Hagen.] 



214 



[Feb. 3. 



For upper ciilmiuatious we have 



z = -f (^ — d) culminatiou south of the zenith 

 z = — ((/> — d) " north " 



and for lower culminations z = 180^ — (^ + '5), hence the corrections for 

 the hour-angle are 



cos ((p — S) 

 cos d 



for upper culm, dt = 2 ^9 



lower 



dt 



2,5 



cos (^. + d) 



If again we compare this correction with the one for the rotation axis 

 not lying parallel to the horizon, we find them coincident, except the con- 

 stant. For if b denotes the elevation of the west end of the rotation axis 

 above the true horizon, we have the usual formula for upper culminations 

 cos (^(p ^) c — for direct image 

 + " reflect. " 



dt = 



cos 8 

 and for lower culminations 



^ _ cos ((f + (?) c — for direct image 

 dt = -+- b -~^ (+ " reflect. " 

 whei'e dt has the same meaning as above. Joining the two corrections and 

 putting 2 Q? -f- b) = d, as before, we find 



For upper culminations. 1 



COS (<p — ()) 



direct image dt 



reflect. 



dt = — (b — d) — 



(f-'^) 



direct image dt = — 



reflect. 



For lower culminations. 



COS (^ -f- (1) 



(11) 



(b-d) 



COS (^ -f 8) 



M^here dt denotes the increment of the hour-angle. We need not consider 

 separately the formulas for lower culmination, as we may deduce them 

 from those for upper culmination at any time by simply substituting 

 180O _ 8 for 8- 



In consequence of these considerations the formulas of Tobias Mayer, 

 Bessel and Hansen are to be modified for observations by reflection as fol - 

 lows : Mayer's formula is the following 



= h 



cos (^ -1- 



+ k 



sin (y> — (5) 



+ 



where r 



^ " cos 8 "^ "" cos 8 ' cos 



dt is the hour-angle east of the meridian, b the elevation of 



