it will be proportional to any one of the homologous lines; hence QR and QT will each ∞ SP;

A body describes an ellipse; to find the law of force tending to the centre.

Let P be the position of the body at any given time, Q a point contiguous to P, PR the tangent at P, QT perpendi

cular to the diameter PCG, DCK the diameter conjugate to PCG, PF perpendicular to DCK, Qv an ordinate to PCG.

COR. 1.

,. CP, since v G = 2CP ultimately,

or F∞ CP.

To find the periodic time in an ellipse described about a force in the centre.

=

Suppose Fμ CP, where μ is a constant quantity depending upon the intensity of the force residing in the centre, and usually called the absolute force of the centre: then, by the preceding proposition, h2 = μ AC2 . BC2.

Let P be the periodic time, that is, the time employed in describing the complete ellipse; then since the area described in a unit of time is and the whole area of the ellipse is

h

2

Hence the periodic time depends solely upon the intensity of the force in the centre.

[COR. 2. Since P is independent of both axes of the ellipse, its value will be the same if we suppose the minor axis to be indefinitely diminished, in which case the motion will approximate to that of a body oscillating in a straight line under the action of an attractive force varying directly as the distance: hence the time of a complete oscillation of a body

moving in the manner described will be

2π

μ

SECTION III.

ON THE MOTION OF A BODY IN A CONIC SECTION, ABOUT A CENTRE OF FORCE IN THE FOCUS.

A body revolves in an ellipse; to find the law of force tending to one of the foci.

Let S be the focus of the ellipse, P the position of the body at any given time, Q a contiguous point in the orbit,

D

T

E

R

PCGK, DC conjugate diameters, PR the tangent at P, QR parallel to SP, Qxv to PR, QT perpendicular to SP, PF to CK, and E the point of intersection of SP and CD.

Qu2 CP. PF'

1 CP2 AC3

vG CD2 CP.PF2 (Conics, Prop. VIII. p. 150.)

[COR. If u be the absolute force of the centre,

where L is the latus rectum of the ellipse. (Conics, Prop. VI. Cor. page 149.)]

PROP. XII. PROB. VII.

A body moves in an hyperbola; to find the law of force tending to one of the foci.

Let S be the focus of the hyperbola, P the position of the body at any given time, Q a contiguous point in the orbit,

PCG, DCK conjugate diameters, PR the tangent at P, QR parallel to SP, Qav to PR, QT perpendicular to SP, PF to CD produced, and E the point of intersection of SP and CD produced.

Then,

2h2 QR

F SP2 QT ultimately.

By similar triangles QTx, PEF,

QT2 PF2 PF2

Q∞2 PE AC2

(Conics, Prop. III. Cor. page 159.)