60 Mr Bowdich on the Measurement of an Eclipse 
The duration of the eclipse will be given by 
in _\A AA ' + AA ') 2 + + A - 2 ) { < r + — a 2 — } 
2 a ( 2 + a , 2 
If we desire the end of the immersion and the beginning of 
the emersion, we make $ = o, and the times will be given by the 
equation 
4 — -aa, — aa,:±:\/(aa + aa ,) 2 -f (a / + a *) (r 2 — a 2 — a 2 ) , 
+ A 2 + A ’ 
being after the middle of the eclipse, if we take the sign -f of 
the radical, and before if we take — . 
Let I represent the duration of the total immersion of the D, 
i. e. the time during which she remains completely invisible, and 
we have 
1T , / (AA y + AA,)* + (A * + A *) (r 2 - A* _ a*) 
a/T^? 
Lastly, we have the hour to which any enlightened part t cor- 
responds, by making e equal to this part, and deducting the cor- 
responding value of t 
In all these equations, we use or repeat nearly the same lo- 
garithms, which very much expedites the calculation. 
Let us suppose that we have measured the chord of distance 
between the two horns of the Moon, which seems to me to ad- 
mit of more precision ; we have only to make the following ad- 
ditions in the original expressions for the elements : 
c= | distance of the horns of }), 
if = mean time of the observation of c, 
61 
60 
(p _j_ f — £) — radius of the section of the conus um- 
brae, 
« = a = A y # -f- a y = ax 
X a, 
tang i — a sin z = /3 cos i = « 
y — (Zaf -f *ij x — ax' — /3 y (Biot, Geom. Anal No. 77.) 
x' 2 + 7 j 2 = r 2 Equation of circle of conus umbrae. 
y 2 +(* i — D) 2 = Equation of circumference of D refer- 
red to 2d axis. See Fig. 5. 
