FREE AND VIBRATORY ENERGIES 23
masses. Yet it must be capable of containing energy. Therefore, it must be, not an ultimate or indivisible unit of mass, but a mass-pair, an elementary mass-pair.
All action between solid bodies in contact, on the other hand, such as is familiar to all engineers in their machines, is not purely mechanical. It is always partly thermodynamic, in so far as heat is being constantly developed by friction, and partly a special case of pure mechanics, in that the body is "constrained" rather than free; that is, it is handling energies which are transient through it from without, which are independent of its own mass, and which are ultra-complex in their nature.
Fig. 4, on the other hand, is entirely general. It displays every possible form of pure and elementary mechanical action between two bodies, supplied with any original store of relative space and relative motion whatever, as at A; and it introduces no foreign element of dependence upon any other mass-system or form of energy whatever. Nor does it introduce any unnatural assumptions. Without stopping now for the proofs, it may be said that any such a case must resolve itself into the mutual revolution of the bodies about each other in an orbit which follows some of the plane conic sections—either the hyperbola, the parabola, the ellipse, the circle or the straight line.
Further, it can be shown that if the original energetic condition of the pair at AM2 be known, by knowledge of the distance d between the two, the velocity v of their relative motion, the angle <£ existent between d and v, and the two masses Mt and Mo, then the nature of the orbit is known, and also its dimensions. Both are best expressed in terms of the distance D between the two when separated by a radius normal to the major axis XX' of the orbit, the velocity U at that point, and the angle a between D and U. The mathematical relationships between all these quantities will be discussed later.
Of all these apparently varied forms of motion only two, the hyperbola and ellipse, are probable forms. For between any two masses, at any given initial distance, there may be an infinite number of directions and magnitudes of velocity which would result in hyperbolic motion, and another infinite number which would result in elliptic motion. But there is only a single direction of motion which would result in a straight-line orbit, such as that of Fig. 2; and there is only one other direction of motion,