1819.] for finding the Longitude. 187 
ena ein. (S —‘A) sin. (S — H) | = cos. ¢ (A), and 
“cos. h.cos, H 
we have by (8) 
cos. D’ = cos. (H’ —h’) — 2 cos. ¢°, or by substituting 1 + cos, 
2 9 for 2 cos. ¢? 
cos. D’ = cos. (H’ — h’) — 1 — cos. 2 ¢ and 
1 + cos. D’ = cos. (W’ — h’) — cos. 2 ¢, which gives 
a H’ —#& 3 H’ — Al 
cos. + D? ='2 sin (? + see) . sin. (0 — HS) 
consequently 
oe ae J sin. ( - — *)) ‘ sin. (¢ — S =~) .. (B) 
We have by (a) 
cos. D —sin.k. sin. H cos. D' —sin. h’. sin, H’ 
cos. A. cos. H i cos, k’cos. 
But 
cos. D — sin. A. sin H 1 cos. D — cos. h . cos. H — sin.h. sin. H 
cos. .cos., H t= aad bani: co eaal 01a alk 
: D+H—A\ . D-—H+Ah 
2 sin. ( ) «sin (+) 
cos. D —cos.(H—h) _ 2 2 I 
cos. hk, cos. H Sai cos. h. cos, H eS < = 
the same manner will be — ees ee {ee 
cos. A! . cos. H’ 
, (> - —) 5 es H! = Af 
2sin.{ ——————_——_ } . sin. “i ) 
9 2 
cos, A’ . cos. i! 
. a - D-—-H+ih 
sin. =) sin, —— 
cos. A. cos. A 
5 /D! + H — fh) c D—H+h 
sin. (———-) sin, -—,—*-) 
cos. h’ . cos, H’ 
sin (S —h). sin. (S — H)cos. h’ . cos. H’ H’— hv 
- D‘ ; 
——_--—— ——_——- = 4) ae A n. 
cos. hk. cos, H ney ( 2 + ( 2 )) * 
D' H’ — i! 
(Fa) vere ee tte ©) 
It may be remarked, that when h’ > oe = His to be sub- 
Hence 
, or 
stituted in the equations B and C. 
The equation (A) gives g, and D’ is found by (B). The term, 
to the left, (of the equation (C)) is the same as cos. ¢° pre- 
viously found, which must be equal to the other term of (C), for 
the calculation of which, D’ is required. If, therefore, D’ as found 
by (B), makes this term equal to cos. 9*, the calculation is 
eH, otherwise not. An example may illustrate the 
whole. 
