The effect of the rotation of the planet will be, of course, that as the satellite advances with its pencil it finds the surface of the sphere being displaced from under it.  The line struck ceases to be the great circle but wanders off in another curve—­which is in fact not a circle at all.

You will readily 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

189

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

190

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 degrees which either sort of curve subtends upon the equator entirely depends upon the velocity of the satellite and the axial velocity of the planet.