ATTENDED BY SEVERAL SATELLITES. 
50 L 
Therefore the equation to the tangent to the steepest path is 
X—x Y—y Z—z 
bzjbx bzjby (bz/bx) 2 + (bz/by) 2 
(23) 
If this steepest path on the energy surface is the path actually pursued by the point 
which represents the configuration of the system, equation (23) must be satisfied by 
X=x J r -8z, Y—y- |--£Sz, Z=z J r 8z 
clz 
And therefore we must have 
bx 
dy 
by 
* S+O 2 ’ * (!)+(£)* 
But these are the values already found in (22) for dx/dz and dy/dz. 
Therefore we conclude that the representative point always slides down a steepest 
path on the energy surface. Hence it only remains to draw the surface, and to mark 
out the fines of steepest slope in order to obtain a complete graphical solution of the 
problem. Since the fines of greatest slope cut the contours at right angles, if we 
project the contours orthogonally on to the plane of x y, and chaw the system of ortho¬ 
gonal trajectories of the contours, we obtain a solution in two dimensions. This 
solution will be exhibited below, but for the present I pass on to more general 
considerations. 
A precisely similar argument might be applied to the case where there are any 
number of satellites, only as space has but three dimensions, a geometrical solution is 
not possible. If there be r satellites, then the problem to be solved may be stated in 
geometrical language thus :— 
It is required to find the path which is inclined at the least angle to the axis of E 
on the locus 
2E={h-$Key-i^ 
This locus is described in space of r- f-1 dimensions. One axis is that of E, and the 
remaining r axes are the axes of the r different ^’s. The solution may be depressed 
so as to merely require space of r dimensions, for we may, in space of r dimensions, 
construct the orthogonal trajectories of the contour loci found by attributing various 
values to E. 
Thus we might actually solve geometrically the case of three satellites. The energy 
locus here involves space of four dimensions, but the contour loci are a family of 
surfaces in three dimensions. If such a system of surfaces were actually constructed, 
it would be possible to pass through them a number of wires or threads which should 
be a good approximation to the orthogonal trajectories. The trouble of execution 
