ral.

a

a

hlap:

a

8; If there were no resistance, the smallest velocity would process we have only learned how to compute the moIts applica- be at the vertex of the curve, and it would immediately tion from the vertex in the descending branch till the tion made

increase by the action of gravity conspiring (in however ball bas acquired a particular direction, and the motion more gene

small degree) with the motion of the body. But in a to the vertex from a point of the ascending branch
resisting medium, the velocity at the vertex is diminish- where the ball has another direction, and all this de-
ed by a quantity to which the acceleration of gravity pending on the greatest velocity which the body can
in that point bears no assignable proportion. It is there- acquire by falling, and the velocity which it has in the
fure diminished, upon the whole, and the point of smal. vertex of the curve. But the usual question is, “ What
lest velocity is a little way beyond the vertex. For the will be the motion of the ball projected in a certain
same reasons, the greatest curvature is a little way be- direction with a certain velocity?"
yond the vertex. It is not very material for our pre-

The mode of application is this: Suppose a trajecto-
sent purpose to ascertain the exact positions of those ry computed for a particular terminal velocity, produced
points.

by the fall a, and for a particular velocity at the vertex, The velocity in the descending branch augments con- which will be characterized by n, and that the velocity tinually: but it cannot exceed a certain limit, if the ve- at that point of the ascending branch where the inclilocity at the vertex has been lees than the terminal velo- nation of the tangent is 30° is 900 feet per second. city; for when the curve is infinite, p is also infinite, and Then, we are certain, that if a ball, whose terminal ve

locity is that produced by the fall a, be projected with
because n in this case is nothing in respect of the velocity of 900 feet per second, and an elevation of

P
P, which is infinite; and because p is infinite, the num- 30°, it will describe this very trajectory, and the velo-

city and time corresponding to every point will be such
ber lyp. log. PX vitp, though infinite, vanishes in

as is bere determined.
comparison with px Vi+po; so that in this case P= Now this trajectory will, in respect to form, answer
fp, and h=a, and v= the terminal velocity.

an infinity of cases : for its characteristic is the propor-
If, on the other hand, the velocity at the vertex has tion of the velocity in the vertex to the terminal velo-
been greater than the terminal velocity, it will diminish city. When this proportion is the same, the number n
continually, and when the curve has become infinite, u will be the same. If, therefore, we compute the tra-
will be equal to the terminal velocity.

jectories for a sufficient variety of these proportions, we In either case we see that the curve on this side will shall find a trajectory that will nearly correspond to any have a perpendicular assymptote. It would require a case that can be proposed : and an approximation suffilong and pretty intricate analysis to determine the place ciently exact will be had by taking a proportional meof this assymptote, and it is not material for our present dium between the two trajectories which come nearest purpose. The place and position of the other assymp- to the case proposed.

87 tote LO is of the greatest moment. It evidently di- Accordingly, a set of tables or trajectories have been computed stinguishes the kind of trajectory from any other. Its computed by the English translator of Euler's Com- tables or position depends on this circumstance, that if p marks mentary on Robins's Gunnery. They are in number 18, trajectothe position of the tangent, n—P, which is the deno. distinguished by the position of the assymptote of the minator of the fraction expressing the square of the ve- ascending branch. This is given for 5°, 10, 15°, &c.

°& locity, must be equal to nothing, because the velocity to 85°, and the whole trajectory is computed as far as it is infinite: therefore, in this place, P = n, or n= can ever be supposed to extend in practice. The followipVi+p++ { log.p + vitp. In order, therefore, ing table gives the value of the number n corresponding

to each position of the assymptote. Through to find the point L, where the assymptote LO cuts the lhe whole horizontal line AL, put P=n, then will AL= y.r pp

OLB

OLB
=aX

P
pa

-P
means they
It is evident that the logarithms used in these expres-

0,00000 45 1,14779 sions are the natural or hyperbolic. But the operations

5 0,08760 50 1,43236

0,17724 55 may be performed by the common tables, by making

1,82207 15 0,27712

60 2,39033 the value of the arch M m of the curve =

0,37185 65 3,29040
25 0,48269 70

4,88425
n+Q
&c. where M means the subtangent of the com-

30 0,60799 75 8,22357

35 0,75382 80 17,54793 mon logarithms, or 0,43429; also the time of descri

40 0,92914

85

67,12291
bing this arch will be expeditiously had by taking a
medium

fe
between the values of

and

Since the path of a projectile is much less incurvated, n+P ntQ' and more rapid in the ascending than in the descending Vā

branch, and the difference is so much the more remarkand making the time = x log.

able in great velocities; it must follow, that the range 86

on a horizontal or inclined plane depends most on the Such then is the process by which the form and mag- ascending branch : therefore the greatest range will not applying nitude of the trajectory, and the motion in it, may be be made with that elevation which bisects the angle of this process determined.

But it does not yet appear how this is to position, but with a lower elevation; and the deviation be applied to any question in practical artillery. In this from the bisecting elevation will be greater as the initial

ries.

of this ar.

I

n

n

ticlef

=

(film) .

n

I

10

a

20

M X log.

T

)

Са с

cos. e. I

COS.

to be very

velocities are greater. It is very difficult to frame an But let us see what advantage we are likely to derive
exact rule for determining the elevation which gives the from it.
greatest range. We have subjoined a little table which In the first place it is very limited in its applica-
gives the proper elevation (nearly) corresponding to tion. There are few circumstances of general coinci-
the different initial velocities.

dence, and almost every case requires an appropriated
It was compnted by the following approximation, calculus. Perhaps the only general rules are the two
which will be found the same with the series used by following:
Newton in his Approximation.

1. Balls of equal density, projected with the same Let e be the angle of elevation, a the height elevation, and with velocities wbich are as the square producing the terminal velocity, h the height produ- roots of their diameters, will describe similar curves.cing the initial velocity, and c the number whose hy. This is evident, because, in this case, the resistance will perbolic logarithm is i (i, e. the number 2,718). be in the ratio of their quantities of motion. Therefore Then,

all the homologous lines of the motion will be in the aa

proportion of the diameters.
y=xx (

tan, et
2h.
2h

2. If the initial velocities of balls projected with the &c. Make y = v, and take the maximum by vary

same elevation are in the inverse subduplicate ratio of

the whole resistances, the ranges, and all the homoinge, we obtain Sine+a sin. e = hyperbol. log. logous lines of their track, will be inversely as those re

2h

sistances. 2 h

These theorems are of considerable use: for by means which gives us the angle e. a sine e

of a proper series of experiments on one ball projected The numbers in the first column, multiplied by the with different elevations and velocities, tables may be terminal velocity of the projectile, give us the initial

constructed which will ascertain the motions of an infivelocity; and the numbers in the last column, being nity of others.

89 multiplied by the height producing the terminal veloci

But when we take a retrospective view of what we shown from ty, and by 2,30 26, give us the greatest ranges. The

have done, and consider the conditions which were as. various conmiddle column contains the elevation. The table is not sumed in the solution of the problem, we shall find that siderations computed with scrupulous exactness, the question not re

much yet remains before it can be rendered of great lide quiring it. It may, however, be depended on within practical use, or even satisfy the curiosity of the man of

science The resistance is all along supposed to be in one part of 2000.

To make use of this table, divide the initial velocity the duplicate ratio of the velocity; but even theory by the terminal velocity w, and look for the quotient in points out many causes of deviation from this law, such the first column. Opposite to this will be found the ele

as the pressure and condensation of the air, in the case vation giving the greatest range ; and the number in the of very swift motions ; and Mr Robins's experiments are last column being multiplied by 2,30 26 xa (the height ceedingly great in such cases. Mr Euler and all sub

sufficient to show us that the deviations must be ex. producing the terminal velocity) will give the range.

sequent writers have allowed that it Table of Elevations giving the greatest Range.

may be three times
greater, even in cases which frequently occur ; and Eu-

ler gives a rule for ascertaining with tolerable accuracy
Initial vel.
Elevation.
Range.

what this increase and the whole resistance may amount
2,3026 a

to. Let H be the height of a column of air whose

weight is equivalent to the resistance taken in the du0,6909 439.40

plicate ratio of the velocity. The wbole resistance will 0,1751

HP 0,7820

0,2169

be expressed by H+ This number 28845 is the
0,8645 42.50
0,2548

28845
1,3817 41 .40 0,4999

beight in feet of a column of air wbose weight balances 1,5641

0,5789

its elasticity. We shall not at present call in question , 1,7291

0,6551

his reasons for assigning this precise addition. They 2,0726 39.50

0,7877

are rather reasons of arithmetical conveniency than of
2,3461
37.20
0,8967

physical import. It is enough to observe, that if this
2,5936 35.50 0,9752

measure of the resistance is introduced into the process
2,7635
35 - 1,0319

of investigation, it is totally changed; and it is not too
3,1281
34 .40 1,1411

much to say, that with this complication it requires the
3,4544

1,2298

knowledge and address of a Euler to make even a par-
34 20
1,2277

tial and very limited approximation to a solution.-
3,9101 33.50 1,3371

Any law of the resistance, therefore, which is more
4,1452 33 :30 1,3901

complicated than what Bernoulli has assumed, namely,
4,3227
33.30 1,4274

that of a simple power of the velocity, is abandoned by
4,6921
31 .50 1,5050

all the mathematicians, as exceeding their abilities; and
88
4,8631 31.50 1,5341

they have attempted to avoid the error arising from the Advantage

assumption of the duplicate ratio of the velocity, either to be de rived from

Such is the solution which the present state of our by supposing the resistance throughout the whole trathe solution mathematical knowledge enables us to give of this cele- jectory to be greater than what it is in general, or of the prob- brated problem. It is exact in its principle, and the ap- they have divided the trajectory into different porlem. plication of it is by no means difficult, or even operose. tions, and assigned different resistances to each, which

43 .20

a

40.20
40.10

34 .20

a

3,4581

ous.

vary, through the whole of that portion, in the dupli- Academy of Paris for 1769) corresponds better with
cate ratio of the velocities. By this kind of patch- the physical circumstances of the case than any other.
work they make up a trajectory and motion which cor- But this process is there delivered in too concise a man-
responds, in some tolerable degree, with what? With an ner to be intelligible to a person not perfectly familiar
accurate theory ? No; but with a series of experiments. with all the resources of modern analysis. We there-
For, in the first place, every theoretical computation fore preferred John Bernoulli's, because it is elementary
that we make, proceeds on a supposed initial velocity; and rigorou
and this cannot be ascertained with any thing approach-

After all, the practical artillerist must rely chiefly on Necessity ing to precision, by any theory of the action of gun- the records of experiments contained in the books of of attend. powder ihat we are yet possessed of. In the next place, practice at the academies, or those made in a more pub-ing to exour theories of the resisting power of the air are en

lic manner.

Even a perfect theory of the air's resis-periments.
tirely established on the experiments on the flights of tance can do him little service, unless the force of gun-
shot and shells, and are corrected and amended till they powder were uniform. This is far from being the case
tally with the most approved experiments we can find. even in the same powder. A few hours of a damp day
We do not learn the ranges of a gun by theory, but the will make a greater difference than occurs in any theory;
theory by the range of the gun. Now the variety and and, in service it is only hy trial that every thing is per-
irregularity of all the experiments which are appealed to formed. If the first shell fall very much short of the
are so great, and the acknowledged difference between mark, a little more powder is added ; and, in cannonad-
the resistance to slow and swift motions is also so great, ing, the correction is made by varying the elevation.
that there is hardly any supposition which can be made We hope to be forgiven by the eminent mathemati-
concerning the resistance, that will not agree in its re- cians for these observations on their theories. They by
sults with many of those experiments. It appears from no means proceed from any disrespect for their labour3.
the experiments of Dr Hutton of Woolwich, in 1784, We are not ignorant of the almost insuperable difficul-
1785, and 1786, that the shots frequently deviated to ty of the task, and we admire the ingenuity with which
the right or left of their intended track 200, 300, and some of them have contrived to introduce into their apa-
sometimes 400 yards. This deviation was quite acci- lysis reasonable substitutions for those terms which would
dental and anomalous, and there can be no doubt but render the equations intractable. But we mușt still
that the shot deviated from its intended and supposed say, upon their own authority, that these are but inge-
elevation as much as it deviated from the intended ver- nious guesses, and that experiment is the touchstone by
tical plane, and this without any opportunity of mea- which they mould these substitutions; and when they

}
suring or discovering the deviation. Now, when we have found a coincidence, they have no motive to make
have the whole range from one to three to choose among any alteration. Now, when we have such a latitude for
for our measure of resistance, it is evident that the con- our measure of the air's resistance, that we may take it
firmations which have been drawn from the ranges of of any value, from one to three, it is no wonder that
shot are but feeble arguments for the truth of any opi- compensations of errors should produce a coincidence ;
nion. Mr Robins finds his measures fully confirmed but where is the coincidence? The theorist supposes the
by the experiments at Metz and at Minorca. Mr ball to set out with a certain velocity, and his theory
Muller finds the same. Yet Mr Robins's measure both gives a certain range; and this range agrees with obser-
of the initial velocity and of the resistance are at least vation-but how? Who knows the velocity of the ball
treble of Mr Muller's ; but by compensation they give in the experiment? This is concluded from a theory in-
the same results. The Chevalier Borda, a very expert comparably more uncertain than that of the motion in a
mathematician, has adduced the very same experiments resisting medium.
in support of his theory, in which he abides by the New- The experiments of Mr Robins and Dr Hutton show,
tonian measure of the resistance, which is about of in the most incontrovertible manner, that the resistance

}
Mr Robins's, and about į of Muller's.

to a motion exceeding 1100 feet in a second, is almost Cause of its What are we to conclude from all this? Simply this, three times greater than in the duplicate ratio to the reinutility. that we have bardly any knowledge of the air's resist- sistance to moderate velocities. Euler's translator, in

ance, and that even the solution given of this problem his comparison of the author's trajectories with experi-
has not as yet greatly increased it. Our knowledge con- ment, supposes it to be no greater. Yet the coincidence
sists only in those experiments, and mathematicians are is very great. The same may be said of the Chevalier
attempting to patch up some notion of the motion of a de Borda's. Nay, the same may be said of Mr Ro-
body in a resisting medium, which shall tally with them. bins's own practical rules : for he makes bis F, which

There is another essential defect in the conditions as- corresponds to our a, almost double of what these au-
sumed in the solution. The density of the air is sup- thors do, and yet his rules are confirmed by practice. Our
posed uniform ; whereas we are certain that it is less observations are therefore well founded.
by one-fifth or one-sixth towards the vertex of the But it must not be inferred from all this, that the The theory
curve,

in many cases which frequently occur, than it is physical theory is of no use to the practical artillerist. is still of at the beginning and end of the flight. This is an- It plainly shows him the impropriety of giving the pro- some use in other latitude given to authors in their assumptions of jectile an enormous velocity. This velocity is of no ef

practice. the air's resistance. The Chevalier de Borda has, with fect after 200 or 300 yards at farthest, because it is so considerable ingenuity, accommodated his investigation rapidly reduced by the prodigious resistance of the air. to this circumstance, by dividing the trajectory into Mr Robins has deduced several practical maxims of the portions, and, without much trouble, has made one greatest importance from what we already know of this equation answer them all. We are disposed to think subject, and which could hardly have been even conjecthat bis solution of the problem (in the Memoirs of the tured without this knowledge. See GUNNERY.

a

a

a

а

90

a

And

2

I 21

428 456

159

93

And it must still be acknowledged, that this branch ral days, not only do the experiments of one day differ and may be of physical science is highly interesting to the philoso- among themselves, but the mean of all the experiments brought to pher; nor should we despair of carrying it to a greater of one day differs from the mean of all the experiments eliter per perfection. The defects arise almost entirely from our of another no less than one-fourth of the whole. The fection.

ignorance of the law of variation of the air's resistance. experiments in wbich the greatest regularity. may be
Experiments may be contrived much more conducive expected, are those made with great elevations. When
to our information here than those commonly resorted the elevation is small, the range is more affected by a
to. The oblique flights of projectiles are, as we have change of velocity, and still more by any deviation from
seen, of very complicated investigation, and ill fitted for the supposed or intended direction of the shot.
instructing us; but numerous and well contrived expe- The first table shows the distance in yards to which
riments on the perpendicular ascents are of great sim- a ball projected with the velocity 1600 will go, wliile
plicity, being affected by nothing but the air's resist- its velocity is reduced one-tenth, and the distance at
ance. To make them instructive, we think that the which it drops 16 feet from the line of its direction.
following plan might be pursued. Let a set of expe- This table is calculated by the resistance observed in Mr
riments be premised for ascertaining the initial veloci- Robins's experiments. The first column is the weight
ties. Then let shells be discharged perpendicularly of the ball in pounds. The second column remains the
with great varieties of density and velocity, and let no- same whatever be the initial velocity; but the third co-
thing be attended to but the height and the time ; even "lumn depends on the velocity. It is here given for the
a considerable deviation from the perpendicular will not velocity which is very usual in military service, and its
affect either of these circumstances, and the effect of use is to assist us in directing the gun to the mark.-
this circumstance can easily be computed. The height If the mark at which a ball of 24 pounds is directed is
can be ascertained with sufficient precision for very va- 474 yards distant, the axis of the piece must be pointed
luable information by their light or smoke. It is evi- 16 feet bigher than the mark. These deflections from
dent that these experiments will give direct informa- the line of direction are nearly as the squares of the di-
tion of the air's retarding force ; and every experiment stances.
gives us two measures, viz. the ascent and descent: and
the comparison of the times of ascent and descent, com-

I. II. III.
bined with the observed height in one experiment made
with a great initial velocity, will give us more informa-

92 420
tion concerning the air's resistance than 50 ranges. If

4 we should suppose the resistance as the square of the ve

9 locity, this comparison will give in each experiment an

18

470 exact determination of the initial and final velocities,

32 272 479
which no other method can give us. These, with ex-
periments on the time of horizontal flights, with known The next table contains the ranges in yards of a 3
initial velocities, will give us more instruction on this pound shot, projected at an elevation of 45°, with the
head than any thing that has yet been done; and till different velocities in feet per second, expressed in the
something of this kind is carefully done, we presume first column. The second column contains the distances
to say that the motion of bodies in a resisting medium to which the ball would go in vacuo in a horizontal
will remain in the hands of the mathematicians as a plane ; and the third contains the distances to which it
matter of curious speculation. In the mean time, the will go through the air. The fourth column is added,
rules which Mr Robins has delivered in his Gunnery to show the height to which it rises in the air; and the
are very simple and easy in their use, and seem to come fifth shows the ranges corrected for the diminution of the
as near the truth as any we have met with. He has air's density as the bullet ascends, and may therefore be
not informed us upon what principles they are founded, called the corrected range.
and we are disposed to think that they are rather em-
pirical than scientific. But we profess great deference 1. 11.

II III. IV. V.
for his abilities and penetration, and doubt not but that
he had framed them by means of as scientific a discus.

416
349

106 360
sion as his knowledge of this new and difficult subject

400 16641121 enabled him to give it.

600

606 1859 94 Tables cal We shall conclude this article, by giving two or three

800 6649 2373

866

2435 culated on tables, computed from the principles established above,

1000 10300 2845 1138 2919 the prece- and which serve to bring into one point of view the

149613259

3259 1378 3343 ding prin- chief circumstances of the motion in a resisting medium. ciples.

1400 20364 3640 1606

3734 Although the result of much calculation, as any person

1600 26597 3950 1814 4050 who considers the subject will readily see, they must not

1800 336634235 1992

4345 be considered as offering any very accurate results ; or

41559 4494

2168 4610 that, in comparison with one or two experiments, the

2200 50286 4720 2348 4842 differences shall not be considerable. Let any person 2400598464917 2460 1 5044 peruse the published registers of experiments which have

2600

51062630 5238 been made with every attention, and he will see such

2800

5293 2760 enormous irregularities, that all expectations of perfect

3000

5455

28625596
agreement with them must cease.

In the experiments
3200

5732
.at Woolwich in 1735, which were continued for seve-

The

200

200

338

1150

3740 1812

I 200

2000

5430