of Edinburgh, Session 1864 - 65 . 
405 
cient to show that such representations depend mainly upon the 
particular nature of the path, and therefore cannot be included in 
any general formula. But the example which follows appeared to 
point out some such general method ; applicable at least to central 
orbits. 
In an elliptic orbit described about the focus, the time is proportional 
to the sectorial area described about one focus, and the action to that 
about the other . — The proof of this theorem is obvious, if we remember 
that the product of the perpendiculars, from the foci, upon the tan- 
gent to an ellipse, is constant. 
This appeared to me to indicate, as a mode of representing the 
action in a central orbit, the seeking for a curve allied to the orbit, 
and in its plane ; such that, if two tangents be drawn to it, the 
area intercepted between them, the curve, and the orbit, shall be 
proportional to the action. In the case of the elliptic orbit, above 
referred to, this curve would evidently become a point, viz., the 
second focus. The following investigation, however, does not give 
a very encouraging result ; — 
Taking the centre of force as origin, let x, y, be the co-ordinates 
of a point in the orbit, rj, those of the corresponding point in the 
allied curve. The equation of the tangent at x, y, is 
and, consequently, the lengths of the perpendiculars drawn to it 
from the origin, and from the point y, are 
respectively. That the elementary triangle, whose vertex is rj, 
and whose base is Ss, may be proportional to the element of the 
action, we must evidently (as we see by referring to the case of the 
ellipse above) have 
pp' = constant. 
Hence our first condition is 
and 
