508 
MR. G. H. DARWIN ON A PLANET 
hyperbolas with the axes of x and y as asymptotes, like the outer curves marked 6, 
6^, 7 in Plate 61. In this case the whole surface would have sloped towards the axes. 
If the momentum had been greater than the smallest, and less than the second 
critical value, the outer contours would have still been like rectangular hyperbolas, 
and the branches which run upwards, more or less parallel to y, would still have 
preserved that character nearer to the axes, whilst the branches more or less parallel 
to x would have had a curve of contrary reflexure, somewhat like that exhibited by 
the curve 7^ in Plate 61, but less pronounced. In this case all the hires of steepest 
slope would approach the axis of x, but some of them in some part of their course 
would recede from the axis of y. 
If the momentum had been greater than the second, but less than the third critical 
value, the contours would still all have been continuous curves, but for some of the 
inner ones there would have been contrary reflexure in both branches, somewhat like 
the curve marked 7^ in Plate 61. There would still have been no closed curves 
amongst the contours. Here some of the lines of greatest slope would in part of 
their course have receded from the axis of x, and some from the axis of y, but the 
same line of greatest slope would never have receded from both axes. 
Finally, if the momentum be greater than the third critical value, we have the case 
exhibited in Plate 61. 
Plate 62, fig. 1, exhibits the lines of greatest slope on the surface. It was constructed 
by making a tracing of Plate 61, and then drawing by eye the orthogonal trajectories of 
the contours of equal energy. The dashed line (-) is the contour corresponding 
to the maximum-minimum point 7*442 of Plate 61. The chain-dot line (-) will 
be explained later. 
One set of lines all radiate from the maximum point 5*529 of Plate 61. The arrows 
on the curves indicate the downward direction. It is easy to see how these lines would 
have differed, had the momentum of the system had various smaller values. 
Plate 62, fig. 2, exhibits the contour lines of the surface 
n = 9 — ri — 2y } 
It is drawn on the same scale as Plate 61 and Plate 62, fig. 1. 
The computations for the energy surface, together with graphical interpolation, gave 
values of x and y corresponding to exact values of n. 
The axis of n is perpendicular to the paper, and the numbers written on the curves 
indicate the various values of n. 
These curves are not asymptotic to the axes, for they all cut both axes. The angles, 
however, at which they cut the axes are so acute that it is impossible to exhibit the 
intersections. 
Hone of the curves meet the axis of x within the limits of the figure. 
The curve n — 3 meets the axis of y when y=2150, and that for n = 3^ when y=1200, 
but for values of n smaller than 3 the intersections with the axis of y do not fall within 
