MOTION OF THE SOLAR SYSTEM. 
101 
distant epochs, a long time must elapse before the deficiency is supplied, and we 
may still say, in the words of Halley, that centuries may be required to discover the 
laws of the proper motions of the stars. 
Method of Calculation. 
The direction of the apparent motion of a star is conveniently defined 
which it makes with the circle of declination. Let S be the place of 
the star found by reducing the place given in the first catalogue to the 
epoch of the second, S' its place given in the second catalogue, and P 
the north pole of the equator. Connecting these points by arcs of great 
circles, the arc SS' represents the proper motion of the star in the 
interval between the epochs, and the angle PSS' is the angle which has 
been denoted by This angle, which gives the direction of the star’s 
motion, is reckoned from left to right all round the circle, from \J/= 0 to -4=360°, and 
is computed from the variations of right ascension and declination indicated by the 
comparison of the catalogues as follows : — 
Let a and S denote respectively the right ascension and declination of the star at 
the mean epoch (1790), and Aa, Ah be the annual variations of those quantities 
arising from proper motion ( Aa being in seconds of arc), and As the annual variation 
of the star’s place in the arc of a great circle, we have then 
As sin 4 = cosoA« 
As cos A=Ah . 
> (I.) 
, , cos SA« 
tan 4=— 
by the angle 
p 
The values of 4 and As calculated from these formulae are given, for all the stars 
under consideration, in the appended table. 
To determine the direction of the parallactic motion, let Q be the 
point towards which the sun’s motion is assumed to be directed, T 
the point diametrically opposite, S the place of a star, P, P' the north 
and south poles of the equator respectively, and let PSP', PQP' and Q 
QST be great circles of the sphere. In consequence of the real 
motion of the sun towards Q, the star, as seen from the earth, will 
appear to move towards T, in the great circle QST, the position of 
which will be given in terms of the angle PST which it makes with 
the declination circle PSP'. Let the angle PST be denoted by 4>', 
and let A and D denote respectively the assumed right ascension and declination of Q. 
The angle \j/' may be computed immediately from the formula 
cot 4 ,= — cos SP' cot QP'S 
sin SP' cot QP' 
sin QP'S 
in which all the quantities are known, since SP is given in terms of S, the star’s de- 
clination, QP' in terms of D, and the angle QPS represents the difference between « 
