RESISTANCE. 
5GD- 
nous ones, is very small in respect of the other 
resistance ; and it also increases in a much 
lower degree, being only as the velocity, 
while the other increases as the square of 
the velocity, and rather more. Hence, then 
the term btv is very small in respect of the 
other term adv 2 ; and consequently the re- 
sistance is nearly as this latter term, or near- 
ly as the square of the velocity. This rule 
has been employed by most authors, and is 
very near the truth in slow motions ; but in 
very rapid ones, it differs considerably from 
the truth, as we shall perceive below ; not 
indeed from the omission of the small term 
ctv, due to the cohesion, but from the want 
of the full counterpressure on the hinder 
part of the body; a vacuum, either perfect or 
partial, being left behind the body in its 
motion ; and also perhaps to some compres- 
• sion or accumulation of the fluid against the 
fore part of the body. 
Resistance and retardation are used indif- 
ferently for each other, as being both in the 
same proportion, and the same resistance 
always generating the same retardation. But 
with* regard to different bodies, the same re- 
sistance frequently generates different retar- 
dations ; the resistance being as the quantity 
of motion, and the retardation as that of the 
celerity. 
The retardations from this resistance may be 
compared together, by comparing the resistance 
with the gravity or quantity of matter. It is 
demonstrated that the resistance of a cylinder, 
which moves in the direction of its axis, is equal 
to the weight of a column of the fluid, whose 
Rase is equal to that of the cylinder, and its al- 
titude equal to the height through which a body 
must fall in vacuo, by the force of gravity, to 
acquire the velocity of the moving body. So 
that, if a denotes the area of the face or end of 
the cylinder, or other prism, v its velocity, and 
n the specific gravity of the fluid; then, the alti- 
tude due to the velocity v being — , the whole 
resistance, or motive force m, will be n X d X 
2 2 
' v anv , . , . 
• — ~ - ; the quantity g being = 16 T V 
4 rt % 
feet, or the space a body falls, in vacuo, in the 
first second of time. And the resistance to a 
globe of the same diameter would be the half 
of this. Let a ball, for instance, of 3 inches 
diameter, be moved in water with a celerity of 
16 feet per second of time : now from experi- 
ments on pendulums, and on falling bodies, it 
has been found, that this is the celerity which a 
body acquires in falling from the height of 4 
feet ; therefore the weight of a cylinder of water 
of 3 inches diameter, and 4 feet high, that is, a 
weight of about 121b. 4oz., is equal to the re- 
flistance of the cylinder; and consequently the 
half of it, or 6 lb. 2oz., is that of the ball. 
anv 1 
Or, the formula — — gives 
% 
.7854 X 9 X 1000 X 16 X 16 
144~X 4 X 16 ~ ~~ 
196 oz., or 12 lb. 4oz , for the resistance of the 
cylinder, or 6 lb. 2 oz. for that of the ball, the 
same as before. 
Let now the resistance, so discovered, be di- 
vided by the weight of the body, and the quo- 
tient will shew the ratio of the retardation to 
the force of gravity, So if the said ball, of 3 
inches diameter, is of cast iron, it will weigh 
nearly 61 ounces, or 3ylb. ; and the resistance 
being 6 lb. 2 oz., or 98 ounces, therefore the 
resistance being to the gravity as 98 to 61, the 
retardation, or retarding force, will be or 
1 . 1 , the force of gravity being 1. Or thus ; be- 
VoL. II. 
cause a the area of a great circle of the ball, is 
z=zpd\ where d is the diameter, and p = .7854, 
therefore the resistance to the ball is m — 
V — ; and because its solid content is zv z=. 
8? 
ifid', and its weight |N^', where N denotes 
its specific gravity, therefore dividing the re- 
sistance or motive force m by the weight zv, gives 
r. 2 
— f the retardation, or retarding 
1 6 M;/iy 
force, that of gravity being 1 ; which is there- 
fore as the square of the velocity directly, and 
as the diameter inversely; and this is the reason 
why a large ball overcomes the resistance better 
than a small one, of the same density. 
RESISTANCE of Fluid Mediums to the Motion of 
F allinn Bodies. A body freely descending in a 
fluid, is accelerated by the relative gravity of 
the bodv, (that is, the difference between its 
own absolute gravity and that of a like bulk of 
the fluid,) which continually acts upon it, yet 
not equably, as in a vacuum : the resistance of 
the fluid occasions a retardation, or diminution 
of acceleration, which diminution increases with 
the velocity of the body. Hence it happens, 
that there is a certain velocity, which is the 
greatest that a body can acquire by falling ; for 
if its velocity is such, that the resistance aris- 
ing from it becomes equal to the relative weight 
of the body, its motion can be no longer acce- 
lerated ; for the motion here continually gene- 
rated by the relative gravity, will be destroyed 
by the resistance; or the force of resistance will 
be equal to the relative gravity, and the body 
forced to go on equably : for, after the velocity 
is arrived at such a degree, that the resisting 
force is equal to the weight that urges it, it will 
increase no longer, and the globe must after- 
ward continue to descend with that velocity 
uniformly. A body continually comes nearer 
and nearer to this greatest celerity, but can ne- 
ver attain accurately to it Now, N and n being 
the specific gravities of the globe and fluid, 
N — ■ n will be the relative gravity of the globe 
in the fluid, anrd therefore zv — jpd' (N — n) is 
the weight by which it is urged downward; also 
m — P—-— is the resistance, as above ; there- 
% 
fore these two must be equal when the velocity 
can be no farther increased, or m ■=. zv, that is 
= If*' (N — «), or nz> 1 = V J S 
(N — n) ; and hence z> = ^ f ‘y dg X — - 
is the said uniform or greatest velocity to which 
the body may attain ; which is evidently the 
greater in the subduplicate proportion of v the 
diameter of the ball. But v is always — */4gfs, 
the velocity generated by any accelerative force 
f in describing the space s ; which being com- 
pared with the former, it gives s = ^d, when / 
is = ; that is, the greatest velocity is 
n 
that which is generated by the accelerating force 
- in passing over the space or of the 
diameter of the ball, or it is equal to the velo- 
city generated by gravity in describing the space 
x td. For example : if the ball is of 
71 3 
lead, which is about Uptimes the density of 
N — n 
water ; then N = 11$, n — 1, N — n zs 
10$ s= *J, and 
X 
= 4 d = 
3 a 
13 $<f.; that is, the uniform or greatest velocity 
4 Cp 
j of a ball of lead, descending in water, is equal 
j to that which a heavy body acquires by falling 
| in vacuo through a space equal to 13-"- of the 
diameter of the ball, which velocity is z> — 2 
y/ ±dg X = 2 = 8 
nearly, or 8 times the root of the same space. 
Hence it appears, how soon small bodies 
come to their greatest or uniform velocity m 
descending in a fluid, as water, and how very 
small that velocity is ; which explains the reason 
of the slow precipitation of mud, and small par- 
ticles, in water ; as also why, in precipitations, 
the larger and gross particles descend soonest, 
and the lowest. 
Farther, where N = «, or the density of the 
fluid is equal to that of the body, then N n 
— 0, consequently the velocity and distance de- 
scended are each nothing, and the body will 
just float in any part of the fluid. 
Moreover, when the body is lighter than the- 
fluid, then N is less than n, and N — n ^ecomes 
a negative quantity, or the force and motion 
tend the contrary way, that is, the ball will as- 
scend up towards the top of the fluid by a mo- 
tive force which is as n — N. In this case, then, 
the body ascending by the action of the fluid, is 
moved exactly by the same laws as a heavier 
body falling in the fluid. Wherever the body 
is placed, it is sustained by the fluid, and car- 
ried lip with a force equal to the difference of 
the weight of a quantity of the fluid of the same 
bulk as the body, from the weight of the body ; 
there is therefore a force which continually acts 
equably upon the body ; by which not only the 
action of gravity of the body is counteracted, so 
as that it is not to be considered in this case ; . 
but the body is also carried upwards by a mo- 
tion equably accelerated, in the same manner as 
a body heavier than a fluid descends by its re- 
lative gravity: but the equability of accelera-. 
tion is destroyed in the same manner by the re- 
sistance, in the ascent of a body lighter than the, 
fluid, as it is destroyed in the descent of a body 
that is heavier. 
For the circumstances of the correspondent 
velocity, space, and time, &c. of a body moving 
in a fluid in which it is projected with a given 
velocity, or descending by its own weight, &c. 
sGe Hr. Hutton’s Select Exercises, prop. 29, 80, 
31, and 32, page 221, &c. 
Resistance of the Air , is the force with which 
the motion of bodies, particularly of projectiles, 
is retarded by the opposition of the air or at- 
mosphere. See Gunnery, Projectiles, &c. 
The air being a fluid, the general laws of the 
resistance of fluids obtain in it ; subject only to 
some variations and irregularities from the dif- 
ferent degrees of density in the different stations 
or regions of the atmosphere. 
The resistance of the air is chiefly of use in 
military projectiles, in order to allow for the 
differences caused in their flight and range by 
it. Before the time of Mr. Robins, it was 
th®ught that this resistance to the motion of 
such heavy bodies as iron balls and shells, was 
too inconsiderable to be regarded; and that the 
rules and conclusions derived from the common 
parabolic theory, were sufficiently exact for the 
common practice of gunnery. But that gentle- 
man shewed, in his New Principles, of Gunnery, 
that, so far from being inconsiderable, it is m 
reality enormously great, and by no means to 
be rejected without incurring the grossest er- 
rors; so much so, that balls or shells which 
range, at the most, in the air, to the distance 
of two or three miles, would in a vacuum range 
to 20 or 30 miles, or more. To determine the 
quantity of this resistance, in the case of dif- 
ferent velocities, Mr. Robins discharged musket- 
balls, with various degrees of known velocity, 
against his balistic pendulums, placed at several 
