Encyclopaedia Britannica > Volume 8, DIA-England

356
DYNAMICS.
i
Second 42. This kind of combination has been called the com-
Law of position of motion ; because, in every point of the motion
'Motion^ really pursued, the two motions are to be found.
Comnosi- ^le fundamental theorem on this subject is this : Two
tion of mo-unif°rm motions in the sides of a parallelogram compose
tion. an uniform motion in the diagonal.
Suppose that a point A (fig. 1) describes AB uniform¬
ly in some given time, while the line AB is carried uni¬
formly along AC in the same time, keeping always paral¬
lel to its first position AB. The point A, by the combi¬
nation of these motions, will describe AD, the diagonal of
the parallelogram ABDC, uniformly in the same time.
Fig. 1.
described in the same time. When the point has got to
E, the middle of AB, the line AB has got into the situa¬
tion GH, half way between AB and CD, and the point E
is in the place e, the middle of GH. Draw EeL parallel
to AC. It is plain that the parallelograms ABDC and
AEeG are similar; because AE and AG are the halves of
AB and AC, and the angle at A is common to botli.
Therefore, by a proposition in the elements, they are
about the same diagonal, and the point e is in the diago¬
nal of AD. In like manner, it may be shown, that when
A has described AF, three fourths of AB, the line AB
will be in the situation IK ; so that AI is three fourths of
AC, and the point f in which A is now found, is in the
diagonal AD. It will be the same in whatever point of AB
the describing point A be supposed to be found. The line
AB will be on a similar point of AC, and the describing
point will be in the diagonal AD.
Moreover, the motion in AD is uniform; for Ae is de¬
scribed in the time of describing AE ; that is, in half the
time of describing AB, or in half the time of describing
AD. In like manner, Kf is described in three fourths of
the time of describing AD, &c. &c.
Lastly, the velocity in the diagonal AD is to the velo¬
city in either of the sides as AD is to that side. This is
evident, because they are uniformly described in the same
time.
This is justly called a composition of the motions AB and
AC, as will appear bj' considering it in the following man¬
ner: Let the lines AB, AC be conceived as two material
lines like wires. Let AB move uniformly from the situa¬
tion AB into the situation CD, while AC moves uniformly
into the situation BD. It is plain that their intersection
will always be found on AD. The point e, for example, is a
point common to both lines. Considered as a point of EL,
it is then moving in the direction eH or AB ; and con¬
sidered as a point of GH, it is moving in the direction eL.
Both of these motions are therefore blended in the motion
of the intersection along AD. We can conceive a small
ring at e embracing loosely both of the wires. This ma¬
terial ring will move in the diagonal, and will really par¬
take of both motions.
Thus we see how the motion of the ship is actually
blended with the motions of the three men ; and the cir-
ehmstance of sameness which is to be found in the four Sect
changes of motion is this motion of the ship, or of the Can
man who was standing still. By composition with each ^
of the three former motions, it produces each of the three
new motions. Now, when each of two primitive motions
is the same, and each of the new motions is the same, the
change is surely the same. If one of the changes has
been brought about by the actual composition of motions
we know precisely what that change is ; and this informs
us what the other is, in whatever way it was produced.
Hence we infer that,
43. When a motion is any how changed, the change is Its im
that motion which, when compounded with the former mo- and ml
tion, will produce the new motion. Now, because we assume sure•
the change as the measure and characteristic of the chan¬
ging force, we must do so in the present instance ; and we
must say,
44. That the changing force is that which will produce in Chang
a quiescent body the motion which, by composition with the force,
former motion of a body, will produce the new motion.
And, on the other hand,
When the motion of a body is changed by the action o/its efi;
any force, the new motion is that which is compounded of the
former motion, and of the motion which the force woidd pro¬
duce in a quiescent body.
When a force changes the direction of a motion, we seeDefbc r
that its direction is transverse in some angle BAG; be-force. ’
cause a diagonal AD always supposes two sides. As we
have distinguished any change of direction by the term
deflection, we may call the transverse force a deflecting
force.
In this way of estimating a change of motion, all the
characters of both motions are preserved, and it expresses
every circumstance of the change ; the mere change of di¬
rection, or the angle BAD, is not enough, because the
same force will make different angles of deflection, accord¬
ing to the velocity of the former motion, or according to
its direction. But in this estimation the full effect of the
deflecting force is seen ; it is seen as a motion; for when
half of the time is elapsed, the body is at e instead of E;
when three fourths are elapsed it is at /instead of F; and
at the end of the time it is at D instead of B. In short, the
body has moved uniformly away from the points at which
it would have arrived independent of the change ; and this
motion has been in the same direction, and at the same
rate, as if it had moved from A to C by the changing
force alone. Each force has produced its full effect; for
when the body is at D, it is as far from AC as if the force
AC had not acted on it; and it is as far from AB as it would
have been by the action of AC alone.
For all these reasons, therefore, it is evident, that if we
are to abide by our measure and character of force as a
mere producer of motion, we have selected the proper
characteristic and measure of a changing force; and our
descriptions, in conformity to this selection, must be agree¬
able to the phenomena of nature, and retain the accuracy
of geometrical procedure; because, on the other hand,
the results which we deduce from the supposed influence
of those forces are formed in the same mould. It is not
even requisite that the real exertions of the natural forces,
such as pressure of various kinds, &c. shall follow these
rules; for their deviations will be considered as new
forces, although they are only indications of the differen¬
ces of the real forces from our hypothesis. We have ob¬
tained the precious advantage of mathematical investiga¬
tion, by which we can examine the law of exertion which
characterizes every force in nature.
45. On these principles we establish the following fun¬
damental elementary proposition, of continual and indis¬
pensable use in all mechanical inquiries.