The “Phantom” Force¹

Abstract

IT might be supposed that permanent and entirely local or “internal” force-pairs of this kind acting on innumerable material couplets in a system would so disturb the individual energies of their motions that no general conclusion as to the total change of energy during the progress of such a system's motion could be drawn; but the simple law that impulses act independently of each other and of existing motions soon shows that the whole gain of energy in the system is the sum of the separate gains in the several mass-couplets due to their absolute or several actions and reactions at every instant of the motion, and that when these abstract force-pairs are all permanent, the above constancy of the sum of their actual and potential energies is possessed by the whole system as perfectly as would be the case by one only of its couplets, or component pairs. That this is not merely an abridged expression for the resulting actual energy in all the possible different phases that such a system may go through, briefly stated for any initial and final configurations and initial motion of the system by means of the negative scale or potential function of all the several component force-pairs supposed known; not merely, that is to say, a logical consequence of arbitrary and fanciful definitions, but a conclusion full of importance and of real natural signification depends, firstly, upon the fact that the thing defined as “impulse”, or the gradient of the scale, which is here independent of the time and depends only on the mutual configuration, is not permanent by a very rare occurrence, but that it is often so, and under very various circumstances; and again that this impulse, or flux of momentum, or gradient of energy, occurs in many other motions with conditions of equal simplicity; and lastly, and above all, on the fact pointed out at the beginning of these reflections, that while we are able to use, and of our free will to call into existence force in innumerable ways, we learn from our experience that this impulse is invariably caused or dictated by a certain special efficacy or compulsion, which our power of exercising it as we please so as either to annul, to modify, or to increase it at will with the consequence of obtaining with it any effective impulses that we desire, shows us to be a different kind of quantity from the impulses that we either thus obtain, or that we see it producing in surrounding nature. Newton's second law of motion in fact recognises this specific difference between the magnitudes of a force and of its effect, when it asserts that forces produce their whole effects (that is to say, remain unaffected in their efficacies) whatever may be the state of rest or of motion of the bodies upon which they act. As it is found that forces or compulsions (measured as they are in statics by additions, subtractions, and oppositions to a standard force) are always proportional to the free impulse, or undisturbed acceleration of a mass-unit which they can produce, so that by taking the impulsive effect and the active compulsion of any one standard force as the units for measuring these quantities respectively, they are then said to be numerically equal to each other in every force; it yet follows from their specific independence that they are not identical in kind as they are in measure. The same is true of the products obtained by multiplying them separately by any small space through which a force acts; and it would be an obvious misstatement to assert that the sum of the works of a compulsion, and of the free impulse which it produces taken negatively, is constant when a force acts freely; because this would be confusing in one sum two different quantities; a result which It seems must arise from the simple fact that our part in mechanical “compulsions” distinguishes and removes them from the category of impulses to which they would otherwise belong, and leads us to regard them as the causes of the impulses which we observe. The language adopted by Newton (and used also by D7'Alembert) in the proposition quoted at the beginning of this letter is that in a proper mechanical system, compulsions equal to the observed impulses reversed, will (as is obvious) arrest in their origin all changes of motion in the system, and will (with the immutable force conditions proper to the system) hold in balance, or give a complete account of all the forces (other than those immutable ones) acting upon it. Using the principle of virtual velocities in this case of equilibrium of balancing forces, Newton expresses the rule for exploring all the mechanical efficacies (superadded to the immutable ones) acting in the system by concluding that the sum of all the similar “works of compulsion”, or of all the “actions” in a short time corresponding to a small motion of the system, when the reversed ones have been introduced, will be a constant quantity. Thus both Newton and D'Alembert agree in this, that they recognise in forces causes which differ from the effects which they produce. By what similar laws of work found to hold true in a proper conservative system the modern science of energy (which deals with the phenomena of causation in a wider and more diversified form) seeks to extend the method of cancelling the counteracting causes, or the principle of energy conservation here laid down by Newton for a mechanical one, to the far larger, but less thoroughly explored and exhausted field of all the onward flowing streams of physical agencies which we perceive following their natural bents or inclinations around us, I will presently endeavour to explain. It should be noticed in connection with this general extension of the principle, that the “work” of a force in a short time, however fixed its efficacy, or its rate of doing work in a short space (or of producing momentum in the short time) may be, is incidental, and not a fixed quality of the agent force like its faculty of tension, since a force as often diminishes as increases actual energy by a momentary action, and thus no fixed rule is drawn from the natural tendency of force to impart momentum, that potential energy necessarily becomes, or even necessarily tends to become, actual energy in every mechanical energy-transformation. The mechanical stress of friction is an example of the opposite tendency; and it also furnishes us with an example of a force whose working power only, and not its motive tendency, is a mechanical “agent” which we can summon up at will; but of which we still regard the motive tendency as similar to that of other mechanical forces, because it can maintain equilibrium with them.