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lias again a mean value, being again at right angles tothe line from S.
(iii.) At B the direction of the body’s motion isinclined at a maximum angle to the line from S, thedistance has its mean value B S, being the arithme-tical mean between A S and S a ; and the velocity haswhat may be entitled its mean value, being the geome-trical mean between the velocities at a and A.
(iv.) At b the same conditions prevail as respectsdistance and velocity as at B, but the direction of thebody’s motion is inclined at a minimum angle to theline from S,*
The relation which we have been considering cor-responds to the first law which Kepler recognized inthe planetary motions; viz., that each planet travelsin an ellipse, the sun being situated at one focus ofthe curve. This law is not strictly true for the planets,or indeed for any known case in nature, since no orbis free to revolve around another quite independently ofextraneous attractions. The law is, however, approx-imately true when any orb is subject almost wholly tothe attraction of a single body; or else, though sub-
* Since any point of the orbit may be regarded as a starting-point, we notice that the same shaped curve is described whether abody is projected as at B on a course making the obtuse angle S B 2with the line from S, or with the same velocity from the equidis-tant point b, on a course making the acute angle S b Z with theline from S. The more general proposition also holds, that inwhatever direction a body be propelled from a given point and witha given velocity, its orbit will have a major axis of constantlength.