hence 12 + + = 23, and √(dx2 + c.x” + b) = √ d× √(x2 + {-{x + '/2) = √ dx √(x2 + — — — 1) = √/d d 4d × √ (z2 ± a2), putting ± a2 = -; hence the given d fluxion becomes d× √(ż2 ±a2), and its fluent as in the last example. dn x2-1x ; assume = %, then the fluxion may be reduced to the form X and the fluent found as before. So far on this subject may suffice on the present occasion. But the student who may wish to see more on this branch, may profitably consult Mr. Dealtry's very methodical and ingenious treatise on Fluxions, lately published, from which several of the foregoing cases and examples have been taken or imitated. CHAPTER XI. ON THE MOTION OF MACHINES, AND THEIR MAXIMUM EFFECTS. ART. 1. When forces acting in contrary directions, or in any such directions as produce contrary effects, are applied to machines, there is, with respect to every simple machine (and of consequence with respect to every combination of simple machines) a certain relation between the powers and the distances at which they act, which, if subsisting in any such machine when at rest, will always keep it in a state of rest, or of statical equilibrium; and for this reason, because the efforts of these powers, when thus related, with regard to magnitude and distance, being equal and opposite, annihilate each other, and have no tendency to change the state of the system to which they are applied. So also, if the same machine have been put into a state of uniform motion, whether rectilinear or rotatory, by the action of any power distinct from those we are now considering, and these two powers be made to act upon the machine in such motion in a similar manner to that in which they acted upon it when at rest, their simultaneous action will preserve it in that state of of uniform motion, or of dynamical equilibrium; and this for the same reason as before, because their contrary effects destroy each other, and have therefore no tendency to change the state of the machine. But, if at the time a machine is in a state of balanced rest, any one of the opposite forces be increased while it continues to act at the same distance, this excess of force will disturb the statical equilibrium, and produce motion in the machine; and if the same excess of force continues to act in the same manner it will, like every constant force, produce an accelerated motion; or, if it should undergo particular modifications when the machine is in different positions, it may occasion such variations in the motion as will render it alternately accelerated and retarded. Or the different species of resistance to which a moving machine is subjected, as the rigidity of ropes, friction, resistance of the air, &c, may so modify a motion, as to change a regular or irregular variable motion into one which is uniform. 2. Hence then the motion of machines may be considered as of three kinds. 1. That which is gradually accelerated, which obtains commonly in the first instants of the communication. 2. That which is entirely uniform. 3. That which is alternately accelerated and retarded. Pendulum clocks, and machines which are moved by a balance, are related to the third class. Most other machines, a short time after their motion is commenced, fall under the second. Now though the motion of a machine is alternately accelerated and retarded, it may, notwithstanding, be measured by a uniform motion, because of the periodical and regular repe tition which may exist in the acceleration and retardation. Thus the motion of a second's pendulum, considered in respect to a single oscillation, is accelerated during the first half second, and retarded during the next: but the same motion taken for many oscillations may be considered as uniform. Suppose, for example, that the extent of each oscillation is 5 inches, and that the pendulum has made 10 oscillations: its total effect will be to have run over 50 inches in 10 seconds; and, as the space described in each second is the same, we may compare the effect to that produced by a moveable which moves for 10 seconds with a velocity of 5 inches per second. We see, therefore, that the theory of machines whose motions are uniform, conduces naturally to the estimation of the effects produced by machines whose motion is alternately accelerated and retarded: so that the problems comprised in this chapter will be directed to those machines whose motions fall under the first two heads; such problems being of far the greatest utility in practice. VOL. III. R Defs. Defs. 1. When in a machine there is a system of forces or of powers mutually in opposition, those which produce or tend to produce a certain effect are called movers or moving powers; and those which produce or tend to produce an effect which opposes those of the moving powers, are called resistances. If various movers act at the same time, their equivalent (found by means of prop. 7, Motion and Forces) is called individually the moving force; and, in like manner, the resultant of all the resistances reduced to some one point, the resistance. This reduction in all cases simplifies the investigation. 2. The impelled point of a machine is that to which the action of the moving power may be considered as immediately applied; and the working point is that where the resistance arising from the work to be performed immediately acts, or to which it ought all to be reduced. Thus, in the wheel and axle, (Mechan. prop. 32), where the moving power P is to overcome the weight or resistance w, by the application of the cords to the wheel and to the axle, B is the impelled point, and a the working point. 3. The velocity of the moving power is the same as the velocity of the impelled point; the velocity of the resistance the same as that of the working point. 4. The performance or effect of a machine, or the work done, is measured by the product of the resistance into the velocity of the working point; the momentum of impulse is measured by the product of the moving force into the velocity of the impelled point. These definitions being established, we may now exhibit a few of the most useful problems, giving as much variety in their solutions as may render one or other of the methods of easy application to any other cases which may occur. PROPOSITION I. If R and r be the distances of the power P, and the weight or resistance w, from the fulcrum F of a straight lever; then will the velocity of the power and of the weight at the end of any time t be R2P RTW gt, and R2P+2w Rrp-r2w ROP+gt, respectively, the R2r+r2w weight and inertia of the lever itself not being considered. If the effort of the power balanced that of the resistance, P would be equal to . Conse TW W quently, the difference between this value of P and its actual value, or P- w, will be the force which tends to move r R the the lever. And because this power applied to the point accelerates the masses P and w, the mass to be substituted T2 for w, in the point. a, must bew, (Mechan. prop. 50) in order that this mass at the distance R may be equally accelerated with the mass w at the distance R. Hence the power P R T2 R2 - w will accelerate the quantity of matter P+W; and the accelerating force F = (P ——w) ÷ (P+ ·w)= w; R T2 But (vol. ii. p. 335) α Ft or is = gtr (g being = 32% feet); which in this case = R2P-RTW .gt, the velocity of r. And, because veloc. of p: veloc. of w :: K: r, therefore veloc. of Cor. 1. The space described by the power in the time t, Cor. 2. If Rr::n 1, then will the force which acce lerates A be = Cor. 3. force p be A will be = If at the same time the inertia of the moving 0, as in muscular action, the force accelerating Cor. 4. If the mass moved have no weight, but possesses inertia only, as when a body is moved along a horizontal plane, the force which accelerates A will be = either of these values may be readily introduced into the investigation. Pn2 + W Cor. 5. The work done in the time t, if we retain the original notation, will be = R2P+r2wgt × W= Cor. 6. When the work done is to be a maximum, and we wish to know the weight when P is given, we must make the fluxion of the last expression = 0. Then we shall have ¿aR3p2 — 2r2R2PW—r1w2=0 and w = PX [√(+3) − = 2]. Cor. 7. If nr::n: 1, the preceding expression will become w = P × [√ (na + 223) — n2]. Cor. 8. When the arms of the lever are equal in length, that is, when n = 1, then is WPX (√2—1) ·414214P, or nearly of the moving force. t 1 2444 Scholium. are in like manner investigate the formulæ relating to expressions correspond exactly. Hence it follows, that when merion on the axis in peritrochio, it will be seen that the obtain a maximum effect on the wheel and axle, (the weight at is required to proportion the power and weight so as to conclusions of cors. 6 and 7 of this prop. And in the exof the machinery not being considered), we may adopt the treme case where the wheel and axle becomes a pulley, the expression in cor. 8 may be adopted. The like conclusions expr be applied to machines in general, if R and 7 represent the distances of the impelled and working points from the of motion; and if the various kinds of resistance arising from friction, stiffness of ropes, &c, be properly reduced to their equivalents at the working points, so as to be comprehended in the character w for resistance overcome. PROPOSITION II. r Given Rundr, the arms of a straight lever, м and m their respective weights, and p the power acting at the extremity of the arm R; to find the weight raised at the extremity of the other arm when the effect is a maximum. In this case 4m is the weight of the shorter end reduced to B, and mr conseq. 2R is the weight which, W applied at A, would balance the shorter end: therefore mr 2R r + w, would sustain both the shorter end and the R weight w in equilibrio. But PM is the power really acting at the longer end of the lever; consequently P+M mr (w), is the absolute moving power. Now 2R R the distance of the centre of gyration of the beam from ** The distance of R, the centre of gyration, from c the centre or axis of motion, in some of the most useful cases, is as below: In a circular wheel of uniform thickness In the periphery of a circle revolving about the diam. rad. In a plane ring formed of circles whose radii are R, †, R4 |