641 



VIRTUAL VELOCITIES. 



VIRTUAL VELOCITIES. 



612 



Before proceeding to give our view of the manner in which this 

 proof may be amended, we shall point out how to proceed when the 

 forces are not equal. In such case they are either commensurable or 

 incommensurable: let them be commensurable, and let them beZp, 

 nip, nr, where I, m, , are integers. Instead of passing the string 

 twice only through each ring, pass it 2 n times through c and N, 2 m 

 times through B and M, 21 times through A and L. Then, the instant 

 the weight J p is applied, there are 2 1 strings in the direction A L, each 

 with the tension 4 P, or altogether there is the force I V, applied to A in 

 the direction A L ; and similarly of the rest. If, then, o, 0, y, be as 

 before, we have 2la + 2m/3 + 2ny, differing only by an infinitely small 

 quantity of the second order from the quantity of string let out or 

 taken in by an infinitely small displacement of the system. The usual 

 methods apply for the extension of this reasoning to the case in which 

 the forces are incommensurable. 



Let A L = a, B H = 6, c N = c : then the whole length of th,e string as 

 far as (13) is 2la + 2mb + 2nc + a constant made up of (4, 5), (8, 9), 

 and (12, 13). Hence 2lda + Zmdb + 2ndc is the infinitely small 

 quantity taken in or let out by an infinitely small displacement ; taken 

 in when positive, let out when negative; so that da, db, dc, answer 

 to o, f>, y, in the preceding. Now 



1. It is established that, equilibrium existing, equilibrium will remain 

 when all the forces take opposite directions. 



2. Neither a, b, e, nor their differential coefficient*, can become 

 infinite in any position of the system ; so that the only way in which 

 2la + 2mb-r2nc can become a maximum or a minimum is by 

 Zlda + 2mdb + 2ndc becoming, in the language of the differential 

 calculus, nothiny, that is, more strictly, an infinitely small quantity of 

 the second order. 



Now let the weight ^p act downwards, and let it draw out all the 

 string possible, and then rest. There must then be equilibrium, for 

 every displacement makes the weight rise ; and the weight has no 

 tendency to take advantage, so to speak, of this power of rising. Con- 

 sequently) there must be equilibrium when 2la + 2mb + 2nc is a 

 minimum, the weight acting downwards ; that is, when 2lda + 2mdb 

 + 2ndc is always positive, and of the second order ; or when fl.a. + 

 rm. f} + fn. y is always negative, and of the second order. And this 

 equilibrium is stable ; for any displacement makes the weight rise, and 

 its tendency is to descend, and restore the former state. Now reverse 

 the direction of the weight, and let the string communicate thrust 

 instead of pull, as before described. Then there is still equilibrium 

 (which is demonstrable independently) because only the directions of 

 the forces are changed ; but since the forces change direction, the 

 virtual velocities change sign, and Tl.a + , etc., is always positive, and 

 of the second order. Here, then, though the weight (we call ^ p weight 

 always, whether it tend upwards or downwards) tends to rise, and (geo- 

 metrically speaking) can rise, it does not rise : observe, also, that the 

 rise would be an infinitely small quantity of the second order. The 

 equilibrium in this case is unstable, for every displacement raises the 

 weight, which does not tend to return. Now let the weight 4 p act 

 upirardt, and let it push in all the string possible, and then rest. 

 There must then be equilibrium, for every displacement makes the 

 weight fall, and the weight has no tendency to take advantage of this 

 power of falling. Consequently, there must be equilibrium when 

 'Ha + , &c. is a maximum, the weight acting upwards ; that is, when 

 2lda + ,Sic. is always negative, and of the second order; or when 

 vl. a. + , 4c. is always negative,* and of the second order ; and this equi- 

 librium is stable, for any displacement makes the weight fall, and its 

 tendency U to rise and restore the former state. Now reverse the 

 direction of the weight, and let the string pull, instead of thrust. 

 There is still equilibrium (because only the directions of the forces are 

 changed) : the virtual velocities change sign, and pi. a + , &c. is always 

 positive, and of the second order ; and in this last case, though the 

 weight tends to fall, and (geometrically speaking) can fall, it does not 

 fall. Observe, also, that the fall would be an infinitely small quantity 

 of the second order; and the equilibrium in this case is unstable, 

 for every displacement lowers the weight, which does not tend to 

 return. 



Collecting these cases, it appears then that whenever Tl.a + , Ac. is, 

 for every infinitely small displacement, an infinitely small quantity of 

 one given sign, there is equilibrium ; stable when that sign is negative, 

 unstable when it is positive. But supposing rl.u +, &c. to be of the 

 second order, sometimes of one sign, and sometimes of the other, 

 according to the displacement, the preceding reasoning does not apply. 

 Nor do we sec how it can be applied without the assumption that an 

 equilibrium, which is produced, though all the displacements of the 

 weight favour the motion which it tends to take, is a fortiori produced 

 when some only do the same. Taking the case in which the weight 

 acts downwards, we have seen that there is equilibrium when the 

 descent of the weight is of the second order, and always downwards ; 

 the circumstance of the descent being of the second order, produces 

 equilibrium, even though its direction is that which the weight can 

 take. Still more must there be equilibrium when all the descents are 

 of the second order at least, and some only downwards. 



Hence, in every case, rl.a +, &c. = (in the common language of 



* When the action of the string is that of a thrust, it will be seen that 

 <// ii = a, &c., since the virtual velocities change sign. 

 AHTS AND SCI. DIV. VOL. VIM. 



the differential calculus) gives a position of equilibrium ; and we have 

 now to prove the converse, namely, that every position of equilibrium 

 gives rl.a-r, &c. = (Lagrange proves this converse first). This 

 converse can be proved, we submit, without taking it for granted, at 

 once, with Lagrange, that if any motion of 4? of the first order 

 were possible, the weight would, by its tendency to descend, take that 

 motion.* 



Supposing the system to be at rest, and the weight to act downwards, 

 it is obviously physically possible that a given finite velocity should be 

 communicated to the weight. Suppose a blow to be given to the 

 weight in a downward direction, such as would communicate a finite 

 velocity ; what would be the effect upon the system at the instant 

 when the weight receives the blow downwards ? An impulsive strain 

 upon the string, which would only communicate forces proportional to 

 those already existing.t and could not disturb the equilibrium. The 

 system then cannot move, neither therefore can the weight move. 

 Now as it is unquestionably physically possible that the weight may 

 take a finite velocity, the impossibility of moving the system must be 

 geometrical ; or a velocity communicated to the system must, be it 

 what it may at the first instant, communicate none to the weight ; 

 and the definition of velocity shows that this can only happen when, 

 the displacements of the system in the time dt bearing a finite ratio to 

 dt, that of the weight is infinitely small compared with dt ; that is, 

 when the displacement of the weight is infinitely small compared with 

 those of the system. From this it follows that 2 la + , &c. is infinitely 

 small as compared with o, &c. 



We do not know how to make the preceding prove its converse, and 

 we object to the mode pursued by Lagrange. Having proved that 

 equilibrium gives 2la +, &c., that is, having proved it on the distinct 

 assumption that the weight cannot descend in the first instant through 

 a quantity comparable to a, &c., he then proceeds as follows : Re- 

 ciprocally 2la + , &c. = 0, gives a case of equilibrium ; for " the weight 

 remaining immoveable under all displacements, the powers which act 

 upon the system remain in the same state, and there is no more reason 

 why they should produce one of the two displacements than the other, 

 of any two in which a, &c. have contrary signs. It is the case of the 

 balance which remains in equilibrium, because there is no more reason 

 why it should incline on one side than the other." Now, first, this 

 reasoning might just as well be applied to prove equilibrium when 

 2la + , 4c. is not = ; secondly, it is not the case of the balanced lever 

 of Archimedes, for there is not that same symmetry, either geometrical 

 or mechanical, which makes it impossible to admit either motion in 

 preference to the other [STATICS ; SUFFICIENT REASON] ; thirdly, 

 there is a mechanical reason why one of the motions should be 

 taken rather than the other, namely, that one in which the displace- 

 ment of the weight (even though supposed of the second order) is 

 positive. This last will appear sufficiently in the sequel. 



We shall now proceed to show that the moment the principle of 

 virtual velocities is granted, a problem of statics becomes one of pure 

 mathematics. This ia all we can undertake to illustrate ; and for this 

 purpose any mathematical result may be taken for granted. First, let 

 the force p be decomposed into three, X, Y, z, in the direction of x, y, 

 and z ; and let the point of application move until the co-ordinates are 

 x + dx, y -h dy, z + dz. Then a force equal and opposite to P (of which 

 the moment is rdp) balances x, Y, and z ; so that the principle gives 

 xdx + fdy + zdz + ( rdp) = 0, or edp = \dx + tdy + zdz. Do the same 

 with each of the forces, and we have 2 (rdp) = 2 (xrfx) + 2 (Yrfy) + 2 ('/.<l: ). 

 If the system be rigid, every virtual motion may be decomposed into 

 two : a motion of translation of any one given point, and a motion of 

 rotation round an axis passing through that point. Let x a , y a , z a , be 

 the co-ordinates of any point which moves with the system, and Jet this 

 jioint move so that its co-ordinates shall become x -r dx a , y + di/,,, 

 z a -t- dz a , at the same time that the system revolves through an angle 

 d<f> about an axis passing through the point (x ot y , z ), and making 

 angles A, ft, v with the three axes. If the consequence of this motion 

 be that the point whose co-ordinates are x, y, z, moves so that its 

 co-ordinates become x + dx, y + dy, z + dz, we have 



dx=dx~ + {COB /* (s-zj cos v (y-y )} d<p 

 dy = dy a + {cos c (x-x a ) cos \ (zz a ) \ dp 

 dz = dz + {cos A (y y a ) - cos n (x - x ) } dp 

 from which we find for 



2 (rdp) or 2 (\dx) + 2 (tdy) + 

 the following expression : 



2x.<fo + (* 2Y y 2z)cos\ (ty> + 3(zy-Y2) .cos 

 + 2v . dy a + (XgSz z 2x) cos n d $ + 2 (xz zx) . cos n dtp 



xy).cosK dp 



It seems to us just as sound to say that if there be any motion of the 

 second order possible, the weight will take that motion, and in an infinitely 

 small time acquire a velocity of the first order, which is exactly what takes 

 ilace in a body falling freely from rest. 



It is here assumed that whatever forces Veep a system at rest, impulses 

 proportional to those forces, and applied in the same manner, will not disturb 

 he equilibrium. 



T T 



