COMSTOCK — STUDIES IN ASTRONOMY 95 



used, e. g. a universal instrument or an engineer's transit. 

 If the makers would furnish a simple means of fastening 

 the striding level which accompanies the better class of 

 transits, with its tube at right angles to the horizontal 

 axis, the efficiency of these instruments would be very 

 greatly increased, but even without this attachment the 

 observation of equal altitudes is the most advantageous 

 mode of employing such an instrument for the determina- 

 tion of either latitude or time. We proceed to develop the 

 equations for the general case in which both of the quan- 

 tities are required. 



Let r, and T.^ denote the observed times at which two 

 stars cross a given almucantar whose (unknown) zenith 

 distance is z, and let a^, ti, at,p, be the right ascensions and 

 polar distances of the northern and southern star, respec- 

 tively. The formulae for the transformation of coordinates 

 furnish for the two stars the equations : 



cos z := sin <p cos n -{- cos q) sin it cos {T -f- r) 

 = sin (p cosp -f- cos (p sm p cos ( T — r) 

 where 



T^r = T,-\- JT-a, T - r = T, -\- JT - a. 



Subtracting the second equation from the first and divid- 

 ing by 



2 sin i (p + TT) sin i {p — tt) cos <p 



we obtain 



tan <p = cot I ip -\- 7t) cos T cos r — cot ^ (p — tt) sin T sin r (2) 



We introduce into this equation the auxiliaries 



I cos A = cot i (p -f TT) cos r I sin A = cot i (p - tt) sin r (3) 



and obtain 



I cos {T — A) = tan cp (4) 



From equations (3) we obtain 



I sin (A - r) ^ \ cot \ {p — n) - cot \{p-{- Tt) \ sin r cos r 



' (5) 



I cos (A — r) == cot hip — Tc) sin'^ r -{- cot i {p — tt) cos'^ r 



