814 
where A/= AÀ — Lo, Aw being the perturbation in the longitude 
of the perigee. Further we have, to the order of accuracy here 
required, d/= dl, — 2e4 cos l. Therefore, neglecting the difference 
between the perturbations 4/ and A/, at the two epochs, and putting 
c = (l—m)c’, we find 
vl, + dl,’ — y’ sin 2F,—Aw—(e—1)A. . . . (7) 
Now we have approximately / = /,'— mA, and also c’—1 differs 
not much from m, therefore, if Aw is neglected, we find from (7) 
v—v' = 1,—l,'—y? sn 2F, or ww = 1,—1,'—y’ sin 2F. 
The term y° sin 2F is the reduction from true opposition to central 
eclipse. Consequently the meaning of these formulas is: The difference 
of the true longitudes of moon and sun at true opposition is equal 
to the difference of the mean longitudes at mean opposition. 
In the expression for A, which only occurs multiplied by the 
small factor c’—1, we can neglect all perturbations except the 
evection. This latter is very easily applied by replacing e, in d/ by 
6 
7% (see e.g. TissERAND III p. 134). We have thus 
pees 
Absil denk 
( 
We must now develop the quantity 
= @ NE esin v 
Kk =| — | coss ——_—_.. 
r 1 He cos v 
where for v we must introduce the value (7). We can take with 
sufficient accuracy 
a Ne 
a = 14-22 ost. 
- 
Further we can take coss=41, and we put 
Ree opens e, Aw = Eu sin a, 
It appears, in fact, on investigation that all perturbations which 
need be considered, are of this form. We then find easily 
ved 6 
K=e, sin Nie — 7 e-)| e,? sin 21, +(e’ +1) ee! sin (Ll, +1,') + Dx sin (1-2). 
The perturbations Se and Aw are not as such contained in the 
existing lunar theories. I have therefore derived them, neglecting all 
perturbations that do not exceed 0.01 e. The only remaining term 
is again the evection. Those terms in the perturbing function, which 
in longitude give rise to the variation, produce a large perturbation 
in e and w, but its argument is a=/+ 2D, and consequently the 
