f. . 



YIKTUAL VELOCITIES. 



VIUTUAL VELOC1T 



be conveyed. It will Ukc aome space to prepare even the mathemi tieal 

 reader, unit** be be already acquainted with the subject, for the recep- 

 tion of this principle M a real and physical consequence of the lawi of 

 matter. So long as it is only treated M a mathematical mode of 

 xpreeaing geometrical conditions, its import is hardly seen, and its 

 value is lessoned 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 bis 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 bo justifiable. 

 We do not like the system of mechanics in which velocity is only 

 tit : dt, moving pressure but a name for tndr : dl, and the principle of 

 virtual velocities nothing but a nickname for 2r</p=0. >or a proper 

 description of real facts, we would rather that nature should abhur a 

 vacuum, that fluid should try to find its level, that the centre of gravity 

 should endeavour 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 result* 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 a 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 so 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 more, it then becomes a 

 question whether a given set of forces will entirely prevent all motion, 

 or will cause one of the possible 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 a second after the 

 forces are applied, rest or motion ? 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 he 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 virtual 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 as) abandons it. When John Bernoulli used this adjective (and 

 we 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 takei 

 place, w are at liberty to suppose that that motion has token plac 

 (which is limply 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. 



When we see a system in equilibrium, experience tells us that there 

 are efforts at motion which are counteracted. Remove any one of the 

 forces, or any part of one of them, and motion immediately begins. 

 It is true that friction and other resistances prevent our having so good 

 a perception of this truth as we otherwise might have ; since, when 

 equilibrating forces are removed in whole or in |>art, friction frequently 

 supplies the place and maintains the equilibrium. A little reflection 

 will however make it apparent that when a system is once in cqui 

 lihrium.no addition nor subtraction of forces can be made without pro 

 ducing motion, unless the forces added or withdrawn be such as by 

 themselves would maintain equilibrium. 



A system, then, at rest, makes efforts to move, which efforts are 

 counteracted ; and the mathematical conditions of equilibrium, what- 

 ever they may be, must express that every force endeavours to produce 

 motion ; must contain, directly or indirectly, a measure of the euergi 

 of that force ; and must show that a complete counteraction of al 

 the effort* at motion takes place. But here arises a question, and one 

 which is of the utmost importance in the comprehension of our prin 

 eipl*. The number of virtual motions is usually infinite : Does any 

 given system of forces make HI effort to produce every one of them, or 



some only .' We know that, if the forces do not produce equilibrium, 

 one of the virtual motions ensue* in the first dt following the applies- 

 ion of the forces, to the exclusion of all the rest ; it ought not, there- 

 ore, to surprise the student if he were told that, for every given set of 

 orces (a given system being always understood), some one in 

 invented is every motion prevented. But in point of fact the direct 

 wntrary is true, in rigid systems at least : generally speaking, there is 

 mt one class of virtual motions which a given set of forces has not a 

 tendency to produce, and any one of the rest may be produced. Our 

 meaning will appear in the following explanation : We have seen 

 "ROTATION] that every infinitely small motion of a rigid system may 

 w produced by a screwlike'motion,* namely, rotation round an axis, 

 accompanied by a slipping up or down that axis. Take any line for an 

 axis, and suppose a screw, fitted to its receiving screw (the latt 

 moveably fixed in space), to be described with that axis : suppose also 

 hat the system to which the forces are applied is fixed to the screw. 

 Here then is every virtual motion prevented, except one ; so that if 

 ihe system begin to move, it must take that one motion. Now apply 

 ,he given set of forces, and resolve them all in directions parallel 

 axis, and in planes perpendicular to it. There must be motion unless 

 ;he former forces destroy each other, and the latter have a resultant or 

 resultants passing through the axis. Consequently, with certain ex- 

 ceptions (which, though infinite in number, are few compared with tin- 

 Test), a given set of forces, acting on a given system, will produce any 

 virtual motion, if others be excluded : but when there are various 

 virtual motions not excluded, the system, if the forces do not balance 

 one another, will take one in preference to any of the rest. The pre- 

 ceding argument ought to be more developed, but we have not room 

 for such an explanation as would be intelligible to every one : most of 

 the difficulty indeed lies in the purely geometrical conception of 

 motion, and is foreign to our arti< lr. 



We are to expect, then, as the condition of equilibrium, a collection 

 of conditions, an infinite number, implying that, of au infinite number 

 of motions, possible a priuri, the given system of forces makes eai i 

 every one impossible. To moke it appear in what the condition may 

 probably consist, look at the following coses : If one point of the 

 system be fixed, forces applied at that point are useless, for they only 

 produce a pressure or strain on the fixed point, and neither promote 

 nor retard any virtual motion. If one point be restrained to m\v 

 upon a given surface or curve, forces applied at that point 

 pendicular to that surface or curve are useless, for a similar reason. 

 Thus suppose one point must be retained on a given horizontal j. 

 any weight added to that point has no effect on the equilibrium ; it is 

 merely equivalent to so much weight t laid upon the plane. Generally, 

 then, a force produces no effect in equilibrium unless the point to 

 which it is applied can move in the direction of that farce ; thus v 

 produces no effect when applied to a point of which all the virtual 

 motions are horizontal. But let the plane be ever so little inclined to 

 the horizon, a point restricted to move upon it has somewl 

 vertical motion : weight applied at that point will have some effect in 

 equilibrium. It would be natural to conclude (and let it be remem- 

 bered that in these d priori views we ore only stating strong probabi- 

 lities) that the more freely a point may move in the direction of the 

 force which acts upon it, the greater the effect of that force in pro- 

 ducing or disturbing equilibrium. Now since it is sufficiently < 

 that, caterit paribut, a force has more or less effect in proportion to its 

 magnitude, for instance, that, under given circumstances, two pounds 

 of pressure produce twice the effect of one pound, it seems that for 

 any given virtual motion, the effect of each force varies jointly as the 

 magnitude of the force, and the length over which, in that virtual 

 motion, the point of application moves in the direction of the force. 

 That is, suppose A to be the point of application of the force, and A Q 



to represent its direction and magnitude. In one virtual or possible 

 motion of the system, let A be transferred to n, infinitely near to A. 

 Draw n s perpendicular upon Q A, then A s is the space moved over in 

 the direction of the force ; and if the force contain p units of pressure, 

 P x A 8 is the product on the value of which the efficiency of the farce 

 seems to have some dependence. Here, however, the motion A s is in 



Simple truncation ii the extreme CIM In which the thread of the ncrcw 

 bccome parallel to the ul> ; and ulntple rotation the other extreme cute In 

 which the sncceeslre colli of the thread coincide. 



t Here stain the common ideal derived from friction mutt be abandoned 

 a weight stuched to luch s point might help, by the friction on the plane, to 

 equilibrate the uritem. 



