258 Butherfurd Photographic Measures. 



We sometimes need the correction for the position angle p instead 

 of that for Z, which is given by equations (c). For that purpose 

 we introduce the parallactic angle q, and write finally : 



<r — s = s k sin if [tan 2 z cos 2 (p — q) -f I ] + A s -\- B s 2 \ (J\ 



<a — p = — k tan 2 z sin (p — q) cos (p — q) — 7c tan 2 sing tan X-^-A' -\- B' s i 



The term 



— k tan z sin q tan 5 



is the usual term introduced for the purpose in question.* It will 



also be noticed that p — q has been substituted for its equivalent I; 



and that the factor sin \" has been introduced in order that we may 



use s expressed in seconds of arc. This last change must also be 



made in equations (c) if we apply them to any practical case. It 



will be noticed that the formulae (d) are very similar in form to 



those of Bessel, from which they differ by the use of k instead of x, 



and z (the apparent zenith distance) instead of £ (the true zenith 



distance). If, following Bessel, we introduce x by means of the 



equation :^ 



dk 

 x tan 2 l=k tan 2 z -f- — tan z 

 dz 



we can write (c) in the form : 



a — s = s x [tan 2 1 cos 2 1 -f 1 ] + 8 (k — ») -f B s 2 



2j — I = — x tan 2 1 sin / cos I -\- B' s 



The following little table gives the values of (k — x) for various 

 values of tan z. 



Log tan 2 (k — x) X Io3 



0.0 -|-- OOI 6 



0.1 +.0023 



0.2 -I-.0028 



0.3 -f.0040 



0.4 +-0059 



It is plain from this table and the preceding ones, that for the 

 reduction of the present Pleiades plates we may neglect the term 

 in k — x , as well as those in B and B' . We may therefore write 

 our formula? : 



a — s = s x [tan 2 £ cos 2 (p — q) -f 1] ) 



X — 1= — x cosec 1" tan 2 1 sin I cos I j 



* Bessel, Astronomische Untersuchungen, vol. i, p. 165. 

 f Loc. cit., p. 157. 



