motion while it is rectilinear, but neither horizontal nor vertical.

(7) Determine the (unresisted) motion of a mass projected vertically at a given point of the earth's surface with a velocity of 7 miles per second.

(8) Apply the principle of varying action to the determination of the (unresisted) motion of a projectile.

(9) Shew that the action and time, in any arc of the ordinary brachistochrone commencing at the cusp, are represented by the area and arc of the corresponding segment of the generating circle.

(10) In the parabolic motion of a projectile, the action is represented by the area included between the curve, the directrix, and the two vertical ordinates: and the time by the intercept on the directrix.

(11) Given a central orbit, and the law of its description, find the differential equation of a curve such that if tangents be drawn to it from any two points of the orbit, the action shall be represented by the area included by these tangents and the two curves.

(12) A particle moves in a given line, under the action of a force =

ds

-us-f; and a given impulse acts on it, alter

dt

nately in opposite directions in its line of motion, at intervals each equal to T. Find the resultant periodic motion. (This is the general case of the pendulum of an electrically-controlled clock.)

APPENDIX.

A. On the integration of the equations of motion about a center of force.

IN general, (Chap. V.) the problem of central forces is

solved by considering the equation connecting u (or):

and 0,

and employing the resulting integrated relation between r and to find in terms of t from the law of equable description of areas. If we try to express and separately, in terms of t, without first determining the form of the orbit, we are led to a host of curious results which may be easily obtained; so easily indeed, that we shall merely notice one or two of them.

From the usual equations for motion about a center, i.e.,

where P is the acceleration due to the central force, we get

dy

de (x dx + y dd) = + d1 (r) = -2 [Pdr - Pr... (1).

dt

dt

dt

This, for any assigned form of P in terms of r, will evidently give us 2 in terms of t.

Now there is a remarkable case in which r2 can be generally expressed as a rational integral function of t. Sup

It

Hence the case in question is that of the inverse third power. may be worth while to find in terms of t, and to obtain, by elimination of t, the equations to the orbits which are possible with such a force.

These are, of course, the results of the integration of the usual equation between u and 0.

(15)).

As another case, suppose in (1),

2 [Pdr

(Compare Chap. V. Ex.

- 2 | Pdr - Pr = mr2 +

с

2

Differentiate, multiply by r3, and integrate, then

n P = − +mr+3 •

· (6).

Hence, in the case of the direct first power, or a combination of this with the inverse third,

which gives, according as m is positive or negative,

By means of (4), these equations give us in terms of t, and, the latter being eliminated, we have the required orbit, which becomes the ellipse or hyperbola as usual when n = 0, it being observed that we have an additional disposable constant introduced by the method employed in obtaining equation (1). It is evident that results of this kind may be multiplied indefinitely. To classify the cases in which the equations for and in terms of t can be completely integrated would be an interesting, but by no means an easy problem.

The method here employed is interesting as being that which is at once suggested by the application of Quaternions to the problem of Central Orbits.