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. 191 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. 192 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.