388 M. Kumker on the Calculation of ParaUa,ves. 
The following is my formula : 
sii 
cos D 
g .sin P cos H cos (b + M) 
V € sin P sin H . ^ 
cos (L — a) tan H = tan M ^ ^ sin (L — a) = sin S 
cos D cos ]\1 cos ^ 
= sm 
f sin (b -h 0 cos 5 == sin /3 
Thence^ tan ^ , c-t^ 
tan 7r=h h. 
DE 
= (n/ 
I cos ( 64 - 
(D + B~-/3).(D--B+~/3) ^. 
cos B . cos /3 
SE + DE^SD. 
Demonstration : 
cos^r 
r . Ti l Sin P . ( sm ;5: . e sm P' = sm p 
Let sm P' be =: Then is -I ^ ^ ^ tt • 
cos D (. cos z . cos M == cos H. sii 
Thence 
but 
Consequently, 
Thence ^ and /3 are easily deduced. 
sin(5-bM) 
tan z. g sin P cos H sin (b 4- M) 
cos M ^ 
tan z. tan (b 4- M) = tan p. cotan 
p sin P. cos H. cos (b 4- M) ^ ^ 
cos M 
Now we have hn —sj ; but k 0 is re- 
^ cos B . cos /S . 
quired. 
A n being a tangent, Cn must cut the circle of latitude of the 
moon in two points, and ko <.kn. The point n where the 
observer on the surface of the earth sees the star immerge or 
emerge, is either hid to the centre of the earth, or falls within 
the disk of the moon there to be seen ; but ko — kn. cos nkoy 
and nko — Cn A=z tc. 
My formula contains nowhere a factor that could become 
The factors are all sines of great, and cosines of small arcs. On 
the contrary, the products are all sines or tangents of small arcs. 
This method can be worked with sufficient - exactness with five 
decimals. I am, &c. 
Hamburgh, 1 
Feb. SL 1821. I 
Rumrer. 
