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Proceedings of the Royal Society 
parallel to Y'CZ. But by this means we also get at once, and 
without analysis, the two well-known and peculiar first integrals, 
in the form 
x = 
hi’ »“s(; + e )> 
h 
which cannot be directly deduced from the equations of accelera- 
tion 
jxx ,, _ fxy 
x — — 
rp 3 
y 
[The equation of the orbit is, of course, 
h = xy — yx = ^(r + ex} y 
from which we see that 
Ji 2 = fxa (1 — e 2 ) J . 
2. The only central orbits whose hodographs also are described 
as central orbits, are those in which the acceleration varies directly 
as the distance from the centre. 
Let S be the centre, P any point in the path, p the correspond- 
ing point in the hodograph, p' that in the hodo- 
graph of the hodograph. Then Sp' is parallel 
to the tangent at p, which again is parallel to 
SP. Hence PSp' is a straight line. Also, 
since p belongs (by hypothesis) to a central 
orbit, the tangent at p' is parallel to Sp, i. e., 
to the tangent at P. Hence the locus of p' is 
similar to that of P, and therefore Sp' is propor- 
tional to SP. But Sp' represents the accelera- 
tion at P. Hence the proposition. 
3. If n be the acceleration in a central orbit, 
n' that required for the description of the hodo- 
graph as a central orbit ; Ji, h\ the moments of 
r , the radii vectores in the two orbits, 
II IP = ~ rr' . 
h 2 
In the figure above let SY = and S y — zr be the perpendi- 
