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ma; hence at the end of that time the particle would be found in
interval it would be found in D, having described the diagonal BD, and so on. In this manner the particle would describe the polygon mBDE; but if the intervals between the successive impulses of the attractive force be indefinitely small, the diagonals mB, BD, DE, &c., will also be indefinitely small, and will coincide with the curve passing through the points m, B, D, E, &c.
64. In this hypothesis, no error can arise from assuming that the particle describes the sides of a polygon with a uniform motion; for the polygon, when the number of its sides is indefinitely multiplied, coincides entirely with the curve.
65. The lines mA, BC, &c., fig. 17, are tangents to the curve in the points, m, B, &c.; it therefore follows that when a particle is moving in a curved line in consequence of any continued force, if the force should cease to act at any instant, the particle would move on in the tangent with an equable motion, and with a velocity equal to what it had acquired when the force ceased to act.
of the circle coinciding with the curve in m.
67. We shall consider the element or differential of time to be a constant quantity; the element of space to be the indefinitely small
space moved over in an element of time, and the element of velocity to be the velocity that a particle would acquire, if acted on by a constant force during an element of time. Thus, if t, s and v be the time, space, and velocity, the elements of these quantities are dt, ds, and dv; and as each element is supposed to express an arbitrary unit of its kind, these heterogeneous quantities become capable of comparison. As a decrement only differs from an increment by its sign, any expressions regarding increasing quantities will apply to those that decrease by changing the signs of the differentials; and thus the theory of retarded motion is included in that of accelerated motion.
68. In uniformly accelerated motion, the force at any instant is directly proportional to the second element of the space, and inversely as the square of the element of the time.
Demonstration.-Because in uniformly accelerated motion, the velocity is only assumed to be constant for an indefinitely small time, v= and as the element of the time is constant, the dif
ds dt' ferential of the velocity is dv=
ing for an indefinitely small velocity, Fdt = dv; hence F
time, produces an indefinitely small d2s
General Equations of the Motions of a Particle of Matter.
69. The general equation of the motion of a particle of matter, when acted on by any forces whatever, may be reduced to depend on the law of equilibrium.
Demonstration.-Let m be a particle of matter perfectly free to obey any forces X, Y, Z, urging it in the direction of three rectangular co-ordinates x, y, z. Then regarding velocity as an effect of force, and as its measure, by the laws of motion these forces will produce in the instant dt, the velocities Xdt, Ydt, Zdt, proportional to the intensities of these forces, and in their directions. Hence when m is free, by article 68,
for the forces X, Y, Z, being perpendicular to each other, each one is independent of the action of the other two, and may be regarded as
if it acted alone. If the first of these equations be multiplied by dx, the second by dy, and the third by dz, their sum will be
Sr, dy, dz, are absolutely arbitrary and independent; and vice versa, if they are so, this one equation will be equivalent to the three separate ones.
This is the general equation of the motion of a particle of matter, when free to move in every direction.
2nd case. But if the particle m be not free, it must either be constrained to move on a curve, or on a surface, or be subject to a resistance, or otherwise subject to some condition. But matter is not moved otherwise than by force; therefore, whatever constrains it, or subjects it to conditions, is a force. If a curve, or surface, or a string constrains it, the force is called reaction: if a fluid medium, the force is called resistance: if a condition however abstract, (as for example that it move in a tautochrone,) still this condition, by obliging it to move out of its free course, or with an unnatural velocity, must ultimately resolve itself into force; only that in this case it is an implicit and not an explicit function of the co-ordinates. This new force may therefore be considered first, as involved in X, Y, Z; or secondly, as added to them when it is resolved into X', Y', Z'.
In the first case, if it be regarded as included in X, Y, Z, these really contain an indeterminate function: but the equations
Now however, there are not enough of equations to determine x, y, z, in functions of t, because of the unknown forms of X', Y', Z'; but if the equation u = 0, which expresses the condition of restraint, with all its consequences du = 0, Su= 0, &c., be superadded to these, there will then be enough to determine the problem. Thus the equations are
u = 0; X
u is a function of x, y, z, X, Y, Z, and t. Therefore the equation u = 0 establishes the existence of a relation
du = pdx + qòy + rdx = 0
between the variations de, dy, dz, which can no longer be regarded as arbitrary; but the equation (6) subsists whether they be so or not, and may therefore be used simultaneously with du 0 to eliminate one; after which the other two being really arbitrary, their co-efficients must be separately zero.
In the second case; if we do not regard the forces arising from the conditions of constraint as involved in X, Y, Z, let du = 0 be that condition, and let X', Y', Z', be the unknown forces brought into action by that condition, by which the action of X, Y, Z, is modified; then will the whole forces acting on m be X+X', Y+Y', Z+Z, and under the influence of these the particle will move as a free particle; and therefore dx, dy, dz, being any variations
and this equation is independent of any particular relation between Sr, dy, dz, and holds good whether they subsist or not. But the condition du 0 establishes a relation of the form pdx+qòy+rdz = 0,
and since this is true, it is so when multiplied by any arbitrary quantity ; therefore,
λ (pdx + qdy + rdz) = 0, or λdu = 0;
du = pdx + qoy + rồz ≈ 0.
If this be added to equation (7), it becomes
Sy + (Z
+ X'dx + Y'dy + Z'sz - λdu,
which is true whatever dx, dy, dz, or λ may be.
Now since X', Y', Z', are forces acting in the direction x, y, z, (though unknown) they may be compounded into one resultant R,, which must have one direction, whose element may be represented
by ds. And since the single force R, is resolved into X', Y', Z', we X'dx + Y'dy + Zdz = R1ds;
and this is true whatever may be.
But A being thus left arbitrary, we are at liberty to determine it by any convenient condition. Let this condition be Rds - Adu=0,
or λ = R,. which reduces equation (8) to equation (6). So
when X, Y, Z, are the only acting forces explicitly given, this equation still suffices to resolve the problem, provided it be taken in conjunction with the equation du 0, or, which is the same thing, por + qoy + rồz = 0,
which establishes a relation between dr, dy, dz.
Now let the condition as.. be considered which determines A.
Since R, is the resultant of the forces X', Y', Z', its magnitude must be represented by √X2 + Y2 + Z by article 37, and since R,ds=λdu, or
X'dx + Y'dy + Z'dz = λ. du 8x +
remain arbitrary, we must
R1 = √X" + Y + Za2 = λ.
be the angles that the normal to the curve or surface makes with the
X' R,. cos a, Y, R,. cos B, Z, R,. cos y.