reducing Lunar Observations. 381 
The third correction is to rectify the error committed in 
assuming one side of a spherical jtriangle equal to the adja- 
cent segment of its base cut off by a perpendicular from the 
vertex of the opposite angle, which error will diminish with 
the angle contained between the above side and the base. 
Suppose a and c to be two sides of a triangle, and } and d 
their adjacent segments, and » = a — 6, then is tan $p 
_ tan} (c—d).tan § (c+d) 
iF ian(O+$u) 
approximations, supposing it in a first one = 0. But 
tan} (c —d).tan}(c+d) = tan’? 3A, if we denote by A the 
tang? 5A 
tang (b+ #) 
Whence we find the following general expression for iy, 
which may be reduced accordingly as circumstances allow : 
Suppose, 
» whence » may be found by 
perpendicular. And tang }p = 
tang 5A Bates 
1+tang? $a — f_tan?3a ve tan22A 
___ | tan d+tan? 4a + | tand+tan?3a_ 
tand (tan d+tan? 3a —tanbf_tan23a q 2+-&e. | tan b-+tan® $A tan b-+tan?$ 
tan 6+tan2 3a tan b+tan2 32 | tand+ tan 6+ 
tan b+tan? 3a tand+tan23a 
tan 6+... tan ...J 
then is: tang b tang $1» = N—N?+ N®— N*4+ N5— 4 — ... 
py. becomes, therefore, negative when > 90, and is = 0 when 
6 ="90°. 
Walbeck, who proposed computing for the time of obser- 
vation, reduced for the estimated longitude from the first meri- 
dian, the apparent distance as seen from the place of obser- 
vation; and to derive, by a comparison of this computed 
distance with the observed one by means of the moon’s ap- 
parent horary motion, the error of the estimated longitude, has 
remarked already the necessity of computing the refraction 
for the points of contact when the altitudes are low, in the 
note to page 15 of his Dissertatio de Modo reducendi Distan- 
tias, Abo 1817: “ Si rigorose calculaveris, refractio non pro 
Centro lunz sed puncto limbi quo distantia capitur sumenda 
est. Inutile vero est, calculum talibus minutiis molestum red- 
dere, que preterea, nisi sit luna vel sol horizonti proximus 
nullius sunt momenti. Ex hac etiam caussa minimee altitu- 
dines evitari debent.” But low altitudes are better than none, 
and cannot always be avoided. 
Walbeck found from latitude, declination and horary angle, 
the altitudes, parallactic angle and the corrections of the alti- 
tudes, and thence the apparent declinations and right ascen- 
