HEAVENS. 49 



The effect of aberration on the right ascension and polar distance of a 

 star will be given by the following formulas, in which a, and % x are the 

 right ascension and polar distance unaffected by aberration, and a, tc, the 

 same quantities affected by aberration. These formulas are due to Dr. G. W. 

 Hill, and are taken from the «Star-Tables of the American Ephemeris», 

 where they are numbered (30). The symbols used in them have their usual 

 signification. The last term in each formula is the effect of diurnal aber- 

 ration. 



n 



a — <x x = — 20.4451 cosec tc, [sin a, sin O -+- cos oc x cos £ cos O] 



0''0009329 cosec 2 «, sin 2 a, cos 2 O 

 0'.'0009295 cosec 2 n x cos 2 a, sin 2 © 

 0''3 11 cos 9 cos (0 — a x ) cosec -k x , 



tc — tc 1 = -+- 20?4451 cos ttj cos a x sin O 



— 20?445 1 cos O [cos 7c, sin a x cos t — sin tc, sin e] 

 0'.'0004648 cot tc, sin 2 a, sin 2 

 [0'.'0000402 — 0'.'0004665 cos 2 a,] cot % i cos 2 O 

 0''3 11 cos © sin (8 — a,) cos tc, . 



(6) 



The effect of proper motion can be computed by means of the follow- 

 ing equations, in which a i and tc 3 are the coordinates of the star unaffect- 

 ed by proper motion. These formulas are taken from the same source as 

 the aberration formulas, and are there numbered (10). 



1 r.. 3„™2.- ....'» ft ._ a <*A+a 



a — a 2 = \Lt -+- jjlol' cot tc 3 t 2 — -=- [[*. 3 cos 2 tc 2 — \lu (1+3 cot 2 r 2 )] f , 



3 



re ~— 77 



(7). 



li* -t- -i- p. 2 sin 27: 2 < 2 H--^fJLV(l -*-2 cos 2 tc 3 ) f. 



2 — r v . r U *« ~ .v 2 „ . g 



The effects of precession and nutation do not usually need to be con- 

 sidered in connection with the problem in hand. For these phenomena 

 change only the fundamental planes which determine the system of spher- 

 ical coordinates used for defining the positions of the stars on the sky. 

 Consequently, if we employ for the known stars on the plate places taken 

 from a catalogue, and correct them only for refraction, aberration and pro- 

 per motion, the places of the unknown stars will come out referred to the 

 equinox and equator of the catalogue of known stars used. It is to be noted, 

 however, that if precession and nutation are disregarded in this way in 

 the reductions, certain difficulties arise in the computation of the refraction. 

 For the refraction ought to be computed with the a and tc of the star re- 

 ferred to the position of the pole at the date of the plate. But when the 



$H3 



9 



