of gunnery; and indeed till this be done, all the labours of mathematicians are of no avail. Their investigations must remain an unintelligible cipher, till this key be supplied. It is to be hoped that Dr Charles Hutton of Woolwich, who has so ably extended Mr Robins's Examination of the Initial Velocities of Military Projectiles, will be encouraged to proceed to this part of the subject. We should wish to see, in the first place, a numerous set of experiments for ascertaining the resistances in moderate velocities; and, in order to avoid all error from the resistance and inertia of the machine, which is necessarily blended with the resistance of the ball, in Mr Robins's form of the experiment, and is separated with great uncertainty and risk of error, we would recommend a form of experiment somewhat different.
Let the axis and arm which carries the ball be connected with wheelwork, by which it can be put in motion, and gradually accelerated. Let the ball be so connected with a bent spring, that this shall gradually compress it as the resistance increases, and leave a mark of the degree of compression; and let all this part of the apparatus be screened from the air except the ball. The velocity will be determined precisely by the revolutions of the arm, and the resistance by the compression of the spring. The best method would be to let this part of the apparatus be made to slide along the revolving arm, so that the ball can be made to describe larger and larger circles. An intelligent mechanician will easily contrive an apparatus of this kind, held at any distance from the axis by a cord, which passes over a pulley in the axis itself, and is then brought along a perforation in the axis, and comes out at its extremity, where it is fitted with a swivel, to prevent it from snapping by being twisted. Now let the machine be put in motion. The centrifugal force of the ball and apparatus will cause it to fly out as far as it is allowed by the cord; and if the whole is put in motion by connecting it with some mill, the velocity may be most accurately ascertained. It may also be fitted with a bell and hammer like Gravelande's machine for measuring centrifugal forces. Now by gradually veering off more cord, the distance from the centre, and consequently the velocity and resistance increase, till the hammer is disengaged and strikes the bell.
Another great advantage of this form of the experiment is, that the resistance to very great velocities may be thus examined, which was impossible in Mr Robins's way. This is the great desideratum, that we may learn in what proportion of the velocities the resistances increase.
In the same manner, an apparatus, consisting of Dr Lynd's Anemometer, described in the article PNEUMATICS, No 311, &c., might be whirled round with prodigious rapidity, and the fluid on it might be made clammy, which would leave a mark at its greatest elevation, and thus discover the resistance of the air to rapid motions.
Nay, we are of opinion that the resistance to very rapid motions may be measured directly in the conduit pipe of some of the great cylinder bellows employed in blast furnaces: the velocity of the air in this pipe is ascertained by the capacity of the cylinder and the strokes of the piston. We think it our duty to point out to such as have the opportunities of trying them methods which promise accurate results for ascertaining this most desirable point.
We are the more puzzled what measure to abide by, because Mr Robins himself, in his Practical Propo-47 tions, does not make use of the result of his own experiments, but takes a much lower measure. We must content ourselves, however, with this experimental measure, because it is yet the only one of which any account can be given, or well-founded opinion formed.
Therefore, in order to apply our formulae, we must reduce this experiment, which was made on a ball of 4½ inches diameter, moving with the velocity of 25 feet per second, to what would be the resistance to a ball of one inch, having the velocity 1 foot. This will evidently give us \( R = \frac{0.24914}{4.5^2 \times 25} \) being diminished in the duplicate ratio of the diameter and velocity. This gives us \( R = 0.00000381973 \) pounds, or \( 3.81973 \times 10^{-6} \) of a pound. The logarithm is 4.58204. The resistance here determined is the same whatever substance the ball be of; but the retardation occasioned by it will depend on the proportion of the resistance to the vis initia of the ball; that is, to its quantity of motion. This in similar velocities and diameters is as the density of the ball. The balls used in military service are of cast iron or of lead, whose specific gravities are 7.207 and 11.37 nearly, water being 1. There is considerable variety in cast iron, and this density is about the medium. These data will give us
| W, or weight of a ball 1 inch in diameter | For Iron. | For Lead. | |------------------------------------------|-----------|-----------| | lbs. 0.13648 | 0.21533 | | Log. of W | 9.13599 | 9.33110 | | E" | 1116"6 | 1761"6 | | Log. of E | 3.04792 | 3.24591 | | u, or terminal velocity | 189.03 | 237.43 | | Log. u | 2.27653 | 2.37553 | | a, or producing height | 558.3 | 882.8 |
These numbers are of frequent use in all questions on this subject.
Mr Robins gives an expeditious rule for readily finding \( a \), which he calls F (see the article GUNNERY), by which it is made 900 feet for a cast iron ball of an inch diameter. But no theory of resistance which he professes to use will make this height necessary for producing the terminal velocity. His F therefore is an empirical quantity, analogous indeed to the producing height, but accommodated to his theory of the trajectory of cannon-shot, which he promised to publish, but did not live to execute. We need not be very anxious about this; for all our quantities change in the same proportion with \( R \), and need only a correction by a multiplier or divisor, when \( R \) shall be accurately established.
We may illustrate the use of these formulae by an example or two.
1. Then, to find the resistance to a 24 pound ball moving with the velocity of 1670 feet in a second, which is nearly the velocity communicated by 16 lbs. of powder. The diameter is 5.603 inches.
Log. But it is found, by unequivocal experiments on the retardation of such a motion, that it is 504 lbs. This is owing to the causes often mentioned, the additional resistance to great velocities, arising from the condensation of the air, and from its pressure into the vacuum left by the ball.
2. Required the terminal velocity of this ball?
Log. R = +4.58204 Log. d^3 = +1.49674
Log. resift. to veloc. r = 6.07878 = a Log. W = 1.38021 = b Diff. of a and b, log u^3 = 5.32143 Log. 447.4 = u = 2.65071
As the terminal velocity u, and its producing height a, enter into all computations of military projectiles, we have inferred the following Table for the usual sizes of cannon-shot, computed both by the Newtonian theory of resistance, and by the resistances observed in Robins's experiments.
| | Newton | Robins | |---|--------|--------| | | Term. Vel. | 2 a. | Term. Vel. | 2 a. | | 1 | 289.9 | 2626.4 | 263.4 | 2168.6 | 1.94 | | 2 | 324.9 | 3208.3 | 295.2 | 2723.5 | 2.45 | | 3 | 348.2 | 3788.2 | 316.4 | 3127.9 | 2.80 | | 4 | 365.3 | 4170.3 | 331.9 | 3442.6 | 3.08 | | 5 | 390.8 | 4472.7 | 355.1 | 3949.7 | 3.52 | | 6 | 418.1 | 5463.5 | 379.9 | 4511.2 | 4.04 | | 7 | 438.6 | 6510.6 | 398.5 | 4962.9 | 4.45 | | 8 | 469.3 | 6883.3 | 426.5 | 5683.5 | 5.59 | | 9 | 492.4 | 7576.3 | 447.4 | 6255.7 | 5.61 | | 10 | 512.6 | 8024.8 | 465.8 | 6780.4 | 6.21 | | 11 | 540.5 | 9129.9 | 491.5 | 7538.3 | 6.75 |
Mr Muller, in his writings on this subject, gives a much smaller measure of resistance, and consequently a much greater terminal velocity: but his theory is a mistake from beginning to end (See his Supplement to his Treatise of Artillery art. 159, &c.) In art. 148, he assumes an algebraic expression for a principle of mechanical argument; and from its consequence draws erroneous conclusions. He makes the resistance of a cylinder one-third less than Newton supposes it; and his reason is false. Newton's measure is demonstrated by his commentators Le Seur and Jaquier to be even a little too small, upon his own principles, (Not. 277, Prop. 36, B. II.) Mr Muller then, without any seeming reason, introduces a new principle, which he makes the chief support of his theory, in opposition to the theories of other mathematicians. The principle is false, and even absurd, as we shall have occasion to show by and by. In consequence, however, of this principle, he is enabled to compare the results with many experiments, and the agreement is very flattering. But we shall soon see that little dependence can be had on such comparisons. We notice these things here, because Mr Muller being 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 recourse to the foreign writers.
We now proceed to consider these motions through their whole course: and we shall first consider them as affected by the resistance only; then we shall consider the perpendicular ascents and descents of heavy bodies through the air; and, lastly, their motion in a curvilinear trajectory, when projected obliquely. This must be done by the help of the abstruse parts of fluxionary mathematics. To make it more perspicuous, we shall, by way of introduction, consider the simply resisted rectilineal motions geometrically, in the manner of Sir Isaac Newton. As we advance, we shall quit this track, and prosecute it algebraically, having by this time acquired distinct ideas of the algebraic quantities.
We must keep in mind the fundamental theorems of varied motions.
1. The momentary variation of the velocity is proportional to the force and the moment of time jointly, and may therefore be represented by \( \frac{d}{dt} v = f \), where \( v \) is the momentary increment or decrement of the velocity, \( f \) the accelerating or retarding force, and \( t \) the moment or increment of the time.
2. The momentary variation of the square of the velocity is as the force, and as the increment or decrement of the space jointly; and may be represented by \( \frac{d}{dt} v^2 = f \). The first proposition is familiarly known. The second is the 39th of Newton's Principia, B. I. It is demonstrated in the article Optics, and is the most extensively useful proposition in mechanics.
These things being premised, let the straight line AC (fig. 5.) represent the initial velocity V, and let only 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 equilateral hyperbola AE, having OF, OCD for its asymptotes; then let the time of the resisted motion be represented by the line CB, C being the first instant of the motion. If there be drawn perpendicular ordinates \( x, y, z \), DB, &c. to the hyperbola, they will be proportional to the velocities of the body at the instants \( x, y, z \), D, &c., and the hyperbolic areas ACe, ACgf, ACDB, &c. will be proportional to the spaces described during the times \( x, y, z, CB, &c. \)
For, suppose the time divided into an indefinite number of small and equal moments, C c, D d, &c. draw the ordinates ac, bd, and the perpendiculars bc, ca. Then, by the nature of the hyperbola, AC : ac = OC : OC; and AC : ac : ac = OC : OC, that is, Aa : ac = Oc : Oc, and Aa : ac = Oc : Oc, = AC : ac : AC : OC; in like manner, BD : bd : BD : BD : BD : OD. Now Dd = Ce, because the moments of time were taken equal, and the rectangles AC:CO, BD:DO, are equal, by the nature of the hyperbola; therefore Aa : ac : BD : bd : but as the points c, d continually approach, and ultimately coincide with C, D, the ultimate ratio of AC : ac to BD : bd is that of AC^2 to BD^2; therefore the momentary decrements of AC...
AC and BD are as AC^2 and BD^2. Now, because the resistance is measured by the momentary diminution of velocity, these diminutions are as the squares of the velocities; therefore the ordinates of the hyperbola and the velocities diminish by the same law; and the initial velocity was represented by AC: therefore the velocities at all other instants, e.g., D, are properly represented by the corresponding ordinates. Hence,
1. Since the abscissae of the hyperbola are as the times, and the ordinates are as the velocities, the areas will be as the spaces described, and AC is to A c g f as the space described in the time C g (1st Theorem on varied motions).
2. The rectangle ACOF is to the area ACDB as the space formerly expressed by 2 a, or E to the space described in the resisting medium during the time CD: for AC being the velocity V, and OC the extinguishing time e, this rectangle is = V, or E or 2 a, of our former disquisitions; and because all the rectangles, such as ACOF, BDOG, &c., are equal, this corresponds with our former observation, that the space uniformly described with any velocity during the time in which it would be uniformly extinguished by the corresponding resistance is a constant quantity, viz., that in which we always had E = E or 2 a.
3. Draw the tangent A x; then, by the hyperbola C x = CO: now C x is the time in which the resistance to the velocity AC would extinguish it; for the tangent coinciding with the elemental arc A a of the curve, the first impulse of the uniform action of the resistance is the same with the first impulse of its varied action. By this the velocity AC is reduced to a e. If this operated uniformly like gravity, the velocities would diminish uniformly, and the space described would be represented by the triangle ACx.
This triangle, therefore, represents the height through which a heavy body must fall in vacuo, in order to acquire the terminal velocity.
4. The motion of a body resisted in the duplicate ratio of the velocity will continue without end, and a space will be described which is greater than any assignable space, and the velocity will grow less than any that can be assigned; for the hyperbola approaches continually to the asymptote, but never coincides with it. There is no velocity BD so small, but a smaller ZP will be found beyond it; and the hyperbolic space may be continued till it exceeds any surface that can be assigned.
5. The initial velocity AC is to the final velocity BD as the sum of the extinguishing time and the time of the retarded motion, is to the extinguishing time alone: for AC : BD = OD (or OC + CD) : OC; or V : v = e : e + t.
6. The extinguishing time is to the time of the retarded motion as the final velocity is to the velocity lost during the retarded motion: for the rectangles AFOC, BDOG are equal; and therefore AVGP and BVCD are equal, and VC : VA = VG : VB; therefore \( t = \frac{V-v}{v} \), and \( e = \frac{v}{V-v} \).
7. Any velocity is reduced in the proportion of m to n in the time \( e \frac{m-n}{n} \). For, let AC : BD = m : n; then DO : CO = m : n, and DC : CO = m : n; and DC = \( \frac{m-n}{n} \) CO, or \( t = \frac{m-n}{n} \). Therefore any velocity is reduced to one half in the time in which the initial resistance would have extinguished it by its uniform action.
Thus may the chief circumstances of this motion be determined by means of the hyperbola, the ordinates determining the relations of the times and ing the velocities, and the areas exhibiting the relations of both motions to the spaces described. But we may render the conception of these circumstances infinitely more easy and simple, by expressing them all by lines, instead of this combination of lines and surfaces. We shall accomplish this purpose by constructing another curve LKP, having the line ML, parallel to OD for its abscissa, and of such a nature, that if the ordinates to the hyperbola AC, e, f, g, BD, &c., be produced till they cut this curve in L, p, n, K, &c., and the abscissae in L, e, h, b, &c., the ordinates s, p, h, b, K, &c., may be proportional to the hyperbolic areas e A C x, f A C g, b A c K. Let us examine what kind of curve this will be.
Make OC : O = Oa : Og; then Hamilton's Conics, IV. 14. Cor.), the areas AC e, e x g f are equal; therefore drawing p s, n i perpendicular to OM, we shall have (by the assumed nature of the curve L p K), M s = s i; and if the abscissae OD be divided into any number of small parts in geometrical progression (reckoning the commencement of them all from O), the axis Vi of this curve will be divided by its ordinates into the same number of equal parts; and this curve will have its ordinates LM, ps, ni, &c., in geometrical progression, and its abscissae in arithmetical progression.
Also, let KN, MV touch the curve in K and L, and let OC be supposed to be to Oc, as OD to Od, and therefore Ce to Cd as OC to OD; and let these lines Ce, Dd be indefinitely small; then (by the nature of the curve) Lo is equal to Kr: for the areas a AC c, b BD d are in this case equal. Also ko is to kr, as LM to KI, because c : d = CO : DO:
Therefore IN : IK = rK : rk IK : ML = rk : ol ML : MV = ol : oL and IN : MN = rK : oL.
That is, the subtangent IN, or MV, is of the same magnitude, or is a constant quantity in every part of the curve.
Lastly, the subtangent IN, corresponding to the point K of the curve, is to the ordinate K as the rectangle BDOG or ACOF to the parabolic area BDCA.
For let f g h n be an ordinate very near to BD K; and let h n cut the curve in n, and the ordinate KI in q; then we have
\[ Kq : gn = KI : IN, \] \[ Dg : gn = DO : IN; \]
but BD : AC = CO : DO;
therefore BD : Dg : AC = CO : IN.
Therefore the sum of all the rectangles BD.Dg is to the sum of all the rectangles AC.gn, as CO to IN; but the sum of the rectangles BD.Dg is the space ACDB; and, because AC is given, the sum of the rectangles AC.qn is the rectangle of AC and the sum of all the lines qn; that is, the rectangle of AC and RL; therefore the space ACDB = AC.RL = CO : IN, and ACDB × IN = AC.CO.RL; and therefore IN : RL = AC.CO : ACDB.
Hence it follows that QL expresses the area BVA, and in general, that the part of the line parallel to OM, which lies between the tangent KN and the curve LρK, expresses the corresponding area of the hyperbola which lies without the rectangle BDQG.
And now, by the help of this curve, we have an easy way of convincing and computing the motion of a body through the air. For the subtangent of our curve now represents twice the height through which the ball must fall in vacuo, in order to acquire the terminal velocity; and therefore serves for a scale on which to measure all the other representatives of the motion.
But it remains to make another observation on the curve LρK, which will save us all the trouble of graphical operations, and reduce the whole to a very simple arithmetical computation. It is of such a nature, that when MI is considered as the abscissa, and is divided into a number of equal parts, and ordinates are drawn from the points of division, the ordinates are a series of lines in geometrical progression, or are continual proportionals. Whatever is the ratio between the first and second ordinate, there is the same between the second and third, between the third and fourth, and so on; therefore the number of parts into which the abscissa is divided is the number of these equal ratios which is contained in the ratio of the first ordinate to the last: For this reason, this curve has got the name of the logistic or logarithmic curve; and it is of immense use in the modern mathematics, giving us the solution of many problems in the most simple and expeditious manner, on which the genius of the ancient mathematicians had been exercised in vain. Few of our readers are ignorant, that the numbers called logarithms are of equal utility in arithmetical operations, enabling us not only to solve common arithmetical problems with astonishing dispatch, but also to solve others which are quite inaccessible in any other way. Logarithms are nothing more than the numerical measures of the abscissa of this curve, corresponding to ordinates, which are measured on the same or any other scale by the natural numbers; that is, if ML be divided into equal parts, and from the points of division lines be drawn parallel to MI, cutting the curve LρK, and from the points of intersection ordinates be drawn to MI, these will divide MI into portions, which are in the same proportion to the ordinates that the logarithms bear to their natural numbers.
In constructing this curve we were limited to no particular length of the line LR, which represented the space ACDB; and all that we had to take care of was, that when OC, Oξ, Oγ were taken in geometrical progression, Mξ, Mγ should be in arithmetical progression. The abscissa having ordinates equal to p, n, etc., might have been twice as long, as is shown in the dotted curve which is drawn through L. All the lines which serve to measure the hyperbolic spaces would then have been doubled. But NI would also have been doubled, and our proportions would have still held good; because this subtangent is the scale of measurement of our figure, as 1: or 2: is the scale of measurement for the motions.
Since then we have tables of logarithms calculated for every number, we may make use of them instead of this geometrical figure, which still requires considerable trouble to suit it to every case. There are two sets of logarithmic tables in common use. One is called a table of hyperbolic or natural logarithms. It is suited to such a curve as is drawn in the figure, where the subtangent is equal to that ordinate τv which corresponds to the side πO of the square πO being between the hyperbola and its asymptotes. This square is the unit of surface, by which the hyperbolic areas are expressed; its side is the unit of length, by which the lines belonging to the hyperbola are expressed; τv is = 1, or the unit of numbers to which the logarithms are suited, and then IN is also 1. Now the square πO being unity, the area BACD will be some number; πO being also unity, OD is some number: Call it x. Then, by the nature of the hyperbola, OB : Oπ = πθ : DB: That is, x : 1 = 1 : 1/x, so that DB is 1/x.
Now calling Dd the area BD db, which is the fluxion (ultimately) of the hyperbolic area, is x/x. Now in the curve LρK, MI has the same ratio to NI that BACD has to πOπ: Therefore, if there be a scale of which NI is the unit, the number on this scale corresponding to MI has the same ratio to 1 which the number measuring BACD has to 1; and this, which corresponds to BD db, is the fluxion (ultimately) of MI: Therefore, if MI be called the logarithm of x, x/x is properly represented by the fluxion of MI. In short, the line MI is divided precisely as the line of numbers on a Gunter's scale, which is therefore a line of logarithms; and the numbers called logarithms are just the lengths of the different parts of this line measured on a scale of equal parts. Therefore, when we meet with such an expression as x/x viz. the fluxion of a quantity divided by the quantity itself, we consider it as the fluxion of the logarithm of that quantity, because it is really so when the quantity is a number; and it is therefore strictly true that the fluent of x/x is the hyperbolic logarithm of x.
Certain reasons of convenience have given rise to another set of logarithms; these are suited to a logistic curve whose subtangent is only 414349 of the ordinate τv, which is equal to the side of the hyperbolic square, and which is assumed for the unit of number. We shall suit our applications of the preceding investigation to both these, and shall first use the common logarithms whose subtangent is 414349.
The whole subject will be best illustrated by taking an example of the different questions which may be proposed.
Recollect that the rectangle ACOF is = 2a, or a²/g, or F, For a ball of cast-iron one inch diameter, and if it has the diameter \(d\), it is \(\frac{u^2d}{g}\), or \(2ad\), or \(Ed\).
I. It may be required to determine what will be the space described in a given time \(t\) by a ball setting out with a given velocity \(V\), and what will be its velocity \(v\) at the end of that time.
Here we have \(NI : MI = ACOF : BDCA\); now \(NI\) is the subtangent of the logistic curve; \(MI\) is the difference between the logarithms of \(OD\) and \(OC\); that is, the difference between the logarithms of \(e+t\) and \(e\);
ACOF is \(2ad\), or \(\frac{u^2d}{g}\), or \(Ed\).
Therefore by common logarithms \(0.43429 : \log_e + t - \log_e = 2ad : S\), space described,
or \(0.43429 : \log_e \frac{e+t}{e} = 2ad : S\),
and \(S = \frac{2ad}{0.43429} \times \log_e \frac{e+t}{e}\),
by hyperbolic logarithms \(S = 2ad \times \log_e \frac{e+t}{e}\).
Let the ball be a 12 pounder, and the initial velocity be 1600 feet, and the time 20 seconds. We must first find \(e\), which is \(\frac{2ad}{V}\).
Therefore, \(\log_2a = 3.03236\)
\[ \begin{align*} \log_d(4.5) & = 0.65321 \\ \log_V(1600) & = 3.20415 \\ \end{align*} \]
Log. of \(3''03 = e = 0.48145\)
And \(e+t\) is \(23''03\), of which the log. is \(1.36229\), from which take the log. of \(e = 0.48145\)
remains the log. of \(\frac{e+t}{e} = 0.88084\)
This must be considered as a common number by which we are to multiply \(\frac{2ad}{0.43429}\).
Therefore add the logarithms of \(2ad = 3.68557\)
\[ \begin{align*} \log_e \frac{e+t}{e} & = 9.94490 \\ \log_0.43429 & = 9.63778 \\ \end{align*} \]
Log. \(S = 9833\) feet \(= 3.99269\)
For the final velocity,
\(OD : OC = AC : BD\), or \(e+t : e = V : v\).
The ball has therefore gone 3278 yards, and its velocity is reduced from 1600 to 210.
It may be agreeable to the reader to see the gradual progress of the ball during some seconds of its motion.
| T | S | Diff. | V | Diff. | |---|---|------|---|------| | 1'' | 1383 | 1073 | 1203 | 397 | | 2'' | 2456 | 880 | 964 | 160 | | 3'' | 3336 | 744 | 804 | 114 | | 4'' | 4080 | 645 | 690 | 86 | | 5'' | 4725 | 569 | 624 | 67 | | 6'' | 5294 | 537 |
The first column is the time of the motion, the second is the space described, the third is the differences of the spaces, showing the motion during each successive second; the fourth column is the velocity at the end of the time \(t\); and the last column is the differences of velocity, showing its diminution in each successive second.
We see that at the distance of 1000 yards the velocity is reduced to one half, and at the distance of less than a mile it is reduced to one-third.
II. It may be required to determine the distance at which the initial velocity \(V\) is reduced to any other quantity \(v\). This question is solved in the very same manner, by substituting the logarithms of \(V\) and \(v\) for those of \(e+t\) and \(e\); for \(AC : BD = OD : OC\), and therefore \(\log_e \frac{AC}{BD} = \log_e \frac{OD}{OC}\), or \(\log_e \frac{V}{v} = \log_e \frac{e+t}{e}\).
Thus it is required to determine the distance in which the velocity 1780 of a 24 pound ball (which is the medium velocity of such a ball discharged with 16 pounds of powder) will be reduced to 1500.
Here \(d\) is 5.68, and therefore the logarithm of \(2ad\) is \(+3.78671\)
\[ \begin{align*} \log_V & = 0.07433, \text{ of which the log. is } +8.87116 \\ \log_0.43429 & = -9.63778 \\ \end{align*} \]
Log. 1047.3 feet, or 349 yards \(= 3.02009\)
This reduction will be produced in about \(\frac{1}{4}\) of a second.
III. Another question may be to determine the time which a ball, beginning to move with a certain velocity, employs in passing over a given space, and the diminution of velocity which it sustains from the resistance of the air.
We may proceed thus:
\[2ad : S = 0.43429 : \log_e \frac{e+t}{e}, = t.\] Then to log.
\[\frac{e+t}{e} \text{ add log. } e,\text{ and we obtain log. } e+t,\text{ and } e+t;\text{ from which if we take } e \text{ we have } t.\] Then to find \(v\), say \(e+t : e = V : v\).
We shall conclude these examples by applying this last rule to Mr Robins's experiment on a musket bullet of an expense of \(\frac{1}{4}\) of an inch in diameter, which had its velocity reduced from 1670 to 1425 by passing through 100 feet of air. This we do in order to discover the resistance which it sustained, and compare it with the resistance to a velocity of 1 foot per second.
We must first ascertain the first term of our analogy.
The ball was of lead, and therefore \(2a\) must be multiplied by \(d\) and by \(m\), which expresses the ratio of the density of lead to that of cast-iron. \(d\) is 0.75, and \(m\) is \(\frac{11.37}{7.21} = 1.577\). Therefore \(\log_2a = 3.03256\)
\[ \begin{align*} d & = 9.87506 \\ m & = 0.19782 \\ \end{align*} \]
Log. \(2adm = 3.10524\)
and \(2adm = 1274.2\).
Now \(1274.2 : 100 = 0.43429 : 0.3408 = \log_e \frac{e+t}{e}\).
But \(e = \frac{2adm}{V} = 0.763\), and its logarithm \(= 9.88252\),
which, added to 0.3408, gives 9.21660, which is the log. of \(e+t = 0.825\), from which take \(e\), and there remains followed the proportion of the hyperbolic areas, we showed the nature of another curve, where lines could be found which increase in the very same manner as the path of the projectile increases; so that a point describing the abscissa MI of this curve moves precisely as the projectile does. Then, discovering that this line is the same with the line of logarithms on a Gunter's scale, we showed how the logarithm of a number really represents the path or space described by the projectile.
Having thus, we hope, enabled the reader to conceive distinctly the quantities employed, we shall leave the geometrical method, and prosecute the rest of the subject in a more compendious manner.
We are, in the next place, to consider the perpendicular ascents and descents of heavy projectiles, where the resistance of the air is combined with the action of gravity: and we shall begin with the descents.
Let \( u \), as before, be the terminal velocity, and \( g \) the accelerating power of gravity: When the body moves with the velocity \( u \), the resistance is equal to \( g \); and in every other velocity \( v \), we must have \( u^2 : v^2 = g : \frac{g}{u^2} \), \( r \), for the resistance to that velocity. In the descent the body is urged by gravity \( g \), and opposed by the resistance \( \frac{g}{u^2} \): therefore the remaining accelerating force, which we shall call \( f \), is \( g - \frac{g}{u^2} \), or \( \frac{g}{u^2} - \frac{g}{v^2} \), or \( \frac{g(u^2 - v^2)}{u^2} \), \( f \).
Now the fundamental theorem for varied motions is
\[ f = u \dot{v}, \quad \text{and} \quad \dot{s} = \frac{v \dot{v}}{f}, \quad \text{or} \quad \frac{u^3}{g} \times \frac{v \dot{v}}{u^2 - v^2}, \quad \text{and} \quad s = \frac{u^3}{g} \times \int \frac{v \dot{v}}{u^2 - v^2} + C. \]
Now the fluent of \( \frac{v \dot{v}}{u^2 - v^2} \) is \( = \) hyperbolic log. of \( \sqrt{\frac{u^2}{u^2 - v^2}} \). For the fluxion of \( \sqrt{\frac{u^2}{u^2 - v^2}} \) is \( \frac{v \dot{v}}{u^2 - v^2} \), and this divided by the quantity \( \sqrt{\frac{u^2}{u^2 - v^2}} \), of which it is the fluxion, gives precisely \( \frac{v \dot{v}}{u^2 - v^2} \), which is therefore the fluxion of its hyperbolic logarithm. Therefore \( S = \frac{u^3}{g} \times L \sqrt{\frac{u^2}{u^2 - v^2}} + C \). Where \( L \) means the hyperbolic logarithm of the quantity annexed to it, and \( \lambda \) may be used to express its common logarithm. (See article Fluxions.)
The constant quantity \( C \) for completing the fluent is determined from this consideration, that the space described is \( o \), when the velocity is \( o \): therefore \( C = \frac{u^3}{g} \times L \sqrt{\frac{u^2}{u^2 - v^2}} = o \), and \( C = \frac{u^3}{g} \times L \sqrt{\frac{u^2}{u^2 - v^2}} \), and the complete fluent \( S = \frac{u^3}{g} \times L \sqrt{\frac{u^2}{u^2 - v^2}} - L \sqrt{\frac{u^2}{u^2 - v^2}} \),
\[ = \frac{u^3}{g} \times L \sqrt{\frac{u^2}{u^2 - v^2}} = \frac{u^3}{g} \times \lambda \sqrt{\frac{u^2}{u^2 - v^2}}, \]
or (putting \( M \) for \( \frac{u^3}{g} \times \lambda \sqrt{\frac{u^2}{u^2 - v^2}} \)).
This This equation establishes the relation between the space fallen through, and the velocity acquired by the fall. We obtain by it \( \frac{gS}{u^2} = L \sqrt{\frac{u^2}{u^2 - v^2}} \), and
\[ \frac{2gS}{u^2} = L \cdot \frac{u^2}{u^2 - v^2}, \text{ or, which is still more convenient for us, } \frac{M \times 2gS}{u^2} = \lambda \cdot \frac{u^2}{u^2 - v^2}, \]
that is, equal to the logarithm of a certain number; therefore having found the natural number corresponding to the fraction \( \frac{M \times 2gS}{u^2} \), consider it as a logarithm, and take out the number corresponding to it: call this \( n \). Then, since \( n \) is equal to \( \frac{u^2}{u^2 - v^2} \), we have \( n u^2 - n v^2 = u^2 \), and \( n u^2 - u^2 = u v^2 \), or \( n v^2 = u^2 \times n - v \), and \( v^2 = \frac{u^2 \times n - 1}{n} \).
To expedite all the computations on this subject, it will be convenient to have multipliers ready computed for \( M \times 2g \), and its half,
viz. 27,794, whose log. is 1.44396 and 13,897 1.14293
But \( v \) may be found much more expeditiously by observing that \( \sqrt{\frac{u^2}{u^2 - v^2}} \) is the secant of an arch of a circle whose radius is \( u \), and whose fine is \( v \), or whose radius is unity and fine \( \frac{v}{u} \): therefore, considering the above fraction as a logarithmic secant, look for it in the tables, and then take the fine of the arc of which this is the secant, and multiply it by \( u \); the product is the velocity required.
We shall take an example of a ball whose terminal velocity is 689 feet, and ascertain its velocity after a fall of 1848 feet. Here,
\[ \begin{align*} u &= 475200 \\ g &= 32 \\ S &= 1848 \\ \end{align*} \]
Then log. 27,794 + 1.44396
\[ \begin{align*} \log. S &= 1.14293 \\ \log. u^2 &= 5.67688 \\ \end{align*} \]
Log. of 0.10809 = log. \( n \) = 9.03378
\( 0.10809 \) is the logarithm of 1,2826 = \( n \), and \( n - 1 = 0.2826 \), and \( \frac{u^2 \times n - 1}{n} = 323.62 = v^2 \), and \( v = 323.6 \).
In like manner, 0.054045 (which is half of 0.10809) will be found to be the logarithmic secant of 28°, whose fine 0.46947 multiplied by 689 gives 324 for the velocity.
The process of this solution suggests a very perspicuous manner of conceiving the law of descent; and it may be thus expressed:
\( M \) is to the logarithm of the secant of an arch whose fine is \( \frac{v}{u} \) and radius \( r \), as 2a is to the height through which the body must fall in order to acquire the velocity \( v \). Thus, to take the same example.
1. Let the height \( h \) be sought which will produce the velocity 323.62, the terminal velocity of the ball being 689.44. Here 2a, or \( \frac{u^2}{g} \) is 14850, and 323.62 = 0.46947, which is the fine of 28°. The logarithmic secant of this arch is 0.05407. Now \( M \) or 0.43429; 0.05407 = 14850 : 1848, the height wanted.
2. Required the velocity acquired by the body by falling 1848 feet. Say 14850 : 1848 = 0.43429 : 0.05407. Look for this number among the logarithmic secants. It will be found at 28°, of which the logarithmic line is 9.67161
Add to this the log. of \( u \) 2.83844
The sum 2.51005 is the logarithm of 323.62, the velocity required.
We may observe, from these solutions, that the acquired velocity continually approaches to, but never equals, the terminal velocity. For it is always expressed by the fine of an arch of which the terminal velocity is the radius. We cannot help taking notice here of a very strange assertion of Mr Muller, late professor of mathematics and director of the royal academy at Woolwich. He maintains, in his Treatise on Gunnery, his Treatise of Fluxions, and in many of his numerous works, that a body cannot possibly move through the air with a greater velocity than this; and he makes this a fundamental principle, on which he establishes a theory of motion in a resisting medium, which he affirms with great confidence to be the only just theory; saying, that all the investigations of Bernoulli, Euler, Robins, Simpson, and others, are erroneous. We use this strong expression, because, in his criticisms on the works of those celebrated mathematicians, he lays aside good manners, and taxes them not only with ignorance, but with dishonesty; saying, for instance, that it required no small dexterity in Robins to confirm by his experiments a theory founded on false principles; and that Thomas Simpson, in attempting to conceal his obligations to him for some valuable propositions, by changing their form, had ignorantly fallen into gross errors.
Nothing can be more palpably absurd than this assertion of Mr Muller. A blown bladder will have but a small terminal velocity; and when moving with this velocity, or one very near it, there can be no doubt that it will be made to move much swifter by a smart stroke. Were the assertion true, it would be impossible for a portion of air to be put into motion through the rest, for its terminal velocity is nothing. Yet this author makes this assertion a principle of argument, saying, that it is impossible that a ball can issue from the mouth of a cannon with a greater velocity than this; and that Robins and others are grossly mistaken, when they give them velocities three or four times greater, and resistances which are 10 or 20 times greater than is possible; and by thus compensating his small velocities by still smaller resistances, he confirms his theory by many experiments adduced in support of the others. No reason whatever can be given for the assertion. Newton, or perhaps Huygens, was the first who observed that there was a limit to the velocity which gravity could communicate to a body; and this limit was found by his commentators to be a term to which it was vastly convenient to refer all its other motions. It therefore became became an object of attention; and Mr Muller, through inadvertency, or want of discernment, has fallen into this mistake, and with that arrogance and self-conceit which mark all his writings, has made this mistake a fundamental principle, because it led him to establish a novel set of doctrines on this subject. He was fretted at the superior knowledge and talents of Mr Simpson, his inferior in the academy, and was guilty of several mean attempts to hurt his reputation. But they were unsuccessful.
We might proceed to consider the motion of a body projected downwards. While the velocity of projection is less than the terminal velocity, the motion is determined by what we have already laid; for we must compute the height necessary for acquiring this velocity in the air, and suppose the motion to have begun there. But if the velocity of projection be greater, this method fails. We pass it over (though not in the least more difficult than what has gone before), because it is of mere curiosity, and never occurs in any interesting case. We may just observe, that since the motion is lighter than the terminal velocity, the resistance must be greater than the weight, and the motion will be retarded. The very same process will give us for the space described \( S = \frac{u^2}{g} \times L \sqrt{\frac{V^2 - u^2}{v^2 - u^2}} \), \( V \) being the velocity of projection, greater than \( u \). Now as this space evidently increases continually (because the body always falls), but does not become infinite in any finite time, the fraction \( \frac{V^2 - u^2}{v^2 - u^2} \) does not become infinite; that is, \( v^2 \) does not become equal to \( u^2 \); therefore although the velocity \( V \) is continually diminished, it never becomes so small as \( u \). Therefore \( u \) is a limit of diminution as well as of augmentation.
We must now ascertain the relation between the time of descent and the space described, or the velocity acquired. For this purpose we may use the other fundamental proposition of varied motions \( f' = v \), which, in the present case, becomes \( \frac{gu'}{v^2} = \frac{v}{u} \); therefore \( i = \frac{u^2}{g} \times \frac{v}{u^2 - v^2} = \frac{u}{g} \times \frac{v}{u^2 - v^2} \), and \( i = \frac{u}{g} \times \int \frac{v}{u^2 - v^2} \).
Now (art. Fluxions) \( \int \frac{v}{u^2 - v^2} = L \sqrt{\frac{u + v}{u - v}} \). Therefore \( i = \frac{u}{g} \times L \sqrt{\frac{u + v}{u - v}} = \frac{u}{g} \times \lambda \sqrt{\frac{u + v}{u - v}} \). This fluent needs no constant quantity to complete it, or rather \( C = 0 \); for \( t \) must be \( = 0 \) when \( v = 0 \). This will evidently be the case: for then \( L \sqrt{\frac{u + v}{u - v}} = L \sqrt{\frac{u}{u}} = L \times 1 = 0 \).
But how does this quantity \( \frac{u}{Mg} \times \lambda \sqrt{\frac{u + v}{u - v}} \) signify a time? Observe, that in whatever numbers, or by whatever units of space and time, \( u \) and \( g \) are expressed, \( v \) expresses the number of units of time in which the velocity \( u \) is communicated or extinguished by gravity; and \( L \sqrt{\frac{u + v}{u - v}} \) or \( M \sqrt{\frac{u + v}{u - v}} \) is always an abstract number, multiplying this time.
We may illustrate this rule by the same example. In what time will the body acquire the velocity \( 323.62 \)? Here \( u + v = 1012.96 \), \( u - v = 365.72 \); therefore \( \lambda \sqrt{\frac{u + v}{u - v}} = 0.22122 \), and \( u \) (in feet and seconds) is \( 21''542 \). Now, for greater perspicuity, convert the equation \( i = \frac{u}{Mg} \times \lambda \sqrt{\frac{u + v}{u - v}} \) into a proportion; thus
\[ M : \lambda \sqrt{\frac{u + v}{u - v}} : : t : 4, \text{ and we have } 0.43429 : 0.22122 = 21''542 : 10''973, \text{ the time required.} \]
This is by far the most distinct way of conceiving the subject; and we should always keep in mind that the numbers or symbols which we call logarithms are really parts of the line \( MI \) in the figure of the logistic curve, and that the motion of a point in this line is precisely similar to that of the body. The Marquis Poleni, in a dissertation published at Padua in 1725, has with great ingenuity constructed logarithmics fitted to all the cases which can occur. Herman, in his Phoronomia, has borrowed much of Poleni's methods, but has obscured them by an affectation of language geometrically precise, but involving the very obscure notion of abstract ratios.
It is easy to see that \( \sqrt{\frac{u + v}{u - v}} \) is the cotangent of the \( \frac{\pi}{2} \) complement of an arch, whose radius is \( r \), and whose sine is \( \frac{v}{u} \): For let \( KC \) (fig. 6.) be \( = u \), and \( BE = v \); then \( KD = u + v \), and \( DA = u - v \). Join \( KB \) and \( BA \), and draw \( CG \) parallel to \( KB \). Now \( GA \) is the tangent of \( \frac{\pi}{2} \) \( BA \), \( = \) complement of \( HB \). Then, by similarity of triangles, \( GA : AC = AB : BK = \sqrt{AD} : \sqrt{DK} = \sqrt{u - v} : \sqrt{u + v} \) and \( \frac{AC}{GA} = \cotan. \)
\[ \frac{BA}{\sqrt{u - v}} ; \text{ therefore look for } \frac{v}{u} \text{ among the natural sines, or for } \log. \frac{v}{u} \text{ among the logarithmic sines, and take the logarithmic cotangent of the half complement of the corresponding arch. This, considered as a common number, will be the second term of our proportion. This is a shorter process than the former.} \]
By reverting this proportion we get the velocity corresponding to a given time.
To compare this descent of 1848 feet in the air Fall of a with the fall of the body in vacuo during the same body in time, say \( 21''542 : 10''973 = 1848 : 1926.6 \), which makes a difference of 79 feet.
Cor. 1. The time in which the body acquires the velocity \( u \) by falling through the air, is to the time of acquiring the same velocity by falling in vacuo, as \( u \) \( L \sqrt{\frac{u + v}{u - v}} \) to \( v \); for it would acquire this velocity in vacuo. vacuo during the time \( \frac{v}{g} \), and it acquires it in the air in the time \( \frac{u}{g} \times \sqrt{\frac{u+v}{u-v}} \).
2. The velocity which the body acquires by falling through the air in the time \( \frac{u}{g} \times \sqrt{\frac{u+v}{u-v}} \) is to the velocity which it would acquire in vacuo during the same time, as \( v \) to \( u \times \sqrt{\frac{u+v}{u-v}} \). For the velocity which it would acquire in vacuo during the time \( \frac{u}{g} \)
\[ L \times \sqrt{\frac{u+v}{u-v}} \text{ must be } u \times L \times \sqrt{\frac{u+v}{u-v}} \text{ (because in any time } \frac{w}{g} \text{ the velocity } w \text{ is acquired.)} \]
In the next place, let a body, whose terminal velocity is \( u \), be projected perpendicularly upwards, with any velocity \( V \). It is required to determine the height to which it ascends, so as to have any remaining velocity \( v \), and the time of its ascent; as also the height and time in which its whole motion will be extinguished.
We have now \( g(u^2+v^2) \) for the expression of \( f \); for both gravity and resistance act now in the same direction, and retard the motion of the ascending body:
\[ \text{therefore } \frac{g(u^2+v^2)}{u^2} = -v \cdot \dot{v}, \text{ and } s = \frac{-u^2}{g} \times \frac{uv}{u^2+v^2}, \]
and \( s = \frac{-u^2}{g} \times \frac{f \cdot uv}{u^2+v^2} + C = \frac{-u^2}{g} \times L \sqrt{u^2+v^2} + C \) (see art. Fluxions). This must be \( = 0 \) at the beginning of the motion, that is, when \( v = V \), that is,
\[ \frac{-u^2}{g} \times L \sqrt{u^2+V^2} + C = 0, \text{ or } C = \frac{-u^2}{g} \times L \sqrt{u^2+V^2}, \]
and the complete fluent will be
\[ s = \frac{-u^2}{g} \times L \sqrt{u^2+V^2} - \frac{L \sqrt{u^2+V^2}}{u^2} = \frac{-u^2}{g} \times L \sqrt{\frac{u^2+V^2}{u^2}} = \frac{-u^2}{g} \times \frac{Mg}{\lambda} \times \sqrt{\frac{u^2+V^2}{u^2}}. \]
Let \( h \) be the greatest height to which the body will rise. Then \( s = h \) when \( v = 0 \); and \( h = \frac{-u^2}{g} \times L \sqrt{\frac{u^2+V^2}{u^2}} \).
We have
\[ \lambda \sqrt{\frac{u^2+V^2}{u^2}} = \frac{mg}{u^2}, \text{ therefore } \lambda \left( \frac{u^2+V^2}{u^2} \right) = \frac{2Mgs}{u^2}. \]
Therefore let \( n \) be the number whose common logarithm is \( \frac{2Mgs}{u^2} \); we shall have \( n = \frac{u^2+V^2}{u^2} \), and \( v^2 = \frac{u^2+V^2}{n} - u^2 \); and thus we obtain the relation of \( s \) and \( v \), as in the case of descents: but we obtain it still easier by observing that \( \sqrt{u^2+V^2} \) is the secant of an arch whose radius is \( u \), and whose tangent is \( V \), and that \( \sqrt{u^2+v^2} \) is the secant of another arch of the same circle, whose tangent is \( v \).
Let the same ball be projected upwards with the velocity 411.05 feet per second. Required the whole height to which it will rise?
Here \( \frac{V}{u} \) will be found the tangent of 30.48°, the logarithmic secant of which is 0.6606. This, multiplied by \( \frac{u^2}{Mg} \) gives 2259 feet for the height. It would have risen 2640 feet in a void.
Suppose this body to fall down again. We can compare the velocity of projection with the velocity projection with which it again reaches the ground. The ascent compared with that descent are equal: therefore \( \sqrt{\frac{u^2+V^2}{u^2}} \), which it reaches the ground.
\[ \sqrt{\frac{u^2}{u^2-v^2}} \text{ the multiplier in the descent. The first is the secant of an arch whose tangent is } V; \text{ the other is the secant of an arch whose sine is } v. \text{ These secants are equal, or the arches are the same; therefore the velocity of projection is to the final returning velocity as the tangent to the sine, or as the radius to the cosine of the arch. Thus suppose the body projected with the terminal velocity, or } V = u; \text{ then } v = \frac{u}{\sqrt{2}}. \text{ If } V = 689, v = 487. \]
We must in the last place ascertain the relation of the space and the time.
Here \( \frac{g(u^2+v^2)}{u^2} = -v \cdot \dot{v}, \text{ and } t = \frac{-u^2}{g} \times \frac{uv}{u^2+v^2} = \frac{-u^2}{g} \times \frac{uv}{u^2+v^2} + C. \)
Now \( \frac{f \cdot uv}{u^2+v^2} \) is an arch whose tangent is \( \frac{v}{u} \) and radius 1; therefore \( t = \frac{-u}{g} \times \text{arc. tan. } \frac{v}{u} + C. \)
This must be \( = 0 \) when \( v = V \), or \( C = \frac{-u}{g} \times \text{arc. tan. } \frac{V}{u} = 0, \text{ and } C = \frac{-u}{g} \times \text{arc. tan. } \frac{V}{u} \), and the complete fluent is \( t = \frac{-u}{g} \times \left( \text{arc. tan. } \frac{V}{u} - \text{arc. tan. } \frac{v}{u} \right) \). The quantities within the brackets express a portion of the arch of a circle whose radius is unity; and are therefore abstract numbers, multiplying \( \frac{u}{g} \), which we have shown to be the number of units of time in which a heavy body falls in vacuo from the height \( a \), or in which it acquires the velocity \( u \).
We learn from this expression of the time, that however great the velocity of projection, and the height ascended, to which this body will rise, may be, the time of its ascent is limited. It never can exceed the time of falling from the height \( a \) in vacuo in a greater proportion than that of a quadrantal arch to the radius, nearly the proportion of 8 to 5. A 24 pound iron ball cannot continue rising above 14 seconds, even if the resistance to quick motions did not increase faster than the square of the velocity. It probably will attain its greatest height in less than 12 seconds, let its velocity be ever so great.
In the preceding example of the whole ascent, \( v = 0 \), and... and the time \( t = \frac{u}{g} \times \text{arc. tan. } \frac{V}{u}, \) or \( \frac{u}{g} \times \text{arc. } 30^\circ.48^\prime. \)
Now \( 30^\circ.48^\prime = 1848^\prime, \) and the radius \( r \) contains 3438; therefore the arch \( = \frac{1848}{3438} = 0.5376; \) and \( \frac{u}{g} = 21^\prime.54^\prime. \)
Therefore \( t = 21^\prime.54^\prime \times 0.5376 = 11^\prime.58, \) or nearly \( 11\frac{1}{2} \) seconds. The body would have risen to the same height in a void in \( 10\frac{1}{2} \) seconds.
Cor. 1. The time in which a body, projected in the air with any velocity \( V, \) will attain its greatest height, is that in which it would attain its greatest height in vacuo, as the arch whose tangent expresses the velocity is to the tangent; for the time of the ascent in the air is \( \frac{u}{g} \times \text{arch}; \) the time of the ascent in vacuo is \( \frac{V}{g}. \) Now
\[ \frac{V}{u} = \tan, \] and \( V = u \times \tan. \) and \( \frac{V}{g} = \frac{u}{g} \times \tan. \]
It is evident, by inspecting fig. 6, that the arch \( AI \) is to the tangent \( AG \) as the sector \( ICA \) to the triangle \( GCA; \) therefore the time of attaining the greatest height in the air is to that of attaining the greatest height in vacuo (the velocities of projection being the same), as the circular sector to the corresponding triangle.
If therefore a body be projected upwards with the terminal velocity, the time of its ascent will be to the time of acquiring this velocity in vacuo as the area of a circle to the area of the circumscribed square.
2. The height \( H \) to which a body will rise in a void, is to the height \( h \) to which it would rise through the air when projected with the same velocity \( V \) as \( M \cdot V^2 \)
\[ u^2 \times \lambda \frac{u^2 + V^2}{u^2} : \text{for the height to which it will rise in vacuo is } \frac{V^2}{2g}, \text{ and the height to which it rises in the air is } \frac{u^2}{Mg} \times \sqrt{\frac{u^2 + V^2}{u^2}}; \text{ therefore } H : h = \frac{V^2}{2g}: \]
\[ \frac{u^2}{Mg} \times \sqrt{\frac{u^2 + V^2}{u^2}}, = V^2 : \frac{u^2}{M} \times 2 \lambda \sqrt{\frac{u^2 + V^2}{u^2}}, = V^2 : \]
\[ \frac{u^2}{M} \times \lambda \frac{u^2 + V^2}{u^2}, = M \cdot V^2 : u^2 \times \lambda \frac{u^2 + V^2}{u^2}. \]
Therefore if the body be projected with its terminal velocity, so that \( V = u, \) the height to which it will rise in the air is \( \frac{30103}{43429} \) of the height to which it will rise in vacuo, or \( \frac{5}{7} \) in round numbers.
We have been thus particular in treating of the perpendicular ascents and descents of heavy bodies through the air, in order that the reader may conceive distinctly the quantities which he is thus combining in his algebraic operations, and may see their connection in nature with each other. We shall also find that, in the present state of our mathematical knowledge, this simple state of the case contains almost all that we can determine with any confidence. On this account it were to be wished that the professional gentlemen would make many experiments on these motions. There is no way that promises so much for afflicting us in forming accurate notions of the air's resistance. Mr. Robin's method with the pendulum is impracticable with great force; and the experiments which have been generally referred to for this purpose, viz. the ranges of shot and shells on a horizontal plane, are so complicated in themselves, that the utmost mathematical skill is necessary for making any inferences from them; and they are subject to such irregularities, that they may be brought to support almost any theory whatever on this subject. But the perpendicular flights are affected by nothing but the initial velocity and the resistance of the air; and a considerable deviation from their intended direction does not cause any sensible error in the consequences which we may draw from them for our purpose.
But we must now proceed to the general problem, of observing the motion of a body projected in any direction, and with any velocity. Our readers will be justified beforehand that this must be a difficult subject, when they see the simplest cases of rectilineal motion abundantly abstruse: it is indeed so difficult, that Sir Isaac Newton has not given a solution of it, and has thought himself well employed in making several approximations, in which the fertility of his genius appears solved by Newton in great lustre. In the tenth and subsequent propositions of the second book of the Principia, he shows what state of density in the air will comport with the motion of a body in any curve whatever: and then, by applying this discovery to several curves which have some similarity to the path of a projectile, he finds one which is not very different from what we may suppose to obtain in our atmosphere. But even this approximation was involved in such intricate calculations, that it seemed impossible to make any use of it. In the second edition of the Principia, published in 1713, Newton corrects some mistakes which he had committed in the first, and carries his approximations much farther, but still does not attempt a direct investigation of the path which a body will describe in our atmosphere. This is somewhat surprising. In prop. 14, &c., he shows how a body, acted upon by a centripetal force, in a medium of a density varying according to certain laws, will describe an eccentric spiral, of which he assigns the properties, and the law of description. Had he supposed the density constant, and the difference between the greatest and least distances from the centre of centripetal force exceeding infinitely small in comparison with the distances themselves, his spiral would have coincided with the path of a projectile in the air of uniform density, and the steps of his investigation would have led him immediately to the complete solution of the problem. For this is the real state of the case. A heavy body is not acted on by equal and parallel gravity, but by a gravity inversely proportional to the square of the distance from the centre of the earth, and in lines tending to that centre nearly; and it was with the view of simplifying the investigation, that mathematicians have adopted the other hypothesis.
Soon after the publication of this second edition of the Principia, the dispute about the invention of the fluxionary calculus became very violent, and the great British promoters of that calculus upon the continent were in foreign habit of proposing difficult problems to exercise the talents of the mathematician. Challenges of this kind frequently passed between the British and foreigners. Dr Keill of Oxford had keenly espoused the claim of Sir Isaac Newton to this invention, and had engaged in a very acrimonious altercation with the celebrated John Bernoulli of Batle. Bernoulli had published in the Acta Eruditorum Lippsiae an investigation of the law of forces, by which a body moving in a resisting medium might describe any proposed curve, reducing the whole to the simplest geometry. This is perhaps the most elegant specimen which he has given of his great talents. Dr Keill propounded to him the particular problem of the trajectory and motion of a body moving through the air, as one of the most difficult. Bernoulli very soon solved the problem in a way much more general than it had been propounded, viz., without any limitation either of the law of resistance, the law of the centripetal force, or the law of density, provided only that they were regular, and capable of being expressed algebraically. Dr Brook Taylor, the celebrated author of the Method of Increments, solved it at the same time, in the limited form in which it was propounded. Other authors since that time have given other solutions. But they are all (as indeed they must be) the same in substance with Bernoulli's. Indeed they are all (Bernoulli's not excepted) the same with Newton's first approximations, modified by the steps introduced into the investigation of the spiral motions mentioned above; and we still think it most strange that Sir Isaac did not perceive that the variation of curvature, which he introduced in that investigation, made the whole difference between his approximations and the complete solution. This we shall point out as we go along. And now proceed to the problem itself, of which we shall give Bernoulli's solution, restricted to the case of uniform density and a resistance proportional to the square of the velocity.
This solution is more simple and perspicuous than any that has since appeared.
**Problem.** To determine the trajectory, and all the circumstances of the motion of a body projected through the air from A (fig. 7.) in the direction AB, and resisted in the duplicate ratio of the velocity.
Let the arch AM be put = z, the time of describing it t, the abscissa AP=x, the ordinate PM=y. Let the velocity in the point M=v, and let MN=z, be described in the moment t; let r be the resistance of the air, g the force of gravity, measured by the velocity which it will generate in a second; and let a be the height through which a heavy body must fall in vacuo to acquire the velocity which would render the resistance of the air equal to its gravity: so that we have \( r = \frac{v^2}{2a} \); because, for any velocity u, and producing height h, we have \( g = \frac{u^2}{2h} \).
Let MM touch the curve in M; draw the ordinate PN=m, and draw MO=o, ON perpendicular to NP and MM. Then we have MN=z, and MO=x, also mo is ultimately = y and MM is ultimately = MN or z. Lastly, let us suppose x to be a constant quantity, the elementary ordinates being supposed equidistant.
The action of gravity during the time t may be measured by mN, which is half the space which it would cause the body to describe uniformly in the time t with the velocity which it generates in that time. Let this be resolved into mN, by which it deflects the body into a curvilinear path, and mn, by which it retards the ascent and accelerates the descent of the body along the tangent. The resistance of the air acts solely in retarding the motion, both in ascending and descending, and has no deflective tendency. The whole action of gravity then is to its accelerating or retarding tendency as mN to mn, or (by similarity of triangles) as mM to mo. Or \( \frac{z}{y} = \frac{g}{\frac{2}{z}} \), and the whole retardation in the ascent will be \( r + \frac{gy}{z} \). The same fluxionary symbol will express the retardation during the descent, because in the descent the ordinates decrease, and \( y \) is a negative quantity.
The diminution of velocity is \( -\dot{v} \). This is proportional to the retarding force and to the time of its action jointly, and therefore \( -\dot{v} = r + \frac{gy}{z} \times \dot{t} \); but the time \( \dot{t} \) is as the space \( \dot{z} \) divided by the velocity \( v \); therefore \( -\dot{v} = r + \frac{gy}{z} \times \frac{\dot{z}}{v} = \frac{r + gy}{v} \), and \( -\dot{v} = \frac{r}{v} - \frac{gy}{2a} = \frac{v^2}{2a} - gy \). Because mN is the deflection by gravity, it is as the force \( g \) and the square of the time \( t \) jointly (the moment action being held as uniform). We have therefore mN, or \( -\dot{y} = g \dot{t} \). (Observe that mN is in fact only the half of \( -\dot{y} \); but \( g \) being twice the fall of a heavy body in a second, we have \( -\dot{y} \) strictly equal to \( g \dot{t} \)). But \( \dot{t} = \frac{\dot{z}}{\dot{v}} \); therefore \( -\dot{y} = \frac{g \dot{z}}{\dot{v}} \), and \( \dot{v} = \frac{g \dot{z}}{-\dot{y}} \), and \( -\dot{v} \dot{y} = g \dot{z} \). The fluxion of this equation is \( -\dot{v} \dot{y} - 2 \dot{v} \dot{y} = 2g \dot{z} \); but, because \( \dot{z} : \dot{y} = mM : mo = mN : mn = y : z \), we have \( \dot{z} = \dot{y} \). Therefore \( 2g \dot{y} = 2g \dot{z} = -\dot{v} \dot{y} - 2 \dot{v} \dot{y} \), and \( -\dot{v} \dot{y} = \frac{v^2}{2a} - gy \). But we have already \( -\dot{v} \dot{v} = \frac{v^2}{2a} - gy \); therefore \( \frac{v^2}{2a} = \frac{v^2}{2a} \), and finally \( \frac{y}{a} = \frac{z}{y} \), or \( ay = zy \), for the fluxionary equation of the curve.
If we put this into the form of a proportion, we have \( a : z = y : y \). Now this evidently establishes a relation between the length of the curve and its variation of curvature; and between the curve itself and its evolution, which are the very circumstances introduced by Newton. Newton into his investigation of the spiral motions. And the equation \( \frac{z}{a} = \frac{y}{b} \) is evidently an equation connected with the logarithmic curve and the logarithmic spiral. But we must endeavour to reduce it to a lower order of fluxions, before we can establish a relation between \( x \), \( y \), and \( z \).
Let \( p \) express the ratio of \( y \) to \( x \), that is, let \( p = \frac{y}{x} \), or \( p \cdot x = y \). It is evident that this expresses the inclination of the tangent at \( M \) to the horizon, and that \( p \) is the tangent of this inclination, radius being unity. Or it may be considered merely as a number, multiplying \( x \), so as to make it \( y \). We now have \( y^2 = p^2 \cdot x^2 \), and since \( x^2 = x^2 + y^2 \); we have \( x^2 = x^2 + p^2 \cdot x^2 = 1 + p^2 \cdot x^2 \), and \( x = \sqrt{1 + p^2} \cdot x \).
Moreover, because we have supposed the abscissa \( x \) to increase uniformly, and therefore \( x \) to be constant, we have \( y = x \cdot p \), and \( y = x \cdot p \). Now let \( q \) express the ratio of \( p \) to \( x \), that is, make \( \frac{p}{x} = q \), or \( q \cdot x = p \).
This gives us \( q \cdot x = p \), and \( x \cdot q = x \cdot p = y \).
By these substitutions our former equation \( ay = z \cdot y \) changes to \( a \cdot x^3 \cdot q = x \cdot \sqrt{1 + p^2} \cdot x \cdot p \), or \( a \cdot y = p \cdot \sqrt{1 + p^2} \), and, taking the fluent on both sides, we have
\[ a \cdot q = f \cdot p \cdot \sqrt{1 + p^2} + C, \]
\( C \) being the constant quantity required for completing the fluent according to the limiting conditions of the case. Now \( \frac{x}{q} = \frac{p}{q} \), and \( \frac{q}{x} = \frac{p}{q} \).
Also, since \( y = p \cdot x \), \( \frac{p}{q} \), we have \( y = f \cdot p \cdot \sqrt{1 + p^2} + C \).
Also \( z = x \cdot \sqrt{1 + p^2} = f \cdot p \cdot \sqrt{1 + p^2} + C \).
The values of \( x \), \( y \), \( z \), give us
\[ x = f \cdot p \cdot \sqrt{1 + p^2} + C, \] \[ y = f \cdot p \cdot \sqrt{1 + p^2} + C, \] \[ z = f \cdot p \cdot \sqrt{1 + p^2} + C. \]
The process therefore of describing the trajectory is \( \frac{1}{2} \). To find \( q \) in terms of \( p \) by the area of the curve whose abscissa is \( p \) and the ordinate is \( \sqrt{1 + p^2} \).
We get \( x \) by the area of another curve whose abscissa is \( p \), and the ordinate is \( \frac{1}{q} \).
We get \( y \) by the area of a third curve whose abscissa is \( p \), and the ordinate is \( \frac{p}{q} \).
The problem of the trajectory is therefore completely solved, because we have determined the ordinate, abscissa, and arch of the curve for any given position of its tangent. It now only remains to compute the magnitude of these ordinates and abscissae, or to draw them magnitude by a geometrical construction. But in this consists the difficulty. The areas of these curves, which expresses the lengths of \( x \) and \( y \), can neither be computed nor exhibited geometrically, by any accurate method yet discovered, and we must content ourselves with approximations. These render the description of the trajectory exceedingly difficult and tedious, so that little advantage has as yet been derived from the knowledge we have got of its properties. It will however greatly assist our conception of the subject to proceed some length in this construction; for it must be acknowledged that very few distinct notions accompany a mere algebraic operation, especially if in any degree complicated, which we confess is the case in the present question.
Let \( B \) in \( NR \) (fig. 8.) be an equilateral hyperbola, of which \( B \) is the vertex, \( BA \) the semitransverse axis, which we shall assume for the unity of length. Let \( AV \) be the foci conjugate axis \( = BA = 1 \), and \( AS \) the asymptote, bisecting the right angle \( BAV \). Let \( PN \), \( PN \) be two ordinates to the conjugate axis, exceedingly near to each other. Join \( BP \), \( AN \), and draw \( B \), \( N \), perpendicular to the asymptote, and \( BC \) parallel to \( AP \). It is well known that \( BP \) is equal to \( NP \). Therefore \( PN = BA^2 + AP^2 \). Now since \( BA = 1 \), if we make \( AP = p \) of our formulae, \( PN = \sqrt{1 + p^2} \), and \( PP = p \), and the area \( BAPNB = f \cdot p \cdot \sqrt{1 + p^2} \). That is to say, the number \( f \cdot p \cdot \sqrt{1 + p^2} \) (for it is a number) has the same proportion to unity of number that the area \( BAPNB \) has to \( BCVA \), the unit of surface. This area consists of two parts, the triangle \( APN \), and the hyperbolic sector \( ABN \). \( APN = \frac{1}{2} AP \times PN = \frac{1}{2} p \cdot \sqrt{1 + p^2} \), and the hyperbolic sector \( ABN = BN \times \beta \), which is equivalent to the hyperbolic logarithm of the number represented by \( A \beta \) when \( A \beta \) is unity. Therefore it is equal to \( \frac{1}{2} \) the logarithm of \( p + \sqrt{1 + p^2} \).
Hence we see by the bye that \( f \cdot p \cdot \sqrt{1 + p^2} = \frac{1}{2} p \cdot \sqrt{1 + p^2} + \frac{1}{2} \) hyperbolic logarithm \( p + \sqrt{1 + p^2} \).
Now let \( AM \) be another curve, such that its ordinates \( VM \), \( PD \), &c. may be proportional to the areas \( ABM \), \( V \), \( ABNP \), and may have the same proportion to \( AB \), the unity of length, which these areas have to \( ABCV \), the unity of surface. Then \( VM : VC = VM : BA : VCBA \), and \( PD : PB = PNBA : VCBA \), &c. These ordinates will now represent \( f \cdot p \cdot \sqrt{1 + p^2} \) with reference to a linear unit, as the areas to the hyperbola represented it in reference to a superficial unit.
Again, Again, in every ordinate make PD : P = P : PO, and thus we obtain a reciprocal to PD, or to \( f \cdot \frac{1}{\sqrt{1 + p^2}} \), or equivalent to \( f \cdot \frac{1}{\sqrt{1 + p^2}} \). This will evidently be \( \frac{x}{a} \), and PO \( \rho \) will be \( \frac{x}{a} \), and the area contained between the lines AF, AW, and the curve GEOH, and cut off by the ordinate PO, will represent \( \frac{x}{a} \).
Lastly, make PO : PQ = AV : AP, = \( \frac{1}{p} \); and then PQ \( \rho \) will represent \( \frac{y}{a} \), and the area ALEQP will represent \( \frac{y}{a} \).
But we must here observe, that the fluents expressed by these different areas require what is called the correction to accommodate them to the circumstances of the case. It is not indifferent from what ordinate we begin to reckon the areas. This depends on the initial direction of the projectile, and that point of the abscissa AP must be taken for the commencement of all the areas which gives a value of \( \rho \) suited to the initial direction. Thus, if the projection has been made from Fig. 7. A (fig. 7.) at an elevation of 45°, the ratio of the fluxions \( x \) and \( y \) is that of equality; and therefore the point E of fig. 8. where the two curves intersect and have a common ordinate, evidently corresponds to this condition. The ordinate EV passes through V, so that AV or \( \rho = AB, = 1 \), tangent 45°, as the case requires. The values of \( x \) and of \( y \) corresponding to any other point of the trajectory, such as that which has AP for the tangent of the angle which it makes with the horizon, are now to be had by computing the areas VEOP, VEQP.
Another curve might have been added, of which the ordinates would exhibit the fluxions of the arch of the trajectory \( \dot{x} = \frac{\rho}{f} \cdot \frac{1}{\sqrt{1 + p^2}} \) and of which the area would exhibit the arch itself. And this would have been very easy, for it is \( \dot{x} = \frac{\rho}{f} \cdot \frac{1}{\sqrt{1 + p^2}} \), which is evidently the fluxion of the hyperbolic logarithm of \( \frac{\rho}{f} \cdot \frac{1}{\sqrt{1 + p^2}} \). But it is needless, since \( \dot{x} = \frac{\rho}{f} \cdot \frac{1}{\sqrt{1 + p^2}} \), and we have already got \( \dot{x} \). It is only increasing PO in the ratio of BA to BP.
And thus we have brought the investigation of this problem a considerable length, having ascertained the form of the trajectory. This is surely done when the ratio of the arch, abscissa, and ordinate, and the position of its tangent, is determined in every point. But it is still very far from a solution, and much remains to be done before we can make any practical application of it. The only general consequence that we can deduce from the premises is, that in every case where the refraction in any point bears the same proportion to the force of gravity, the trajectory will be similar. Therefore, two balls, of the same density, projected in the same direction, will describe similar trajectories if the velocities are in the subduplicate ratio of the diameters. This we shall find to be of considerable practical importance. But let us To determine the velocity in the different points of the trajectory, and the time of describing its several portions.
Recollect, therefore, that \( v^2 = \frac{-g}{y} \), and that \( \dot{x} = \frac{\rho}{f} \cdot \frac{1}{\sqrt{1 + p^2}} \) and \( \dot{y} = \frac{\rho}{f} \cdot \frac{1}{\sqrt{1 + p^2}} \). This gives \( v^2 = \frac{-g}{q} \cdot \frac{1}{\sqrt{1 + p^2}} \).
But \( \dot{p} = q \cdot \dot{x} \). Therefore \( v^2 = \frac{-g}{q} \cdot \frac{1}{\sqrt{1 + p^2}} \), \( = \frac{-g}{q} \cdot \frac{1}{\sqrt{1 + p^2}} + C \), and \( v = \sqrt{\frac{-g}{q} \cdot \frac{1}{\sqrt{1 + p^2}}} \).
Also \( i \) was found \( = \frac{\dot{x}}{v} = \frac{\dot{x}}{v} \cdot \frac{1}{\sqrt{1 + p^2}} \), \( = \frac{\dot{p}}{q} \cdot \frac{1}{\sqrt{1 + p^2}} \).
If we now substitute for \( v \) its value just found, we obtain \( i = \frac{\dot{p}}{\sqrt{-gq}} \), and \( t = \int \frac{\dot{p}}{\sqrt{-gq}} \).
The greatest difficulty still remains, viz. the accommodating these formulae, which appear abundantly firm, of example, to the particular cases. It would seem at first sight, that all trajectories are similar; since the ratio of the fluxions of the ordinate and abscissa corresponding to any particular angle of inclination to the horizon seems the same in them all: but a due attention to what has been hitherto said on the subject will show us that we have as yet only been able to ascertain the velocity in the point of the trajectory, which has a certain inclination to the horizon, indicated by the quantity \( \rho \), and the time (reckoned from some assigned beginning) when the projectile is in that point.
To obtain absolute measures of these quantities, the term of commencement must be fixed upon. This will be expressed by the constant quantity \( C \), which is assumed for completing the fluent of \( \frac{\rho}{f} \cdot \frac{1}{\sqrt{1 + p^2}} \), which is the basis of the whole construction. We there found \( q = \frac{\rho}{f} \cdot \frac{1}{\sqrt{1 + p^2}} \). This fluent is in general \( q = \frac{C}{a} + \frac{\rho}{f} \cdot \frac{1}{\sqrt{1 + p^2}} \), and the constant quantity \( C \) is to be accommodated to some circumstances of the case. Different authors have selected different circumstances. Euler, Euler, in his Commentary on Robins, and in a dissertation in the Memoirs of the Academy of Berlin published in 1753, takes the vertex of the curve for the beginning of its abscissa and ordinate. This is the simplest method of any, for C must then be so chosen that the whole fluent may vanish when \( p = 0 \), which is the case in the vertex of the curve, where the tangent is parallel to the horizon. We shall adopt this method.
Therefore, let \( AP \) (fig. 9.) \( = x \), \( PM = y \), \( AM = z \). Put the quantity \( C \) which is introduced into the fluent equal to \( \frac{n}{a} \). It is plain that \( n \) must be a number; for it must be homologous with \( \sqrt{1 + p^2} \), which is a number. For brevity's sake let us express the fluent of \( \sqrt{1 + p^2} \) by the single letter \( P \); and thus we shall have \( x = a \times f \frac{P}{n + P} \), \( y = a \times f \frac{P}{n + P} \), \( z = a \times f \frac{P}{n + P} \). And \( v = -\frac{ag(1 + p^2)}{n + P} \).
Now the height \( h \) necessary for communicating any velocity \( v \) is
\[ \frac{v^2}{2g} = \frac{-ag(1 + p^2)}{2g(n + P)} = \frac{-\frac{a}{2}(1 + p^2)}{n + P} \]
And lastly,
\[ t = \frac{\sqrt{a}}{\sqrt{g}} \int \frac{P}{\sqrt{n + P}} \]
These fluents, being all taken so as to vanish at the vertex, where the computation commences, and where \( p = 0 \) (the tangent being parallel to the horizon), we obtain in this case \( h = \frac{a}{n} \), \( v = \frac{a}{2n} \), and \( n = \frac{a}{2h} \).
Hence we see that the circumstance which modifies all the curves, distinguishing them from each other, is the velocity (or rather its square) in the highest point of the curve. For \( h \) being determined for any body whose terminal velocity is \( u \), \( n \) is also determined; and this is the modifying circumstance. Considering it geometrically, it is the area which must be cut off from the area \( DMAP \) of fig. 8. in order to determine the ordinates of the other curves.
We must farther remark, that the values now given relate only to that part of the area where the body is descending from the vertex. This is evident; for, in order that \( y \) may increase as we recede from the vertex, its fluxion must be taken in the opposite sense to what it was in our investigation. There we supposed \( y \) to increase as the body ascended, and then to diminish during the descent; and therefore the fluxion of \( y \) was first positive and then negative.
The same equations, however, will serve for the ascending branch \( CNA \) of the curve, only changing the sign of \( P \); for if we consider \( y \) as decreasing during the ascent, we must consider \( q \) as expressing \( \frac{p}{x} \), and therefore \( P, or f \frac{p}{\sqrt{1 + p^2}}, which is \( \frac{q}{a} \), must be taken negatively. Therefore, in the ascending branch, we have \( AQ \) or \( x \) (increasing as we recede from \( A \)) — \( a \times f \frac{p}{n - P}, QN \) or \( y = a \times f \frac{p}{n - P}, AN \) or \( z = \)
\[ a \times f \frac{p}{n - P}, t = \frac{\sqrt{a}}{\sqrt{g}} \times f \frac{p}{\sqrt{1 - P}}, \text{ and the height producing the velocity at } N = \frac{\frac{a}{2}(1 + p^2)}{n - P}. \]
Hence we learn by the bye, that in no part of the ascending branch can the inclination of the tangent be such that \( P \) shall be greater than \( n \); and that if we suppose \( P \) equal to \( n \) in any point of the curve, the velocity in that point will be infinite. That is to say, there is a certain affinable elevation of the tangent which cannot be exceeded in a curve which has this velocity in the vertex. The best way for forming a conception of this circumstance in the nature of the curve, is to invert the motion, and suppose an accelerating force, equal and opposite to the resistance, to act on the body in conjunction with gravity. It must describe the same curve, and this branch \( ANC \) must have an asymptote \( LO \), which has this limiting position of the tangent. For, as the body descends in this curve, its velocity increases to infinity by the joint action of gravity and this accelerating force, and yet the tangent never approaches to near the perpendicular position as to make \( P = n \). This remarkable property of the curve was known to Newton, as appears by his approximations, which all lead him to curves of a hyperbolic form, having one asymptote inclined to the horizon. Indeed it is pretty obvious; for the resistance increasing faster than the velocity, there is no velocity of projection so great but that the curve will come to deviate so from the tangent, that in a finite time it will become parallel to the horizon. Were the resistance proportional to the velocity, then an infinite velocity would produce a rectilineal motion, or rather a deflection from it less than any that can be assigned.
We now see that the particular form and magnitude of this trajectory depends on two circumstances, \( a \) and its form and \( n \). \( a \) affects chiefly the magnitude. Another circumstance might indeed be taken in, viz. the diminution of the accelerating force of gravity by the statical effect of the air's gravity. But, as we have already observed, this is too trifling to be attended to in military projectiles.
\[ \frac{y}{x} \text{ was made equal to } \frac{p}{x}. \text{ Therefore the radius of curvature, determined by the ordinary methods, is } \frac{(1 + p^2)(\sqrt{1 + p^2})}{p}, \text{ and, because } \frac{p}{x} \text{ is } \frac{a}{n + P} \text{ for the descending branch of the curve, the radius of curvature at } M \text{ is } \frac{a \times (\sqrt{1 + p^2})}{n + P}, \text{ and, in the ascending branch at } N, \text{ it is } \frac{a \times (\sqrt{1 + p^2})}{n - P}. \]
On both sides therefore, when the velocity is infinitely great, and \( P \) by this means supposed to equal or exceed \( n \), the radius of curvature is also infinitely great. We also see that the two branches are unlike each other, and that when \( p \) is the same in both, that is, when the tangent is equally inclined to the horizon, the radius of curvature, the ordinate, the abscissa, and the arch, are all greater in the ascending branch. This is pretty obvious. For as the resistance acts entirely in diminishing the velocity, and does not affect the deflection occasioned by gravity, it must allow gravity to curvate the path so much the more (with the same inclination of its line of action) as the velocity is more diminished. The curvature, therefore, in those points which have the same inclination of the tangent, is greatest in the descending branch, and the motion is swiftest in the ascending branch. It is otherwise in a void, where both sides are alike. Here \( u \) becomes infinite, or there is no terminal velocity; and \( n \) also becomes infinite, being
\[ \frac{a}{2h} \]
It is therefore in the quantity \( P \), or \( f \rho \sqrt{1+\rho^2} \),
that the difference between the trajectory in a void and in a resisting medium consists; it is this quantity which expresses the accumulated change of the ratio of the increments of the ordinate and abscissa. In vacuo the second increment of the ordinate is constant when the first increment of the abscissa is \( fo \), and the whole increment of the ordinate is \( 1+\rho \). And this difference is so much the greater as \( P \) is greater in respect of \( n \). \( P \) is nothing at the vertex, and increases along with the angle \( MTP \); and when this is a right angle, \( P \) is infinite. The trajectory in a resisting medium will come therefore to deviate infinitely from a parabola, and may even deviate farther from it than the parabola deviates from a straight line. That is, the distance of the body in a given moment from that point of its parabolic path where it would have been in a void, is greater than the distance between that point of the parabola from the point of the straight line where it would have been, independent of the action of gravity. This must happen whenever the resistance is greater than the weight of the body, which is generally the case in the beginning of the trajectory in military projectiles; and this (were it now necessary) is enough to show the inutility of the parabolic theory.
Although we have no method of describing this trajectory, which would be received by the ancient geometers, we may ascertain several properties of it, which will assist us in the solution of the problem. In particular, we can assign the absolute length of any part of it by means of the logistic curve. For because \( P = f \rho \sqrt{1+\rho^2} \), we have \( \dot{\rho} \sqrt{1+\rho^2} = \dot{P} \), and therefore \( z \), which was \( a \times f \rho \sqrt{1+\rho^2} + C \), or \( a \times f \rho \sqrt{1+\rho^2} \), may be expressed by logarithms; or \( z = a \times \text{hyp. log. of } \frac{n+P}{n} \), since at the vertex \( A \), where \( z \) must be \( = a \), \( P \) also \( = 0 \).
Being able, in this way, to ascertain the length \( AM \) of the curve (counted from the vertex), corresponding to any inclination \( \rho \) of the tangent at its extremity \( M \), we can ascertain the length of any portion of it, such as \( MM \), by first finding the length of the part \( Am \), and then of the part \( AM \). This we do more expeditiously thus: Let \( \rho \) express the position of the tangent in \( M \), and \( q \) its position at \( m \); then \( AM = a \times \log. \frac{n+P}{n} \) and \( Am = a \times \log. \frac{n+Q}{n+P} \), and therefore \( MM = a \times \log. \frac{n+Q}{n+P} \). Thus we can find the values of a great number of small portions, and the inclination of the tangents at their extremities. Then to each of these portions we can assign its proportion of the abscissa and ordinate, without having recourse to the values of \( x \) and \( y \).
For the portion of abscissa corresponding to the arch \( MM \), whose middle point is inclined to the horizon in the angle \( b \), will be \( MM \times \cos b \), and the corresponding portion of the ordinate will be \( MM \times \sin b \). Then we obtain the velocity in each part of the curve by the equation \( h = \frac{3}{2} a \times \frac{1+\rho^2}{\sqrt{n+P}} \); or, more directly the velocity \( v \) at \( M \) will be \( \sqrt{a \times \frac{1+\rho^2}{\sqrt{n+P}}} \). Lastly, divide the length of the little arch by this, and the quotient will be the time of describing \( MM \) very nearly. Add all these together, and we obtain the whole time of describing the arch \( AM \), but a little too great, because the motion in the small arch is not perfectly uniform. The error, however, may be as small as we please, because we may make the arch as small as we please; and for greater accuracy, it will be proper to take the \( \rho \) by which we compute the velocity, a medium between the \( \rho \) for the beginning and that for the end of the arch.
This is the method followed by Euler, who was one of the most expert analysts, if not the very first, in Euclid's method. It is not the most elegant, and the methods of some other authors, who approximate directly to the areas of the curves which determine the values of \( x \) and \( y \), have a more scientific appearance; but they are not ultimately very different: For, in some methods, these areas are taken piecemeal, as Euler takes the arch; and by the methods of others, who give the value of the areas by Newton's method of describing a curve of the parabolic kind through any number of given points; the ordinates of these curves, which express \( x \) and \( y \), must be taken singly, which amounts to the same thing, with the great disadvantage of a much more complicated calculus, as any one may see by comparing the expressions of \( x \) and \( y \) with the expression of \( z \). As to those methods which approximate directly to the areas or values of \( x \) and \( y \) by an infinite series, they all, without exception, involve us in most complicated expressions, with coefficients of sines and tangents, and ambiguous signs, and engage us in a calculation almost endless. And we know of no series which converges fast enough to give us tolerable accuracy, without such a number of terms as is sufficient to deter any person from the attempt. The calculation of the arches is very moderate, so that a person tolerably versant in arithmetical operations may compute an arch with its velocity and time in about five minutes. We have therefore no hesitation in preferring this method of Euler's to all that we have seen, and therefore proceed to determine some other circumstances which render its application more general. If there were no resistance, the smallest velocity would be at the vertex of the curve, and it would immediately increase by the action of gravity confining (in however small degree) with the motion of the body. But in a resisting medium, the velocity at the vertex is diminished by a quantity to which the acceleration of gravity in that point bears no assignable proportion. It is therefore diminished, upon the whole, and the point of final left velocity is a little way beyond the vertex. For the same reasons, the greatest curvature is a little way beyond the vertex. It is not very material for our present purpose to ascertain the exact positions of those points.
The velocity in the descending branch augments continually; but it cannot exceed a certain limit, if the velocity at the vertex has been less than the terminal velocity; for when the curve is infinite, \( \rho \) is also infinite, and
\[ h = \frac{a}{P}, \quad \text{because } n \text{ in this case is nothing in respect of } P, \]
which is infinite; and because \( \rho \) is infinite, the number hyp. log. \( \rho \times \sqrt{1 + \rho^2} \), though infinite, vanishes in comparison with \( \rho + \sqrt{1 + \rho^2} \); so that in this case \( P = \frac{1}{2} \rho^2 \), and \( h = a \), and \( v \) is the terminal velocity.
If, on the other hand, the velocity at the vertex has been greater than the terminal velocity, it will diminish continually, and when the curve has become infinite, \( v \) will be equal to the terminal velocity.
In either case we see that the curve on this side will have a perpendicular asymptote. It would require a long and pretty intricate analysis to determine the place of this asymptote, and it is not material for our present purpose. The place and position of the other asymptote LO is of the greatest moment. It evidently distinguishes the kind of trajectory from any other. Its position depends on this circumstance, that if \( \rho \) marks the position of the tangent, \( n - P \), which is the denominator of the fraction expressing the square of the velocity, must be equal to nothing, because the velocity is infinite; therefore, in this place, \( P = n \), or \( n = \frac{3}{2} \rho \sqrt{1 + \rho^2} + \frac{1}{2} \log \rho + \sqrt{1 + \rho^2} \). In order, therefore, to find the point L, where the asymptote LO cuts the horizontal line AL, put \( P = n \), then will \( AL = x - \frac{y_n}{y} = a \times \left( \int \frac{\rho}{n - P} - \frac{1}{\rho} \int \frac{\rho}{n - P} \right) \).
It is evident that the logarithms used in these expressions are the natural or hyperbolic. But the operations may be performed by the common tables, by making the value of the arch \( Mm \) of the curve \( = \frac{a}{M} \times \log \frac{n + Q}{n + P} \), &c., where \( M \) means the subtangent of the common logarithms, or 0.43429; also the time of describing this arch will be expediently had by taking a medium \( \mu \) between the values of \( \frac{\sqrt{1 + \rho^2}}{\sqrt{n + P}} \) and \( \frac{\sqrt{1 + \rho^2}}{\sqrt{n + Q}} \), and making the time \( = \frac{\sqrt{a}}{M \times \sqrt{g}} \times \log \frac{n + Q}{n + P} \).
Such then is the process by which the form and magnitude of the trajectory, and the motion in it, may be determined. But it does not yet appear how this is to be applied to any question in practical artillery. In this process we have only learned how to compute the motion from the vertex in the defending branch till the ball has acquired a particular direction, and the motion to the vertex from a point of the ascending branch where the ball has another direction, and all this depending on the greatest velocity which the body can acquire by falling, and the velocity which it has in the vertex of the curve. But the usual question is, "What will be the motion of the ball projected in a certain direction with a certain velocity?"
The mode of application is this: Suppose a trajectory computed for a particular terminal velocity, produced by the fall \( a \), and for a particular velocity at the vertex, which will be characterized by \( n \), and that the velocity at that point of the ascending branch where the inclination of the tangent is 35° is 900 feet per second. Then, we are certain, that if a ball, whose terminal velocity is that produced by the fall \( a \), be projected with the velocity of 900 feet per second, and an elevation of 35°, it will describe this very trajectory, and the velocity and time corresponding to every point will be such as is here determined.
Now this trajectory will, in respect to form, answer an infinity of cases: for its characteristic is the proportion of the velocity in the vertex to the terminal velocity. When this proportion is the same, the number \( n \) will be the same. If, therefore, we compute the trajectories for a sufficient variety of these proportions, we shall find a trajectory that will nearly correspond to any case that can be proposed; and an approximation sufficiently exact will be had by taking a proportional medium between the two trajectories which come nearest to the case proposed.
Accordingly, a set of tables or trajectories have been computed by the English translator of Euler's Commentaries on Robin's Gunnery. They are in number 18, distinguished by the position of the asymptote of the ascending branch. This is given for 5°, 10°, 15°, &c., to 85°, and the whole trajectory is computed as far as it can ever be supposed to extend in practice. The following table gives the value of the number \( n \) corresponding to each position of the asymptote.
| OLB | n | OLB | n | |-----|-----|-----|-----| | 0 | 0.0000 | 45 | 1.14779 | | 5 | 0.08760 | 50 | 1.43236 | | 10 | 0.17724 | 55 | 1.82207 | | 15 | 0.27712 | 60 | 2.39033 | | 20 | 0.37185 | 65 | 3.20940 | | 25 | 0.48269 | 70 | 4.88425 | | 30 | 0.60799 | 75 | 8.22357 | | 35 | 0.75382 | 80 | 17.51793 | | 40 | 0.92914 | 85 | 67.12291 |
Since the path of a projectile is much less incurvated, and more rapid in the ascending than in the descending branch, and the difference is so much the more remarkable in great velocities; it must follow, that the range on a horizontal or inclined plane depends most on the ascending branch: therefore the greatest range will not be made with that elevation which bisects the angle of position, but with a lower elevation; and the deviation from the bisecting elevation will be greater as the initial velocities... velocities are greater. It is very difficult to frame an exact rule for determining the elevation which gives the greatest range. We have subjoined a little table which gives the proper elevations (nearly) corresponding to the different initial velocities.
It was computed by the following approximation, which will be found the same with the series used by Newton in his Approximation.
Let \( e \) be the angle of elevation, \( a \) the height producing the terminal velocity, \( h \) the height producing the initial velocity, and \( c \) the number whose hyperbolic logarithm is \( i \) (i.e., the number 2718).
Then,
\[ y = a \times \left( \tan e + \frac{a}{2h \cdot \cot e} \right) - \frac{a^2}{24} \left( \frac{\cot e}{2h} - 1 \right), \]
&c. Make \( y = v \), and take the maximum by varying \( e \), we obtain \( \sin^2 e + \frac{a}{2h \cdot \cot e} = \text{hyperbol. log.} \)
\[ \left( 1 + \frac{2h}{a \cdot \cot e} \right), \]
which gives us the angle \( e \).
The numbers in the first column, multiplied by the terminal velocity of the projectile, give us the initial velocity; and the numbers in the last column, being multiplied by the height producing the terminal velocity, and by 23026, give us the greatest ranges. The middle column contains the elevation. The table is not computed with scrupulous exactness, the question not requiring it. It may, however, be depended on within one part of 2000.
To make use of this table, divide the initial velocity by the terminal velocity \( u \), and look for the quotient in the first column. Opposite to this will be found the elevation giving the greatest range; and the number in the last column being multiplied by 23026 \( \times a \) (the height producing the terminal velocity) will give the range.
**Table of Elevations giving the greatest Range.**
| Initial vel. | Elevation | Range | |--------------|-----------|-------| | 0.6909 | 43° 40' | 0.1751 | | 0.7820 | 43° 20' | 0.2169 | | 0.8645 | 42° 50' | 0.2548 | | 1.3817 | 41° 40' | 0.4999 | | 1.5641 | 40° 20' | 0.5789 | | 1.7201 | 40° 10' | 0.6551 | | 2.0726 | 39° 50' | 0.7877 | | 2.3461 | 37° 20' | 0.8967 | | 2.5936 | 35° 50' | 0.9752 | | 2.7635 | 35° | 1.0319 | | 3.1281 | 34° 40' | 1.1411 | | 3.4544 | 34° 20' | 1.2298 | | 3.4581 | 34° 20' | 1.2277 | | 3.9101 | 33° 50' | 1.3371 | | 4.1452 | 33° 30' | 1.3901 | | 4.3227 | 33° 30' | 1.4274 | | 4.6921 | 31° 50' | 1.5050 | | 4.8631 | 31° 50' | 1.5341 |
Such is the solution which the present state of our mathematical knowledge enables us to give of this celebrated problem. It is exact in its principle, and the application of it is by no means difficult, or even operose.
But let us see what advantage we are likely to derive from it.
In the first place, it is very limited in its application. There are few circumstances of general coincidence, and almost every case requires an appropriated calculus. Perhaps the only general rules are the two following:
1. Balls of equal density, projected with the same elevation, and with velocities which are as the square-roots of their diameters, will describe similar curves.—This is evident, because, in this case, the resistance will be in the ratio of their quantities of motion. Therefore all the homologous lines of the motion will be in the proportion of the diameters.
2. If the initial velocities of balls projected with the same elevation are in the inverse subduplicate ratio of the whole resistances, the ranges, and all the homologous lines of their track, will be inversely as those resistances.
These theorems are of considerable use: for by means of a proper series of experiments on one ball projected with different elevations and velocities, tables may be constructed which will ascertain the motions of an infinity of others.
But when we take a retrospective view of what we have done, and consider the conditions which were assumed in the solution of the problem, we shall find that deviations much yet remains before it can be rendered of great practical use, or even satisfy the curiosity of the man of science. The resistance is all along supposed to be in the duplicate ratio of the velocity; but even theory points out many causes of deviation from this law, such as the pressure and condensation of the air, in the case of very swift motions; and Mr. Robins's experiments are sufficient to show us that the deviations must be exceedingly great in such cases. Mr. Euler and all subsequent writers have allowed that it may be three times greater, even in cases which frequently occur; and Euler gives a rule for ascertaining tolerable accuracy what this increase and the whole resistance may amount to. Let \( H \) be the height of a column of air whose weight is equivalent to the resistance taken in the duplicate ratio of the velocity. The whole resistance will be expressed by \( H + \frac{H^2}{28845} \). This number 28845 is the height in feet of a column of air whose weight balances its elasticity. We shall not at present call in question his reasons for assigning this precise addition. They are rather reasons of arithmetical convenience than of physical import. It is enough to observe, that if this measure of the resistance is introduced into the process of investigation, it is totally changed; and it is not too much to say, that with this complication it requires the knowledge and address of a Euler to make even a partial and very limited approximation to a solution.—Any law of the resistance, therefore, which is more complicated than what Bernoulli has assumed, namely, that of a simple power of the velocity, is abandoned by all the mathematicians, as exceeding their abilities; and they have attempted to avoid the error arising from the assumption of the duplicate ratio of the velocity, either by supposing the resistance throughout the whole trajectory to be greater than what it is in general, or they have divided the trajectory into different portions, and assigned different resistances to each, which vary,
vary through the whole of that portion, in the duplicate ratio of the velocities. By this kind of patchwork they make up a trajectory and motion which corresponds, in some tolerable degree, with what? With an accurate theory? No; but with a series of experiments. For, in the first place, every theoretical computation that we make proceeds on a supposed initial velocity; and this cannot be ascertained with any thing approaching to precision, by any theory of the action of gunpowder that we are yet possessed of. In the next place, our theories of the resisting power of the air are entirely established on the experiments on the flights of shot and shells, and are corrected and amended till they tally with the most approved experiments we can find. We do not learn the ranges of a gun by theory, but the theory by the range of the gun. Now the variety and irregularity of all the experiments which are appealed to are so great, and the acknowledged difference between the resistance to slow and swift motions is also so great, that there is hardly any supposition which can be made concerning the resistance, that will not agree in its results with many of those experiments. It appears from the experiments of Dr Hutton at Woolwich, in 1784, 1785, and 1786, that the shots frequently deviated to the right or left of their intended track 222, 320, and sometimes 400 yards. This deviation was quite accidental and anomalous, and there can be no doubt but that the shot deviated from its intended and supposed elevation as much as it deviated from the intended vertical plane, and this without any opportunity of measuring or discovering the deviation. Now, when we have the whole range from one to three to choose among for our measure of resistance, it is evident that the confirmations which have been drawn from the ranges of shot are but feeble arguments for the truth of any opinion. Mr Robins finds his measures fully confirmed by the experiments at Metz and at Minorca. Mr Muller finds the same. Yet Mr Robins's measure both of the initial velocity and of the resistance are at least twice of Mr Muller's; but by compensation they give the same results. The Chevalier Borda, a very expert mathematician, has adduced the very same experiments in support of his theory, in which he abides by the Newtonian measure of the resistance, which is about \( \frac{1}{2} \) of Mr Robins's, and about \( \frac{1}{2} \) of Muller's.
What are we to conclude from all this? Simply this, that we have hardly any knowledge of the air's resistance, and that even the solution given of this problem has not as yet greatly increased it. Our knowledge confines only in these experiments, and mathematicians are attempting to patch up some notion of the motion of a body in a resisting medium, which shall tally with them.
There is another essential defect in the conditions assumed in the solution. The density of the air is supposed uniform; whereas we are certain that it is less by one-fifth or one-sixth towards the vertex of the curve, in many cases which frequently occur, than it is at the beginning and end of the flight. This is another latitude given to authors in their assumptions of the air's resistance. The Chevalier de Borda has, with considerable ingenuity, accommodated his investigation to this circumstance, by dividing the trajectory into portions, and, without much trouble, has made one equation answer them all. We are disposed to think that his solution of the problem (in the Memoirs of the Academy of Paris for 1769) corresponds better with the physical circumstances of the case than any other. But this process is there delivered in too concise a manner to be intelligible to a person not perfectly familiar with all the resources of modern analysis. We therefore preferred John Bernoulli's, because it is elementary and rigorous.
After all, the practical artillery must rely chiefly on the records of experiments contained in the books of practice at the academies, or those made in a more public manner. Even a perfect theory of the air's resistance can do him little service, unless the force of gunpowder were uniform. This is far from being the case even in the same powder. A few hours of a damp day will make a greater difference than occurs in any theory; and, in service, it is only by trial that every thing is performed. If the first shell fall very much short of the mark, a little more powder is added; and, in cannonading, the correction is made by varying the elevation.
We hope to be forgiven by the eminent mathematicians for these observations on their theories. They by no means proceed from any disrespect for their labours. We are not ignorant of the almost insuperable difficulty of the talk, and we admire the ingenuity with which some of them have contrived to introduce into their analysis reasonable substitutions for those terms which would render the equations intelligible. But we must still say, upon their own authority, that these are but ingenious guesses, and that experiment is the touchstone by which they mould these substitutions; and when they have found a coincidence, they have no motive to make any alteration. Now, when we have such a latitude for our measure of the air's resistance, that we may take it of any value, from one to three, it is no wonder that compensations of errors should produce a coincidence; but where is the coincidence? The theorist supposes the ball to let out with a certain velocity, and his theory gives a certain range; and this range agrees with observation—but how? Who knows the velocity of the ball in the experiment? This is concluded from a theory incomparably more uncertain than that of the motion in a resisting medium.
The experiments of Mr Robins and Dr Hutton show, in the most incontrovertible manner, that the resistance to a motion exceeding 1100 feet in a second, is almost three times greater than in the duplicate ratio to the resistance to moderate velocities. Euler's translator, in his comparison of the author's trajectories with experiment supposes it to be no greater. Yet the coincidence is very great. The same may be said of the Chevalier de Borda's. Nay, the same may be said of Mr Robins's own practical rules: for he makes his \( F \), which corresponds to our \( a \), almost double of what these authors do, and yet his rules are confirmed by practice. Our observations are therefore well founded.
But it must not be inferred from all this, that the theory is of no use to the practical artillery. It plainly shows him the impracticability of giving the projectile an enormous velocity. This velocity is of no effect in practice, feet after 200 or 300 yards at farthest, because it is so rapidly reduced by the prodigious resistance of the air. Mr Robins has deduced several practical maxims of the greatest importance from what we already know of this subject, and which could hardly have been even conjectured without this knowledge. See Gunnery. And it must still be acknowledged, that this branch of physical science is highly interesting to the philosopher; nor should we despair of carrying it to greater perfection. The defects arise almost entirely from our ignorance of the law of variation of the air's resistance. Experiments may be contrived much more conducive to our information here than those commonly referred to. The oblique flights of projectiles are, as we have seen, of very complicated investigation, and ill fitted for instructing us; but numerous and well contrived experiments on the perpendicular ascents are of great simplicity, being affected by nothing but the air's resistance. To make them instructive, we think that the following plan might be pursued. Let a set of experiments be premised for ascertaining the initial velocities. Then let shells be discharged perpendicularly with great varieties of density and velocity, and let nothing be attended to but the height and the time; even a considerable deviation from the perpendicular will not affect either of these circumstances, and the effect of this circumstance can easily be computed. The height can be ascertained with sufficient precision for very valuable information by their light or smoke. It is evident that these experiments will give direct information of the air's retarding force; and every experiment gives us two measures, viz. the ascent and descent: and the comparison of the times of ascent and descent, combined with the observed height in one experiment made with a great initial velocity, will give us more information concerning the air's resistance than 50 ranges. If we should suppose the resistance as the square of the velocity, this comparison will give in each experiment an exact determination of the initial and final velocities, which no other method can give us. These, with experiments on the time of horizontal flights, with known initial velocities, will give us more instruction on this head than any thing that has yet been done; and till something of this kind is carefully done, we presume to say that the motion of bodies in a resisting medium will remain in the hands of the mathematicians as a matter of curious speculation. In the mean time, the rules which Mr Robins has delivered in his Gunnery are very simple and easy in their use, and seem to come near the truth as any we have met with. He has not informed us upon what principles they are founded, and we are disposed to think that they are rather empirical than scientific. But we profess great deference for his abilities and penetration, and doubt not but that he had framed them by means of as scientific a discussion as his knowledge of this new and difficult subject enabled him to give it.
We shall conclude this article, by giving two or three tables, computed from the principles established above, and which serve to bring into one point of view the chief circumstances of the motion in a resisting medium. Although the result of much calculation, as any person who considers the subject will readily see, they must not be considered as offering any very accurate results; or that, in comparison with one or two experiments, the differences shall not be considerable. Let any person peruse the published registers of experiments which have been made with every attention, and he will see such enormous irregularities, that all expectations of perfect agreement with them must cease. In the experiments at Woolwich in 1735, which were continued for several days, not only do the experiments of one day differ among themselves, but the mean of all the experiments of one day differs from the mean of all the experiments of another no less than one fourth of the whole. The experiments in which the greatest regularity may be expected, are those made with great elevations. When the elevation is small, the range is more affected by a change of velocity, and still more by any deviation from the supposed or intended direction of the shot.
The first table shows the distance in yards to which a ball projected with the velocity 1600 will go, while its velocity is reduced one-tenth, and the distance at which it drops 16 feet from the line of its direction. This table is calculated by the resistance observed in Mr Robins's experiments. The first column is the weight of the ball in pounds. The second column remains the same whatever be the initial velocity; but the third column depends on the velocity. It is here given for the velocity which is very usual in military service, and its use is to affix it in directing the gun to the mark.—If the mark at which a ball of 24 pounds is directed is 474 yards distant, the axis of the piece must be pointed 16 feet higher than the mark. These deflections from the line of direction are nearly as the squares of the distances.
| I. | II. | III. | |----|-----|------| | 2 | 92 | 420 | | 4 | 121 | 428 | | 9 | 159 | 436 | | 18 | 200 | 470 | | 32 | 272 | 479 |
The next table contains the ranges in yards of a 2 pound shot, projected at an elevation of 45°, with the different velocities in feet per second, expressed in the first column. The second column contains the distances to which the ball would go in vacuo in a horizontal plane; and the third contains the distances to which it will go through the air. The fourth column is added, to show the height to which it rises in the air; and the fifth shows the ranges corrected for the diminution of the air's density as the bullet ascends, and may therefore be called the corrected range.
| I. | II. | III. | IV. | V. | |----|-----|------|-----|----| | 200| 416 | 349 | 106 | 360| | 400| 1664| 1121 | 338 | 1150| | 600| 3740| 1812 | 606 | 1859| | 800| 6649| 2373 | 866 | 2435| | 1000|10300| 2845 | 1138| 2919| | 1200|14961| 3259 | 1378| 3343| | 1400|20364| 3640 | 1606| 3734| | 1600|26597| 3950 | 1814| 4050| | 1800|33663| 4235 | 1992| 4345| | 2000|41559| 4494 | 2168| 4610| | 2200|50286| 4720 | 2348| 4842| | 2400|59846| 4917 | 2460| 5044| | 2600|70376| 5106 | 2630| 5238| | 2800|81870| 5293 | 2762| 5430| | 3000|94340| 5455 | 2862| 5596| | 3200|10780| 5732 | | | The initial velocities can never be pushed as far as we have calculated for in this table; but we mean it for a table of more extensive use than appears at first sight. Recollect, that while the proportion of the velocity at the vertex to the terminal velocity remains the same, the curves will be similar: therefore, if the initial velocities are as the square-roots of the diameters of the balls, they will describe similar curves, and the ranges will be as the diameters of the balls.
Therefore, to have the range of a 12 pound shot, if projected at an elevation of 45°, with the velocity 1500; suppose the diameter of the 12 pounder to be \(d\), and that of the 24 pounder \(D\); and let the velocities be \(v\) and \(V\). Then say, \(\sqrt{d} : \sqrt{D} = 1500\), to a fourth proportional \(V\). If the 24 pounder be projected with the velocity \(V\), it will describe a curve similar to that described by the 12 pounder, having the initial velocity 1500. Therefore find (by interpolation) the range of the 24 pounder, having the initial velocity \(V\). Call this \(R\). Then \(D : d = R : r\), the range of the 12 pounder which was wanted, and which is nearly 3380 yards.
We see by this table the immense difference between the motions through the air and in a void. We see that the ranges through the air, instead of increasing in the duplicate ratio of the initial velocities, really increase slower than those velocities in all cases of military service; and in the most usual cases, viz. from 800 to 1600, they increase nearly as the square-roots of the velocities.
A set of similar tables, made for different elevations, would almost complete what can be done by theory, and would be much more expeditious in their use than Mr Euler's Trajectories, computed with great labour by his English translator.
The same table may also serve for computing the ranges of bomb-shells. We have only to find what must be the initial velocity of the 24 pound shot which corresponds to the proposed velocity of the shell. This must be deduced from the diameter and weight of the shell, by making the velocity of the 24 pounder such, that the ratio of its weight to the resistance may be the same as in the shell.
That the reader may see with one glance the relation of those different quantities, we have given this table, expressed in a figure (fig. 10). The abscissa, or axis \(DA\), is the scale of the initial velocities in feet per second, measured on a scale of 400 equal parts in an inch. The ordinates to the curve \(ACG\) express the different quantities in it. The ordinates to the curve \(AXY\) express (by the same scale) the height to which the ball rises in the air.
The ordinate \(BC\) (drawn through the point of the abscissa which corresponds to the initial velocity 2000) is divided in the points 4, 9, 12, 18, 24, 32, 42, in the ratio of the diameters of cannon-shot of different weights; and the same ordinate is produced on the other side of the axis, till \(BO\) be equal to \(BA\); and then \(BO\) is divided in the subduplicate ratio of the same diameters. Lines are drawn from the point \(A\), and from any point \(D\) of the abscissa, to these divisions.
We see distinctly by this figure how the effect of the initial velocity gradually diminishes, and that in very great velocities the range is very little increased by its augmentation. The dotted curve \(APQR\), shows what the ranges in vacuo would be.
By this figure may the problems be solved. Thus, to find the range of the 12 pounder, with the initial velocity 1500. Set off 1500 from \(B\) to \(F\); draw \(FH\) parallel to the axis, meeting the line \(12A\) in \(H\); draw the ordinate \(HK\); draw \(KL\) parallel to the axis, meeting \(24B\) in \(L\); draw the ordinate \(LM\), cutting \(12B\) in \(N\). \(MN\) is the range required.
If curves, such as \(ACG\), were laid down in the same manner for other elevations, all the problems might be solved with great dispatch, and with much more accuracy than the theory by which the curves are drawn can pretend to.
Note, that fig. 10, as given on Plate CCCCXLII, is one-half less than the scale according to which it is described; but the practical mathematician will find no difficulty in drawing the figure on the enlarged scale to correspond to the description.