to 10%. This circumstance therefore agrees nearly with the theory. 2. The resistance to the same surface, is nearly as the square of the velocity; but gradually increasing more and more above that proportion, as the velocity increases. This is manifest from all the columns, And therefore this circumstance also differs but little from the theory, in small velocities. 3. When the hinder parts of bodies are of different forms, the resistances are different, though the fore parts be alike; owing to the different pressures of the air on the hinder parts. Thus, the resistance to the fore part of the cylinder, is less than that on the flat base of the hemisphere. or of the cone; because the hinder part of the cylinder is more pressed or pushed, by the following air, than those of the other two figures. 4. The resistance on the base of the hemisphere, is to that on the convex side, nearly as 2 to 1, instead of 2 to 1, as the theory assigns the proportion. And the experimented resistance, in each of these, is nearly part more than that which is assigned by the theory. 5. The resistance on the base of the cone is to that on the vertex, nearly as 2 to 1. And in the same ratio is radius to the sine of the angle of the inclination of the side of the cone, to its path or axis. So that, in this instance, the resistance is directly as the sine of the angle of incidence, the transverse section being the same, instead of the square of the sine. 6. Hence we can find the altitude of a column of air whose pressure shall be equal to the resistance of a body, moving through it with any velocity. Thus, Let the area of the section of the body, similar to any of those in the table, perpendicular to the direction of motion ; r = the resistance to the velocity, in the table; and the altitude sought, of a column of air, whose base is a, and its pressure r. x= Then ax = the content of the column in feet, and fax or fax its weight in ounces; therefore tax =r, and x = × is the altitude sought in a feet feet, namely, of the quotient of the resistance of any body ៖ divided by its transverse section; which is a constant quantity for all similar bodies, however different in magnitude, since the resistance r is as the section a, as was found in art. 1, When a of a foot, as in all the figures in the foregoing table, except the small hemisphere: then, x = 웅 X a gr, where r is the resistance in the table, to becomes = the similar body. If, for example, we take the convex side of the large hemisphere, whose resistance is 634 oz. to a velocity of 16 feet per second, then r -- ·634, and x = 2.3775 feet, is the altitude of the column of air whose pressure is equal to the resistance on a spherical surface, with a velocity of 16 feet. And to compare the above altitude with that which is due to the given velocity, it will be 322: 162:: 16:4, the altitude due to the velocity 16; which is near double the altitude that is equal to the pressure. And as the altitude is proportional to the square of the velocity, therefore, in small velocities, the resistance to any spherical surface is equal to the pressure of a column of air on its great circle, whose altitude is 2 or 594 of the altitude due to its velocity. But if the cylinder be taken, whose resistance r = 1·526 ; then x = r = 572; which exceeds the height, 4, due to the velocity in the ratio of 23 to 16 nearly. And the difference would be still greater, if the body were larger; and also if the velocity were more. 7. Also, if it be required to find with what velocity any flat surface must be moved, so as to suffer a resistance just equal. to the whole pressure of the atmosphere : of The resistance on the whole circle whose area is of a foot, is 051 oz. with the velocity of 3 feet per second; it is 051, or 0056 oz. only, with a velocity of 1 foot. But 2 X 13600 × = 7555 oz. is the whole pressure of the atmosphere. Therefore, as 0056: 7556 :: 1: 1162 nearly, which is the velocity sought. Being almost equal to the velocity with which air rushes into a vacuum. 8. Hence may be inferred the great resistance suffered by military projectiles. For in the table, it appears, that a globe of 63 inches diameter, which is equal to the size of an iron ball weighing 361b, moving with a velocity of only 16 feet per second, meets with a resistance equal to the pressure of of an ounce weights and therefore, computing only according to the square square of the velocity, the least resistance that such a ball would meet with, when moving with a velocity of 1600 feet, would be equal to the pressure of 417 lb, and that independent of the pressure of the atmosphere itself on the fore part of the ball, which would be 487lb more, as there would be no pressure from the atmosphere on the hinder part, in the case of so great a velocity as 1600 feet per second. So that the whole resistance would be more than 900lb to such a velocity. 9. Having said, in the last article, that the pressure of the atmosphere is taken entirely off the hinder part of the ball moving with a velocity of 1600 feet per second; which must happen when the ball moves faster than the particles of air can follow by rushing into the place quitted and left void by the ball, or when the ball moves faster than the air rushes into a vacuum from the pressure of the incumbent air; let us therefore inquire what this velocity is. Now the velocity with which any fluid issues, depends on its altitude above the orifice, and is indeed equal to the velocity acquired by a heavy body in falling freely through that altitude. But, supposing the height of the barometer to be 30 inches, or 2 feet, the height of a uniform atmosphere, all of the same density as at the earth's surface, would be 2 × 136 × 833 or 28333 feet; therefore ✔ 16 : √ 28333 : : 32:8 √/28333 1346 feet, which is the velocity sought. And therefore, with a velocity of 1600 feet per second, or any velocity above 1346 feet, the ball must continually leave a vacuum behind it, and so must sustain the whole pressure of the atmosphere on its fore part, as well as the resistance arising from the vis inertia of the particles of air struck by the ball. 10. On the whole, we find that the resistance of the air, as determined by the experiments, differs very widely, both in respect to its quantity on all figures, and in respect to the proportions of it on oblique surfaces, from the same as determined by the preceding theory; which is the same as that of Sir Isaac Newton, and most modern philosophers. Neither should we succeed better if we have recourse to the theory given by Professor Gravesande, or others, as similar differences and inconsistencies still occur. We conclude therefore, that all the theories of the resistance of the air hitherto given, are very erroneous. And the preceding one is only laid down, till further experiments, on this important subject, shall enable us to deduce from them another, that shall be more consonant to the true phænomena of nature. ON 416 ON THE MOTION OF MACHINES, AND THEIR ART. I. 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 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 con- . stant 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 A 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 repetition 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 se conds; 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. 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 is to overcome the weight or resistance w, by the application of the cords to the wheel and to the axle, в is the impelled point, and a the working point. VOL. II. Hhh 3. The |