-318 Proceedings of the Poyal Society of Edinburgh. [Sess. 
XIX. — Elliptic Expansions. By Mr R. T. A. Innes. 
(MS. received February 1, 1922. Read February 20, 1922.) 
(Abstract.) 
In seeking the expansion of the equation of the centre, and the logarithm 
of the radius vector in a planetary orbit, in terms of the mean anomaly, 
the following expressions are obtained : — 
Equation of centre = - (A^- + B^- + CA sin ig, 
Log r\a 
= y 4( B^-Cdcos^(7, 
where g is the mean anomaly, and 
00 00 
1 1 
/3={1 — y^(l = e^)/e, and ^i{x) is Bessel’s function. 
Mr Innes remarks : “ It is shown for the first time, I believe, that the 
expansion for the true anomaly in terms of the mean (equation of the 
centre) can be derived quite easily from the addition of the two simple 
expansions 
True in terms of eccentric anomaly, 
Mean „ „ „ , 
and the two series have most simple recurrence expressions.” 
The MS. containing the proofs of these expressions, with further 
developments, is in the hands of the Society. 
{Issued sejoarately August 10, 1922.) 
