510 
MR. G. H. DARWIN ON A PLANET 
either n — fl x or n — Sl y changes sign in passing from one region to another. In these 
figures a line drawn at 45° to the axes through the origin divides the space into 
two parts; in the upper region y is greater than x, and in the lower x is greater 
than y. Hence configurations, for which the greater or ^-satellite is exterior to the 
lesser or a>satellite, are represented by points in the upper space and those in which 
the lesser satellite is exterior by the lower space. 
In the figures of which I have been speaking hitherto the abscissas and ordinates 
are the -f power of the distances of the two satellites ; now this is an inconveniently 
high power, and it is not very easy to understand the physical meaning of the result. 
I have therefore prepared another figure in which the abscissas and ordinates are the 
actual distances. In Plate 63, fig. 3, the curves are no longer lines of steepest slope. 
The reduction from Plate 62, fig. 1 , to Plate 63, fig. 3, involved the raising of all the 
ordinates and abscissas of the.former one to the -§- power. This process was rather 
troublesome, and Plate 63, fig. 3, cannot claim to be drawn with rigorous accuracy ; 
it is, however, sufficiently exact for the hypothetical case under consideration. If we 
had to treat any actual case, it would only be necessary to travel along a single line of 
change, and for that purpose special methods of approximation might be found for 
giving more accurate results. 
In this figure the numbers written along the axes denote the distances of the 
satellites in mean radii of the planet—the radius of the planet having been chosen as 
the unit of length. 
The chain-dot curves, as before, enclose the region for which the orbital angular 
velocities of the satellites are less than that of the planet’s rotation. The line at 45° 
to the axes marks out the regions for which the larger satellite is exterior or interior 
to the smaller one. , 
Let us consider the closed space, within which L2. r and n y are less than n. 
The corner of this space is the point of maximum energy, from which all the curves 
radiate. 
Those curves which have tangents inclined at more than 45° to the axis of x denote 
that, during part of the changes, the larger satellite recedes more rapidly from the 
planet than the smaller one. 
If the curve cuts the 45° line, it means that the larger satellite catches up the 
smaller one. Since these curves all pass from the lower to the upper part of the 
space, it follows that this will only take place when the larger satellite is initially 
interior. According to the figure, after catching up the smaller satellite, the larger 
satellite becomes exterior. In reality there would probably either be a collision or 
the pair of satellites would form a double system like the earth and moon. After this 
the smaller satellite becomes almost stationary, revolves for an instant as though 
rigidly connected with the planet, and then slower than the planet revolves (when the 
curve passes out of the closed space) ; the smaller satellite then falls into the planet, 
whilst the larger satellite maintains a sensibly constant distance from the planet. 
