182 SECULAK VARIATIONS OP THE ELEMENTS OF 



If, in these equations, we neglect quantities of the order sin 2 q>", they assume a 

 very convenient form for computation, and at the same time possess sufficient 

 accuracy for computations extending over a period of several hundred years, 

 except for stars situated very near the pole, in which cases some of the preceding 

 equations must be employed if we wish to obtain accurate results. 



X =% +^'+<?>" tan /? cos &—d") \ ,_. 



(¥=p —$" sin a— 6") ; J l J 



p'=P— ^sin(A- 0"); 



2 =X — 4*'— q>" tan /3' cos (X—d"—^') 1 ff .- q . 

 (3 =/3'+<2>" sin (X— &'—$). J ( ^ 



In these equations <p" is to be expressed in seconds of arc. 



9. For the reduction of right ascensions and declinations all the necessary data 

 depending on the motion of the equinox are contained in Table X. The quantities 

 in the table are adapted to computation by the following formulae, in which a and 8 

 denote the mean right ascension and declination of a star at the epoch'of 1850; 

 and a' and 8' denote the same co-ordinates referred to the mean equinox of any 

 time t before or after the epoch. 



cos 8' sin (a' — z' — $')=cps 8 sin (ce-f-z — $) ~\ 



cos 8' cos (a'—z'—^')—cos 8 cos cos (a+z— 3)— sin 8 sin I (580) 

 sin 8' =cos 8 sin © cos (a-\-z — SO-f-sin 8 cos J 



For reducing to the mean equinox of 1850 these equations take the following 

 form : — 



cos 8 sin (a+z — 3)=cos 8' sin (a' — z' — 3') ~\ 



cos 8 cos (a-\-z— 3)=cos 8' cos cos (a — z'— S')+sin sin 8' > (581) 

 — sin 8 =cos 8' sin cos (a — z' — 3') — cos sin 8' J 

 The first two of equations (580) will very readily give 

 a'=a+(z+z4-$'— 3)+[~arc tan = 



| tan 8 sin sin (a+z— SQ+2 sin 2 § sin (a- \-z— $■) cos (a+z— 3) ) "j _ _~ 

 \ 1 —tan 8 sin cos (a+z— 3)— 2 sin 2 § ©cos 2 (a+z— 3) J J ( ° . 



Here the term z-j-z'-|-3' — 3- appears as the general precession in right ascension 

 common to, all the stars, and the last term of the equation gives the correction 

 depending on the place of each particular star. 



In like manner the first two of equations (581) will give 

 a=a' — (z-|~z'+3' — S-) — jarc tan = 



f tan 8' sin sin (a — z 7 — $')— 2 sin 2 \ sin ( a'— z*— SQ cos (a'— z!— 3Q 1 -| 



1 l+tanS'sin@cos( a '— z 1 — $')— 2 sin 2 § cos 2 (a'— z'— 3') J J ( ' 



For stars situated near the pole, equations (580) and (581) are preferable to 

 (582) and (583), because when 8 or 8' is equal to 90° the terms depending on tan 

 8 or tan 8' become infinite, and the equations (582) and (583) assume an indeter- 

 minate form. But this is not the case with equations (580) and (581) ; for if 5=90° 

 equations (580) will give sin 6'=cos0, and then we shall find cos ( a —z'— $•')=— 

 sin ©-hcos 8'— — 1, whence a'— z'— $'= 180°, from which a is easily determined. 



