815 



VIRTUAL VELOCITIES. 



VIRTUAL VELOCITIES. 



(116 



affective forces, together with tho system (R) : for (p) substitute (Q) 

 and (B), and the effect upon the system, in the infinitely small time 

 following the moment of which we speak, is what it would have been 

 if (p) had continued. But that effect is precisely what is produced by 

 (Q) ; for (Q) was nothing but the pressures necessary to produce the 

 actual effect of which we aru speaking. Therefore (H) has no effect, 

 and would of itself equilibrate the system : to suppose that (R) would 

 not equilibrate, or would produce some motion, while (q) is actually 

 calculated to produce all that is to take place, is to suppose that the 

 system will, in the infinitely small time next ensuing, have another 

 motion besides that which (Q) would produce, which is absurd. Con- 

 sequently (B) is a system of equilibrating forces, which is expressed by 

 saying that the forces lost and gained balance one another : for if 

 in, p,, the force impressed on nt,, be equivalent to 411,3, and m, B,, of 

 which niiQ, is enough to produce what takes place, it is obvious that 

 'in, a,, so far as the molecule m 1 is concerned, is lost. It would be 

 better to say that m l R, is transferred, and that all the forces trans- 

 Lnvd balance one another. Again, sinua (B) is wholly without effect, 

 it follows that (p) is equivalent to (Q) ; or, at every instant of the 

 motion, the impressed forces are a set of equivalent statical powers 

 with the effective forces : so that if either set were applied to the 

 system at rest, and also the opposites of all the forces in the other set, 

 there would be equilibrium. Or the impressed forces balance the 

 ! i 1 :; forces with their signs changed. Now the effective forces on 



m., in the directions of x, ?/, and z, are m, '^i,m -JL', MI, L-L', and 



aft it I- a P 



similarly for the rest ; while, if we decompose the rate of acceleration 

 r,,into x,,v,, z,, in the directions of -r, y, and .-, the impressed 

 pressures in these directions are m,!,, >,,, and m,z r And [VABIA- 

 TIONS, CALCULUS OP,] to distinguish tho virtual motion which thu 

 problem of equilibrium requires, from the actually coming motion 

 in terms of which the effective forces are expressed, we may use o".r, 

 instead of dx, in the former, and so on. Hence, changing the signs of 

 the impressed forces and combining them, so changed, with the 

 effective forces, we have, for the fundamental equation of every 

 dynamical problem 



\ f(Py \ ItT-z 



*(d?-*) mt * + 2 (s? - ) "^ + 2 V5P 



From which are obtained, as in a preceding process*, the following six 

 equations of motion, abbreviating dfx : dt- into x" , and no on 



{ > (fy - /*) } = 2 { m (ty - \z) 

 2 { (*"- a"*) j = S { m (xs-xx) } 

 = S m(*x-xy) 



2 (mx") = 2 (m x) 

 2 (my") = 2 (Y) 



These equations express the property already mentioned [TBANBLA- 

 TIOX], namely, tliat the centre of gravity moves as it would do if all 

 the masses were collected there, and all the pressures applied there. 

 We shall merely enumerate the steps of the proof of this proposition. 

 The co-ordinatea of the centre of gravity being .r u , y^ t , we have .r c) 

 2m = 2 (mx), 4o., whence .r,,"2i = 2(w.r") = 2(mx), Ac., which are 

 precisely the equations for the motion of a molecule of the mass 2m, 

 and to which the force Z(mx) is applied. With regard to the initial 

 velocity which ought to be given to the centre of gravity when the 

 molecules are there collected, observe that x ' ~ 2(>nx') 4- 2m = 

 {A + 2(/mxrf() } -j- 2m, where A is the initial value of 2 (mx 1 ). 



Consequently, at the commencement of the motion a-,,' should have the 

 game value as 2(mx') -f- 2m), or we should have j?,,'Z m = (m;r/) at the 

 outlet ; that is, the momentum of the collected mass, in the direction 

 of .<, should be the same as the sum of the momenta of the molecules 

 in the system, and the same of the other co-ordinates. Again, let {, ), 

 (, be the co-ordinates, referred to the centre of gravity, of the point 

 whose original co-ordinates are x, y, z. 



We have then x = x y + l,y = y a + il,z = z a + (; also S.mf = 0,5wi) 

 = 0, 2 in C = 0- Stil tftitution gives 



whence 2 {m(f i)-"0} = 2 {m(Ki)-yO}- 



which, with the two other equations similarly deduced, are precisely 

 those which would determine the motion if the centre of gravity were 

 fixed and the forces then applied. We must refer to works on the 

 subject for further development of these conditions, and shall proceed 

 to cases more illustrative of the principle under consideration. 



Among the virtual motions, one of course is the motion the s.vstem 

 is actually about to take. In this case &x is dx, &c., and the funda- 

 mental equation becomes 



2{m (x"dx + y"dy + z"dz)} = x{m(xrlx + Yrfy + zdz)}. 



Now the 6rst side of this equation is nothing but the differential 

 with respect to the time of J2 ^ m (x' 2 + y'' + /')(, or 42mv', r,, s ,&c., 

 being the actual velocities of the molecules at the end of the time t. 

 Hence we have 



2/r' = A -i- 2 { m/(xdx 4 ydy + zefe) } 



where A is the value of 2 m ir at the commencement of the motion, 

 and the integral also begins at that commencement. Suppose the 

 system to be at rest at the commencement of the motion, then A = 0, 

 since each of the incipient velocities is nothing ; consequently at the 

 end of the first infinitely small element dt, 2mi> 3 has changed from 

 to m(x.dx+ fdy + zdz). But this is precisely thu sum of the 

 moments of the impressed forces in the principle of virtual velocities ; 

 and 2mt> ! being m, v l 1 +m 1 v 1 a + &c., must be a positive quantity. 

 Hence the sum of the moments must be positive, for the virtual 

 motion which the system actually tends to take : and this is the 

 principle of which we have forestalled the use in completing the cor- 

 rect enunciation of the principle of virtual velocities. This might 

 be suspected beforehand from the following consideration : The 

 forces which have positive moments are those which tend, so far as 

 they go, to produce the virtual motion in question ; and those which 

 have negative motions to hinder it. Whatever motion the system 

 takes, it must be one in which tho forces tending to produce that 

 motion predominate over those which tend to hinder it : or the forces 

 with positive moments must have those moments together larger than 

 the forces with negative moments. 



The choice which the system * makes among all the virtual motions, 

 in which to begin its motion, is that in which the sum of all the 

 moments of the forces is a maximum, in tho sense which will pre- 

 sently be explained. Since every motion of a system can be reduced 

 to a translation of the centre of gravity and rotation round an axis 

 passing through that centre, let us reduce the virtual motion to terms 

 of the motion of and round the centre of gravity. If2mx, &c., be 

 p, Q, B, and if 2{m (zy yj) j. &c., be L, M, N, it follows from what has 

 been shown respecting the motion of this centre that its first direction 

 of translation (the system starting from rest) is such that d.r a , dy ia dz a 

 are in the proportion of pjq, B, and that tho axis round which tho 

 fy.-ti'in begins to turn makes angles with the axes of x, y, and z, whose 

 cosines are in the proportion of L, M, and N. Now suppose any motion 

 of and round the centre of gravity, and returning to the expressions 

 in which the sum of the momenta is given in terms of those motions, 

 observe that we must write x for x, &n., because the pressures are 

 now represented bymx, &c., which were then represented by x, &c. 

 Moreover z 2; Y y 2mz, and the other terms corresponding, all 

 vanish, because 2 =2?z-i-2m, &c, We lavo then for tjie sum pf the 

 moments, 



fdx a + <jrfy + nrfz + (L cos A. + M cos n + y cos ti) d<p. 



Let the displacement of the centre of gravity be du, we have then 

 rfu= ^(drj+dyj + dzj). Now the theorem is, that for given values 

 of du and rf#, for a given amount cf translation and rotation, the 

 direction of translation and the position of the axis of rotation, in the 

 virtual motion which the initial effect of the forces actually causes, are 

 such as to make the preceding expression a maximum. 



First, it must be shown by the common methods that for a con- 

 stant value of p' + if+t*, the expression up + nq + cr, if tlte.n 

 , is a maximum when p, q, r are in. the proportion of A, B, c. 

 Now In the actual motion of the system, pd.r u + Q d y n + u d?,,, 

 and ( r. cos A. -j- &c. )d$, are fiotitire yuanlilici : for the first is the 

 initially obtained value of J5iD- when the system is all collected 

 in the centre of gravity and all the forces are then applied : 

 and the second is the same when the centre of gravity is fixed and the 

 system begins to move about it. And since the variables of the first 

 and second are entirely independent of each other, the sum of the two 

 is a maximum when each separately is a maximum. In the first i/.r ,,-' i 

 &c., is a constant, being du-, and therefore the first is a maximum 

 when dx a , dy a , and dz a are in the proportion of p, Q, B, But in the 

 second, COS'A + cosV -f cos*>' = 1, whence the second is a maximum 

 when cos A, cos ft, cos v are In the proportion of L, M, M. But those 

 two seta of conditions put together precisely represent the motion 

 which at the outset the system does take from the impressed forces. 

 Whence the theorem is true, as asserted, 



We may now treat the exception of which we have spoken in a 

 preceding part of this article. Suppose that the moments in all the 

 directions in which the system can move are equal, or else that there 

 is among them a set which are equal, and each of them greater than 

 any of the rest. Which of all the virtual motions having these 

 moments is the system to take? It cannot prefer either, and will 

 remain in equilibrium. As an instance, let the end of a string be 

 attached to a curve on which it can slide freely, while the string sup- 



ports a weight. Let the curve have a cusp pointing upwards, with its 

 tangent vertical, and let the end of the string be placed at tho cusp, as 



We confine ourselves here to a rigid sytem, though the proposition is (ruo 

 univmally. But the universal proof would be too long. 



