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 from

X

dex dt2

_day =

=0; X- =0; Z- = 0

dt2

dez
dt2

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 dx, 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

Xdr + Ydy + Zdz=

dxdx

dydy + dzdz
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

parallel to the axis x.

forces resolved in a direction In the same manner, Y = Σ.FY; Z = Σ.FZ are the sums of the forces

resolved in a direction parallel to the axes y and z, so that

Xdx+Ydy + Zdz = 2.Fxdx+ydy+ zdz = 2. 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 f(Xdx + Ydy + Zdz), or f (x, y, z,) a function of x, y, z; and by dx2+dy2+dz integration the second is which is evidently the half dt3

m

M

D

fig. 19

of the square of the velocity; for if any curve MN, fig. 19, be represeented by s, its first differential ds or Am is

√AD2 + Dm2 = √dx2 + dy2; hence, ds2dx2 + dy2 when the curve

is in one plane, but when in space it is ds dx2 + dy2 + dz:

[the element of the space divided by the element of the

c being an arbitrary constant quantity introduced by integration. 77. This equation will give the velocity of the particle in any point of its path, provided its velocity in any other point be known: for if A be its velocity in that point of its trajectory whose co-ordinates are a, b, c, then

and

Ac+ 2f (a, b, c),

v2 - A2 = 2ƒ (x, y, z) - 2f (a, b, c);

whence v will be found when A is given, and the co-ordinates a, b, c, x, y, z, are known.

It is evident, from the equation being independent of any particular curve, that if the particle begins to move from any given point with a given velocity, it will arrive at another given point with the same velocity, whatever the curve may be that it has described.

78. When the particle is not acted on by any forces, then X, Y, and Z are zero, and the equation becomes v2 = c. The velocity in this case, being occasioned by a primitive impulse, will be constant; and the particle, in moving from one given point to another, will always take the shortest path that can be traced between these points, which is a particular case of a more general law, called the principle of Least Action.

fig. 20.

Principle of Least Action.

79. Suppose a particle beginning to move from a given point A, fig. 20, to arrive at another given point B, and that its velocity at the point A is given in magnitude but not in direction. Suppose also that it is urged by accelerating forces X, Y, Z, such, that the finite value of Xdx + Ydy + Zdz can be obtained. We may then determine v the velocity of the particle in terms of x, y, z, without knowing the curve described by the

B

particle in moving from A to B. If ds be the element of the curve, the finite value of vds between A and B will depend on the nature of the path or curve in which the body moves. The principle of Least Action consists in this, that if the particle be free to move in every direction between these two points, except in so far as it obeys the action of the forces X, Y, Z, it will in virtue of this action, choose the path in which the integral svds is a minimum; and if it be constrained to move on a given surface, it will still move in the curve in which fvds is a minimum among all those that can be traced on the surface between the given points.

To demonstrate this principle, it is required to prove the variation of fvds to be zero, when A and B, the extreme points of the curve are fixed.

By the method of variations &fvds=fd.vds: for the mark of integration being relative to the differentials, is independent of the variations.

Now 8.vds dv. ds + vôds, but v =

hence

and therefore

ds

or ds = vdt;

dt

Sv. ds vdvdt = dtd.v2,

8.vds= dt. 8.v2 + v.§.ds.

The values of the two last terms of this equation must be found separately. To find dt. d.v. It has been shown that

v = c + 2/ (Xdr + Ydy + Zda),

its differential is vdv (Xdr + Ydy + Zdz),

and changing the differentials into variations,

đô.v = Xôr + Yông+Zôz.

Ifd.v2 be substituted in the general equation of the motion of a particle on its surface, it becomes

But du does not enter into this equation when the particle is free; and when it must move on the surface whose equation is u = 0, du is also zero; hence in every case the term Adu vanishes; there

A value of the

first term required.

second term v.8.ds must now be found. Since dsdx2 + dy + dz3,

its variation is ds. dds dr. ddx+dy.ddy + dz.ddz, but ds = vdt,

which is the value of the second term; and if the two be added,

as may easily be seen by taking the differential of the last member of this equation. Its integral is

If the given points A and B be moveable in space, the last member of this equation will determine their motion; but if they be fixed points, the last member which is the variation of the co-ordinates of these points is zero: hence also dvds = 0, which indicates either a maximum or minimum, but it is evident from the nature of the problem that it can only be a minimum. If the particle be not urged by accelerating forces, the velocity is constant, and the integral is vs. Then the curve s described by the particle between the points A and B is a minimum; and since the velocity is uniform, the particle will describe that curve in a shorter time than it would have done any other curve that could be drawn between these two points.

80. The principle of least action was first discovered by Euler : it has been very elegantly applied to the reflection and refraction of light. If a ray of light IS, fig. 21, falls on any surface CD, it

fig. 21.

S

-D

will be turned back or reflected in the direction Sr, so that ISA=rSA. But if the medium whose surface is CD be diaphanous, as glass or water, it will be broken or refracted at S, and will enter the denser medium in the direction SR, so that the sine of the angle of incidence ISA will be to the sine of the angle of refraction RSB, in a constant ratio for

any one medium. Ptolemy discovered that light, when reflected from any surface, passed from one given point to another by the shortest path, and in the shortest time possible, its velocity being uniform.