209 



[lagen. 



or introducing the angle -i instead of C by the formula 



tan fl =: tan C sin z 

 we have 



(1 4- cos d .\) tant /3 = tan z sin d A, 



or simpler, 



d J = 2 ;3 cofez, (1) 



which is the correction of the itzimuth, for observations by reflection. 

 There the azimuth is to be reckoned from south to west etc., and [i riglit- 

 liand of the observer. 



The correction of the hour-amjle may be derived from formula (1) by 

 means of the well-knov.-n difierential formula, 



_ s in z 



'^^ — cos 5 cos p^^^' 

 where p denotes the parallactic angle and 8 the declination of the observed 

 object. Thus we find 



For the sake of verification, this last formula may also be derivedln the 

 following way. Considering the great circle C Z S^ as the meridian and 

 joining S with the north pole N we have in the triangle S S^ N 



sin S SI 

 sm dt = „^^ ^ . 

 cos (J 



But in the triangle S S^ C we have in like manner 



tan S S' = tan C sin 3 z, 



since Z S' may be put equal to z and finally we have as above 



tan ^5 =^ tan C sin z, 



hence. 



sin 2z 

 tan S S^ = tan y? . = 2 tan y3 cos z 



and consequently by combining the first and last equation and supposing 

 dt and ,5 to be very small angles 

 cos z 



2. Ike azimuth instruments. 



The correction of the azimuth for the observation by reflection 

 d J = 2 ^3 cot z (1) 



has the meaning, that in such observations the actual reading of the azi- 

 muth is by d A too small, as long as /5 is positive right-hand of the observer. 



