ON HANSENS LUNAR THEORY. 
cla = cos i dd, 
d© = (1 — cos i) dd. 
Moreover v, = <I> + a, which gives (assuming that the differentiations are performed with 
respect to t, only in so far as it enters through the variable elements) dv / = d<f> + da 
= d<I> + cos i dd, i.e. dv / = 0, an equation which might have been assumed for the purpose 
of defining the departure point; the equation, in fact, expresses that the departure 
v / is measured from a point not actually fixed, but such that the increment of v / in 
the interval of time dt is the angular distance between two consecutive positions of the 
moon. 
We have, as above noticed, da = cos i dd, and thence and from what has preceded 
j. na cot dil 
dl = V( 1 - e 2 ) ~di ’ 
na cot i d£l 
V(1 — e 2 ) di 
Now the position of the moon can be determined by means of the quantities 
r, v,, ©, a, i\ hence fi (which has been considered as a function of r, <f>, d, i) may, 
if we please, be considered as a function of r, v n ©, a, i and from the differential 
relations 
dr = dr, 
dv, = + cos i dd, 
d® = (1 — cos i) dd, 
da — da) + cos i dd, 
di = di, 
dQ . fdCl 
— = cos i -j—H —j— 
V dv, da / 
+ (1 — cos i 
d© dff> cos i dd ’