637 



VIRTUAL VELOCITIES. 



VIRTUAL VELOCITIES. 



ess 



the direction of the force, and the force helps to produce that motion ; 

 for it is obviously easier, caelerii paribus, that the point A should 

 mo%-e in the direction AK, when the force acts in the direction 

 A Q than it would have been if the same force had acted in the 

 opposite direction A T. But suppose that another virtual motion 

 might bring A to v. Draw vw perpendicular to AT; then AW 

 is the space moved over in the direction of the force, and p x AW is 

 the product on which the efficiency of the force seems to depend. But 

 here the motion AW is in the direction opposite to that of the force, 

 and it is obviously less easy that the point A should move in the 

 direction AV, when the force acts in the direction AQ, than it would 

 have been if the force had acted in the opposite direction AT. Hence, to 

 what has preceded, we may probably add that the efficiency of a force, 

 in promoting, or preventing one given kind of virtual motion, is to be 

 considered as of one kind or another according, as, for that motion, 

 the virtual motion of the point of application, estimated in the line 

 of action of the force, is with the direction of the force, or opposite 

 to it. . 



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, B, Ac , each in an assigned direction. Let one of the 

 virtual (that is, possible) motions which the system may undergo in 

 the infinitely small 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, Ac., 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 fdp + <)dq + 

 Rdr + Ac., is a quantity on which it depends whether the given 

 virtual motion can actually take place or not. If fdp + <jrf{ + Rdr + 

 Ac., = 0, that motion cannot be the result of the applied forces : but 

 if fdp + Qdq + ndr + Ac. be not = 0, that motion may take place. 

 And there is equilibrium, that ia, no one of the possible motions can 

 actually take place when prfp -r Qdq + Rrfr + Ac. is always = 0, for 

 every virtual motion ; and there is not equilibrium when one or more 

 virtual motions can be assigned, for which rdp + qdq + lirfr + Ac. is 

 not =0. This is the principle of virtual vdocitia, 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 virtua 

 motion of the system to be actually performed in the time dt, then 

 the velocities of the points of application, in the directions of the 

 several forces, are dp : dt,dq : dt, dr : dt, Ac., and the principle above 

 stated may be affirmed of 



dp dq i/i- 



dx dv dz 



cos \= _, cos n = -i, cos v = 



AB AB AB 



bw the line AB decomposed in the direction of P, gives ABXCOS 

 angle made by p with A B) or 



AB (cos \ . cos o + cos /*. . cos J3 + cos v . cos 7) 

 or cos a . rfx + cos ft . dy -fcos y . dz=dp. 



tence the moment of the force p is p cos o . dx + p cos /3 . dy + P cos 

 . dz, and the sum of the moments of all the forces is dx 2 (P cos o) 

 + dy 2 (P cos 0) + dz 2 (p cos 7), where 2 (P cos a) stands for p cos a + 

 ' cos a' + &c., and so on. But when there is equilibrium 2 (p cos o) = 

 , since p cos o, p' cos o', Ac., are the components of the several forces 

 n the direction of .r. For similar reasons 2 (P cos /3) = 0, 2 (P cos 7) 

 = 0, whence pdp + r"dp' + Ac., is = for every motion of which the 

 x>int is capable. 



Let there be any number of points, and let each of them be acted 

 upon by any number of forces : but as all the forces which act upon a 

 ;iven point may be reduced to one, let n, be the force which acts on 

 !he first, and o,, j8,, 7,, be the angles it makes with the axes : let 

 B., o 2 , 0,, 7,, stand in the same relations to the second point ; and so 

 on. Let x t ,y lt i t , be the co-ordinates of the first point, and so on. 

 ^et any of the points be connected by rigid bars without weight : and 

 suppose A to be one of the points, and A B the bar connecting it with 

 another point B. The point A, then, besides other forces, is acted on 

 >y a pressure called the tension of the bar, either in the direction A B 

 or B A : while B, besides the other forces, is acted on by the same ten- 

 sion, but in a contrary direction. Supposing A B to receive one of its 

 virtual motions, and to come into the position M K (which need not be 



instead of pdp + Q'/? * Rrir + Ac. But the latter is the more con 

 venient of the two. The product fdp is called the moment of th 

 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 her 

 depart from established usage) that pdp should be called the measur 

 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 motio 

 which is in the line of p's action. No perfectry general proof 

 this principle has been given ; indeed to apply it demonstratively 

 the cases of fluid and gaseous systems would require a knowledge o 

 the constituent parts of matter, and of then- connection with 

 other, which we do not possess. But the cases in which it can 

 strictly shown are very extensive : all cases whatsoever in which th 

 conditions of equilibrium can be established admit of the truth of th 

 principle being shown ft posteriori, with certain exceptions, the reason 

 of which will presently appear ; and when it is assumed, it alway 

 leads to results which are consistent with the other known principle 

 of mechanics. In the demonstration which we give, we shall confin 

 ourselves to the case of forces which act upon points, which are eithe 

 independent of each other, or some or all of which are connected b 

 rigid rods without weight : and our limits require us to speak bu 

 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 formulas of algebraic geometry of three 

 dimensions. 



First, let there be a single point A, the co-ordinates of which are 

 x, y, and z. Let there act upon this point the forces P, in a direction 

 which makes with the axes, angles a, (8, 7 ; P", the direction of which 

 makes angles o', ff, /; r", the direction of which makes angles a", 0", y", 

 Ac. Let the point A move to B. the co-ordinates of which are x + dx, 

 y + dy, z + dz, and let A B make the angles A, /, v with the axes. Then, 



A B = \/(dn? + cfy 2 + dz"), 



Either of the word activity, efficiency, energy, would do as well ; anything 

 bat moment, which has other meanings. 



in the same plane with A B), then if M c and s i> be drawn perpendi- 

 cular to A B, and if the position M s be infinitely near to A B, so that 

 M c and N D need not be distinguished (so far as small quantities of the 

 first order are concerned) from arcs of circles with the centres B and A 

 it follows that A c may be considered as the diminution of the line 

 if A only changed place, and came to M, while B D may be considered 

 as the quantity by which it would be lengthened, if B only changed 

 place, and came to N. Hence, since the bar remains of the same 

 length, we have A c = B D, or at least the two only differ by an infinitely 

 email part of either. But A C gives the virtual velocity of the tension 

 at A, and B D that of the tension at B, and these lines being equal, and 

 the tensions equal, their moments are equal ; but these moments have 

 different signs, one of the virtual velocities being in the direction of 

 its force, and the other in the opposite direction. Hence the sum of 

 these two moments is = ; and the same follows for the two moments 

 of any other of the tensions, exerted by any other of the connecting 

 bars. Let T, be the sum of the moments of the tensions which act on 

 the first point, T,, T s , Ac., of those which act on the second, third, &c., 

 points ; then, taking the principle as established above, for each point 

 separately, we have B, rfr, + T, = 0, R 2 rfr a + T 3 = 0, Ac.; by summing 

 which we have n l dr l + K t dr.+ Ac., + T, + T 2 + Ac. =0. But T t + 

 T, + Ac. = 0; for, as shown, every term in each of T,, T 3 , &c., finds an 

 equal and contrary term in one of the others. Hence R, rfr, + n a dr, + 

 Ac. = 0, or the principle is established for any system consisting of 

 forces applied to points connected by rigid bars, and this whether 

 there be connections enough to ensure complete stability of form 

 or not. 



Various other cases may be examined in which the same conclusion 

 as the last will be arrived at, namely, that the principle of virtual 

 velocities is true of the external forces only, and that those which 

 arise from the internal forces of the system may be neglected. If, for 

 example, one of the points to which a force is applied slide upon a 

 string, in the manner of a bead, the ends of the string being attached 

 to other points of the system, the two tensions are the same on both 

 sides of the bead, and any virtual motion of the bead alone shortens 

 one part of the string as much as it lengthens the other. Those parts, 

 by which one side is lengthened and the other shortened, are, when the 

 motion is infinitely small, the spaces from which the virtual velocities 

 of the tensions are obtained, and they are of contrary signs. The 

 moments of the tensions are therefore equal and contrary; or the 

 principle is true independently of those tensions. Again, suppose one 

 of the points of the system is restrained to move upon a given sur- 

 face or curve ; being tied in such a manner as to slip freely upon the 

 surface or curve, without being able to leave it. The force which 

 retains the point thus attached is perpendicular to the surface or 

 curve, but every virtual motion of that point is (when infinitely small) 

 in the tangent plane of the surface or tangent of the curve : so that 

 there is no component in the direction of the force, and the moment 

 of the force vanishes. 



When questions occur in which friction n an element, the principle 

 of virtual velocities is not of very easy application. Even in the 

 ordinary modes of solving such problems, the formula; which must 

 vanish when there is no friction, are not required to vanish, but must 

 lie between certain positive and negative limits, depending on the 



