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covered with the usual imaginary lines of latitude and longitude.
The orbit of a supposed satellite is shown by a line crossing the
sphere at some assumed angle with the equator. Along this line
the satellite always moves at uniform velocity, passing across
and round the back of the sphere and again across. If the sphere
is not turning on its polar axis then this satellite, which we
will suppose armed with a pencil which draws a line upon the
sphere, will strike a great circle right round the sphere. But
the sphere is rotating. And it is to be expected that at
different times in a planet's history the rate of rotation varies
very much indeed. There is reason to believe that our own day was
once only 2½ hours long, or thereabouts. After a preliminary rise
in velocity of axial rotation, due to shrinkage attending rapid
cooling, a planet as it advances in years rotates slower and
slower. This phenomenon is due to tidal influences of the sun or
of satellites. On the assumption that satellites fell into Mars
there would in his case be a further action tending to shorten
his day as time went on.
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
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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 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.
But of these two velocities that of the satellite may be taken as
sensibly invariable, when close enough to use his pencil. This
depends upon the law of centrifugal force, which teaches us that
the mass of the planet alone decides the velocity of a satellite
in its orbit at any fixed distance from the planet's centre. The
other velocity--that of the planet upon its axis--was, as we have
seen, not in the past what it is now. If then Mars, at various
times in his past history, picked up satellites, these satellites
will describe curves round him having different spans which will
depend upon the velocity of axial rotation of Mars at the time
and upon this only.
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In what way now can we apply this knowledge of the curves
described by a satellite as a test of the lunar origin of the
lines on Mars?
To do this we must apply to Lowell's map. We pick out preferably,
of course, the most complete and definite curves. The chain of
canals of which Acheron and Erebus are members mark out a fairly
definite curve. We produce it by eye, preserving the curvature as
far as possible, till it cuts the equator. Reading the span on
the equator we find' it to be 255 degrees. In the first place we
say then that this curve is due to a retrograde satellite. We
also note on Lowell's map that the greatest rise of the curve is
to a point about 32 degrees north of the equator. This gives the
inclination of the satellite's orbit to the plane of Mars'
equator.
With these data we calculate the velocity which the planet must
have possessed at the time the canal was formed on the hypothesis
that the curve was indeed the work of a satellite. The final
question now remains If we determine the curve due to this
velocity of Mars on its axis, will this curve fit that one which
appears on Lowell's map, and of which we have really availed
ourselves of only three points? To answer this question we plot
upon a sphere, the curve of a satellite, in the manner I have
described, assigning to this sphere the velocity derived from the
span of 255 degrees. Having plotted the curve on the sphere it
only remains to transfer it to Lowell's map. This is easily
done.
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This map (Pl. XXII) shows you the result of treating this, as
well as other curves, in the manner just described. You see that
whether the fragmentary curves are steep and receding far from
the equator; or whether they are flat and lying close along the
equator; whether they span less or more than 180 degrees; the
curves determined on the supposition that they are the work of
satellites revolving round Mars agree with the mapped curves;
following them with wonderful accuracy; possessing their
properties, and, indeed, in some cases, actually coinciding with
them.
I may add that the inadmissible span of 180 degrees and spans
very near this value, which are not well admissible, are so far
as I can find, absent. The curves are not great circles.
You will require of me that I should explain the centres of
radiation so conspicuous here and there on Lowell's map. The
meeting of more than two lines at the oases is a phenomenon
possibly of the same nature and also requiring explanation.
In the first place the curves to which I have but briefly
referred actually give rise in most cases to nodal, or crossing
points; sometimes on the equator, sometimes off the equator;
through which the path of the satellite returns again and again.
These nodal points will not, however, afford a general
explanation of the many-branched radiants.