264 Joty—On the Origin of the Canals of Mars. 
not the case with the greater number of his curves. The most careful exami- 
nation of the others fails to reveal any curves satisfying this condition. A certain 
number of the curves rise from the equator along lines of longitude, apparently. 
It will presently be seen that this is no proof that they are parts of great 
circles. Meanwhile there is some prima facie evidence in the “ rifts” observed 
by Mr. Lowell in the polar cap that the lines in question are not indeed 
great circles. It is reasonable to suppose that these rifts are the continuations 
of such lines. Now the definite rifts very plainly do not cross the pole, and there- 
fore do not represent the continuation of lines following lines of longitude. Fur- 
ther than this, however, these polar rifts fail to enlighten us as to their course. 
In addition to the great circle, curves, corresponding to axial rotations of the 
planet at a less velocity than the present, are, as already remarked, excluded. But 
general considerations attending the nature of the curves claim our attention 
before proceeding. 
In Plate xy., a simple mode of plotting the intersection of the radius vector 
of a satellite with the surface of its planet is explained. We assume the 
satellite, supposed to be moving on the surface of the planet, to start at any 
instant from 0° longitude, and pursue the path inclined to the equator as marked 
out by the full straight line 0a. This path is described in space by the satellite 
at uniform velocity. Ifthe planet is supposed rotating in the direction of the 
arrow drawn upon the equator, this satellite is a direct satellite. Let the planet be 
turning at such an angular velocity that while the satellite accomplishes the 
distance 0a the point 4 upon the planet is transported the distance da and so carried 
into the path of the satellite. The ratios of the angular distances 0a and ba 
represent the assumed ratios of the angular velocities of satellite and planet. 
Preserving this ratio, we similarly determine at sufficiently close degrees of 
latitude points which lie upon the curve. The curve formed by joining these 
points is that which this ratio of angular velocities gives rise to. It is seen 
dotted where it is carried round to its second intersection with the equator. 
The distance in degrees of longitude between these points of intersection we call 
the ‘‘ span” of the curve. 
It is apparent that this span at once determines the ratios of angular velocities 
of planet and satellite. In fact, as the remote point of intersection with the 
equator of the curve in question is that which will be carried under the path of 
the satellite when this has described 180° of its path, we may express the 
ratio of the angular velocities in terms of the span as follows :— 
= a 2) 
where V, and JV, are the angular velocities of planet and satellite. 
