298 ON DR MORRISON'S PAPER (ON KEPLER'S PROBLEM). [39 



The numerical coefficient of the term in (xy) which involves 



e m sin (m '2n) g 



- 

 ' 



2 



where m is a positive integer, and n is either zero or a positive integer 

 less than - , and (1 . 2...) is to be put =1, when /j = 0. 



i 



The expressions for x and for the sines of multiples of x are developed 

 to the 12th power of c by Schubert in the appendix to Bode's Jahrlmch 

 for 1820. In the same appendix Schubert likewise gives the development 

 of the true anomaly in terms of the mean to the 13th power of e. 



Oriani had already given this last-mentioned development to the llth 

 power of e in the appendix to the Milan Eplicmeris for 1805. 



The numerical coefficients which he finds differ in four cases from those 

 given by Schubert, but I have recomputed the coefficients in these cases, 

 and find that Schubert's results are correct. 



There is a misprint, however, in Schubert's expression for the true 

 anomaly at the foot of p. 230, where the coefficient of e K s'ml'2g should be 



7218065 . 7218065 



instead of 



2". 3. 7. 11 2".3Mr 



Delambre's formula is copied from Oriani's, and is therefore affected by 

 the same errors, together with some additional typographical ones. 



I have verified Schubert's result for (v), the true anomaly in terms 

 of the mean, by the consideration that when </ = 0, the value of 



dv (1+e) 2 



-j becomes - S 

 dg (l-e 2 )* 



27 . 15 . 65 , 35 . 595 , 315 , 1323 

 ---- 



. 



693 , 5775 , , 3003 



- 



