PHOTOGRAPHIC RESEARCHES NEAR THE POLE OP THE HEAVENS. 79 



K + » 2 ) (X) sin 2 V — dm (7) sin l" — dn cos (it), 



f = H-m(X)sinl", 



d*Y 



dt* 



d*X 

 dt* 



m* (T) sin 2 1 " -+- tfw (X) sin l" — mit cos (w) sin 1", 



m (m 2 -h n 2 ) (F) sin 3 l" — 3 {mdm -+- ndn) (X) sin 2 1 



n 



n (m 2 -+- w 3 ) cos (it) sin 



2 





m (m 2 -*- w 2 ) (X) sin 3 1" — 3 mdm (F) sin 2 l" 



(2 wdm -+- mdn) cos (ic) sin l". 



The corresponding polar coordinates for the beginning of the year of 

 observation, which we will call (a ) and (icj, are computed by the formulas : 



sin O ) cos (a ) = (Z ) sin 1 ", | 

 sin (« ) sin (a ) = (T Q ) sin l". { 



(31) 



We shall now write: 



f 



\ a = («o) — (°0> A, ir = (ic ) — (tc). 





Then the ordinary reductions from mean to apparent place can be com- 



puted by the usual Besselian formulas: 



Eed. (L = aA-t'bB-+-cC-i-dD-+-E-t-ii.T 



where : 



Bed . « as — a '^4 — &'Z? — c'C — d'-D — |t't 



(32) 



a = m -+- n sin (a ) cot (r ), a' = n cos (o^), 



& = cos (a ) cot (tc ), }> — — sin (a ), 



c = cos (a ) cosec (ir ), c = sin (tt ) tan i — sin (a ) cos (ir ), 



d = sin (a ) cosec (ic ), d' = cos fo) cos (it ), 



and A, B, C, D are the day-numbers from the American Ephemeris or Ber- 

 liner Jahrbuch. The complete corrections required by (a ) and (ir ) to allow 

 for nutation, aberration, precession, and proper motion are computed by the 

 following formulas, in which we have put B a and B K for the quantities 

 just computed by the Besselian formulas, and called «Red. a» and «Red. tc»: 



-R-v sin 1 



tanA a a= ! _ C ot Jig JJ* sin i 7 » 



1 L T, ♦ l" 



A 8 tt = B K -*- tan (w ) tan ¥ A 2 a fl x sin 1 



*H3.-M aT . cTp. 78. . 39 



