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, b, a correction, ± /i, for 

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

 rection, 



s. 

 - 0.021 coscp .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: 



s. 



q), a^, a.,, 5,, do, yi = 0.031 cos (p, p., = 90'^ - 8.^ 



Data given by the observations: -S", S, b, K. 



t = (cTi - n:,) + (S - S) 

 /i = 1 -)- tan (5g cot S^ cos t 



1 = 1 — tan I P2 cot S^ cos t 



C = h tan i Po + i tan q) 



C = 15 / sec q> 



cot 8. tan '?.> sin t 



tan U = ~ — ; — ^ T 



1 — cot d J tan ., cus t 



— sin m' = tan q) cot 8^ sin {t -\- U) 

 tan a' = tan m' cosec <p 

 ^T-i-Cc = a, - (5 + [^ + m' + b sec (p - Ch) 

 M — C c = K -\- a' -\- b tan cp — Cx 



The computation of these formulas 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 stn X = log X ^ log tan x = log x -\- 2 — — 



where M denotes the modulus of the common system of 

 logarithms. Putting 



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



{16) 



