ATTENDED BY SEVERAL SATELLITES. 
509 
the figure. The thickness, which it is necessary to give to the lines in drawing, 
obviously prevents the possibility of showing these facts, except in a figure of very 
large size. 
On the side remote from the origin of the curve marked 0, n is negative, on the 
nearer side positive. 
Since ili = 50, 
/2, —20/ad and /2 y =20 jip 
Hence the lines on the figure, for which fl x is constant, are parallel to the axis of y, 
and those for which f2, y is constant are parallel to the axis of x. 
The points are marked off along each axis for which S2 X or fl y are equal to 3/, 3, 2\, 
2, 1-|, 1. The points for which they are equal to ^ fall outside the figure, 
Now, if we draw parallels to y through these points on the axis of x, and parallels 
to x through the points on the axis of y, these parallels will intersect the n curves of 
the same magnitude in a series of points. For example, S2 X = llr, when x is about 420, 
and the parallel to y through this point intersects the curve n= 1 -|, where y is about 
740. Hence the first or ^-satellite moves as a rigid body attached to the planet, 
when the first satellite has a distance (420)/ and the second a distance (740)/ In 
this manner we obtain, ; a curve shown as chain-clot (-) and marked fl x =n for 
every point on which the first satellite moves as though rigidly connected with the 
planet; and similarly there is a second curve (-) marked 12 y —n for every 
point on which the second satellite moves as though rigidly connected with the planet. 
This pair of curves divides space into four regions, which are marked out on the figure. 
The space comprised between the two, for which J2 X and S2 y are both less than n, is 
the part which has most interest for actual planets and satellites, because the satellites 
of the solar system in general revolve slower than their planets rotate. 
If the sun be left out of consideration, the Martian system is exemplified by the 
space f2 x >n, f2 y <n, because the smaller and inner satellite revolves quicker than the 
planet rotates, and the larger and outer one revolves slower. 
The little quadrilateral space near O is of the same character as the external space 
f2 x >n, f2 y >n, but there is not room to wnite this on the figure. 
These chain-dot curves are marked also on Plates 61 and 62, fig. 1. In Plate 61 the 
line I2 x =n passes through all those points on the contours of energy whose tangents 
are parallel to x, and the line J2 y =n passes through points whose tangents are parallel 
to y. 
The tangents to the lines of greatest slope are perpendicular to the tangents to the 
contours of energy; hence in Plate 62, fig. 1, f2 x =n passes through points whose 
tangents are parallel to y, and il y —n through points whose tangents are parallel to x. 
Within each of the four regions into which space is thus divided the lines of slope 
preserve the same character; so that if, for example, at any part of the region they 
are receding from x and y, they do so throughout. 
This is correct, because dx/dt changes sign with n — J2 X and dy/dt with n — Sl y ; also 
3 u 2 
