ON HANSENS LUNAR THEORY. 
or reducing 
Y P sm y 
' r sin / 
sin/(1 + e cos (f>) 
Write for a moment 
1 'sm/- sin cf> + esin(/- (f>)) (^ ~ sin (/- <f>) del. 
n 9 //-1 9\ n nae sm / t> a (1 - e-) 
P = na 2 V(1 - e 2 ), Q =-77- P= T -^ S, 
" * V(1 — e 2 ) 1 + e cos/ 
so that 
P- 
a (1 - e 2 ) = —3, 
n-a 3 
1 02 1 P 4 2 P 2 
n 4 a 6 ^ n 4 a 6 P 2 n 2 a 3 P 
We have therefore 
da 2ede 2 dP 
a 1 — e* r na‘{L-e-) 
/ 1 9 p3 o P\ i / 1 P 4 2 P 2 \ 
ede=(4- ß PQ 2 --f~ 6 dP + -j— P 2 QdQ + I- -f- 6 ^ + 4 1 dP, 
\?i 4 a ; c n 4 a 6 P 2 n 2 a 3 P/ n 4 a 4 ^ 4 " > L* 3 w- 2 «- 8 K-i 
n 4 a 6 P 2 n 2 a 3 P 
which, after reduction, becomes 
1 
na 2 (1 — e 2 ) 
and substituting these values, 
p sin 4> y _ 1 
' r sin/ ^ 1 + e cos $ 
e sin 2 / + 2 (cos/ + e cos 2 /)^ dP+ ^ V(1 — e 2 ) sin/dQ — ^ + a (l^e 2 ) 003 ^ 
'2 - 2cos(/-« + csin/sin(/- *)) dP 
-o(l-c>)sin (/-*) 
(1+e cos/) 5 cot/sin (/-(#>) 1 ,„| 
V(i-<0 
aa 2 V(1 — e 2 ) 
or substituting for X, X /y P, Q, P, their values 
p sin <b pe sin <p 
dlp ~ 7sSf dlr +i/-V) {df - d<t>) = 
/_ ^ ( 2 -2 cos (/- 4>) + esin/sin (/- (/.)) n(t! ,/ 1 —¿,) d,M s V(1 - e=) 
nae sin / 
- p sin (/- 0) 
pa /(1 - e 2 ) 
,7 
na 2 V(1 — e 2 ) V(1 - e 2 ) 
1 
cot /sin (/— d>) 
na 2 /(1 — e 2 ) 
dr. 
r->4 $-> te m a 
fer as a, f rá svis . sí 
p V S'# *