376 Mr. Rumker’s new Method of 
 H, the greater, / the less, apparent altitude of centres ; 
H’, the greater, 4! the less, apparent altitude of the points 
of contact ; 
7, the semidiameter parallel to the horizon (augmented if 
the moon’s) of the upper body ; 
e—7, the correction of the upper altitude ; 
oe! —z’, the correction of the lower altitude ; 
§, the true distance of the limbs, to which the equatorial ho- 
rizontal semidiameters must be added to obtain the true di- 
stance of the centres. 
Let A be the distance of the middle of the apparent di- 
stance of the centres, and A’ the distance of the middle of the 
apparent distance of the limbs from the highest point in the 
apparent distance or its prolongation, that is, from that point 
in a great circle drawn through sun and moon where the 
effect of parallax and refraction is 0,—then is 
tan 4 (H—A) Sas 
tani (H + %) tan} D and A’=id+r4+A-—iD 
whence we obtain the apparent altitudes of the points of con- 
tact, 
sin H! = 
tan A = 
cos (1d — A’) sinH gil eprgpmeteos (4 d+ A’) cosh 
cos (1 D—A) ~ cos (4 D4-A) 
which are also found by H’/—H =, tan HD tan (} D—A) and 
h'—h =r tanh tan(} D+A) and adding f’/—H and /'—h 
to the apparent altitudes of the centres above the sensible hori- 
zon and subtracting from the sum the change of refraction from 
H’ to Hand /' to 4. But if A>3 D, H'—H is to be sub- 
tracted from the greater altitude, and the change of declina- 
tion to be added thereto; the same method is to be followed 
when the distance of the moon’s remote limb from a star is 
observed. It being, however, not so much an error in the al- 
titude as an error in the refraction that materially affects the 
calculation, and this refraction not being sensibly altered by 
a few seconds of difference of altitude, the change of refraction 
may safely be neglected. For the apparent altitudes of the 
points of contact of the sun and moon above the horizon, 
compute strictly the refractions g and e’ with regard to baro- 
meter and thermometer, and add the sum of these refractions 
to the observed distance of the limbs. From the same appa- 
rent altitudes of the points of contact above the sensible hori- 
zon, find, by applying thereto the above-stated reduction, the 
altitudes H’ and /’ with respect to the true zenith, and de- 
duct from each the corresponding refraction found before, and 
compute for the rest the parallaxes in altitude z and 7’, and 
