of which, by the 12th form, is the hyp. log. of ≈ +√(x2+a2) = hyp. log. of r±a + √(x2± 2ax), the fluent required. Er. 2. To find now the fluent of xx √(x2+2ar), having given, by the above example, the fluent of suppose. Assume √(x2 + 2ax) xx √(x2+2ax) = A √(x2+2ar) = y; then its fluxion is √(x2+2a1) the fluent of which is yan= √(x2+2ax) — a▲, the fluent sought. Ex. 3. Thus also, this fluent of xx being given, √(x2+2ax) 12x √(x2+2ax) and so on for any other will be found, the flu. of the next in the series, or xn-- - 1x √(x2+2ax) be given = c; then, by assuming x”-1、'(x2 + 2ax) = y, the fluent Ex. 4. In like manner, the fluent of given, as in the first example, that of √(x2-2ax) √(x2-ax) may be found; and thus the series may be continued exactly as in by pa. 321 vol. 2, is —–—× circular arc to radius a and versed sine x, the fluents of may be assigned by the same method of continuation. Thus, Ex. 1. For the fluent of y; the required fluent will be found (2ax − x2)+▲ or arc to radius a and vers. x. where a denotes the arc mentioned in the last example. Ex. 3. And in general the fluent of —x"1 √(2αx − x2), where c is the fluent of n the next preceding term in the series. 33. Thus = 33. Thus also, the fluent of (x − a) being given, (x-a), by the 2d form, the fluents of xx(x-a), x2x√(x-a), &c...x" x√√(x − a), may be found. And in general, if the fluent of "√(x-a) = c be given; then -I by assuming "(x-a)=y, the fluent of x"√(x- a) is found = 2na 2n+3 C. 1 m + 1 34. Also, given the fluent of (r− a)"x, which is (ra)+1 by the 2d form, the fluents of the series (x − a)TMxx, (x− a)TMx2x, &c. . . (x —a)”x”x can be found. And in general, the fluent of (ra)" being given = c; then (x a)"r"-1 by assuming (x —− a)"+1x" = y, the fluent of (x- a)TMx”x is x^(x− a)" + 1 + nac found = m+n+1 Also, by the same way of continuation, the fluents of x'x✅ (a ≈ x) and of x*x(a + x)" may be found. 35. When the fluxional expression contains a trinomial quantity, as (b + cx + x2), this may be reduced to a binomial, by substituting another letter for the unknown one , connected with half the coefficient of the middle term with its sign. Thus, put z=x+c: then z2 = x2+cx+¡c2; theref. z2-4c2 = x2 + cx, and z2 + b-c2 = x2 + cx + b the given trinomial; which is = x2 + a2, by putting a2 = b- c2. Ex. 1. To find the fluent of 3x Here z = + 2; then z2 = x2 + 4x + 4, and z2 + 1 = 5 + 4x + x2, also ; theref. the proposed fluxion ré 3z √(1+x3) duces to (+); the fluent of which, by the 12th form in the 2d vol. is 3 hyp. log. of x +√(1 + z) = 3 hyp. log. x+2+√(5+ 4x + x2). Ex. 2. To find the fluent of (b + cx + dx2) = √/dx √ √ 2/2 + 1/2 x + x2). Here assuming +2=z; then = 2, and the proposed flux. reduces to ż√✓d× √(x2 + - - _—_—_ )=ż √ d× √(z2+a2), ს putting a2 for d A and the fluent will be found by a similar process to that employed in ex. 1 art. 27. - I Ex. 3. In like manner, for the flu. of "~1x√(b + cx” + dx2”), assuming ♫” + —=z, nx”—! ¿=¿, and a”—1÷==ż; -I hence 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 |