Encyclopaedia Britannica > Volume 8, DIA-England

364
DYNAMICS.
Second therefore e?A is equal to the same sum, and in equilibrio
Law of wjfh them.
Motion.
Secondly, Let them be reduced to one plane, EFGH,
and let a/S, ad, be the reduced forces. The lines Dd,
Aa, B/3, Cx, LX, are all parallel, being perpendicular to
the plane ; therefore the planes AB, /Sa, and CL, Xx, are
parallel, and a/3, xX, are parallel. For similar reasons,
/3X, ax, are parallel; therefore afiXx is a parallelogram.
Also, because the lines D§, Aa, Lx, are parallel, and DA
is equal to AL; therefore da is equal to aX. But be¬
cause afiXx is a parallelogram, the forces afi, ax, are equi¬
valent to aTi; and ad is equal and opposite to aX, and will
balance it; and therefore will balance afi and ax, which
are the reductions of AB and AC to the plane EFGH,
while ad is the reduction of AD; therefore the proposi¬
tion is demonstrated.
The most The most usual and the most useful mode of reduction
useful is to estimate all forces in the directions of three lines
mode of re-drawn from one point, at right angles to each other, like
to their co-^e three plane angles of a rectangular chest, forming the
ordinates, length, the breadth, and the depth of the chest. These
are commonly called the three co-ordinates. The result¬
ing force will be the diagonal of this parallelepiped. This
process occurs in all disquisitions in which the mutual ac¬
tion of solids and fluids is considered, and when the oscil¬
lation or rotation of detached free bodies is the subject of
discussion.
Relative 67. The only other general theorem that remains to be
motions of deduced from this law of motion is, that if a number of
affected In bodies are moving in an7 manner whatever, and an equal
any extra^ ^orce. act on every particle of matter in the same or paral-
neous " directions, their relative motions will suffer no change ;
equal and Fi 14
parallel 1 ’S'
force.
for the motion of any body A (fig. 14), relative to another
body B, which is also in motion, is compounded of the
real motion of A, and the opposite to the real motion of
B; for let A move uniformly from A to C, while B de¬
scribes BD uniformly; draw AB, also draw AE equal and
parallel to BD ; join EC, DC, ED. The motion of A, re- Sec
lative to B, consists in its change of position and distance. La* >
Had A described AE, while B described BD, there would Mot
have been no change of relative place or distance; but aH
is now at C, and DC is its new direction and distance.
The relative or apparent motion of A therefore is Ec!
Complete the parallelogram ACFE; it is plain that the
motion EC is compounded of EF, which is equal and pa¬
rallel to AC, the real motion of A, and of EA, the equal
and opposite to BD, the real motion of B.
Now let the motions of A and B sustain the same chano-e;
let the equal and parallel motions AG, BH, be compound¬
ed with the motions AC and BD ; or let forces act at
once on A and B, in the parallel directions AG, BH, and
with equal intensities ; in either supposition, the resulting
motions will be Ac, ¥>d, the diagonals of the parallelograms
AGcC and BHc?D. Construct the figure as before, and
we see that the relative motion is now ec, and that it is
the same with EC both in respect of magnitude and po¬
sition.
Here we still see the constant analogy between the
composition of motions and the composition of forces. In
the first case, the relative motions of things are not chan¬
ged, whatever common motion be compounded with them
all; or, as it is usually but inaccurately expressed, al¬
though the space in which they move be carried along
with any motion whatever. In the second case, the rela¬
tive motions and actions are not changed by any external
force, however great, when equally exerted on every par¬
ticle in parallel directions.
Thus it is that the evolutions of a fleet in a uniform
current are the same, and produced by the same means,
as in still water. Thus it is that we walk about on the
surface of this globe in the same manner as if it neither
r'evolved round the sun, nor turned round its axis. Thus
it is that the same strength of a bow will communicate a
certain velocity to an arrow, whether it is shot east, or
west, or north, or south. Thus it is that the mutual ac¬
tions of sublunary bodies are the same, in whatever direc¬
tions they are exerted, and notwithstanding the very great
changes in their velocities by reason of the earth’s rota¬
tion and orbital revolution. The real velocity of a body
on the earth’s equator is about 3000 feet per second great¬
er at midnight than at mid-day. For at midnight the mo¬
tion of rotation nearly conspires with the orbital motion,
and at mid-day it nearly opposes it. The difference be¬
tween the velocities at the beginning of January and the
beginning of July is vastly greater. And at other times
of the day, and other seasons of the year, both motions of
the earth are transversely compounded with the easterly
or westerly^ motion of an arrow or cannon bullet; yet we
can observe no change in the effects of the mutual actions
of bodies.
68. This is an important observation, because it proves This
that forces are to be measured by no other scale than by fords e*
the motions which they produce. We have had repeated1?01181
occasions to mention the very different estimation of mov-11™,'1 L
ing forces by Mr Leibnitz; and have shown how, by a [ioMi 0f
very partial consideration of the action of those natural movii
powers called pressures, he has attempted to prove that forces
moving forces are proportional to the squares of the velo-1}16111
cities; and we showed briefly in what manner a right
consideration of what passes when motion is produced by^^
measurable pressures, proves that the forces really exert¬
ed are as the velocities produced. But the most copious
proof is had from the present observation, that, in fact, the
mutual actions of bodies depend on their relative motions
alone.
69. The Leibnitzian measure of moving force is altoge¬
ther incompatible with the universal fact now mentioned,