479] 
511 
479. 
THE SECOND PART OF A MEMOIR ON THE DEVELOPMENT 
OF THE DISTURBING FUNCTION IN THE LUNAR AND 
PLANETARY THEORIES. 
[From the Memoirs of the Royal Astronomical Society, vol. xxxix. (1872), pp. 55—74. 
Read January 12, 1872.] 
The present communication is a sequel to my paper, “The First Part of a Memoir 
on the Development of the Disturbing Function in the Lunar and Planetary Theories,” 
Memoirs R.A.S., vol. xxvm. (1859), pp. 187—215, [214], and I have therefore entitled it 
as above, but it, in fact, relates only to the Planetary Theory. In the First Part, I gave 
in effect, but not explicitly, an expression for the general coefficient D(j, j') in terms 
of the coefficients of the multiple cosines of 6 in the expansions of the several powers 
(r 2 + r' 2 _ 2r/ cos d)~ s ~%, or say (a 2 + a' 2 — 2aa cos 6f s ^; viz., at the foot of page 208 
I speak of the term involving cos(jU+j'U') as having a certain given value; the 
term in question is J) (j, j') cos (jU +j'U'); and consequently the expression for 
D (j, j ) is 
D (j, j’) = S II ‘g~-t ) ; 
the omission was, however, a material one, inasmuch as this expression for the general 
coefficient serves to connect my formulae with Leverrier’s development, Annales de I’Ohserv. 
de Paris, t. I. (1855), pp. 275—330 and 358—383, and I resume the question for the 
purpose of applying it. 
Formula for the general Coefficient JD (j, j'). 
In the First Part, the reciprocal of the distance of the two planets, or function 
[r 2 + r' 2 — 2rr' (cos Ucos U' + sin Usin U' cos d>)} - *