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Fig. 7.

Fig. 8.

77

A

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Consequences of

knowing the form

of the trajectory.

1+p, or equivalent to

I

المعزول

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This

will evidently be, and POopwill be, and the area

a

ap
contained between the lines AF, AW, and the curve
GEOH, and cut off by the ordinate PO, will represent

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But p=q. Therefore v×1+p2
-a gi+p2
I+P2 + C

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rection to accommodate them to the circumstances of fp √ 1 + p¦ + C°

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 abscis-

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sa AP must be taken for the commencement of all the+p
areas which gives a value of p suited to the initial di-
rection. Thus, if the projection has been made from
A (fig. 7.) at an elevation of 45°, the ratio of the just found, we obtain i
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 p AB, I, tangent 45°, as the case re-
quires. 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
ap√1+p
trajectory =
and of which the area
would exhibit the arch itself. And this would have
p√ 1 + p3

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=

v

now substitute for its value

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To deter mine the velocity in

points.

79

The greatest difficulty still remains, viz. the accom- Difficulty modating these formula, which appear abundantly sim- of accom ple, to the particular cases. It would seem at first modating sight, that all trajectories are similar; since the ratio of the formsthe fluxions of the ordinate and abscissa corresponding to lato parany 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 inclina

which is evidently the fluxion of the hyperbolic loga- tion to the horizon, indicated by the quantity p, and the

rithm of f√1+p. But it is needless, since

P

+p, and we have already got x. It is only in--
creasing PO in the ratio of BA to BP.

And thus we have brought the investigation of this
problem to considerable length, having ascertained the
form of the trajectory. This is surely done when the
ratio of the arch, absciss, 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
we can make any practical application of it.
The only general
the premises is, that in every case where the resistance
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,

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 p√1+p, which is the basis of the whole construction. basis of the whole construction. We there found q =

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done before weral consequence that we can deduce from C++, and the constant quantity C is to

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ticular

cases.

So

Euler's method the simplest.

Fig. 9.

Euler, in his Commentary on Robins, and in a disserta-
tion in the Memoirs of the Academy of Berlin publish-
ed in 1753, takes the vertex of the curve for the begin-
ning of his 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 paral-
lel to the horizon. We shall adopt this method.

Therefore, let AP (fig. 9.)r, PM=y, AM=%.
Put the quantity C which is introduced into the fluent

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equal to It is plain that ʼn must be a number; for

a

it must be homologous with p1+p, which is a
number. For brevity's sake let us express the fluent of
P+p by the single letter P; and thus we shall

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have a X

I + p1 n+P

p
n+P

xf pr

y= ax

n+P

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Hence we learn by the bye, that in no part of the Remarkascending branch can the inclination of the tangent be able prosuch that P shall be greater than n; and that if we sup- perty of the

curve or

pose P equal to n in any point of the curve, the velo- trajectory.

city in that point will be infinite. That is to say, there
is a certain assignable 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 assymptote
LO, which has this limiting position of the tangent.

And v——ag (1+p"). Now the For, as the body descends in this curve, its velocity

n+P

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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 further 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 as expressing

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increases to infinity by the joint action of gravity and
this accelerating force, and yet the tangent never ap-
Pn.
proaches so 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, ha-
ving one assymptote 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 pa-
rallel 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.

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83

Several

vious. For as the resistance acts entirely in diminishing

=ax log.

n+Q

n

n+Q

and Am

and therefore M m is a x log.

Thus we can find the values of a great num

the velocity, and does not affect the deflection occasioned 7 its position at m; then AM = a × log. "+P
by gravity, it must allow gravity to incurvate 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 de-
scending 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

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ber 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 absciss corresponding to the arch Mm, whose middle point is inclined to the horizon in the angle b, will be M m x cosine b, and the corresponding

It is therefore in the quantity P, or fp1+, portion of the ordinate will be M'm x sin. b. Then we

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 absciss. In vacuo the
second increment of the ordinate is constant when the
first increment of the abscissa is so, and the whole
increment of the ordinate is as 1+p. And this diffe-
rence 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 para-
bola, and may even deviate farther from it than the
parabola deviates from a straight line. That is, the di-
stance 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 gra-
vity. This must happen whenever the resistance is great-
er than the weight of the body, which is generally the
case in the beginning of the trajectory in military pro-
jectiles; 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
properties geometers, we may ascertain several properties of it,
of it ascer- 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
1+p, we have
we have p√1+p3 — P, and there-

tained.

=

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n+P

n

> hyp. log. of since at the vertex A, where ≈
must be 0, P is also = o.

Being able, in this way, to ascertain the length AM
of the curve (counted from the vertex), corresponding
to any inclination p 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 A m, and
then of the part AM. This we do more expeditiously
thus. Let p express the position of the tangent in M, and

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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 p by which we compute the velocity, a medium between the p for the beginning and that for the end of the arch.

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This is the method followed by Euler, who was one Euler's me of the most expert analysts, if not the very first, in Eu-thod prerope. It is not the most elegant, and the methods offerred. 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 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 expressions of %. 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 au arch with its velocity and time in about five minutes. We have therefore no besitation 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

85

tion made more general.

If there were no resistance, the smallest velocity would Its applica- be at the vertex of the curve, and it would immediately increase by the action of gravity conspiring (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 smal lest velocity is a little way beyond the vertex. For the same reasons, the greatest curvature is a little beyond the vertex. It is not very material for our present purpose to ascertain the exact positions of those points.

Through the whole of this ar

ticle f

way

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, p is also infinite, and ap h= because n in this case is nothing in respect of P, which is infinite; and because p is infinite, the number hyp. log. p× √1+p, though infinite, vanishes in comparison with px VI+p; so that in this case P p', and h=a, and v 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 assymptote. It would require a long and pretty intricate analysis to determine the place of this assymptote, and it is not material for our present purpose. The place and position of the other assymptote LO is of the greatest moment. It evidently distinguishes the kind of trajectory from any other. Its position depends on this circumstance, that if p 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, Pn, or n= p√1+p+ log.p+I+p. In order, therefore, to find the point L, where the assymptote LO cuts the horizontal line AL, put Pn, then will AL-xI PP =ax -P P n-P

y.r

means flu- Y

ent.

86

process we have only learned how to compute the motion from the vertex in the descending 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 30° 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 30°, 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.

87

Accordingly, a set of tables or trajectories have been Computed computed by the English translator of Euler's Com-tables or mentary on Robins's Gunnery. They are in number 18, trajectodistinguished by the position of the assymptote of the is. ascending branch. This is given for 5°, 10°, 15°, &c. to 850, 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 assymptote.

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20

0,37185 65

3,29040

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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 M m of the curve = n+ Q &c. where M means the subtangent of the comn+ P' mon logarithms, or 0,43429; also the time of describing this arch will be expeditiously had by taking a medium μ between the values of

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and

1+p1 √n+P √n+Q

n+ Q Ma√g x log. n+P Lode of Such then is the process by which the form and magpplying nitude of the trajectory, and the motion in it, may be his process determined. But it does not yet appear how this is to practice. be applied to any question in practical artillery. In this

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88

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 elevation (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 1 (i. e. the number 2,718). Then,

y = x × ( (tan.

Advantage to be de

rived from

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tan. e+ 2 h. cos. e &c. Make yv, and take the maximum by vary

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I 3

we obtain Sin. ea sin. e

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hyperbol. log.

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 2,3026, 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 2,3026 Xa (the height producing the terminal velocity) will give the range. TABLE of Elevations giving the greatest Range.

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Such is the solution which the present state of our the solution mathematical knowledge enables us to give of this celeof the prob-brated problem. It is exact in its principle, and the application of it is by no means difficult, or even operose.

lem.

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 re

sistances.

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.

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But when we take a retrospective view of what we show have done, and consider the conditions which were assumed in the solution of the problem, we shall find that siden 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 ceedingly great in such cases. Mr Euler and all subsufficient to show us that the deviations must be exsequent writers have allowed that it may be three times greater, even in cases which frequently occur; and Euler gives a rule for ascertaining with 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 du plicate ratio of the velocity. The whole resistance will H be expressed by H+ This number 28845 is the 28845 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 conveniency 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

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