61 
of the Moon with a Sextant , or Refecting Circle. 
r 2 — y 2 + a/ 2 — 2Dy + D 2 = r 
/2 _ D 4 -fr 4 +y 4 + SD 2 r 2 — 2D 2 r — 2r 2 r 
a — 4U 2 ’ 
-2 2 — D 4 — r 4 — ?' 4 + 2DV + 2D 2 r + gr 2 J' 2 
«/ 2 = r 2 — ^ 2 = • 
making y — c> we have 
4D 2 c 2 — D 4 — 2D 2 r 2 — 2D 2 y 2 =: — r 4 — y 4 -4 2r 2 y 2 , 
D 4 + 2 D 2 ( 2 c 2 — r 2 — y 2 ) — r 1 — y 4 + 2 r 2 y 2 , 
D 2 — — (2c 2 — r 2 — ^ 2 )dz V4c 4 -{-r 4 -fy 2 — 4cV 2 — 4c 2 y 2 -f 4£ /2 r 2 , 
D 2 — — (c + r) (c — r) — (c — (c + ^) Hh 2*/c 4 — c 2 r 2 — c 2 ^ 2 +^ /2 r 2 , 
c 4 — c 2 r 2 — c 2 y 2 +y 2 r 2 = c 2 (c 2 — y 2 ) — r 2 (c 2 — 2 /2 ) = 
(c 2 — r 2 ) (c 2 — V 2 ) =(r + c) (r — * c) (y + c) (y — c), 
D 2 = (r -f c) (r — c) -f (y 4 c) (y — c) Hh */(r + c) (r — c) Q' 4- c) (y — c), 
A' = (r4- c) (r— c) B' = (yq- c) (y— c) D 2 =A'4- B'+ 2VAB? 
D 2 = « 2 + /3 2 = (A, 2 + A, 2 )* 2 -f 2(AA, 4- AA,)* 4- A 2 4- A*> 
whence, (making — AA, — aa, == A A 2 4- a, 2 = B) 
2 A t D 2 — a 2 — a 2 
B 
B 
The 4 roots of the following Equation, 
— 0)> gi ye the beginning of the eclipse. 
The end of the immersion, 
The beginning of the emersion, 
The end of the eclipse. 
/ /a\ 2 (j — t— yt 2 a 2 ~D 
The duration of the eclipse is = \b) + B 
The shortest distance between the centre of the Moon and of 
the section of the conus umbrae (occurring when t is equal to 
