374 Mr. Rumker’s new Method of 
the two triangles formed by the apparent zenith distances and 
distance, and the true zenith distances and distance, and only 
differ by slight variations in the manner of finding thence the 
true distance, or side of the latter triangle opposite to the angle 
at the zenith common to both triangles, the latter methods 
unite all in computing in the apparent triangle the angle at 
the sun and moon, and the sides adjacent thereto in two right- 
angled triangles having for hypothenuses the corrections of 
the sun’s and moon’s altitudes. These sides are the corre- 
sponding corrections of the distance. 
Let D designate the apparent distance of the centres, S and 
M the above angles atthe sun and moon, e—7 the difference 
between parallax and refraction or correction of altitude, 
then is the true distance of centres = D + cosine S.(e—7) 
+ cosine M (@!—7’) +a (e—7')? + (el —w)? + 4+. 
The moon’s parallax being greater than her refraction, the 
second correction becomes negative; and when one of the 
angles is obtuse, its cosine takes the opposite sign. ‘The fun- 
damental formula of all approximative methods is, 
( sin k — sin H cos D ipso 
cos H sin D ) g 
Ee H-—sin 4 cos D 
cos A sin D 
where g and = refer to that altitude 2 or H which stands first 
and by itself in the parenthesis. Lyons obtains by executing 
the division, 
D + (sin 2 sec H cosec D— tan H cot D) (e—7) 
+ (sin Hsec 4 cosec D— tan h cot D) (e—7)+ +. 
sin k—sin H cos D 
If we make ( ent tustallk ) (y—7) 
sin h—sin H cos D 
Se tien 
* 2cos$}(D+H+h) sin} (D+ H—A2) 
=(1 Bh cos H. sin D ) (e—"), 
we obtain the one part of Mendoza Rios’ approximative for- 
mula to which the other is analogous; and by separating in 
Lyons’s method the moon’s parallax from the refraction, thence 
results the correction of the distance for the moon’s parallax 
sin € alt. sin © alt. 
only = hor. par. x [_sap~ ee ie i 
which is identical in Elford’s, Thomson’s, and Lynn’s tables, is 
found graphically by Kelly, and by Norie in his linear tables, 
and which Thomson finds with his lunar scale; and it is only in 
true distance = D + 
) (—2") + + ty ots 
