406 Proceedings of the Boyal Society 
or, in a somewhat more convenient form, 
dx 
^ ds 
-p + 
P 
( 1 ). 
Also, by the nature of the construction we are attempting, it is 
obvious that the line joining the points x,y, and ^,17, must be a 
tangent to the locus of tj. This gives us at once the second 
condition 
d^ ^ V - y 
d^ i — X 
( 2 .) 
Since, by the equation of the central orbit, which is supposed to be 
known, we have x and y in terms of s, we may take s as indepen- 
dent variable, and we have the equations (1) and (2) necessary and 
sufficient for the determination of ^ and y in terms of s. 
Eliminating y by differentiation and substitution, we find for ^ 
the equation 
dx d^ 
p ds ds 
(i-x) 
r /dx d“^y 
pp L^Ws ds“^ 
, /dx d?y dy d'^x\ 
dy d^x\ ^ dx d^x~i I 
with, of course, a similar equation for y. 
If we take x as the independent variable, and put 
a given function of x, 
it is easy to reduce this equation to 
/ Y \ Y ^ Py 
dx\x — (x - iy dx^^ 
which seems to he the simplest form to which it can be brought, 
unless special relations between x and y are introduced, and with- 
out such it seems to be quite intractable. 
The remainder of the paper refers to a subject which, though 
allied to action, is so distinct from the investigation above that I 
give the following abstract of it under a distinct title. 
