These conjectures, for they are nothing more, will show of the prin ciple of virtual velocities, the moment it is announced, that it is a highly reasonable and probable principle. It may be announced as follows :—Let the forces which are applied to a system, at different points, be P, Q, R, &c , each in an assigned direction. Let one of the virtual (that is, possible) motions which the system may undergo in the infinitely email time dt succeeding the moment of application of the forces, be supposed to be given, upon trial. Decompose the several motions of the points of application of the forces each into two, one in the line of the applied force, the other perpendicular to that line ; let dp, dq, dr, &c., be the resolved motions in the lines of the forces, and let those be reckoned positive which are in the directions of the forces, and negative which are in the contrary directions. Then rdp + Qdq + Rdr + &c., is a quantity on which it depends whether the given virtual motion can actually take place or not. If rdp + + Rdr + &c., =0, that motion cannot be the result of the applied forces : but if rdp + Qdq + + &c. be not = 0, that motion may take place. And there is equilibrium, that is, no one of the possible motions can actually take place when rdp t Qdq + Rdr + &c. is always -= 0, for every virtual motion ; and there is not equilibrium when one or more virtual motions can be assigned, for which Pdp + Qdq + ndr + &c. is not =0. This is the principle of virtual reloeittes, as to which perhaps the first thing that will strike the reader is that the word velocity does not occur in the explanation of it. But if we suppose the virtual motion of the system to be actually performed in the time di, then the velocities of the points of application, in the directions of the several forces, are dp : di, dq : dr : dt, &c., and the principle above stated may be affirmed of instead of rdp + Qdq + Rdr + &o. But the latter is the more con venient of the two. The product rdp is called the moment of the force P, which is not a well-chosen term, since " moment " is used in other senses. It would be much better (though we shall not here
depart from established usage) that Nip should be called the measure of the equilibrating power of the force P, or, in one word, the power* of the force r : with reference, of course, to the promotion or hindrance of those virtual motions only, in which dp is the part of the motion which is in the line of is action. No perfectly general proof of this principle has been given ; indeed to apply it demonstratively to the cases of fluid and gaseous systems would require a knowledge of the constituent parts of matter, and of their connection with each other, which we do not possess. But the cases In which it can be strictly shown are very extensive : all cases whatsoever in which the conditions of equilibrium can be established admit of the troth of this principle being shown h posteriori, with oertain exceptions, the reasons of which will presently appear; and when it is assumed, it always leads to results which are consistent with the other known principles of mechanics. In the demonatmtion which we give, we shall confine ourselves to the case of forces which act upon points, which are either independent of each other, or some or all of which are connected by rigid rods without weight : and our limits require us to speak but briefly ,of all the steps which are purely mathematical. Ordinary works on mechanics give the simple illustrations which the beginner wants : and it is impossible to read anything like a general de monstration without being well acquainted with the infinitesimal calculus and with the principal formulae of algebraic geometry of three dimensions.
First, let there be a single point A, the co-ordinates of which are x, y, and r. Let there act upon this point the forces r, in a direction which makes with the axes, angles a, 8, ; the direction of which makes angles a', S', 7'; P", the direction of which makes angles a", fr, 7", &c. Let the point A move to a, the co-ordinates of which are x+ dx, y +dy, z + dz, and let A B make the angles A, e, v with the axes. Then, dy dz COS , cos 14 = cos y