reducing Lunar Observations. 377 
subtract their sum ++’ from the observed distance augmented 
by the refractions, and call the remainder d + ¢ + oe’ —2— 7! 
Find also for each altitude the corrections g—z and ¢/ —7'; 
then is 
gees COS (H'+3d—A’).(e—7) __ cos (2'+3d+A').(e!—7') 
cos H’cos(id—A') . cosh’ cos(1d+A ) 
4 00s (H’ +3 4d—A’) cos [H'— (3 d—A')] (g—z)?. sin 1” 
2 tan d cos* H’ cos? (1 d— A’) 
4, 88 (A! +4 d+A) cos [h'—($ d+ A')] (g!—2z!)?. sin 1! me 
2 tan d cos* h’ cos? (1 d— A’) ae 
If we call the first correction «, the second £, then is: 
y sin 1/ 
baba BH (pe baat (fa 3 A ay 
to which both horizontal semidiameters are to be added to 
find the true distance of the centres. The square of the sun’s 
correction can always be neglected, and that even of the moon’s 
correction disappears when d is near 90°, or when the moon’s 
correction is small. All four corrections disappear when the 
distance is the supplement of the sum of the altitudes ; but 
when the distance is equal to the difference of the altitudes, 
all other corrections vanish except the first one, which be- 
comes = 2 (p—7). 
The first two corrections only, however, need be computed, 
as the others can be taken as a small third correction from a 
table contained in most nautical works, with its sign, which 
becomes negative when 6>90; so that this method has the ad- 
vantage that no difference of cases need be attended to, as in 
that of Witchell’s, whose example I have followed in the com- 
putation of it. For if we call, of the two first corrections, the 
one proceeding from the sun S and that from the moon M, 
then is, 
§ = 6— S+M+ third correction, 
which latter one, in the absence of the tables, may be found by 
the formula ((’s corr. —} M).M cot 6 sin 1", 
To the semidiameters, which are to be added to the ob- 
served distance to obtain the apparent distance of the centres, 
a correction should be applied on account of inclination to 
the horizon, for which Mendoza Rios has given a table. This 
inclination to the horizon is found by sin. inclination = 
sin Hsec(4D— A). In the subsequent example this incli- 
nation is = 90°; so that the vertical semidiameters have been 
used. As however the distance of the centres enters only into 
Third Series. Vol. 8. No. 48. May 1836. 2Q 
