[ 601 ] 
XXVI. Researches in Physical Astronomy . By J. W. Lubbock, Esq. V. P. 
and Treas. R.S. 
Read June 21, 1832. 
On the development of R. 
In the following method of developing the disturbing function, the coeffi- 
cients of the inequalities corresponding to any given order are expressed in 
terms of the coefficients of the inferior orders ; so that, for example, the coeffi- 
cients of the terms in the disturbing function multiplied by the squares of the 
eccentricities, are given analytically by means of the coefficients of those inde- 
pendent of the eccentricities, and of those multiplied by their first powers. 
As the theorems to which this method gives rise, are of great simplicity, I 
trust they will not be thought unworthy attention. By their means and with 
the assistance of the table given in my Lunar Theory, the expressions may be 
obtained, which are necessary for the development of R, as far as the fourth 
powers of the eccentricities inclusive ; it may easily be carried to any extent, 
and the expressions given by Burckhardt in the Memoires de l’lnstitut, 
1808, may be verified without difficulty. This method is peculiarly advan- 
tageous in the lunar theory, and for the terms in R dependent on powers of 
the eccentricities above the squares ; for the expression thus obtained for the 
coefficients of the terms dependent on the squares and products of the eccen- 
tricities in the planetary theory, is by no means so simple or so convenient for 
numerical calculation as that given in the Phil. Trans. 1831, p. 295. A simi- 
lar method is applicable to the terms dependent on the inclinations. 
Let R — R 0 4* e~ R 0 r 4~ R 0 11 + &c. 
+ { i?! + e a - Ri + e, 2 R," + &c.} cos (int — in,t ) 
[1] 
+ { + e- R ' + ef R 2 " 4- &c.} e cos {nt — zs) 
[2] 
