VIRTUAL VELOCITIES. 



V1HTUAL VELOCITIES. 



Ml 



friction. A similar change most b predicated of the sum of the 

 momenta of the impressed force* ; but aa even Lagrange does not 

 appear to have thought the principle before ua conveniently applicable 

 to friction problcma, we may well dispense with the consideration of 

 them here. When elastic bodies are in question the principle can be 

 applied, but only on condition that the elasticities of the several parts 

 of the system are considered aa external, not internal forces : an hypo- 

 thesis rendered necessary by our ignorance of the molecular constitu- 

 tion which gives rise to elasticity. It may also be said that in its 

 application to hydrostatics there are mathematical conventions 

 (expressive, no doubt, of truths, but foreign to the mere enunciation 

 of the principle) which represent our ignorance of the molecular con- 

 stitution of a fluid. On this point we should recommend the student 

 who has enough of mathematics to have recourse to the ' Mecaniqtie 

 Analytique ' of Lagrange, the standard work on virtual velocities : the 

 demonstrations, so called, given by all the elementary writers we know 

 of (even Poisaon, see his ' Mccanique,' vol. ii., p. 612. 2nd edition) are 

 mere illustrations conducted upon the most limited suppositions. 

 These are more than excusable, considered with reference to the sup- 

 posed mathematical knowledge of the reader ; but it is not right to 

 make him believe that be is considering a subject generally, when 

 nothing bat a limited case is presented to him. The great fault of the 

 mathematical writers of our day is the mint of avmml of incompletenea: 

 and any one who looks in Poisson's table of contents for 'Ddmon- 

 stration du Principe des Vitesse* Virtuelles dans 1'Kquilibre d'un 

 Liquide,' and compares it with the article indicated, will see a notable 

 instance. 



When we look at the preceding demonstration of the principle, we 

 see that it depends upon knowledge of the mode of compounding and 

 decomposing forces ; but there is an d priori proof of a most singular 

 character, aa extensive as can be given by the mode already used. 

 This proof was prefixed by Lagrange to the ' Me'canique Analytique,' 

 and judging from the slight degree of notice which it has obtained 

 from succeeding writers on mechanics, we should suppose that it was 

 disputed or thought unsound. We have ourselves strong objections to 

 the form given by Lagrange ; but we believe that a sound and sufficient 

 method of proof does exist in the principle which he has used ; and 

 this we shall endeavour to developc. 



Suppose, first, that all the forces which are applied are equal to one 

 another; the case of unequal forces will follow very readily. As 

 an instance, suppose three equal forces applied at the points A, B, c, in 

 the directions AL, DM, cjr; ABC being a solid triangle without weight. 

 At A, B, o, attach rings * to the triangular system, and at L, M, N, 

 attach rings to a solid frame unconnected with the system, except by 

 the flexible string now to be mentioned. Let this string be made fast 

 to the ring u at 1, from whence let it be carried through the ring c, 



-ii 



and *g* in through the ring ii at 4, from whence it is passed through 

 M at 5 ; being nowhere attached to the frame except at 1. Its course 

 is then denoted by the numbers 1, 2, 3, 4, Ac ; and. when it emerges 

 at 14, let a weight be attached, equal to the half of the force which is 

 required to act at each of the points A. B, C ; this force being P, the 

 weight is 4 P. The tension of the string being everywhere the same 



Palltra, In Lafrange ; bat the wheel In the pulley to only a friction-wheel 

 sad, we are t liberty to clinpeow with friction In oar thought*, we may alo 

 depttvs Uw polity of its wheel. 



t is everywhere equal to 4 T : and at the point A each of the parts of 

 he string (10, 0), and (11, 12), applies a force equal to j r, so that the 

 ores r is, from the two strings, applied at A. The same may be said 

 of the points B and c. If then, at the outset, the system ABC were so 

 >laced that forces r, P, P, applied at the three poinU in the given 

 tirectiona would produce equilibrium, it follows that there will be no 

 motion when the weight |p is made to act on the string; for e niili- 

 brating forces will at that instant be applied to the system : and the 

 weight IP cannot move unless the triangle move. 



Now it is obvious d priori that if any forces keep a system in equi- 

 ibriuin, forces exactly opposite to those forces will also keep it in 

 equilibrium : if r, <j, R, keep a system in equilibrium : so will r, - Q, 

 and B, forces equal and opposite to the tirst three. For it is obvious 

 hat all the six, r, P, Q. Q. n, n, keep it in equilibrium, being three 

 sets of equilibrating forces. Take away the set p, Q, B, which, by hypo- 

 .hesis, equilibrate the system, and the remaining set, p, <J, - n, will 

 .hen equilibrate it. But here it must be noticed that when the 

 nvcrsion of directions is made, the inversion of the tensions must also 

 je possible : a force which before the inversion pulls by a string, must, 

 after the inversion, be supposed to push that string: that is to say, 

 the string must have the property of a rigid bar as to pushing or 

 pulling being indifferent. The reader of theoretical mechanics must 

 iccustorn himself to the idea of a string which, though laterally 

 lexible, can transmit a push or thrust in the direction of its length. 

 Imagine the direction * of gravity to be changed in the machine, so 

 ihat 4P acts upwards, the string being capable of transmitting the 

 thrust through the whole of its length. Nothing is then changed 

 except the directions of the forces acting at A, B, c, in such manner 

 that, if the original position be one of equilibrium, the weight Jp cannot 

 ascend, any more than it could descend in the first supposition. 



When there is equilibrium, then, the weight 4i', whether it be 

 supposed to pull downwards or thrust upwards, cannot either ascend 

 or descend. But what is to hinder JP from descending in the first case, 

 or ascending in the second ? The weight is only counterbalanced at 

 1, which is made fast to the ring at y, and if more string can be drawn 

 out beyond (13) by the descent of 4P. or pushed in by its ascent, there 

 is no mechanical reason why such drawing out or pushing in should 

 not take place. The reason why the ascent or descent cannot take 

 place must be of a geometrical character, and Lagrange reasons as 

 follows : It will be sufficient that any infinitely small displacement 

 of the triangle ABC should produce no displacement of the weight ; 

 and this will also be necessary : for if any possible infinitely small 

 displacement of the system could let out string and give motion 

 to the weight, the tendency of the weight to descend would pro- 

 duce that small displacement. But (implies Lagrange) it is enough 

 that any infinitely small displacement of the system should only 

 produce a displacement of the weight which is of an inferior order : 

 or it is enough that the second displacement should be an infinitely 

 small part of the first Here we cannot follow the reasoning : why 

 should the weight not be capable of descending because the first 

 infinitely small motion of A B c is attended by one of au inferior order 

 in the weight 4?? We could name any number of cases in which con- 

 tinued motions begin in this manner. We can only understand 

 Lagrange's argument to this extent: if there be a position of equili- 

 brium at all, it must be that in which a given infinitely small dis- 

 placement produces the smallest effect upon the weight ; so that, if 

 there be one position in which every displacement produces relatively 

 an infinitely small displacement of the weight, that position, or none, 

 must be the position of equilibrium. We shall, however, proceed with 

 Lagrange's reasoning, and shall then endeavour to show that it may be 

 saved from the preceding objection at least, if not rendered absolutely 

 rigorous. Let s be the fixed ring to which D (moveable with the 

 system) belongs ; and let the latter, in a certain infinitely email dis- 

 placement of the system, be removed to E. If s D be greater than 8 K, 

 the string B D is shortened by the removal, and drawing the arc E K and 

 the perpendicular zk, the virtual velocity of the force acting in the 

 direction OS is to (and is positive), while the quantity by which each 

 string is shortened is K D ; but if R E be longer than s D, the virtual 

 velocity is negative, and the string is lengthened. Hence, if a, |3, 7, be 

 the virtual velocities of the forces in their own directions, the expres- 

 sion 2u + 2/3 + 2? is, if positive, the quantity of string let out by the 

 displacement; if negative, the quantity taken in. Or rather we should 

 say that 2o + 2ft + 2y differs from the quantity let out or taken in by 

 an infinitely small quantity of the second order ; for tr> and K D, even 

 when is infinitely near to D, are not equal, but differ by a small 

 quantity of the second order. Lagrange, then, confounding an iufi- 

 nitoly small quantity of the second order with absolute Mtftw, com- 

 pared with one of the first order, takes 2o + 2/J + 2y = 0, or, multiplying 

 by Jp, i>a + p/B + r7 = 0, as the condition that an infinitely small dis- 

 placement of the system will allow no displacement whatever of the 



weight; from which, by the aid of the mathematical consideration 

 already alluded to, he completes what he gives aa the proof that 



Pa+ P/3+ py=0 is the condition of equilibrium : which is for this case 



the enunciation of the principle of virtual velocities. 



* I-Sfrrange avoldi thli woond COM by an appeal to mathematics, which not 

 only dctioy the elementary character of the proof, uut It of a character 

 Incongruous with the other parts of it, and ia moreover not always cornet. 



