212 Cincinnnti SocAety of Natural History. 



Let T = the observed time of transit of a given star over tlie mean 

 wire, 



a = the right ascension of the star, 



8 = the declination of the star, . 



r = the hour angle of the star east of the meridian at the time 



of observation, 

 90° — m = the hour angle of the point in which the horizontal 

 axis of the Instrument produced toward the west meets the 

 celestial sphere. 

 d and h = the declination and altitude of that point, 

 g = the angular distance of the mean wire frora the coUimation 



axis, 

 t = the clock correction = a — T — - 

 Then, putting n = tan d, and c = sin g sec ??, and neglecting b, 

 since it may be applied directly to the reading of the clock, we have 



sin m = — n tan <p, (1 ) 



sin (r — in) r= n tan ^ -|- c sec <5, (2) 



where c is usually very small. For a star whose declination is (V, 



sin (r'^?n) = n. tan «5' -|- c sec <5'. (3) 



Adding (1) to each of the equations (2) and (3), and differentiating, 

 we have 



cos (t — m) d T -\- [cos m — cos (t — ?>?,)] dm = (tan ') — tan <p) d)i 



-\- sec d d c, (4) 



cos (t' — m) d r' -\- [cos m — cos (t' — m) ] dm. = (tan (V — tan ^) dn 



-\- sec 8' d c, (5) 



where if the stars are observed near the zenith, and the instrument is 



not too far from the meridian, we may assume 



cos m = cos (r — m) = cos (r' — m) = 1. 



Assuming also, 



sin mi = — ??i tan cp, (0) 



sin (ti — mi) = 7?i tan 8, (7) 



sin (r,' — Wi) = », tan 8', (8) 



where m,, n^, - and r,' are approximate values of m, ti, r and r', we 



have 



T — T, = (n — )?,) (tan 8 — tan ^)-f- c sec 8, (9) 



t'- — T,'=(n — ??, ) (tan 8'- — tan <f)-{- c sec 8'. (1*^) 



