[ 427 ] 
L. A Demonstration of Sir W. R. Hamilton’s Theorem of the 
Isochronism of the Circular Hodograph. By Anrtuur Cay- 
LEY, Esq.* 
Se a body moving i plano under the action of a cen- 
tral force, and let h denote, as usual, the double of the area 
described in a unit of time; let P be any point of the orbit, then 
measuring off on the perpendicular let fall from the centre of 
force O on the tangent at P to the orbit, a distance OQ equal 
or proportional to / into the reciprocal of the perpendicular on 
the tangent, the locus of Q is the hodograph, and the points 
P, Q are corresponding points of the orbit and hodograph. 
It is easy to see that the hodograph is the polar reciprocal of 
the orbit with respect to a circle having O for its centre, and 
having its radius equal or proportional to “A. _ And it follows 
at once that Q is the pole with respect to this circle of the tan- 
gent at P to the orbit. 
In the particular case where the force varies inversely as the 
square of the distance, the hodograph is a circle. And if we 
consider two elliptic orbits described about the same centre, 
under the action of the same central force, and such that the 
major axes are equal, then (as will be presently seen) the com- 
mon chord or radical axis of the two hodographs passes through 
the centre of force. ; 
Imagine an orthotomic circle of the two hodographs (the 
centre of this circle is of course on the common chord or radical 
axis of the two hodographs), and consider the ares intercepted 
on the two hodographs respectively by the orthotomic cirele ; 
then the theorem of the isochronism of the circular hodograph 
is as follows, viz. the times of hodographic description of the 
intercepted arcs are equal; in other words, the times of de- 
scription in the orbits, of the arcs which correspond to the 
intercepted arcs of the hodographs, are equal. It was remarked 
by Sir W. R. Hamilton, that the theorem is im fact equivalent 
to Lambert’s theorem, that the time depends only on the chord 
of the described arc and the sum of the two radius vectors. And 
this remark suggests a mode of investigation of the theorem. 
Consider the intercepted are of one of the hodographs: the tan- 
gents to the hodograph at the extremities of this are are radii 
of the orthotomic circle; 7. e. the corresponding arc of the orbit 
is the are cut off by the polar (in respect to the directrix circle 
by which the hodograph is determined) of the centre of the 
orthotomic circle ; the portion of this polar intercepted by the 
orbit is the elliptic chord, and this elliptic chord and the sum of 
the radius vectors at the two extremities of the elliptic chord, 
* Communicated by the Author. 
