72 
MR. RUMKER’S OBSERVATIONS 
For the observations of the equal altitudes of the stars I made use of the 
repeating- circle. The level was during the whole month kept invariably in 
the same position towards the division of the great circle, which by means of 
the level was maintained in the same position to the horizon. Thus the equal 
altitudes of any number of stars could be observed together with their superior 
and inferior culminations. In order to derive some benefit from one set of 
altitudes in case that clouds should prevent the corresponding one, I had deter- 
mined the point of the division answering to the zenith in the manner described 
in page 7 ; so that each observation could be reduced to the culmination by 
means of the hour-angle; and in that view I had also constructed a table of the 
hour-angles for every five minutes of altitude, corrected for the refraction 
answering to the mean height of the barometer and thermometer during that 
period, separately for the morning and evening set. 
These observations having been also chiefly made in the day-time, it was ex- 
pedient to be provided with a table for finding the stars more readily. 
Suppose <p the colatitude, <5 the polar distance, r and t the hour-angles, z and 
£ zenith distances, Z the meridional zenith distance, D the difference of alti- 
tudes, A difference of azimuths, x = ^ (r — t) half difference of hour-angles, 
and N an auxiliary angle : then is 
cot <p tan h = cos t, 
cos <p sec <5 = cos £, 
cos <p sin S = sin azimuth; 
, . . v. , , . 2 sin 8 sin x sin N 
and sin x sin S = sin \ D, sin A = — s in (g + D) — 
where tan N = cos § tan x. 
With D and A by simple addition or subtraction a table of altitudes and azi- 
muths may be constructed for every five minutes of the hour-angle. 
Stars observed on one side of the meridian become often visible on the other 
side, only at a greater distance from it ; so that it is sometimes necessary to 
combine unequal altitudes, which is not difficult with stars, if the same dif- 
ferences of altitudes are observed on each side. 
sin £ D 
The formula sin x = 
serves to reduce these observations, x is 
sin <p in azimuth 
the quantity to be applied to the middle time to reduce it to the time of culmi- 
