be assumed.
43. Application of Pendulum Principles
The movements of the pendulum masses described in the previous article have been regarded as carried out over the surface of an imaginary sphere. Let us now proceed to consider the phenomena of a similar movement of material over the surface of an actual spherical mass. The precise dimensions of the sphere are of little moment in the discussion, but for the purpose of illustration, its mass and general outline may be assumed to correspond to that of the earth or other planetary body. This spherical mass A ([Fig. 12]) rotates with uniform angular velocity about an axis NS through its centre. Associated with the rotating sphere are four auxiliary spherical masses, M1, M2, M3, M4, also of solid material, which are assumed to be placed symmetrically round its circumference as shown. These masses form an inherent part of the spherical system; they are assumed to be united to the main body of material by the attractive force of gravitation in precisely the same fashion as the atmosphere or other surface material of a planet is united to its inner core (§ [34]); they will therefore partake completely of the rotatory motion of the sphere about its axis NS, moving in paths similar to those of the rotating pendulum masses already described (§ [42]). The restraining action of the pendulum arms is, however, replaced in this celestial case by the action of gravitation, which is the central force or influence of the system. Opposite masses are thus only united through the attractive influence of the material of the sphere. The place of the springs, both central and radial, in our pendulum system is now taken by this centripetal force of gravitative attraction, which therefore forms the restraining influence or determining factor in all the associated energy processes. While the auxiliary masses M1 M2, &c., partake of the general motion of revolution of the main spherical mass about NS, they may also be assumed to revolve simultaneously about the axis WE, perpendicular to NS, and also passing through the centre of the sphere. Each of these masses will thus have a peculiar motion, a definite velocity over the surface of the sphere from pole to pole—about the axis WE—combined with a velocity of rotation about the central axis NS. The value of the latter velocity is, at any instant, directly proportional to the radius of the circle of latitude of the point on the surface of the sphere where the mass happens to be situated at that instant in its rotatory motion from pole to pole; this velocity accordingly diminishes as the mass withdraws from the equator, and becomes zero when it actually reaches the poles of rotation at N and S; and the energy of each mass in motion, since its linear velocity is thus constantly varying, will be itself a continuously varying quantity, increasing or diminishing accordingly as the mass is moving to or from the equatorial regions, attaining its maximum value at the equator and its minimum value at the poles. Now, since the masses thus moving are assumed to be a material and inherent portion of the spherical system, the source of the energy which is thus alternately supplied to and returned by them is the original energy of motion of the system; this original energy being assumed strictly limited in amount, the increase of the energy of each mass as it moves towards the equator will, therefore, be productive of a retardative effect on the revolution of the system as a whole. But, in a precisely similar manner, the energy thus gained by the mass would be fully returned on its movement towards the pole, and an accelerative effect would be produced corresponding to the original retardation. In the arrangement shown ([Fig. 12]), the moving masses are assumed to be situated at the extremities of diameters at right angles. With this symmetrical distribution, the transformation and return of energy would take place concurrently. Retardation is continually balanced by acceleration, and the motion of the sphere would, therefore, be approximately uniform about the central axis of rotation. It will be clear that the movements thus described of the masses will be very similar in nature to those of the pendulum masses in the experimental system previously discussed. The fact that the motion of the auxiliary masses over the surface of the sphere is assumed to be completely circular and not vibratory, as in the pendulum case, has no bearing on the general energy phenomena. These are readily seen to be identical in nature with those of the simpler system. In each case every movement of the masses implies either an expenditure of energy or a return, accordingly as the direction of that movement is to or from the regions of high velocity.
Fig. 12
The paths of the moving auxiliary masses have been considered, so far, only as parallel to the surface of the sphere, but the general energy conditions are in no way altered if they are assumed to have in addition some motion normal to that surface; if, for example, they are repelled from the surface as they approach the equatorial regions, and return towards it once more as they approach the poles. Such a movement of the masses normal to the spherical surface really corresponds to the movement against the radial springs in the pendulum system; it would now be made against the attractive or restraining influence of gravitation, and a definite expenditure of energy would thus necessarily be required to produce the displacement. Energy, formerly stored in the springs, corresponds now to energy stored as energy of position (§ [20]) against gravitation. If this energy is obtained at the expense of the inherent rotatory energy of the sphere, then its conversion in this fashion into energy of position will again be productive of a definite retardative effect on the revolution of the system. It is clear, however, that if each mass descends to the surface level once more in moving towards the poles, then in this operation its energy of position, originally obtained at the expense of the rotatory energy of the sphere, will be gradually but completely returned to that source. In a balanced system, such as we have assumed above, the descent of one mass in rotation would be accompanied by the elevation of another at a different point; the abstraction and return of the energy of rotation would then be equivalent, and would not affect the primary condition of uniformity of rotation of the system. In the circumstances assumed, the whole energy process which takes place in the movement of the masses from poles to equator and normal to the spherical surface would obviously be of a cyclical nature and completely reversible. It would be the working of mechanical energy in a definite material machine, and in accordance with the principles already outlined (§ [20]) the maximum amount of energy which can operate in this machine is strictly limited by the mass of the material involved in the movement. The energy machine has thus a definite capacity, and as the maximum energy operating in the reversible cycle is assumed to be within this limit, the machine would be completely stable in nature (§ [25]). The movements of the auxiliary masses have hitherto also been considered as taking place over somewhat restricted paths, but this convention is one which can readily be dispensed with. The general direction of motion of the masses must of course be from equator to pole or vice versa; but it is quite obvious that the exact paths pursued by the masses in this general motion is of no moment in the consideration of energy return, nor yet the precise region in which they may happen to be restored once more to the surface level. Whatever may be its position at any instant, each mass is possessed of a definite amount of energy corresponding to that position; this amount will always be equal to the total energy abstracted by that mass, less the energy returned. The nature of the energy system is, however, such that the various energy phases of the different masses will be completely co-ordinated. Since the essential feature of the system is its property of uniformity of rotation, any return of energy in the rotational form at any part of the system—due to the descent of material—produces a definite accelerating effect on the system, which effect is, however, at once neutralised or absorbed by a corresponding retardative effect due to that energy which must be extracted from the system in equivalent amount and devoted to the upraising of material at a different point. For simplicity in illustration only four masses have been considered in motion over the surface of the sphere, but it will be clear that the number which may so operate is really limited only by the dimensions of the system. The spherical surface might be completely covered with moving material, not necessarily of spherical form, not necessarily even material in the solid form (§ [13]), which would rise and fall relative to the surface and flow to and from the poles exactly in the fashion already illustrated by the moving masses. The capacity of the reversible energy machine—which depends on the mass—would be altered in this case, but not the general nature of the machine itself. If the system were energised to the requisite degree, every energy operation could be carried out as before.