ly see how we find this curve. Suppose the sphere
to be rotating at such a speed that while the satellite is
advancing the distance _Oa_, the point _b_ on the

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sphere will be carried into the path of the satellite. The pencil
will mark this point. Similarly we find that all the points along
this full curved line are points which will just find themselves
under the satellite as it passes with its pencil. This curve is
then the track marked out by the revolving satellite. You see it
dotted round the back of the sphere to where it cuts the equator
at a certain point. The course of the curve and the point where
it cuts the equator, before proceeding on its way, entirely
depend upon the rate at which we suppose the sphere to be
rotating and the satellite to be describing the orbit. We may
call the distance measured round the planet's equator separating
the starting point of the curve from the point at which it again
meets the equator, the "span" of the curve. The span then depends
entirely upon the rate of rotation of the planet on its axis and
of the satellite in its orbit round the planet.

But the nature of events might have been somewhat different. The
satellite is, in the figure, supposed to be rotating round the
sphere in the same direction as that in which the sphere is
turning. It might have been that Mars had picked up a satellite
travelling in the opposite direction to that in which he was
turning. With the velocity of planet on its axis and of satellite
in its orbit the same as before, a different curve would have
been described. The span of the curve due to a retrograde
satellite will be greater than that due to a direct satellite.
The retrograde satellite will have a span more than half

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way round the planet, the direct satellite will describe a curve
which will be less than half way round the planet: that is a span
due to a retrograde satellite will be more than 180 degrees,
while the span due to a direct satellite will be less than 180
degrees upon the planet's equator.

I would draw your attention to the fact that what the span will
be does not depend upon how much the orbit of the satellite is
inclined to the equator. This only decides how far the curve
marked out by the satellite will recede from the equator.

We find then, so far, that it is easy to distinguish between the
direct and the retrograde curves. The span of one is less, of the
other greater, than 180 degrees. The number of d