Hageu. 



216 [Feb. 



which determination will be the more exact, the greater cos {<p — li), i. e. the 

 nearer the observed star passed by the zenith. 



The collimation constant is found in the usual way either by reversing 

 the axis, or by using two horizontal coUimating telescopes, and the con- 

 stant n by observations of the upper and lowfer culmination. If then, we 

 suppose the times of transit already corrected as to the errors arising from 

 c and n, we find from Bessel's formula 



for direct image a = T + J T + m 



cos ((p — ()) 



"reflect. " a = T^ + J T + m — d — ^^^ — 

 and from Hansen's formula 



for direct image a = T + J T + b sec ^ 

 " reflect. " a = T^ + J T + (b — d) sec ^. 

 By these formulas it is made evident, that neither m nor b can be found inde- 

 pendently of the dock correction. But if this is known, Bessel's formula 

 will give the constant m, or Hansen's formula b. The azimuth constant 

 k may be determined by observations of upper and lower transits or be 

 computed from (13). Thus, b being found, we may finally determine 



,?=4-b. 



i. e. the west inclination of the apparent to the true horizon. 

 4. The Transit Instrument in the Prime Vertical. 

 From the general formula 

 cos z 



we shall obtain the formula for the transit instrument in the prime vertical 

 by finding the value of cos p for the azimuth A ^= 90° and substituting it 

 in the above formula. We have in general 



cos p sin z = cos ^ sin ^ — sin 5 cos ^ cos t. 

 But for the prime vertical we have the three special equations 

 sin z = cos 5 sin t 

 cos ip cos z 

 ^««'^= cost 

 sin <) = sin <p cos z. 

 Substituting these quantities successively into the three members of the 

 general equation we find 



cos p cos d = sin <p cos ^ cos z tan t. 

 But from the three formulas for the prime vertical follows 



tan z 

 t^'^* = cTs"^ 

 consequently, 



cos p cos d =■ sin cp sin z, 



