1882] ^^^ [Hagen 



Hence, with the two observations by reflection mentioned above, the read- 

 ings 7} and z"* are to be diniinislied by 2 yj tan p cos z, in order to have in 

 all the four equations the same true zenith distance belonging to the same 

 azimuth. If the observation by reflection is taken in the meridian, where 

 tan p is very small, this correction may be omitted as small of the second 

 order. The same value of dz may also be found by the usual diff'erential 

 formula 



dz = cos ,J sin p dt 



and the following formula, which was developed above 



cos z 



dt — 2 /? 



Ul — -- /3 jjog ^ COS p" 



If for brevity's sake we denote the apparent zenith point, corrected as to 

 flexure, by Z^ and put 



Zi = I8OO + N — ai + a" — a"i + . . . 

 our four equations mentioned several times will become 



^ = z + a^ cos z + a" cos 2 z -(- a"^ cos 3 z + . . . 1 



-f b^ sin z + b^^ sin 2 z + b"i sin 3 z + . . . 



— Z, + «„. 

 180° — ^ = zi — (ai — 2 a/) cos z + (a" — 2 a,") cos 2 z — . 



_|- (bi — 2 b/) sin z — (b" — 2 bj") sin 2 z + . 



— Zj — 2 q + «„ — 2/5 tan p cos z. 

 360° — ^ r^ z" + ai cos z + a" cos 2 z + a"i cos 3 z + . . . 



— bi sin z — b" sin 2 z — b"i sin 3 z — ... 



— Zi - G„. 

 180° + ^ = z"i — (ai + 2 a/) cos z + (a" + 2 a,ii) cos 2 z — . 



— (W + 2 bji) sin z + (b" + 2 b^") sin 2 z — . 



— Zi + 2 q — flo — 2 ^ tan p cos z. 



These equation are sufficient to find the probable values of the constants 

 a, b, aj and bj by observations of different stars. The constants a however 

 can be eliminated, so that, to find zenith distances, we need not know but 

 the constants b and q. For we find 



^^180o = i(z — zii)+Vsinz-f b"sin 2 z + bi"sin3z + . . . + «„ (19) 

 The b being found by this equation, the constants a, may be found by 

 the following one 



— ^ = i (z^ — z"i) + 2 a^^ cos z — 2 a/^ cos 2 z + . . . 



+ bi sin z — b" sin 2 z + . . . — 2 q + «,. 

 The constants a may be determined from 



I8OO = i (z + z") + a} cos z + a" cos 2 z + . . . — Zi 

 and afterwards also the b^ from 

 I8OO = i (z^ -f z"i) — a^ cos z + a" cos 2 z — . . . 



— 2 bji sin z + 2 bj" sin 2 z — . . . — Zj — 2 ^3 tan p cos z. 



The equations (18) and all the others developed from them show, that 



(18) 



