VIRTUAL VELOCITIES. 



VIUTUAL VELOCITIES. 



Ml 



and in the 



ot equilibrium tliU a always - 0. Since rfr <(, c/,,, 

 fc, *** Mh rtoWv* any value w* r,l*a*e lnde- 

 pmdenUy of the mt, the preceding can only vani*h when the six 

 following conditions are fulfilled : 



2s -o ST = o si = o 



which are the six well-known equation* of equilibrium of a rigid 

 system. 



Wo might give more example*, but our limits require us at once to 

 outer u|xm a point which will require fuller explanation, because the 

 ttucleut will nut find it in any elementary work. " When Z(n/;<) U 

 for every virtual motion, there mutt be equilibrium :" but the 

 convene, namely, that " when there it equilibrium the equation 2 ( nip) 

 _ mut be true for every virtual motion," has not the same univer- 

 sality as the direct proposition. If we look carefully at the proof, we 

 shall see that, taking any particular instance of virtual motion, the 

 only reason why we want 2 (rrfy<) to be - for that particular motion, 

 is, that the forces may not be able to make the system set olf with that 

 motion : or that the incapability might exist even though that motion 

 were the only on* which the system could take. If then there be in 

 the nature of the system itself any reason why a particular cose of 

 virtual motion should be unattainable by the action of those forces, we 

 have no longer any reason to say that S(Prf/>) must be = in that 

 rafln 



As a general rule, if P, <J, B, Ac. be the acting forces, and vJp + Qrfj 

 + *dr+ Ac , the sum of their moment* ; and if in one virtual motion 

 <l}i = a, tlq-P, tiff, *., that one virtual motion baa its opposite, in 

 which rfp = a, </7 f, d r y, Ac. And we shall presently 

 see that if that one motion and its opposite be by proper restrictions 

 made the only one* which the system can take, the system will begin 

 to take the first motion ifpo-t-<l/3+B7+Ac. be positive, and the 

 opposite if r(- a) + Q (- /3) + B ( - y) + Ac. be positive. In fact, a 

 system must set off from rest in such manner that the sum of the 

 initial moment* is positive : and it is clear enough that either Po + 

 Ac , or f ( a) + Ac. must be positive unless both vanish. As a 

 general rule then, p + Ac. must vanish ; for if not, either the virtual 

 motion first named, or it* opposite, ha* a positive sum of moments, and 

 can be, and (if no other motion can take place) will be, an initial 

 motion of the system. But if ever it should happen that there are 

 rjira in which a virtual motion is possible, but iU opposite motion U 

 impossible, then all that is requisite U that for the posnible one of the 

 pair, j rdp should be or negative, not pontivr. There is another 

 exception of a remarkable character, for which it will be better to 

 wait until we come to see the meaning of the sum of the moments 

 in a dynamical point of view. Excluding this for the present, let a 

 virtual motion which has ita opposite be called a double motion, and 

 one which ha* not it* opposite, a single motion : then the true state 

 raent of the principle of virtual velocities is a* follows : 



If ZrWp be nothing for every double virtual motion, and nothing or 

 negative for every single one, there is equilibriuui : and if there be 

 equilibrium, then S(i',V;, )ia nothing for every double virtual motion, 

 aud nothing or negative for every single one. 



We might easily have incorporated the consideration of these ex- 

 ceptional case* of single virtual motions in thu general proof. Wo 

 shall now give a simple instance. Let a weight bo fastened to the 

 inicidl* of a string, at the end of which are two rings ; these rings 

 slide upon curve* which have cusps a* in the diagram. The weight is 



in equilibrium, ami the weight H the only external outing force : bu 

 it* moment is not nothing (that is, is not of the second order will 

 reference to the displacement of the rings), but is negative. Th 

 virtual motion* of the rings are single, and can only be upwards. 1'h 

 reader who compares the preceding omission in the statement of th 

 principle'of virtual velocities with VAIIIATIOXB, CAU ut.us or, will se 

 a remarkable likeness between the caw* : in fact, these errors am 

 several others depend upon the tame sort of omission, which may be 

 stated as follows : If there be a proposition (A) which is true on con 

 ilition that the quantity B u never positive ; and if, generally speaking 

 every negative value of B be accompanied by a corresponding pomtiv 

 one, then, generally speaking, (A) cannot be true if B be negative : tba 

 is, th* truth of (A) requires u-U. But if there be exceptional 



singular oases in which negative values of B are not accompanied by 

 corresponding positive ones, then B =0 is no longer necessary ; it is 

 jh that B should be negative. Now the error which has run 

 irough the result* of the differential calculus from book to book, 

 rom country to country, and from century to century, consist* in 

 taking the usual and general oas* for universal, and forgetting the 

 xoeption. 



Thu principle of virtual velocities 1* applied to dynamic* by menus 

 t the celebrated principle which goes by the name of D'Alcmbert, 

 ropounded by him In his treatise on dynamic*, published in 17' ; 

 **e have touched upon thi* principle in FORCES, IMTIIMSH 

 ImcrrvK, but we liavo referred the complete development of it to 

 1* present article. 



It will do for our present purpose to suppose a system of point* con- 

 ectcd together, each point being considered as a certain mass of in 

 Whatever may be th* faults of the system of Cavalieri f CAVALIKRI, in 

 lioo. Div.j for geometrical deduction, it is sound enough mechanically 

 onsidered : a point may not be taken to be one ot ' ;< n'- 



art* of a length, but there is no difficulty in considering it as end 

 with weight aud impenetrability, or as rigidly connected with other 

 joint*. If we imagine a moss of matter to be divided into an infinite 

 umber of infinitely small element*, each of which Is an extended man, 

 liough we may not, for geometrical purposes, suppose each of them 

 lements to have ita bait collected in any one of ita points, there is no 

 ilHculty in supposing it* man to be so collected, if then we begin 

 ith the consideration of a finite number of points, having various 

 lasses, we may, by increasing the number of our points and diminishing 

 heir mimes, approach as near as we please to the case of a continuous 

 eometrical solid, all the part* of hich have weight, and of which the 

 ensity varies according to any law. Again, when a system in 

 and when the law of its motion is known, we can determine, at any 

 one instant [VKi.ocrrr], the velocity of any one point in any one 

 lirection, and the acceleration (or retardation, vryatirt acceleration) at 

 hat one instant : that is to ay, the ratn per second at which the 

 motion is receiving acceleration at the moment named. From this 

 acceleration, as in the place last cited, we can determine the pressure 

 which the mass of the point in question is actually exjieriencing at thu 

 moment ; for on one mass there is but one pressure which can produce 

 acceleration at one given rate. In this way then we can determine the 

 treasures which the various point* (or molecules *) of the system are 

 undergoing: and this determination is made in term* of the mvti'm, 

 hat is, in terms of tlir velocities and accelerations of the molecules, 

 he pressures being derived from th* accelerations by reference to the 

 mown masses of the molecules. Th* pressures so obtained are called 

 tfattrr farce*, a sufficient and expressive name. But it by no meant 

 ollows that the forces aji/i/iitl at the different molecules arc those 

 which are tftctite on those molecules. Two molecules are inseparably 

 joined by a rigid bar without weight, and thrown into vacuous ajnce, 

 [f these molecules were thrown separately, each would describe a 

 parabola ; but as the case stands, the centre of gravity of the molecules 

 iescrib** a parabola, and the bar revolves round its centre of gravity 

 THANSI.ATIUS and ROTATION] : the effective forces are very different 

 from the impressed force*. Now IJ'Alembert's principle is the 

 expression of this simple law, that force it never lot! t nor gained. If a 

 force applied to any molecule of the system be not wholly effect: 

 that molecule, the port which is not effective on the moleculu of 

 application is effective elsewhere ; and if the motion gained by or rate 

 of acceleration shown by any given molecule be greater than is due to 

 the force impressed on that molecule, some other molecule of the 

 system must have less than is due to its impressed force. Thus the 

 motion of a system of connected molecules involve* a collection of 

 debtor and creditor accounts, the balances of which cannot show, when 

 put together, the smallest amount of momentum in any direction, 

 except what the system either had at the beginning or has received 

 from the impressed forces during the motion. The consideration of 

 the third law of motion [ MOTION, LAWS OF,] would make such a result 

 appear extremely probable, If not necessary; but a specific demonstra- 

 tion of the truth of the principle can be given. 



Let the molecules have the masses m,, m.,, Ac., and let the impressed 

 forces be such as, in their directions, would give the rate* of accelera- 

 tion r,, P,, Ac., if these molecules were free and unconnected. Then 

 [FoRCKj MASS; VAIUATION ; VIMX-ITI] w,p,, m.i',, represent the 

 pressure* Impressed, on the condition that the unit of pressure is 

 that which produces a unit of acceleration in tli<> unit of mass. Let 

 the effective pressure*, derived from the velocities in the direction* 

 of the co-ordinates of x, y, and *, and compounded into one force 

 fur each molecule, be such a* wuuld produce the rates of accele- 

 ration Q,, Q,, Ac. ; no that the effective pressures arc *,<),, 01,9,, Ac, 

 Win u two forces net on a ]xi!nt, cither is equivalent to the other 

 with a certain thin! force j let m,p, be equivalent to m.q, and 

 M, it, ; let m, P, be equivalent to OT, Q, and m t u,, aud so on. Then the 

 system (p) of impressed forces is equivalent to the system (q) of 



* A molecule, In ftomettiesl mechanic', meant a point, endowed with the 

 properties of s maw of matter, finite or infinitely mnU, a* the oue may be. 



t We here exclude friction and rcoiiUnccs, but only on account of our 



the notion of three force*. The forcca loit (that it, 1 

 rnpect to the M>tcm) arc here communicated to other substances, to the HUM 

 in contact or to the air. 



