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, Sy, Sz, 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

still subsist; and therefore also equation (6).

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, du= 0, &c., be superadded to these, there will then be enough to determine the problem. Thus the equations are

dez
= 0.
dt2

u is a function of x, y, z, X, Y, Z, and t. Therefore the equation u = 0 establishes the existence of a relation

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+qdy+rdz = 0,

and since this is true, it is so when multiplied by any arbitrary quantity ; therefore,

because

λ (pdx + qoy + rdz) = 0, or λdu = 0;

du= pdx + q8y + rz = 0.

If this be added to equation (7), it becomes

0=(X

d'x

dt

(Y - dy) by + (Z.

+ X'dx + Y'dy + Zdz - λdu,

which is true whatever dx, dy, dz, or λ may be.

d2z

dt

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

must have

X'dx + Y'sy + Z'd≈ = R‚ds ;

so that the preceding equation becomes

But A being thus left arbitrary, we are at liberty to determine it by any convenient condition. Let this condition be R ̧ds — λdu — 0, or λ = R,.. which reduces equation (8) to equation (6). So " δι

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, pdx + qoy + rdz 0,

=

which establishes a relation between dr, dy, Sz.

Now let the condition a=s. be considered which determiner A.

12

ds δι

Since R, is the resultant of the forces X', Y', Z', its magnitude must be represented by √X^2 + Y + Z2 by article 37, and since R1ds = λdu, or

therefore, in order that dx, dy, dz, may remain arbitrary, we must

dy

=K; then if a, B, Y,

be the angles that the normal to the curve or surface makes with the

co-ordinates,

and

=

X' R,. cosa, Y, R,.cos B, Z, R,. cos y.

Thus if u be given in terms of x, y, z; the four quantities λ, X', Y', and Z', will be determined. If the condition of constraint expressed by u = 0 be pressure against a surface, R, is the re-action.

Thus the general equation of a particle of matter moving on a curved surface, or subject to any given condition of constraint, is proved to be

70. The whole theory of the motion of a particle of matter is contained in equations (6) and (10); but the finite values of these equations can only be found when the variations of the forces are expressed at least implicitly in functions of the distance of the moving particle from their origin.

71. When the particle is free, if the forces X, Y, Z, be eliminated

by functions of the distance, these equations, which then may be integrated at least by approximation, will only contain space and time; and by the elimination of the latter, two equations will remain, both functions of the co-ordinates which will determine the curve in which the particle moves.

72. Because the force which urges a particle of matter in motion, is given in functions of the indefinitely small increments of the coordinates, the path or trajectory of the particle depends on the nature of the force. Hence if the force be given, the curve in which the particle moves may be found; and if the curve be given, the law of the force may be determined.

73. Since one constant quantity may vanish from an equation at each differentiation, so one must be added at each integration; hence the integral of the three equations of the motion of a particle being of the second order, will contain six arbitrary constant quantities, which are the data of the problem, and are determined in each case either by observation, or by some known circumstances peculiar to each problem.

74. In most cases finite values of the general equation of the motion of a particle cannot be obtained, unless the law according to which the force varies with the distance be known; but by assuming from experience, that the intensity of the forces in nature

varies according to some law of the distance and leaving them otherwise indeterminate, it is possible to deduce certain properties of a moving particle, so general that they would exist whatever the forces might in other respects be. Though the variations differ materially, and must be carefully distinguished from the differentials dr, dy, dz, which are the spaces moved over by the particle parallel to the co-ordinates in the instant dt; yet being arbitrary, we may assume them to be equal to these, or to any other quantities consistent with the nature of the problem under consideration. Therefore let dr, dy, dz, be assumed equal to dx, dy, dz, in the general equation of motion (6), which becomes in consequence

Xdc +Ydy + Zdz = dxdx + dyd'y + dzď z

dt2

75. The integral of this equation can only be obtained when the first member is a complete differential, which it will be if all the forces acting on the particle, in whatever directions, be functions of its distance from their origin.

Demonstration.-If F be a force acting on the particle, and s

the distance of the particle from its origin, F is the resolved portion parallel to the axis x; and if F', F", &c., be the other forces acting on the particle, then X . F will be the sum of all these forces resolved in a direction parallel to the axis x. In the same y manner, Y = Σ.F ; Z=Σ.F are the sums of the forces

resolved in a direction parallel to the axes y and z, so that Xdx+Ydy + Zdz = Σ. Fxdx+ydy+ zdz

= Σ. F sds = Σ. Fds,

which is a complete differential when F, F', &c., are functions of s. 76. In this case, the integral of the first member of the equation is (Xdx + Ydy + Zdz), or f(x, y, z,) a function of x, y, z; and by dx2+dy+dz integration the second is which is evidently the half dts