16 
ON HANSENS LUNAR THEORY. 
Consider the orbit as an ellipse, then putting 
а, the mean distance, 
,r . • /k (d/ + K) 
n, the mean motion = a / ——r— , 
V a 3 
e, the excentricity, 
c, the mean anomaly at epoch, 
co, the distance of perigee from node, 
б, the longitude of node, 
i, the inclination, 
d>, the distance from node, 
M*, the reduced distance from node, = L — 6, 
U, the excentric anomaly, 
f the true anomaly, =<!> — <«, 
the elements of the orbit are a, e, c, co, 6, i, and we have from the theory of 
elliptic motion, 
nt + c = TJ —e sin U, 
T1 
in 1 
in 
ele 
f— tan 
V(1 — e 2 ) sin U 
cos U — e 
/i a (1 — e 2 ) 
r = a (1 — e cos U) = 
' 1 + e cos ; 
cos/' 
Moreover i is the angle at the base of a right-angled spherical triangle, the base, 
perpendicular, and hypothenuse of which are 'i r , A, <1>, hence 
tan ^ = cos i tan <E> , 
sin A = sin i sin O , 
sin ^ = cot i tan A , 
cos = cos A cos 
Considering the elements as constant, we have 
dr _ nae sin f 
dt V(1 — e 2 ) ’ 
df _ no? V(1 — e 2 ) 
dt r 2 
_ no? V(1 — e~) 
dt r 2 
d'P _ cos i no? V(1 — e 2 ) 
dt cos 2 A r 2 
dL _ cos i na? V(1 — e 2 ) 
dt cos 2 A r 2 
no? V(1 - e?)