21 
[163 
163] 
ON HANSENS LUNAR THEORY. 
rmed with 
= ddt? q- da 
le purpose 
departure 
t of v / in 
)ns of the 
receded 
quantities 
I i) may, 
lifferential 
and effecting the substitutions, and collecting the results, 
d no? \/(l — e 2 ) = n 2 a 3 ^ dt, 
dr 
= 0, 
d nae sin f 
V(1 - e 2 ) 
= war 
,<m 
dr 
dt, 
di 
do 
na cot i íd£l 1 — cos i dCl 
+ 
V(1 — e ¿ ) \dcr cos i d% 
nacoti dfL 7 , 
dt, 
dt, 
V(1 — e 2 ) di 
where il is considered as a function of r, v /} ©, a, i. 
Instead of a, i we may introduce the new quantities p, q defined by the equations 
P = sin i sin er, 
q = sin i cos <7 ; 
this gives sin 2 i — p 2 + q 2 , cr = tan -1 ~ and retaining in the formulas the sine and cosine 
of i, to avoid the introduction of irrational functions of p 2 + q 2 , we have 
d© = (1 — cos i) d6 = -—da = ——, C0S /. (qdp — pdq), 
v ' cos% cosisnri 
i. e. 
d© = 
qdp — pdq 
cos i (1 + cos i) ’ 
which determines © by means of p and q. We have moreover 
dp = sin i cos a do + cos i sin a- di, 
dq = — sin i sin a do + cos i cos <7 di, 
dCL sin o- dfl ^ cos o- dfl 
dp cos i di sin i da ’ 
dCL cos a dfl sin a dfl 
dq cos i di sin i da ’ 
from which equations and the foregoing values of di and da we find the values of 
dp and dq; the other equations of the system remain unaltered, and we have 
therefore 
dVL 
d na 2 a/(1 — e 2 ) = n 2 a 3 dt, 
dr =0, 
7 nae sin f 
d W^) 
= irar 
dSl 
dr 
dt,