86 BULLETIN OF THE UNIVERSITY OF WISCONSIN 



quality of pivots and diurnal aberration has been neg- 

 lected. These quantities may, however, be taken into ac- 

 count, as in the case of meridian observations, by applying 

 to the observed level constant, &, a correction, p, for 

 the first two sources of error, and by applying to S a cor- 

 rection, 



s. 

 - 0.021 cos (p. C 



for the aberration. 



The formulae requisite for the reduction of observations 

 in the vertical of the pole star may now be collected, 

 slightly simplified and arranged as follows : 



Data known independently of the observations: 



8. 



<p, a 19 a 2 , 8j, 8 2 , K 0.021 cos q>, p a = 90 - S 2 

 Data given by the. observations: S", S, 6, K. 



t = (a i - or,) + (S - S') 

 h = I -f- tan S s cot d l cos t 

 1 = 1 tan \p z cot 8 l cos t 

 C = h tan p 2 + / tan cp 

 C' = 15 I sec <p 



tan U = 



cot 8 tan d s sin t 



1 cot d t tan 6 2 cos t 

 sin m' = tan g> cot 8 t sin (t -f U} 



tan a' = tan m' cosec (p 



A T + Cc = <x a - (S + U + m' -f b sec q> - CK) 

 M - C c = K -f a' + b tan <p - C'x 



The computation of these formulae may be somewhat fa- 

 cilitated by an algebraic device upon which Dollen places 

 great stress. From the ordinary development of sin x and 

 tan x in series, we have, when x is small, 



log sin x = log x ~- log tan x log x -f 2 -~- 



where M denotes the modulus of the common system of 

 logarithms. Putting 



6 = $ MOJ 



we may tabulate 6 with x or log x as argument, and such a 



