Hagen.] 212 [Feb. 3, 



level or b}- observations of the direft and reflected image, is applied to 

 three different azimuths, dividing the circle into three equal parts, the three 

 constants i, i^ and A 2 may be found by these formulas, and hence also the 

 constant ^d may be computed for any azimuth by the formula 



id = i — i, cos U — A^) • (8) 



Thus we see, that b cannot be obtained in the usual way, before the col- 

 limation constant c has been found. But if the time is linown, we may 

 succeed in finding c in the following way: Let be the sidereal time, 

 when the direct image passes over any azimuth, and 0^ the time, when the 

 same star passes over the same azimutli of the reversed instrument, then 

 we have the two equations 



ar= A -\- A A — b cot z — c cosec z 

 a} := A -\- J A — b cot z^ + c cosec z^ 

 If again the star in the moment of observation did not pass very near 

 the zenith, the quantity b (cot z — cot z^) may be neglected as small of the 

 second order, hence M'e find by subtraction of the two equations 



c ^ J (a^ — a) sin z^ 

 where Zo is a mean value of z and z^ and may be computed from the dec- 

 lination, the latitude and the mean hour-angle. Again we have 





d^ 



where ^ denotes the variation of the azimuth in the unit of time for the 

 moment ^ ((9^ + 0). 



If we now suppose the reading of the azimuth corrected as to the colli- 

 mation constant, equation (4) becomes 



a = ^ + J .4 — b cot z. (4>) 



Again, if we observe the time of transit over the same azimuth for different 

 stars, any two observations will afford an equation of this form, 

 a^ — a sin z^ sin z 



cot z — cot z^ >* ^^ sin (z^ — z)' 

 The factor of (a^ — a) will turn out very small, consequently, b will be 

 found with great exactness, if any star near the zenith is combined with 

 any near the horizon. The quantities a and z may be computed from the 

 hour-angle t by the formulas 



sin z sin a = cos d sin t 



sin z cos a = — cos ^ sin d + sin ^ cos d cos t, 

 where 8 denotes the declination of the star and <p the latitude of the place. 

 The latter equation may be changed into the following form, more con- 

 venient for logarithmic computation : 



sin z cos a = — m cos ( <p -{- M), 



if we put 



sin = rn cos M, cos d cos t = m sin M. 



