Let be the normal distance at any point between the consecutive surfaces

where Sx, Sy, Sz are the relative co-ordinates of any two contiguous points on the two surfaces. If p be the length of the line joining these points, its inclination to the normal (i.e. the line of motion) this may evidently be written

vp cos =V@= = SC,

since p cos is the normal distance between the surfaces.

Thus, the distance between consecutive equiactional surfaces is, at any point, inversely as the velocity in the corresponding path.

This may be seen at once as follows; the element of the action is vds (where ds, being an element of the path, is the normal distance between the surfaces) and must therefore be equal to &C.

255. To deduce, from the principle of Varying Action, the form and mode of description of a planet's orbit.

In this case it is obvious that the force of gravity (-)

Hence the right hand member of § 250 (1)

may be written 2 (H+).

Let us take the plane of xy as that of the orbit, then the equation § 250 (1) becomes

It is not difficult to obtain a satisfactory solution of this equation; but the operation is very much simplified by the use of polar co-ordinates. With this, (1) becomes

2

a

וי

.(4).

2H+

-

A = a + dr
+ √ dr √2 (11 + 1) − = *

The final integrals are therefore, by § 253,

These equations contain the complete solution of the problem, for they involve four constants, A, a, H, e. (5) gives the equation of the orbit, and (6) the time in terms of the radius-vector.

256. To illustrate the subject farther, we will deduce the ordinary results of Chaps. V. and VI. from these formulæ. Thus, let 0, ro denote the polar co-ordinates of any fixed point in the path, from which the action is to be reckoned. We have, by (4),

2

To integrate (7), remark that (§ 140) < in an elliptic orbit, and that thus H is negative by § 255 (1).

By employing the same substitutions as in last section, it is easy to bring this expression into the form

258. By the process of § 152 we see that while

e sin o is proportional to the area described about the center of force, and therefore proportional to the time; +e sin & is proportional to the area described about the other focus, and is, by § 256, proportional to the action. Thus the time is measured by the area described about one focus, and the action by that about the other.

An easy verification of this curious result is as follows. With the usual notation we have

dAvds,

But in the ellipse or hyperbola, p' being the perpendicular from the second focus,

which expresses the result sought.

It is easy to extend this to a parabolic orbit, for which, indeed, the theorem is even more simple.

259. When a particle moves in any curve, it has been shewn (§§ 16, 17), that the acceleration along the radius of absolute curvature of the path is; that is, a force

v2

ρ

mv2

ρ

is re

quired to deflect the particle from the tangent, which is the path it would take if left to itself.

T. D.

18

From this, or by the formula in § 135, we see that if a particle revolve at distance r, with angular velocity w, about à point, a force mrw2 to that point is requisite to maintain the distance r unaltered. This tendency to move in the tangent, which arises from the inertia of matter, was formerly supposed to be due to a force, called Centrifugal Force, generated in the particle by its rotation about the point.

We have seen that when the motion of a particle in any path is referred to polar co-ordinates in a plane, the acceleration along the radius vector is

de to it r ; hence the first term of the above is the acceleration dt of the velocity along the radius vector, and the other is the de dt

so-called centrifugal force due to a velocity r in a circle

of radius r. The idea of this so-called force is useful, as we have already seen (§ 208), in enabling us to form the equations of motion of a particle in particular cases.

260. Given the path of a particle, and the manner of its description, to find the requisite forces.

If X, Y, Z be the required forces for unit of mass, we must have

with similar expressions for Y and Z. But as the path is given, and the manner of its description, that is v in terms of the co-ordinates, the value of the above expressions is completely known.