n m n m n n 2 a un time E in which the velocity 1 is extinguished creased in the proportion of i to 1, the retarding by the uniform action of the corresponding re force of the resistance increases in the same prosistance, or by 2 a, which is the space uniformly portion : hence we easily deduce these general described during this time, with the velocity 1. expressions. And E and 2 a must be expressed by the same number. It is a number of units, of time, or of. The terminal velocity = - Vam length. Thus, having ascertained these leading circum- ✓ 2 gad stances for a unit of velocity, weight, and bulk, we proceed to deduce the similar circumstances for any other magnitude ; and, to avoid unneces The producing fall in vacuo = ad sary complications, we shall always suppose the The retarding power of resistance to any velobodies to be spheres, differing only in diameter and density. First, then, lėt the velocity be in- city = ? 2 adm creased in the ratio of 1 to v. va The resistance will now be The extinguishing time for any velocity v = 2 a Edm E Thus we see that the chief circunstances are and e v = 2 a; so that the rule is general, that regulated by the terminal velocity, or are convio the space along which any velocity will be ex- niently referred to it. tinguished by the uniform action of the corre To communicate distinct ideas, and render the sponding resistance is equal to the height neces- deductions from these premises perspicuous, it sary for communicating the terminal velocity to will be proper to assume some convenient units, that body by gravity. For e v is twice the space by which all these qualities may be measured ; through which the body moves while the ve- and, as this subject is chiefly interesting in the locity vis extinguished by the uniform resistance case of military projectiles, we shall adapt our 2dly, Let the diameter increase in the propor- units to this purpose. Therefore let a second be tion of 1 to d. The aggregate of the resistance the unit of time, a foot the unit of space and vechanges in the proportion of the surface similarly locity, an inch the unit of diameter of a ball or resisted, that is, in the proportion of 1 to d?. But shell, and a pound avoirdupois the unit of presthe quantity of matter, or number of particles sure, whether of weight or of resistance : thereamong which this resistance is to be distributed, fore g is thirty-two feet. The great difficulty is changes in the proportion of 1 to do. Therefore to procure an absolute measure of r, or u, or a; the retarding power of the resistance changes in any one of these will determine the others. Sir Isaac Newton attempted to determine r by the proportion of 1 toWhen the diameter theory, and employed a great part of the second book of the Principia in demonstrating, that the was 1 the resistance to a velocity 1 was It resistance to a sphere moving with any velocity is to the force which would generate or destroy must now be The time in which this its whole motion in the time that it would uni formly move over eight-thirds of its diameter diminished resistance will extinguish the velocity with this velocity as the density of the air is to 1 must increase in the proportion of the diminu- the density of the sphere. This is equivalent to tion of force, and must now be Ed, or 2 ad, demonstrating, that the resistance of the air to a and the space uniformly described during this sphere, moving through it with a velocity, is time with the initial velocity 1 must be 2 ad; equal to half the weight of a column of air havand this must still be twice the height necessary ing a great circle of the sphere for its base, and for communicating the terminal velocity w to for its altitude the height from which a body wa this body. We must still have must fall in vacuo to acquire this velocity. This 8 2 ad appears from Newton's demonstration ; for, let therefore w' = 2g ad, and w = 2 gad, = the specific gravity of the air be to that of the ball ✓ 2ga vd. But u = n2ga. Therefore as 1 to m; then, because the times in which the same velocity will be extinguished by the uniform the terminal velocity w for this body is =und; action of different forces are inversely as the and the height necessary for communicating it is forces, the resistance to this velocity would exad. Therefore the terminal velocity varies in tinguish it in the time of describing eight-thirds the subduplicate ratio of the diameter of the ball, md, d being the diameter of the ball. Now 1 is and the fall necessary for producing it varies in to m as the weight of the displaced air to the the simple ratio of the diameter. The extin- weight of the ball, or as two-thirds of the diame Ed. guishing time for the velocity v must now be ter of the ball to the length of a column of air of equal weight. Call this length a; a is there3dly, If the density of the ball be :creased in fore equal to two-thirds md. Suppose the ball the proportion of 1 to m, the number “ticles to fall from the height a in the time t, and acamong which the resistance is to be distributed quire the velocity u. If it moved uniformly with is increased in the same proportion, and there- this velocity, during this time, it would describe fore the retarding force of the resistance is equally a space = 2a, or four-thirds md. Now its diminished; and, if the density of the air is in weight would extinguish this velocity, or destroy Vol. XVIII. L 1 2 a 1 2 ad ; and this motion, in the same time, that is, in the time sumes in his investigation. He describes with of describing four-thirds md; but the resistance the greatest precision the state of the fluid ia of the air would do this in the time of describing which the body must move, so as that the deeight-thirds md; that is, in twice the time. The monstrations may be strict, and leaves it to others resistance therefore is equal to half the weight of to pronounce whether this is the real constitution the ball, or to half the weight of the column of of our atmosphere It must be granted that it is air whose height is the height producing the ve- not; and that many other suppositions have locity. But the resistance to different velocities been introduced by his commentators and foiare as the squares of the velocities; and there- lowers to suit his investigation (for little or fore as their producing heights, and, in general, nothing has been added to it) to the circumthe resistance of the air to a sphere moving with stances of the case. any velocity, is equal to the half weight of a co Sir Isaac Newton himself, therefore, attemptlumn of air of equal section, and whose altitude ed to compare his proportions with experiment. is the height producing the velocity. Some were made by dropping balls from the The result of this investigation has been ac- dome of St. Paul's cathedral ; and all these quiesced in by all Sir Isaac Newton's commen showed as great a coincidence with his theory as tators. Many faults have indeed been found they did with each other: but the irregularities with his reasoning, and even with his principles; were too great to allow him to say with precision and it must be acknowledged that although this what was the resistance. It appeared to follow investigation is by far the most ingenious of any the proportion of the squares of the velocities in the Principia, and sets his acuteness and ad- with sufficient exactness; and, though he could dress in the most conspicuous light, his reasoning not say that the resistance was equal to the weight is liable to serious objections, which his most of the column of air having the height necessary ingenious commentators have not completely re- for communicating the velocity, it was always moved. Yet the conclusion has been acquiesced equal to a determinate part of it; and might be in, but as if derived from other principles, or by stated=na, n being a number to be fixed by more logical reasoning. The reasonings or as. numerous experiments. One great source of sumptions, however, of these niathematicians are uncertainty in his experiments seems to have no better than Newton's; and all the causes of escaped his observation: the air in that dome is deviation from the duplicate ratio of the veloci- almost always in a state of motion. In summer ties, and the causes of increased resistance, which there is a very sensible current of air downwards, the latter authors have valued themselves for dis- and frequently in winter it is upwards: and this covering and introducing into their investigations, current bears a very great proportion to the vewere actually pointed out by Sir Isaac Newton, locity of the descents. Sir Isaac takes no nobut purposely omitted by him to facilitate the tice of this. He made another set of experiments discussion in re difficillima (See Schol. prop. with pendulums; and pointed out some very 37. b. 2). curious and unexpected circumstances of their The weight of a cubic foot of water is 624 lbs. motions in a resisting medium. There is hardly and the nedium density of the air is sto of water; any part of his noble work in which his address, therefore let a be the height producing the velo- his patience, and his astonishing penetration, apcity (in feet), and d the diameter of the ball (in pear in greater lustre. It requires the utmost inches), and a the periphery of a circle whose intenseness of thought to follow him in these diameter is 1; the resistance of the air will be disquisitions. Their results were much more 62} uniform, and confirmed his general theory; and Х Х 840 4 pounds, it has been acquiesced in by the first mathemati2 144 4928) va 22 d? cians of Europe. very nearly, = But the deductions from this theory were so 4928) X 64' 315417 inconsistent with the observed motions of milipounds. tary projectiles, when the velocities are prodiErample.—A ball of cast iron weighing twelve gious, that no application could be made which pounds is four inches and a half in diameter. Sup- could be of any service for determining the path pose this ball to move at the rate of 251, feet in a se- and motion of cannon shot and bombs; and alcond. The height which will produce this velocity though John Bernouilli gave, in 1718, a most in a falling body is 97 feet. The area of its great elegant determination of the trajectory and mocircle is 0·11044 feet, or content of one foot. Sup- tion of a body projected in a fluid which resists pose water to be 840 times heavier than air, the in the duplicate ratio of the velocities (a problem weight of the air incumbent on this great circle, which even Newton did not attempt), it has reand 97 feet high, is 0.081151 lbs. half of this is mained a dead letter. Mr. Benjamin Robins 00405755 or or nearly gk of a pound. was the first who suspected the true cause of the This should be the resistance of the air to this imperfection of the usually received theories; motion of the ball. and in 1737 he published a small tract, in which It is proper, in all matters of physical discus- he showed clearly that even the Newtonian sion, to confront every theoretical conclusion theory of resistance must cause a cannon ball, with experiment. This is particularly necessary discharged with a full allotment of powder, to in the present instance, because the theory on deviate farther from the parabola, in which it which this proposition is founded is extremely would move in vacuo, than the parabola deviates uncertain. Newton speaks of it with the most from a straight line. But he farther asserted, cautious diffidence, and secures the justness of from good reasoning, that in such great velocities the conclusions by the conditions which he as- the resistance must be much greater than this 1 a ad Х 405755 or theory assigns; because, besides the resistance resulting from Mr. Robins's experiments nearly arising from the inertia of the air which is put in in the proportion of seven to ten. Chev. de motion by the ball, there must be a resistance Borda made experiments similar to those of Mr. arising from a condensation of the air on the an Robins, and his results exceeded those of Robins terior surface of the ball, and a rarefaction be- in the proportion of five to six. hind it: and there must be a third resistance, We must content ourselves, however, at prearising from the statical pressure of the air on sent with the experimental measure mentioned its anterior part, when the motion is so swift that above. To apply to our formulæ, therefore, we there is a vacuum behind. Even these causes of reduce this experiment, which was made on a disagreement with the theory had been foreseen ball of four inches and a half diameter, moving and mentioned by Newton (see the Scholium to with the velocity of twenty-five feet and one-fifth prop. 37, Book II. Princip.); but the subject per second, to what would be the resistance to a seems to have been little attended to. Some ball of one inch, having the velocity a foot. authors, however, such as St. Remy, Antonini, 0:04919 and Le Blond, have given most valuable collec- This will give R = being dimin 4:52 X 25•2?' tions of experiments, ready for the use of the ished in the duplicate ratio of the diameter and profound mathematician. velocity. This gives R=0.00000381973 pound, Sect. IV.-OBSERVATIONS BY MR. ROBINS, ON 3.81973 VELOCITY AND RESISTANCE. of a pound. The ogarithm is, 1000000 Two or three years after the appearance of his 4-58204. The resistance here determined is the first publication, Mr. Robins discovered that in same whatever substance the ball be of; but the genious method of measuring the velocities of retardation occasioned by it will depend on the military projectiles which has handed down his proportion of the resistance to the vis insita of name to posterity with great honor: and, having the ball; that is, to its quantity of motion. This ascertained these velocities, he discovered the in similar velocities and diameters is as the denprodigious resistance of the air, by observing the sity of the ball. The balls used in military serdiminution of velocity which it occasioned. vice are of cast iron, or of lead, whose specific This made him anxious to examine what was the gravities are 7-207 and 11.37 nearly, water being real resistance to any velocity whatever, in order 1. There is considerable variety in cast iron, to ascertain what was the law of its variation; and tnis density is about the medium. These and he was equally fortunate in this attempt data will give us, likewise. From his Mathematical Works, vol. For Iron. For Lead. i. p. 205, it appears that a sphere of four inches w, or weight of a ball one inch in diameter and a half in diameter, moving at the rate of lbs. 0.13648 0.21533 9.13509 9.33310 twenty-five feet one-fifth in a second, sustained Log. of W E“ a resistance of 0.04914 lb. or I'd of a pound. 1116“,6 1761",6 This is a greater resistance than that of the New: Log. of E 304790 3.24591 189,03 237.43 tonian theory, which gave 1868 al in the proportion 4, or terminal velocity of 1000 to 1211, or very nearly in the proportion Log. u 2.27653 2.37553 of five to six in small numbers. And we may a, or producing height 558,3 880,8 adopt as a rule, in all moderate velocities, that These numbers are of frequent use in all questhe resistance to a sphere is equal to one of the tions on this subject. Mr. Robins gives an exweight of a column of air having the great circle peditious rule for readily finding a, which he of the sphere for its base, and for its altitude the calls F, by which it is made 900 feet for a castheight through which a heavy body must fall in iron ball of an inch diameter. But no theory of vacuo to acquire the velocity of projection. The resistance which he professes to use will make importance of this experiment is great, because this height necessary for producing the terminal the ball is precisely the size of a twelve pound velocity. His F, therefore, is an empirical quanshot of cast iron; and its accuracy may be de- tity, analogous indeed to the producing height, pended on. There is but one source of error. but accommodated to his theory of the trajectory The whirling motion must have occasioned some, of cannon-shot, which he promised to publish, whirl in the air, which would continue till the but did not live to execute. We need not be ball again passed through the same point of its very anxious about this; for all our quantities revolution. The resistance observed is therefore change in the same proportion with R, and need probably somewhat less than the true resistance only a correction a multiplier or divisor, when to the velocity of twenty-five feet one-fifth, be- R shall be accurately established. cause it was exerted in a relative velocity which The use of these formulæ may be illustrated was less than this, and is, în fact, the resistance by an example or two. competent to this relative and smaller velocity. Ex. 1. To find the resistance to a twenty-four Accordingly, Mr. Smeaton places great confi- pound ball moving with the velocity of 1670 dence in the observations of Mr. Rouse of Lei- feet in a second, which is nearly the velocity comcestershire, who measured the resistance by the municated by sixteen pounds of powder. The effect of the wind on a plane properly exposed diameter is 5603 inches. to it. He does not tell us how the velocity of Log. R + 4:58204 the wind was ascertained ; but our opinion of + 1.49674 his penetration and experience leads us to be Log. 16702 + 6.44548 lieve that this point was well determined. The resistance observed by Mr. Rouse exceeds that Log. 3344 lbs. Er 2.52426 Log. d Log. R But it is found, by unequivocal experiments on tions geometrically, in the manner of Sir Isaac the retardation of such a motion, that it is 504 Newton. As we advance, we shall quit this lbs. This is owing to the above causes, the ad- track, and prosecute it algebraically, having by ditional resistance to great velocities, arising from this time acquired distinct ideas of the algebraic the condensation of the air, and from its pressure quantities. into the vacuum left by the ball. We must remember the fundamental theorems Ex. 2. Required the terminal velocity of this of varied motions. ball ? 1. The momentary variation of the velocity is + 4:58204 proportional to the force and the moment of time Log. da + 1.49674 jointly, and may therefore be represented by +ürft, where v is the momentary increment Log. resist. to veloc. 1 6.07878= or decrement of the velocity v, f the accelerating Log. W 1.38021 = b or retarding force, and i the moment or increDift. of a and b, =log. u? 5.30143 ment of the time t. Log. 4474 2.65071 2. The momentary variation of the square of We proceed lo consider these motions through the velocity is as the force, and as the increment their whole course : and we shall first consider or decrement of the space jointly; and may be them as affected by the resistance only; then we represented by Evv=fs . The first proposhall consider the perpendicular ascents and de- sition is familiarly known. The second is the scents of heavy bodies through the air; and, 39th of Newton's Principia, B. I. It is delastly, their motion in a curvilineal trajectory, monstrated in the article Optics, and is the when projected obliquely. This must be done most extensively useful proposition in mechanics. by the help of the abstruser parts of Auxionary Having premised these things, let the straight mathematics. To make it more perspicuous, we line AC (ig. 2) represent the initial velocity V, shall consider the simply resisted rectilineal mo- and let C 0, perpendicular to AC, be the time Fig. 2. F 1 G MM NV tli in which this velocity would be extinguished by taken equal, and the rectangles AC-CO, BD.DO, the uniform action of the resistance. Draw are equal by the nature of the hyperbola ; therethrough the point A an equilateral hyperbola fore A a : BB=AC: ac: B Dbd: but as the A e B having 0 F, OCD, for its assymptotes; points c, d, continually approach, and ultimately then let the time of the resisted motion be rep- coincide with C, D, the ultimate ratio of ACresented by the nine C B, C being the first instant ac to B Dbd is that of A C? to B D’; thereof the motion. If there be drawn perpendicu- fore the momentary decrements of A C and BD lar ordinates ke, fg, DB, &c., to the hyperbola, are as A C and B D. Now, because the resisthey will be proportional to the velocities of thé tance is measured by the momentary diminution body at the instant; k,g, D, &c., and the hyper- of velocity, these diminutions are as the squares bolic areas A Cre, AC, F8, A CD B, &c., will of the velocities; therefore the ordinates of the be proportional to the spaces described during hyperbola and the velocities diminish by the same the times Cr, Cg, CB, &c. For suppose the law; and the initial velocity was represented by time divided into an indefinite number of small AC; therefore the velocities at all the other and equal moments, C c, Dd, &c., draw the or instants <,g, D, are properly represented by the dinates ac, bd, and the perpendiculars bb, aa. corresponding ordinates. Hence, Then, by the nature of the hyperbola, A C:ac 1. Ås the abscissa of the hyperbola are as the = 00:0 C. and AC-ac:ac=00—0C times, and the ordinates are as the velocities, the :00, that is, A a:ac=Cc:0C, and A a : areas will be as the spaces described, and AC Ccrac:00,=AC.ac: AC0C; in like ke is to A c gf as the space described in the manner, BB: Dd=BD.6 D:BDO D. Now time Ck to the space described in the time Cg Dd=C c, because the moments of time were (first theorem on varied motions). 2. The rectangle ACOF is to the area ACDB times and velocities, and the areas exhibiting the as the space formerly expressed by 2 a, or E to relations of both to the spaces described., But the space described in the resisting medium we may render the conception of these circumduring the time CD; for AC being the velocity stances much more easy and simple, by expresV, and OC the extinguishing time e, this reci- sing them all by lines, instead of this combinaangle is = eV, or E, or 2 a, of our former dis- tion of lines and surfaces. We shall accomplish quisitions; and because all the rectangles such as this purpose by constructing another curve LKP, ACOF, BDOG, &c., are equal, this corre- having the line M Lè, parallel to OD for its sponds with our former observation, that the space abscissa, and of such a nature that if the ordiuniformly described with any velocity during the nates to the hyperbola A Cek, fg, BD, &c. time in which it would be uniformly extinguished be produced till they cut this curve in L, p, n, by the corresponding resistance is a constant K, &c., and the abscissa in L, e, h, o, &c., the quantity, viz. that in which we always had v=E, ordinates e, p, h, n, d, K, &c., may be proporor 2 a. tional to the hyperbolic areas e A ck, f A cg, o 3. Draw the tangent A «; then, by the hyper- A c K. Let us examine what kind of curve bola Cr=CO: now C « is the time in which this will be. Make OC : 0x=0x : 0g; the resistance to the velocity A C would extin- then (Hamilton's Conics, IV. 14. Cor.) the areas guish it ; for the tangent coinciding with the ele- A Cre, exgf are equal : therefore drawing ps, mental arc A a of the curve, the first impulse of nt, perpendicular to OM, we shall have (by the the uniform action of the resistance is the same assumed nature of the curve Lp K), Ms=st; with its first impulse of its varied action. By and if the abscissa 0 D be divided into any numthis the velocity A C is reduced to ac. If this ber of small parts in geometrical progression, operated uniformly, like gravity, the velocities (reckoning the commencement of them all from would diminish uniformly, and the space de- O), the axis V i of this curve will be divided by scribed would be represented by the triangle its ordinates into the same number of equal AC *. This triangle, therefore, represents the parts; and this curve will have its ordinates height through which a heavy body must fall LM, ps, nt, &c., in geometrical progression, in vacuo, in order to acquire the terminal ve- and its abscissæ in geometrical progression. locity. Also, let KN, MV, touch the curve in K and L, 4. The motion of a body resisted in the du- and let OC be supposed to be to Oc, as O D tó plicate ratio of the velocity will continue with- Od, and therefore Cc to D d as 0 C to OD; out end, and a space will be described which is and let these lines Cc, Dd, be indefinitely greater than any assignable space, and the ve- small; then (by the nature of the curve) Lo is locity will grow less than any that can be as- equal to Kr; for the areas a AC 6, 6 B D d are signed; for the hyperbola approaches continually in this case equal. Also lo is to kr, as L M to to the assymptote, but never coincides with it. KI, because c C:dD=CO:DÓ: There is no velocity B D so small, but a smaller Therefore IN:IK=rKirk Z P will be found beyond it; and the hyper IK:ML=rk:01 bolic space may be continued till it exceeds any ML:MV=01:0L surface that can be assigned. and IN:MN=rK:0 L. 5. The initial velocity A C is to the final ve- That is the subtangent IN, or M V, is of the locity B D as the sum of the extinguishing time same magnitude, or is a constant quantity in and the time of the retarded motion is to the every part of the curve. extinguishing time alone; for AC:BD=OD Lastly, the subtangent IN, corresponding to (or OC XCD):0C:or V:v=e:ex t. the point K of the curve, is to the ordinate K 8 6. The extinguishing time is to the time of as the rectangle BDOG or ACOF to the the retarded motion as the final velocity is to the parabolic area BDCA. For let fghn be an velocity lost during the retarded motion : for the ordinate very near to BD 8 K; and let hn cut rectangles AF OČ, B DOG, are equal; and the curve in'n, and the ordinate K I ing; then therefore A V. G F and BVCD are equal and we have e V-V VC:VA=VG: VB; therefore t= K9:on=KI:IN, or Dg:qn=DO:IN; but BĎ: AC=CO: DO; and ezt V therefore BD.Dg:AC. .on=CO:IN: 7. Any velocity is reduced in the proportion is to the sum of all the rectangles A C.qn, as Therefore the sum of all the rectangles B D.Dg of m to n in the time e For, let AC: CO to IN; but the sum of the rectangles BD:Dg is the space AC DB; and, because BD=m:n; then DO: CO=m:n, and AC is given, the sum of the rectangles AC.qn DC: CO=m-n: n, and DC = is the rectangle of A C, and the sum of all the lines qn; that is, the rectangle of A C and RL; CO, ort=e Therefore any velocity is therefore the space ACDB:AC.RL=có : IN, aud ACDBX IN=AC.CO·RL; reduced to one-half in the time in which the and therefore I N:RLI AC.CO: ACD B. initial resistance would have extinguished it by Hence it follows that QL expresses the area its uniform action. BV A, and, in general, that the part of the line The chief circumstances of this motion may parallel to OM, which lies between the tangent thus be determined by the hyperbola, the ordi- K N and the curve Lp K, expresses the correnates and abscissæ exhibiting the relations of the sponding area of the hyperbola which lies with mn n mn n mn n |