10 



with the star: its polar distance should be the arc PO (see the 

 figure) but 



tang. P0= — cotang. PE 

 PO = 90° +^ 



therefore it is known, and any error which affects the instrumental 

 determination of the star's PD, affects this arc equally. 



Lastly it is necessary to estimate the errors which arise from the 

 declination axis not being perpendicular to the polar axis, and from 

 the latter not being parallel to that of the earth. 



In the first case the observed PD is the side of the right angled 

 triangle whose hypolhenuse is the true, and calling the error of the 

 axis n, we obtain the ordinary reduction, 



sin. A' = 2 sin.* „ 



sin. n' = 



tang. D 

 sin. n 



tang. D. 



But this error is easily corrected, and the adjustment is not liable 

 to vary. 



The error of the polar axis is much more unmanageable ; it 

 changes with every variation of temperature, and the instrument 

 can scarcely be touched without affecting it. Suppose its upper ex- 

 tremity raised e seconds, and moved westward m seconds ; to find the 

 values of A and 11, thence arising, we have 



COS. D = COS. z. COS. a -\- sin z. sin. a x cos. Azimuth, 



differentiating on the hypothesis of z constant, 



,•„ n 7 r» (do X (cos. «sin. a — sin. zcos. a. cos.A.) 1 



sin. L). a JJ zz < , j a • • a r 



\ -\- a A. sm. z. sin. a. sin. A. j 



Substituting for d a,d D, and d A, their equivalents, e, A and -^j^. 

 »nd for the coeflBcient of d a its value cos. P x sin. D, we derive 



A = e. cos. P-\-m. sin. P. (7) 



