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Again, the Periodic Time in a circle whose radius is

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540. Let the distance of the vertex of the parabola from

its focus be a, then the velocity in the curve is

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Again, if ẞ be the velocity with which the system moves, and y' the ordinate of the curve-in-space at the end of the time t, the whole increment of y' during dt is

dy Bdt dy

according as the velocity to the axis arising from the motion in the curve, is in the same or opposite direction to that of the system.

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Hence, putting a + 1 = A, and ‚√ (1 + a + — x + x = X

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a

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1. (2x + A +
+ 2X)} + C

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541. radius = r, is

x2)

The velocity in a circle, when the force is F, and

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where is the absolute force, and or that same function of the radius () according to which the central force attracts at different distances.

Hence, at the same distance, and for the same law of force denoted by 4, we have

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the law required.

543. periodic time.

Let a, b be the given semi-axes, and T the given Then since the force tends to the focus, if and be the polar co-ordinates, it is easily shewn (see Newton, Prop. 1.) that the areas described by ę are proportional to the times of deв scription; and because the area of the ellipse is ab, we have

T: rab:: dt:

2

ds

e2 de

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2rab

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2rab ds
Tę do

T

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which gives the velocity at any point of the ellipse.

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c2 dp F=

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545.

If v and v'denote the velocity in the curve, and that

in a circle at the same distance, we have generally (438)

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g) be the space described in a time towards

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