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:—

where g is the mean anomaly, and

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

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.