IN PHYSICAL ASTRONOMY. 
57 
By means of this equation, the equations 
s — zu = n l, e'- = e-{ 1 — sin 2 1 sin 2 (v — or) } 
and the equations given byM. de Ponte couLANT,vol.i. p.328, the values of ~ 
Tt y d7 5 and ^ ma y eas ^^ obtained in terms of the quantities a, w, e, s. 
,, ,, and the partial differential coefficients (if), (f|), (if), (if), (if) 
and (it)- 
Substituting in the equations of p. 40, for q n their values and neglecting 
e>, e, 2 , e e,, and sin 2 when a is less than a / ; 
7 = 1 “ p {|* K ° ~ $ S> -‘} + e { 1 “ + 
vij n°- f 2 n / of _ \ + 
+ p, (2« — «,) n, L (n — re,) Va, 2 cr / ‘’7 a, 2 
+ 0- b i,x ~ ho ~ \ b i ,°- ) } cos (» < — »,< + * — O [ 1] 
m, n* f 2n a_ b ^ # 
iu, (3)1-2^) («-2 ?i ( )1(b- 1!,) a ( 1,2 
— ^ (J- &3,2 — 5 fe 3,l ~ 2^3,3 ) } C0S ( 2w < ~ 2w t t + 2 £ — 2 £ ( ) [2] 
m, wf f 2n b 
[L (4 w — 3 «,) (211-3 7 L (w — w,) a ; 1,3 
-£(^- hs - i h> - h,*) } cos (3 n t - 3 n, t + 3 £ - 3 £,) [3] 
+ ^ * ( 2 n -“b^ 
I \h (5 n — 4 rc,) (3 k — 4 n t ) l (« — «,) a, 
- ^ ^3.4 - 5 *3,3 - 2 &3, 5 ) } COS (4 » t - 4 11, t + 4 £ + 4 £,) 
k 2 f 2« « ^ 
,u (6 k — 5 w ( ) (4n — 5 w ; ) L (k — n t ) a t 1,5 
~ a* (a~ 63,5 “ ^ 63,4 “ ^ ^- 6 ) } cos ( 5 n 1 ~ 5 n i 1 + 5 £ ~ 5 O 
* In the terms multiplied by ^ the quantities a & a, e & e may be used indifferently. 
f* 
MDCCCXXXI. I 
[4] 
+ m i 
[ 5 ] 
Expression 
for the reci- 
procal of the 
radius vector, 
when a < a r 
