IN PHYSICAL ASTRONOMY. 
43 
A, = m X — 2 m e sin (c A — zv) -f 2 e, sin (c m A — t?,) -f &c. 
The equation 
* d!=dx {' + 7/( ! s!) rl<u } 
gives t in terms of X, and by the reversion of series X may afterwards be obtained 
in terms of t. The equation for determining the inequalities of latitude is 
+ Jlfi*) ( d ^) = o 
h- \ ds J h 2 \ dr / h* \d A / \ d s) 
d R 3 tn r 2 f , , /o \ vl d s , , x 
d7 = + C0S ( 2A -2\)| — =gycos(gA-v) 
I have given these equations, (which are to be found in various works *,) 
for the convenience of reference. 
On the Planetary Theory. 
In a former paper I have shown how the coefficients of the terms in the dis¬ 
turbing function multiplied by the cubes of the eccentricities in some particular 
examples may be reduced by means of some transformations applied to the 
coefficients of the same function multiplied by the squares of the eccentricities. 
The form of the disturbing function thus obtained is I think simpler than that 
of the Mec. Cel. in the terms multiplied by the cubes of the eccentricities, 
although the advantage obtained by these reductions is not so great as in the 
case of the terms multiplied by the squares of the eccentricities. I have now 
given the general form of the transformations required, in case any one should 
think it worth while to extend to the cubes of the eccentricities the general 
expression for the disturbing function given in the Philosophical Transactions 
for 1831, p. 295. 
The coefficient of e e, cos (2 nt — 4 n t t + zs + w,) or e, cos (3 t — x -f- z) 
3 a 
4.4a, 2 
^ 3,2 + 
3 a h , 3.3 a 2 A , 3 (3 a- — a") sa, t 
2.4a, 2 3)4 + 2.4.2a, 3 3,1 + 274 a,» 
3.7a 2 / 
2.4 a, 3 5>e 
* See The Mechanism of the Heavens, by Mrs. Somerville, p. 427. 
G 2 
