COMSTOCK — STUDIES IN ASTRONOMY 83 



sin (r — to) = tan n tan d -f sin c sec n sec S (1) 



together with the equations 



tan n = sin b see n cosec q> — sin in cot q> {2) 



cos a tan m = tan b cos (p -\- sin q) sin a (5) 



furnished by the spherical triangle, P Z A, formed by the 

 pole, the zenith and the point in which the rotation axis 

 of the instrument, produced toward the west, intersects 

 the celestial sphere. The sides and angles of this triangle 

 have the following values: 



PZ = m^ - <p PA = 90'' - n ZA = W - b 

 P = 90^ - VI Z - 90^ + a 



The symbol z" represents the east hour angle of the star 

 at the instant of transit over the middle thread, and we 

 have obviously the relation 



r = a - S - AT [i) 



Since each star observed furnishes an equation of the 

 types (i) and (4), it appears that if the instrumental con- 

 stants h and c are known an observation of the transits of 

 a circum-polar star and a southern star suffice for the de- 

 termination of the unknown quantities aT, m, n, a, and our 

 problem consists solely in so transforming the preceding 

 equations as to facilitate the determination of aT and o. 



Denoting by the subscripts 1 and 2, respectively, quanti- 

 ties pertaining to the polar and the southern star, we 

 write equation (1) for each of these stars as follow^s: 



sin (r^ — m — ^) = tan S^ tan n ] 1 + cosec d^ cosec n sw (c + x^) i 



(5) 

 sin {to — m — S) = tan d^ tan n • 1 + cosec 8.^ cosec n sin (c 4- cr,) [ 



where 5, oc^, and x, are small arbitrary quantities subject 

 only to the condition that they must be so determined as 

 to satisfy the equations. Since this is equivalent to only 

 two relations among the three quantities we are at liberty 

 to impose a third relation, for which we choose 



sin {c-\- Xi) sin do = sin (c + Xo) sin 5, 



