140 On Machines in General. 



That is to say, by calling F each of these forces*, u the 

 velocity which the point where it is applied will have at 

 the first instant, if we make the machine assume a geome- 

 trical movement, and z the angle comprehended between 

 the directions of F and of u, it must prove that wt 

 shall have for the whole system s F a cosine 2 = 0. Now 

 this equation is precisely the equation (AA) found (XXX), 

 which is nothing else in the eml but the same fundamental 

 equation (F), presented under another form. 



Il is easy to perceive that this general principle is, properly 

 speakmg, nothinc; el.^e than that of Descartes, to which 

 a sufficient extension is to he given, in order that it may con- 

 tain not only ail the condiiioas of the equilibrium between 



* It will not perhaps be useless to anticipate an objection which might 

 occur to those who have not paid sufficient attention to what has been 

 said (XXX) upon the true meaning we ought to attach to the word force: 

 Let us imagine, for instance, they will say, a wheel and axle to the cylin- 

 der of which weights are suspended by means of cords ; if there be equili- 

 brium, or if the movement be uniform, the weight attached to the wheel will 

 be to that of the cylinder as the radius of the cylinder is to the radius of the 

 wheel; which is conformable to the proposition. But the case is not the same 

 when the machine assumes an accelerated or a retarded movement : it seems, 

 ' therefore, that here the forces are not in reciprocal ratio of their velocities es- 

 timated in the direction of these forces, as would follow from the proposition. 

 The answer to that is that in the case where this movement is not uniform, 

 the weights in question are not the only forces exercised in the system ; for 

 the movement of each body changing continually, it also opposes at each 

 instant, by its vis inerlicc, a resistance to this change of state : we must, 

 therefore, keep an account of this resistance. We have already said (XXX, 

 see the note,) how this force should be estimated, and we shall see further 

 on (XLl), how we should make it enter into the calculation. In the mean time 

 it is sufficient to remark, that the forces applied to the machine in question 

 are not the weights, but the quantities of movement lost by these weights 

 (XXX), which should be estimated by the tensions of the cords to which 

 they are suspended : now whether the machine be at rest or in motion, 

 whether this motion be uniform or not, the tension of the cord attached to 

 the wheel is to that of the cord attached to the cylinder, as the radius of th^ 

 cylinder is to the radius of the wheel, i. e. these tensions aie always in reci- 

 procal ratio of the velocities of the weights they support : this agrees with 

 the proposition. But these tensions are not equal to the weights ; they are 

 (XXX. see the note) the results of these weights and of their vis inerliee, 

 which are themselves (XXX. see the note) the results of the actual move- 

 ments of these bodies, and of the movements equal and direaly opposed to 

 those which they will really assume the instant afterwards. 



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