276 Prof. Encke on Transits. 



Assuming that a star whose dedination is = 5 is in the real 

 hne of vision in s, and calling t the hour-angle which is to be 

 added to the observed one, in order to obtain the time of the 

 star's passage over the meridian, we have in the triangle Vsp 

 this equation. 



(I) sin c = — sin 8 . sin 7i + cos 8 . cos n .sin (t — m) 



by which t is to be determined. This equation, from which all 

 the others are derived, is true both for the superior and inferior 

 culmination, provided, for the latter, S is counted from the same 

 semicircle of the equator ASO as for the former; passing, there- 

 fore, through the pole, 8 is to be increased above 90°. This is 

 usually expressed by the rule, that for the lower culminations, 

 instead of the real declination, its supplement is to be taken. 

 From equation (I) we obtain this: 



sin (t — 7«) . cos n = sin 7i . tang 8 + sin c . sec 8 

 or by adding sin m . cos n . on both sides : 

 (A) 2 sin ^ T . cos (5 T—m) cos nz= sin m . cos n + sin n . tang 8 

 -H sin c . sec 8, from which the formula of Bessel, in which p 

 is only referred to the pole, is immediately derived. It is as 

 follows : 



T = m + 71 . tang 8 + c sec 8. 



As n may be found by observations of circumpolar stars, but 

 jn cannot be found by any direct method, it becomes necessary 

 to combine with this equation the 4th or 8th of the above rela- 

 tions, according as i can be more accurately found by a level, 

 or ^ by a meridian-mark. Those relations give 

 sin m . cos 71 = sin i . sec <f> — sin 71 . tang <p 



= cos i . sin /t . cosec .f + sm7i. cotang ^. 



The factor, cos (^t— m) cos n is the cosine of the angle at 

 which the great circle bisecting t intersects the circle S^', and 

 cos 7n . cos 71 is the angle of Sp' and the meridian, at their 

 point of intersection Q. The distance AQ is obtained by the 



k /-\ sin ?n 



equation tang AQ = - ■^^^. 



If we substitute on the right-hand side of (A) the first and 

 second relations, we have 



„^ . , ,, . . . cos f*— S) 



(B) 2 sm ^ T . cos (^ T—m) . cos 71 = sini — ^^-| — 



• , • sin ((p— S) . 5. 



4- sm k . cos I ~-^ [- sm c. sec 8 



cos d 



answering to Mayer's formula : 



T = i cos 3 . sec 8 + k sin 2; . sec 8 + c. sec 8, in which p is 

 referred to the zenith only. 



If the relative positions to the pole and the zenith are both 



to 



