Encyclopaedia Britannica > Volume 15, PLA-RAM

567
P R O J E C
all t!ie homologous lines of the motion will be in the
proportion of the diameters.
2. If the initial velocities of balls proje&ed with the
fame elevation are in the inverfe fubduplicate ratio of
the whole refiftances, the ranges, and all the homo¬
logous lines of their track, will be inverfely as thofe re-
filtances.
Thefe theorems are of confiderable ufe : for by means
of a proper feries of experiments on one ball proje&ed
with different elevations and velocities, tables may be
conftru&ed which will afcertain the motions of an infi-
89 nity of others.
Shown But when we take a retrofpe&ive view of what we
from va- ^ave done> and confider the conditions which were af-
ftd-n'i-v's fumed i'1 the folution of the problem, we (hall find that
to be very much yet remains before it can be rendered of great
little. practical ufe, or even fatisfy the curiofity of the man of
fcicnce. The refinance is all along fuppofed to be in
the duplicate ratio of the velocity ; but even theory
points out many caufes of deviation from this law, fuch
as the preffure and condenfation of the air, in the cafe
of very fwift motions ; and Mr Robins’s experiments are
fufficient to fhow us that the deviations mull be ex¬
ceedingly great in fucb cafes. Mr Ruler and all lub-
fequent writers have allowed that it may be three times
greater, even in cafes which frequently occur ; and Eu¬
ler gives a rule for afcertaining with tolerable accuracy
what this increafe and the whole refiltance may amount
to. Let H be the height of a column of air whofe
weight is equivalent to the refiftance taken in the du¬
plicate ratio of the velocity. The whole refillance will
H *
be expreffed by H + ThIs numberaSBqjis the
height in feet of a column of air whofe weight balances
its elafticity. We (hall not at prefent call in queftion
his reafons for affigning this precife addition. They
are rather reafons of arithmetical conveniency than of
phyfical import. It is enough to obferve, that if this
meafure of the refiftance is introduced into the procefs
of inveftigation, it is totally changed ; and it is not too
much to fay, that with this complication it requires the
knowledge and addrefs of a Euler to make even a par¬
tial and very limited approximation to a folution.—
Any law of the refiftance, therefore, which is more
complicated than what Bernoulli has aflumed, namely,
that of a fimple power of the velocity, is abandoned by
all the mathematicians, as exceeding their abilities ; and
they have attempted to avoid the error arifing from the
affumption of the duplicate ratio of the velocity, either
by fuppofing the refiftance throughout the whole tra-
je&ory to be greater than what it is in general, or
they have divided the trajedfory into different por¬
tions, and affigned different refiftances to each, which
vary, through the whole of that portion, in the dupli¬
cate ratio of the velocities. By this kind of patch-
work they make up a trajectory and motion which cor-
refponds, in fome tolerable degree, with what ? With an
accurate theory ? No ; but with a feries of experiments.
Tor, in the firft place, every theoretical computation
that we make, proceeds on a fuppofed initial velocity ;
and this cannot be afeertained with anything approach¬
ing to precifion, by any theory of the attion of gun¬
powder that we are yet pofleffed of. In the next place,
our theories of the refilling power of the air are en-
TILES.
tirely eftablifhed on the experiments on the flights of
ftiot and fhells, and are corredted and amended till they
tally with the moft 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 fo great, and the acknowledged difference between
the refiftance to flow and fwift motions is alfo fo greatr
that there is hardly any fuppofition which can be made
concerning the refiftance, that will not agree in its re-
fults with many of thofe experiments. It appears from
the experiments of I)r Hutton of Woolwich, in I7^4»
1785, and 1786, that the fliots frequently deviated to
the right or left of their intended track 200, 3 .0, and
fometimes 4.00 yards. This deviation was quite acci¬
dental and anomalous, and there can be no doubt but
that the Ihot deviated from its intended and
elevation as much as it deviated from the intended ver¬
tical plane, and this without any opportunity of mea-
furing or difeovering the deviation. Now, when we
have the whole range from one to three to choofe among
for our meafure of refiftance, it is evident that the con¬
firmations which have been drawn from the ranges of
{hot are but feeble arguments for the truth of any opi¬
nion. Mr Robins finds his meafures fully confirmed
by the experiments at Metz and at Minorca. Mr
Muller finds the fame. Yet Mr Robins’s meafure both
of the initial velocity and of the refiftance are at kaft
treble of Mr Muller’s ; but by compenfation they give
the fame refults. The Chevalier Borda, a very expert
mathematician, has adduced the very fame experiments
in fupport of his theory, in which he abides by the
Newtonian meafure of the refiftance, which is about -f
of Mr Robins’s, and about \ of Muller’s. vv.
What are we to conclude from all this ? Simply this, Caufes of
that we have hardly any knowledge of the air’s refill- its inuti-
ancej and that even the folution given of this problem has !lty*
not as yet greatly increafed it. Our knowledge confifts
only in thofe experiments, and mathematicians are at¬
tempting to patch up fome notion of the motion of a
body in a refilling medium, which fhall tally with
them.
There is another effential defedft in the conditions af-
fumed in the folution. The denfity of the air is fup¬
pofed uniform; whereas we are certain that it is lefs
by one fifth or one fixth towards the vertex of the
curve, in many cafes which frequently occur, than it is
at the beginning and end of the flight. This is ano¬
ther latitude given to authors in their affumptions of
the air’s refiftance. The Chevalier de Borda has, with
confiderable ingenuity, accommodated his inveftigation
to this circumftance, by dividing the trajedlory into
portions, and, without much trouble, has made one
equation anfwer them all. We are difpofed to think
that his folution of the problem (in the Memoirs of the
Academy of Paris for 1769) correfponds better with
the phyfical circumftances of the cafe than any other.
But his procefs is there delivered in too concife a man¬
ner to be intelligible to a perfon not perfectly familiar
with all the refources of modern analyfis.. We there¬
fore preferred John Bernoulli’s, becaufe it is elementary
and rigorous.
After all, the praftical artillerift muft rely chiefly on
the records of experiments contained in the books of
practice