ROC 
there can be no doubt; but we much question whether that 
determined by Dr. Hutton, viz. r = .00002576v ! — .00388 v, 
although the most accurate of any yet found, is as correct as 
could be wished. Indeed, when we consider the infinitude 
of different forms under which the function of a single 
variable may appear, it seems much too confined a scale to 
attempt to reduce it to the simple form a v 2 + b v ; viz. to 
limit the dimension of the function, and attempt every 
accommodation by means of the co-efficients a and b. With 
regard to the rocket, the case is very different: the very 
medium, which in the former instance is the great impedi¬ 
ment to an accurate theory, is here the principal agent in 
producing the motion; and moreover, we are here, from the 
nature of the weapon, enabled to ascertain all the successive 
energies of the propelling power, and the resisting force, 
which, in the other case, are only determinable at two or 
three different distances: on which account, it is to be pre¬ 
sumed that more advantage may be here expected to be 
gained from experiment, than in the cases above referred to. 
Instead of a ball impinging on the ballistic pendulum, at 
the distance of 60 or 100 yards, as practised in gunnery 
experiments, a rocket might be fixed to the same pendulum, 
and its whole energy observed with the greatest accuracy; 
or, in case such experiment should be thought inconclusive, 
for want of that partial vacuum which has place behind the 
rocket when in flight, it might be attached to some wheel, 
or revolving body, and its successive energies measured by 
the motion of some weight attached to the revolving axis of 
the machine. This is a most important advantage attending 
experiments on the momentum of rockets, which it is im¬ 
possible to accommodate to other projectiles. 
No such-experiments have yet been undertaken; and, 
therefore, all our investigations on the flight, momentum, 
&c. of rockets, must necessarily be hypothetical. In fact, 
we have two distinct theories of the motion of rockets, the 
one by Mariotte, and the other by Desaguliers; the latter 
attributing their motion to the momentum of combustion, 
and the other to the elastic nature of the gas generated by the 
combustion and the resistance of the air. Desaguliers illus¬ 
trates his hypothesis as follows: “Conceive the rocket to 
have no vent at the choke, and to be set on fire; the conse¬ 
quence will be, either that the rocket will burst in the weak¬ 
est place, or if all its parts be equally strong, and able to 
sustain the impulse of the flame, the rocket would burn out 
immoveable. Now as the force of the flame is equable, 
suppose its action downwards, or that upwards sufficient to 
lift 40 pounds; as these forces are equal, but their directions 
contrary, they will destroy each other's action. Imagine 
then the rocket opened at the choke; by this means, the 
action of the flame downwards is taken away, and there 
remains a force equal to 40 pounds acting upwards, to carry 
up the rocket and stick.” Although there is some ingenuity 
and plausibility in the above reasoning, we are by no means 
inclined to admit its accuracy. The action of the flame or 
gas within the rocket, when closed, as supposed above, we 
conceive to arise wholly from the elastic nature of the gas, 
and the re-action it experiences against the ends and sides of 
the rocket-case; the whole of which ceases, as soon as a free 
vent is given to the flame; and, therefore, if a rocket could 
be fired in a vacuum, as the flame would, in that case, expe¬ 
rience no resistance, there would be no re-action, and conse¬ 
quently no motion would ensue. In order to submit the 
above supposition to experiment, take a strong piece of 
whale-bone, and bend it into the form of a bow, by means 
of a bit of thread or silk fastened to each extremity: then 
if this bow be suspended by its middle, and two pieces of 
board, or two books, be set up on their edges, each touching 
one end of the bow, and the string by which it is bent be 
cut, both books will, from the elastic nature of the whale¬ 
bone, be thrown down with considerable force. Now repeat 
the experiment, but set up only one book, leaving the other 
end of the bow entirely free; then cut the string as before, 
and it will be found that, for want of the re-action of the 
other book, no effect, or very little, is produced on the stand¬ 
ing book: it may be a little disturbed, but it will not fall. 
K E T. 175 
This we consider to be a very similar case to the action of 
the gas on the rocket, when shut up and opened, as sup¬ 
posed by Desaguliers; and if so, it shews very distinctly the 
inaccuracy of his hypothesis. 
As a mere matter of mathematical investigation, it cer¬ 
tainly reduces the theory to the most simple form; because 
here it is not essential, so far as regards the propelling power, 
what may be the velocity of the rocket; which power is, 
therefore, supposed uniform during the whole time of com¬ 
bustion : whereas, in Mariotte’s theory, which attributes the 
motion of the rocket to the resistance or re-action of the air, 
the propelling force will decrease as the velocity increases, 
in consequence of the partial vacuum left behind the rocket 
in its flight; so that the velocity becomes, as it were, both a 
datum and quexsitum; and the correct solution of the pro¬ 
blem necessarily involves the integration of partial differences 
of the highest orders. 
As the problem under this formidable shape would be one 
of extreme difficulty, we shall prefer availing ourselves of a 
few problems relative to the motion and flight of rockets in 
non-resisting mediums, as given by Mr. Moore, of the Royal 
Military Academy, in his Treatise on the Motion and Flight 
of Rockets, who has adopted the hypothesis of Desaguliers, 
by supposing the motion of the rocket to arise from the 
momentum of the ignited composition. We shall also 
suppose the rocket and stick perfectly free the moment 
after being fired; for, without this, it is obvious that the 
angle of elevation of the rocket’s direction, and that of its 
actual discharge, will be essentially different. The first 
motion of the rocket, like all other motions not produced 
by a great momentary impulse, is slow; and before the stick 
is clear of the frame, gravity has been acting upon the rocket, 
and depressed it below its natural position, while the stick is 
prevented from being equally depressed, by the top of the 
frame; so that the angle of projection is in fact considerably 
less than the angle of the frame, or slope of the rocket’s first 
position. In consequence of this, the rocket has the appear¬ 
ance of falling the moment after the projection; and, for 
this reason also, the angle for producing the greatest range 
of a rocket exceeds very considerably that which gives the 
extreme range of a shell projected from a mortar. 
The strength or first force of the gas from the inflamed 
composition of a rocket being given, as also the weight and 
quantity of the composition, the time of its burning, and 
the weight and dimensions of the case and stick; to find 
the height to which it will ascend, when projected perpen¬ 
dicularly upwards. 
It is obvious here, that the principal point of investigation 
is the height to which the rocket will rise, and the velocity 
it will have acquired, at the moment when the composition 
is all expended; as the determination of its farther ascent, 
with these data, depends upon well-known and established 
principles. We shall, therefore, only consider the former 
case. For this purpose, put 
•w = the weight of the rocket-case and stick. 
c — the weight of the composition. 
a = the time in which it will be consumed. 
n — the medium pressure of the atmosphere. 
sn — the assumed force of the inflamed composition. 
d = the diameter of the rocket’s base, and p d 2 its area. 
x = the space described. And 
v = velocity acquired in any indeterminate time t. 
Then snp d 2 is the constant impelling force of the compo¬ 
sition. 
Now the weight of the quantity of rocket-matter that is 
consumed in the time t. is — ; therefore, c — — is the 
a a 
weight of the part unconsumed; and w + c— —, or 
a 
c t , 
in —-- (making m = w + c), is the weight of the whole 
mass at the end of the time t. 
Hence 
