1882.] '^^" ^ [Hagen. 



By adding the above equations we tind 



k == sin ^ [i {0 + ^0 — HT + Ti) — J T], 

 or as ^ {fl -|- (9^) = a is the star's right ascension 



k = sin ^. [a — HT + T') — J T]. 



Part II. — lafluence of the iiidiiiation a on Altitude Obsermtioiis. 

 By a ■^e have denoted that component of the inc]ination Z Z^ of tlie 

 apparent to the true horizon, which lies in the vertical plane of the instru- 

 ment used. "With large instruments part of this component may be 

 caused by the instrument and its piers, and is, therefore, as was explained 

 in the beginning, depending on the zenith distance of the object observed. 

 The other part of « is according to former notations [see formula (6)] 



q == i, cos (.-j — A,) (15) 



and is caused by the constant local irregularities in the figure and density 

 of the earth. The first part of « will have an effect on altitude observa- 

 tions quite analogous to the flexure of the instrument. This latter correc- 

 tion is generally represented by the series 



a^ cos z -f a" cos 2 z -\- a"^ cos 3 z + . . . 

 -j- b^ sin z -f b" sin 2 z -j- V" sin 3 z -{- . . . 

 and its sign is understood so, that if z is the reading of the zenith dis- 

 tance of a star 



z -j- a^ cos z -J- . . . -f b^ sin z -^ 



represents the true zenith distance freed from flexura. If for instance N 

 denotes the reading of the Nadir point (for which z = 180°,) 



N — ai -f a" — a"i + • . . 

 will represent the true nadir freed from flexure. 

 By a similar formula the component a may be represented this way 

 a = q + ai^ cos z -f ai" cos 2 z -f aj'" cos 3 z -f . . . ■) ^g^ 

 + b/ sin z -f bi'i sin 2 z + b/" sin 3 z + . . . / 

 For the nadir (z = 180°) we have 



a„ = q — a^i -f a," — a^'" + . • . 

 Now let z denote the reading of the instrument, ^ the true zenith dis- 

 tance of the object S observed, and N the reading of the nadir, then we 

 shall have for direct observations (Fig. 2). 



z -j- a^ cos z + a" cos 2 z -(- a"i cos 3 z + . . 



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

 — (N -f 180O — ai -f a" — a"i +...) + ^-o = ? 

 Again let z^ be the reading of an observation by reflection and we shall 

 have 



zi — a^ cos z -f- a" cos 2 z — a"' cos 3 z -j- . . . 

 -f bi sin z — b" sin 2 z -f b^" sin 3 z — . . . 

 — (N -f 180O — a^ + all — a"i -f . . .) -f Co = ISO^ — ^ -f 2 a 



PRaC. AMER. PHII.03. SOC. XX. 111. 2b. PRINTED MAY 18, 1832. 



