VIRTUAL VELOCITIES. The name of the principle of virtual velocities, which is given to what is perhaps the moat important gene ralisation in mechanics, is very ill-fitted to express the idea which is to be conveyed. It will take some space to prepare even the mathematical reader, unless ho be already aoqusintod with the subject, for the recep tion of this principle as a real and physical consequence of the laws of matter. So long as it is only treated at a mathematical mode of expressing geometrical conditions, its import is hardly seen, and its value is lessened by a want of perfect conviction.
Our works on mechanics are now written in so very cold a style, and mathematical deduction has so completely taken the place of every thing else, that little apace is given even to interpretation of results, and none to illustration of first principles. The consequence is a strong leaning to purely mathematical definitions, which, though they place the student in the smallest possible time at the beginning of his career of deduction, nevertheless make it difficult for him ever to con nect his first principles (first equations we ought rather to call them) with the actual properties of the matter around him, and with the phraseology which sight and touch make him feel to be justifiable. We do not like the system of mechanics in which velocity is only ds : dt, moving pressure but a name for vide : dt, and the principle of virtual velocities nothing but a nickname for Iedp O. For a proper description of real facts, we would rather that nature should abhor a vacuum, that fluid should try to find its level, that the centre of gravity should cndearour to descend as low as possible, and so on. Of such language the mathematician must allow the use, if the learner be to feel the truth of the results of mechanics : and in no case is such permission of more importance than in the illustration of the principle before us.
When we say that any system whatever is in equilibrium under the action of forces, it is obvious that the word equilibrium is only used for • state for rest, as opposed to one of motion ; which last is possible to be imagined, and might actually take place, if it were not that the impressed forces mutually counteract each other's efforts. If a system could not move, if to many of its points were fixed that, consistently with those points remaining fixed, no geometrical possibility of motion was left, it would be useless to ask whether any given set of forces would keep that system in equilibrium or not. For the answer would
be that the system must be in equilibrium, forces or no forces. But when it is left possible that a system may move, it than becomes a question whether a given set of forces will entirely prevent all motion, or will cause one of the posaiblo motions to begin : and the alternative may be restricted by the use of as small a portion of time as we please. What will take place during the first millionth of is second after the forces are applied, rest or motion I And instead of the millionth of a second, any smaller fraction may be used ; so that we may say the question of rest or motion, the settlement which of the two is to take place, may be considered as one which involves but an infinitely small portion of time. We shall throughout this article use the language of the infinitesimal calculus, leaving it to the reader to reduce it to the stricter form, if lie think that there is such a thing.
Now all the different infinitely small motions of which it is possible that a system may take any one during the infinitely small time dt which elapses after forces are applied to it—are called virtue/ motions. This word is not used in the meaning which it commonly bears, as when we say that a man who does not prosecute a claim virtually (as good u) abandons it. When John Bernoulli used this adjective (and wo can find none prior to him who did so) it was in a sense which it will not now bear : by a virtual velocity he meant any infinitely small velocity, or increase of velocity. But in modern times, virtual is used in the sense of potential, or possible : a virtual motion is one which a system might take, whether it take it or not : thus if forces keep a system at rest, it will not take any one whatsoever of the virtual (or possible) motions; but if they do not keep it at rest, it will, in the time dt which elapses after the forces are applied, take some one of the virtual motions, to the exclusion of the rest. Nevertheless, so long as it is geometrically possible that any one given motion might have taken place, we are at liberty to suppose that that motion has taken place (which is simply making an arbitrary displacement of the system), if by so doing, and noting the displacements which the different parts re ceive. we can draw any conclusions as to the conditions of equilibrium.