ATTENDED BY SEVERAL SATELLITES. 
507 
There is also another surface to be considered, namely 
n=9—x* — 2y* 
which gives the rotation of the planet corresponding to any values of x and y. 
The equations 
n=fl x , n—i1 y 
have also to be exhibited. 
The computations requisite for the illustration were laborious, as I had to calculate 
values of z and n corresponding to a large number of values of x and y, and then by 
graphical interpolation to find the values of x and y, corresponding to exact values of 
z and n. 
The surface of energy will be considered first. 
Plate 61 shows the contour-lines (that is to say, lines of equal energy) in the positive 
quadrant, z being either positive or negative. 
I speak below as though the paper were held horizontally, and as though positive z 
were drawn vertically upwards. 
The numbers written along the axes give the numerical values of x and y. 
The numbers written along the curves are the corresponding values of —2z. Since 
the numbers happen to be all negative, smaller numbers indicate greater energy than 
larger ones; and, accordingly, in going down hill we pass from smaller to greater 
numbers. 
The full-line contours are equidistant, and correspond to the values 9, 8|, 8, 1\, 7, 
6l>, and 6 of — 2z; but since the slopes of the surface are very gentle in the central 
part, dotted lines (....) are drawn for the contours 7§ and 7\. 
The points marked 5'529 and 7’442 are equidistant from x and y, and therefore 
correspond to the case when the two satellites have the same distance from the planet, 
or, which amounts to the same thing, are fused together. The former is a maximum 
point on the surface, the latter a maximum-minimum. 
The dashed line (--) through 7*442 is the contour corresponding to that value 
of — 2z. 
The chain-dot lines (-) through the same point will be explained below. 
An inspection of these contours shows that along the axes of x and y the surface 
has infinitely deep ravines ; but the steepness of the cliffs diminishes as we recede 
from the origin. 
The maximum point 5*529 is at the top of a hill bounded towards the ravines by 
very steep cliffs, but sloping more gradually in the other directions. 
The maximum-minimum point 7*442 is on a saddle-shaped part of the surface, for 
we go up hill, whether proceeding towards 0 or away from O, and we go down hill in 
either direction perpendicular to the line towards 0. 
If the total angular momentum of the system had been less than the smallest 
critical value, the contour lines would all have been something like rectangular 
3 u 
MDCCCLXXXI. 
