be = 211 35 made ge I 2 a 20 I 20 2a 2a by the same number. It is a number of units, of time, Moreover, the times in which the same ve or of length. locity will be extinguished by different forces, acting Having ascertained these leading circumstances for The comupiformly, are inversely as the forces, and gravity would an unit of velocity, weight, and bulk, we proceed to de-parison duce the similar circumstances for any other magnitude; neral. extinguish the velocity I in the time = (in these mea and, to avoid unnecessary complications, we shall always g' suppose the bodies to be spheres, differing only in diasures) to Therefore we have the following meter and density. First, then, let the velocity be increased in the ratio of i to v. proportion (=R)(=)= 2a, and 2 a is The resistance will now be equal to E, the time in which the velocity I will be ex E tinguished by the uniform action of the resistance com The extinguishing time will be = 0,= and = petent to this velocity. ev=2a; so that the rule is general, that the space The velocity I would in this case be extinguished af- along which any velocity will be extinguished by the ter a motion uniformly retarded, in which the space de uniform action of the corresponding resistance, is equal scribed is one-half of what would be uniformly describ to the beight necessary for communicating the terminal ed during the same time with the constant velocity 1: velocity to that body by gravity. For ev is twice the Therefore the space thus described by a motion which space through which the body moves while the velocity begins with the velocity 1, and is uniformly retarded v is extinguished by the uniform resistance. by the resistance competent to this velocity, is equal to In the 2d place, let the diameter increase in the prothe height through which this body must fall in vacuo portion of i to d. The aggregate of the resistance in order to acquire its terminal velocity in air. changes in the proportion of the surface similarly resistAll these circumstances may be conceived in a man. ed, that is, in the proportion of 1 to do. But the quanner which, to some readers, will be more familiar and tity of matter, or number of particles among which this palpable. The terminal velocity is that where the re resistance is to be distributed, changes in the proporsistance of the air balances and is equal to the weight tion of I to ds. Therefore the retarding power of of the body. The resistance of the air to any particulace the resistance changes in the proportion of I to body is as the square of the velocity; therefore let R be When ď the whole resistance to the body moving with the velo the diameter was 1, the resistance to a velocity I was city 1, and r the resistance to its motion with the terminal velocity u; we must have r=R xa', and this must It must now be The time in wbich this 2ad be =W the weight. Therefore, to obtain the terminal velocity, divide the weight by the resistance to the ve diminished resistance will extinguish the velocity I locity 1, and the quotient is the square of the terminal must increase in the proportion of the diminution of W force, and must now be Ed, or 2 a d, and the space velocity, or i =u*: And this is a very expeditious me uniformly described during this time with the initial ve locity i must be 2 ad; and this must still be twice the thod of determining it, if R be previously known. Then the common theorems give a, the fall necessary height necessary for communicating the terminal veloci wa for producing this velocity in vacuo = and the time ty w to this body. We must still have 28 therefore w* = 2g ad, and w=2gad=zgadd. of the fall ==e, and eu=2a, = the space uniformly But i= n2ga. Therefore the terminal velocity w described with the velocity u during the time of the fall, for this body is =u'vā; and the height necessary for or its equal, the time of the extinction by the uniform communicating it is a d. Therefore the terminal veloaction of the resistancer; and, sincer extinguishes it in city varies in the subduplicate ratio of the diameter of the time e, R, which is uø times smaller, will extinguish the ball, and the fall necessary for producing it varies in it in the time u’e, and R will extinguish the velocity 1, the simple ratio of the diameter. The extinguishing which is u times less than u, in the time u e, that is, in Ed time for the velocity v. must now be the time 2 a; and the body, moving uniformly during the time 2a, =E, with the velocity 1, will describe If, in the 3d place, the density of the ball be increased 'the space 2a; and, if the body begin to move with the in the proportion of i to m, the number of particles velocity 1, and be uniformly opposed by the resistance among which the resistance is to be distributed is inR, it will be brought to rest when it has described the creased in the same proportion, and therefore the retardspace a ; and the space in which the resistance to the ing force of the resistance is equally diminished; and if velocity 1 will extinguish that velocity by its uniform the density of the air is increased in the proportion of 1 action, is equal to the height through which that body to n, the retarding force of the resistance increases in the must fall in vacuo in order to acquire its terminal velo same proportion : hence we easily deduce those general city in air. And thus every thing is regulated by the expressions. time E in which the velocity 1 is extinguished by the The terminal velocity = a Jam-= 1/2g a di uniform action of the corresponding resistance, or by 2a, which is the space uniformly described during this time, with the velocity 1. And E and 2 a must be expressed The producing fall in vacuo =ad" Vol. XVII. Part II. + 3 E The I 2a 2 ad ; and 38 2 adm n OUS. υη is 32 feet. I a ada The retarding power of resistance to any velocity = have indeed been found with his reasoning, and even with his principles; and it must be acknowledged that His resu'a 90', = although this investigation is by far the most ingenious just, iut of any in the Principia, and sets his acuteness and ad-lus reaso. ing erfos Edm dress in the most conspicuous light, his reasoning is liable The extinguishing time for any velocity n= to serious objections, which his most ingenious commes tators have not completely removed. However, the And thus we see that the chief circumstances are re conclusion has been acquiesced in, as we have already gulated by the terminal velocity, or are conveniently stated, but as if derived from other principles, or by 36 referred to it. more logical reasoning. We cannot, however, say that Units ne To render the deductions from these premises perspi- the reasonings or assumptions of these mathematicians cessary by and for communicating distinct notions or ideas, which the cuous, are much better than Newton's: and we must add, that quantities it will be proper to assume some convenient units, by all the causes of deviation from the duplicate ratio of the may be which all these quantities may be measured ; and, as velocities, and the causes of increased resistance, which measured. this subject is chiefly interesting in the case of military the latter authors have valued themselves for discovering projectiles, we shall adapt our units to this purpose. and introducing into their investigations, were pointed Therefore, let a second be the unit of time, a foot the out by Sir Isaac Newton, but purposely omitted by bim, unit of space and velocity, an inch the unit of diameter in order to facilitate the discussion in re difficillima. (See of a ball or shell, and a pound avoirdupois the unit of Schol. prop. 37. book ii.). pressure, whether of weight or of resistance; therefore g It is known that the weight of a cubic foot of water is 62 pounds, and that the medium density of the air The great difficulty is to procure an absolute measure is in of water; therefore, let a be the height produof r, or u, or a; any one of these will determine the cing the velocity (in feet), and d the diameter of the 37 others. ball (in inches), and a the periphery of a circle whose Sir Isaac Sir Isaac Newton has attempted to determine r by 62 Newton's theory, and employs a great part of the second book of diameter is 1; the resistance of the air will be = endeavours 840 in this way the Principia in demonstrating, that the resistance to a -X X-Xd? xd= pounds, very nearly, = Х would generate or destroy its whole motion in the time 144 4928 ve da that it would uniformly move over of its diameter -a',= pounds. We may take an example. A ball of cast iron weighit with any velocity, is equal to balf the weight of a co ing 12 pounds, is 4 inches in diameter. Suppose this lumn of air having a great circle of the sphere for its ball to move at the rate of 2516 feet in a second (the base, and for its altitude the height from which a body reason of this choice will appear afterwards). The must fall in vacuo to acquire this velocity. This appears height which will produce this velocity in a falling body from Newton's demonstration ; for, let the specific gra is 93 feet. The area of its great circle is 0.11044 feet, vity of the air be to that of the ball as i to m; then, 1844 of one foot. Suppose water to be 840 times because the times in which the same velocity will be ex heavier than air, the weight of the air incumbent on tinguished by the uniform action of different forces are this great circle, and of feet high, is 0.081151 pounds: inversely as the forces, the resistance to this velocity balf of this is 0.0405755 or 186786, or nearly 1s of would extinguish it in the time of describing & m d d a pound. This should be the resistance of the air to being the diameter of the ball. Now i is to m as the this motion of the ball. weight of the displaced air to the weight of the ball, or In all matters of physical discussion, it is prudent to Necessity as of the diameter of the ball to the length of a co confront every theoretical conclusion with experiment. of experilumo of air of equal weight. Call this length a; a is This is particularly necessary in the present instance, be- svent. therefore equal to 1 m d. Suppose the ball, to fall cause the theory on which this proposition is founded is from the height a in the time t, and acquire the velo- extremely uncertain. Newton speaks of it with the city u. If it moved uniformly with this velocity during most cautious diffidence, and secures the justness of the this time, it would describe a space = 2a, orm d. conclusions by the conditions which he assumes in his Now its weight would extinguish this velocity, or de investigation. He describes with the greatest precision stroy this motion, in the same time, that is, in the time the state of the fluid in which the body must move, so of describing 4 md; but the resistance of the air would as that the demonstrations may be strict, and leaves it to do this in the time of describing , md; that is, in twice others to pronounce, whether this is the real constituthe time. The resistance therefore is equal to half the tion of our atmosphere. It must be granted that it is weight of the ball, or to balf the weight of the column not; and that many other suppositions bave been introof air whose height is the beight producing the velocity. duced by his commentators and follo vers, in order to But the resistances to different velocities are as the suit bis investigation (for we must assert that little or squares of the velocities, and therefore, as tbeir produ- nothing has been added to it) to the circumstances of cing heights; and, in general, the resistance of the air the case. to a sphere moving with any velocity, is equal to the Newton himself, therefore, attempted to compare bis Netton's half weight of a column of air of equal section, and propositions with experiment. Some were made by experiwhose altitude is the height producing the velocity dropping balls from the dome of St Paul's cathedral ments The result of this investigation bas been acquiesced in by and all these showed as great a coincidence with bió all Sir Isaac Newton's commentators. Many faults theory as they did with each other: but the irregulari. or 39 40 43 N° 9, 4914 we ties were too great to allow him to say with precision this matter, were generally too deficient in mathemati. nage of sovereigns are little more than prolix extensions height necessary for communicating the velocity, it was of the same theory of Galileo. Some of them, howalways equal to a determinate part of it; and might be ever, such as St Remy, Antonini, and Le Blond, have stated =na, n being a number to be fixed by nume given most valuable collections of experiments, ready rous experiments. for the use of the profound mathematician. One great source of uncertainty in his experiments Two or three years after this first publication, Mr Observa. seems to have escaped his observation : the air in that Robins hit upon that ingenious method of measuring tions of w's Robins on dome is almost always in a state of motion. In the sum the great velocities of military projectiles, which has velocity mer season there is a very sensible current of air down handed down his name to posterity with great honour. and resist. wards, and frequently in winter it is upwards: and this And having ascertained these velocities, he discovered ance, He made another set of experiments with pendulums; him anxious to examine wbat was the real resistance to have said above, it has been acquiesced in by the first which gave 7*667785 in the proportion of 100s to 1211, or very nearly in the proportion of five to six in Inutility of But the deductions from this theory were so inconsist small numbers. And we may adopt as a rule in all mothe theory ent with the observed motions of military projectiles, derate velocities, that the resistance to a sphere is equal in practice. when the velocities are prodigious, that no application to 67 of the weight of a column of air having the could be made which could be of any service for ileter- great circle of the sphere for its base, and for its alli42 mining the path and motion of cannon shot and bombs ; tude the height through which a heavy body must full The at and although Mir John Bernoulli gave, in 1718, a most in vacuo to acquire the velocity of projection. tempts of elegant determination of the trajectory and motion of a This experiment is peculiarly valuable, because the various mathemati body projected in a fluid which resists in the duplicate ball is precisely the size of a 12 pound sliot of cast iron; cians, &c. ratio of the velocities (a problem which even Newton and its accuracy may be depended on. There is but , of 4.5 3 E 2 47 46 3.81973 of 1000000 of gunnery; and indeed till this be done, all the labours to such as have the opportunities of trying them, methods evidently give us R= _0.04914 A new form being diminished in 4,5X 25.237 of experi. nected with wheelwork, by which it can be put in mo the duplicate ratio of the diameter and velocity. This ment re tion, and gradually accelerated. Let the ball be 30 commend connected with a bent spring, that this shall gradually gives us R=0,00000381973 pounds, or ed. compress it as the resistance increases, and leave a mark a pound. The logarithm is 4.58204. The resistance here determined is the same whatever substance the ball apparatus be screened from the air except the ball. The be of; but the retardation occasioned by it will depend velocity will be determined precisely by the revolutions on the proportion of the resistance to the vis insita of of the arm, and the resistance by the compression of the the ball; that is, to its quantity of motion. This in spring. The best method would be to let this part of similar velocities and diameters is as the density of the the apparatus be made to slide along the revolving arm, ball. The balls used in military service are of cast iron so that the ball can be made to describe larger and lar or of lead, whose specific gravities are 7,207 and 11,37 ger circles. An intelligent mechanician will easily contrive an apparatus of this kind, held at any distance pearly, water being !: There is considerable variety from the axis by a cord, which passes over a pulley in These data will give us in cast iron, and this density is about the medium. the axis itself, and is then brought along a perforation in the axis, and comes out at its extremity, where it is For Iron. For Lead fitted with a swivel, to prevent it from snapping by be W, or weight of a ball i inch in ing twisted. Now let the machine be put in motion. diameter lbs. 0.13648 0.21533 The centrifugal force of the ball and apparatus will Log. of EV 9.13509 9.33310 cause it to fly out as far as it is allowed by the chord ; E" 1116".6 and if the whole is put in motion by connecting it with Log. of E 3.04790 3.24591 some mill, the velocity may be most accurately ascer u, or terminal velocity 189.03 237:43 tained. It may also be fitted with a bell and hammer Log. u 2.27653 2.37553 like Gravesande’s machine for measuring centrifugal a, or producing height 558.3 880.8 forces. Now by gradually veering off more cord, the distance from the centre, and consequently the velocity These numbers are of frequent use in all questions on and resistance increase, till the hammer is disengaged this subject. and strikes the bell. Mr Robins gives an expeditious rule for readily findAnother great advantage of this form of the experi- ing a, which he calls F (see the article GUNNERY), by ment is, that the resistance to very great velocities may which it is made goo feet for a cast iron ball of an inch be thus examined, which was impossible in Mr Robins's diameter. But no theory of resistance which he proway. This is the great desideratum, that we may learn fesses to use will make this height necessary for produin what proportion of the velocities the resistances in cing the terminal velocity. His F therefore is an em. pirical quantity, analogous indeed to the producing In the same manner, an apparatus, consisting of Dr beight, but accommodated to his theory of the trajecLynd's Anemometer, described in the article PNEUMA tory of cannon-shot, which he promised to publish, but TICS, N° 311, &c. might be whirled round with pro did not live to execute. We need not be very anxious digious rapidity, and the fluid on it might be made about this; for all our quantities change in the same clammy, which would leave a mark at its greatest ele proportion with R, and need only a correction by a vation, and thus discover the resistance of the air to ra multiplier or divisor, when R shall be accurately estapid motions. blished. Nay, we are of opinion that the resistance to very illustrate the use of these formulæ by an es. rapid motions may be measured directly in the conduit ample or two. pipe of some of the great cylinder bellows employed in 1. Then, to find the resistance to a 24 pound ball baserte blast furnaces : the velocity of the air in this pipe is moving with the velocity of 1670 feet in a second, ei ikil.* ascertained by the capacity of the cylinder and thie which is nearly the velocity communicated by 16lbs. strokes of the piston. We think it our duty to point out, of powder. The diameter is 5,603 inches. 1761".6 crease. We may Log. R con Log. +7.58204 bled to compare the results with many experiments, and Log.d +1.49674 the agreement is very flattering. But we shall soon see Log. 1670 +0.44548 that little dependence can be bad on such comparisons. We notice these things here, because Mr Muller being 2.52426 head of the artillery school in Britain, his publications have become a sort of text-books. We are miserably deficient in works on this subject, and must have re- 52 We now proceed to consider these motions through The moresistance to great velocities, arising from the condensation of the air, and from its pressure into the vacuum their whole course ; and we shall first consider them as tions left by the ball. affected by the resistance only; then we shall consider sidered, through the air ; and, lastly, their motion in a curvili-course. 2. Required the terminal velocity of this ball ? neal trajectory, when projected obliquely. This must +4.58204 be done by the help of the abstruser parts of fluxionary by way of introduction, consider the simply resisted rec- 6.07878=a tilineal motions geometrically, in the manner of Sir Log. W. 1.38021=b Isaac Newton. As we advance, we shall quit this track, 5.30143 and prosecute it algebraically, having by this time ac- 53 50 Table of We must keep in mind the fundamental theorems of Prelin inary As the terminal velocity u, and its producing height varied motions. observaterminal a, enter into all computations of military projectiles, relocity tions. we have inserted the following Table for the usual sizes 1. The momentary variation of the velocity is proaccording of cannon-shnt, computed both by the Newtonian theo- portional to the force and the moment of time jointly, to Newton and Robins. of resistance, and by the resistances observed in Ro and may therefore be represented by +=fi, where v is the momentary increment or decrement of the velo- moment or increment of the time t. 2. The momentary variation of the square of the ve- of the space jointly; and may be represented by Ev =fs. The first proposition is familiarly known. The 54 second is the 39th of Newton's Principia, B. I. It is the mo 1.94 demonstrated in the article Optics, and is the most ex. tions as af fected by 4 365.3 4170.3 331.9 3442.6 | 3.08 These things being premised, let the straight line resistance 6 390.8 only. 4472.7 355. I 3940.7 3.52 AC (fig. 5) represent the initial velocity V, and let CO, perpendicular to AC, be the time in which this velocity would be extinguished by the uniform action of the resistance. Draw through the point A an equi- lateral hyperbola A e B, having OF, OCD for its as- represented by the line CB, C being the first instant of xe, gf, DB, &c. to the hyperbola, they will be pro- Mr Muller, in his writings on this subject, gives a portional to the velocities of the body at the instants ler's theory much smaller measure of resistance, and consequently a %,g, D, &c. and the hyperbolic areas AC x e, AC gf, altogether much greater terminal velocity : but his theory is a ACDB, &c. will be proportional to the spaces describ- his Treatise of Artillery, art. 150, &c.). In art. 148. he For, suppose the time divided into an indefinite num- the ordinates a c, b d, and the perpendiculars b B, a d. : a c=Cc: OC, and A a: Cc=ac:OC=AC.ac: were taken equal, and the rectangles AC.CO, BD:DO, introduces a new principle, which he makes the chief are equal, by the nature of the hyperbola ; therefore support of his theory, in opposition to the theories of A« : BB=AC.ac: BD.bd: but as the points c, d other mathematicians. The principle is false, and even continually approach, and ultimately coincide with C, absurd, as we shall have occasion to show by and by. D, the ultimate ratio of AC•ac to 'BD • bd is that of Jo consequence, however, of this principle, he is ena AB' to BD’; therefore the momentary decrements of AC Lb. Ball. 2 a. 2 a. 3788.2 3127.9 2.80 Fig. 5. 418.1 438.6 512.6 |