of the Moon with a Sextant ) or Reflecting Circle. 59 
/ 
and making r—{p-\-p'< — T) — V *, we have, see Fig. 4. 
^ — * A A, — aa' Hz sj (a a, -J- aa,) 2 (A 2 -}- a 2 ) | ( r + s) 2 i ^ a2 }• 
“ " A 2 + A , 2 ~ 
Thus, this problem serves to determine the longitude, when 
we have observed s, and the relative time at the place ; for, de- 
signating this time by T', that of the calculation of the elements 
by T (which must be taken near the opposition), we have the lon- 
gitude expressed in time by the formula l—t — T, being west 
if l is positive, and east if it be negative. 
The discussion of this formula shews, that if we make the 
quantity under the radical = 0, i. e. 
(AA, + aa,) 2 + (a; + a *) { (r + ,)* — A* - X* } = 0, 
and derive the value of (r -f e) therefrom, 
r 4- g — (AA / -f- aa,) 2 
A , 2 + A , 2 
it will be the value of the shortest distance between the centres 
of the conus umbrae and the D. 
The time will be given by the equation 
t — 
AA, AA, 
+ T, 
K + V 
in which t indicates the middle of the eclipse. 
If we would have the beginning and the end of the eclipse* 
we make s = 2^, and the time of each will be given by the equa- 
tions, 
— AA, — AA,ii:v(AA,-f- ^ , +(A»+\*)pr+ay)»-A»-Aq 
+ T 
A/ + A / 
taking the sign — for the beginning, and + for the end. 
* Or rather r = (p +p' — $)—S\ “ Pour faire ces calculs de demi-duree, on 
; ‘ ' 1 \ 
61 
fait ordinairement le rayon de l’ombre — (cr -f- x — <1), parce qu’on a remarquE 
60 
qtie les durees observees etaient toujours plus longues que le dulses calculecs, ce qu’on 
attribue a l’atmosphere de la terre, qui intercepte la lumiere du soldi, et fdit le 
mfme effet que produirait une augmentation d’environ dans le rayon de la 
terre. Cette Evaluation parait bien considerable.— Delambre, Astronomic, p. 350. 
